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Typogenetics : a logic of artificial propagating entities Morris, Harold Campbell 1989

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TYPOGENETICS: A LOGIC OF ARTIFICIAL PROPAGATING ENTITIES A.B. , (Honors), The U n i v e r s i t y o-f Miami, 1975 J.D. , The U n i v e r s i t y o-f Idaho, 1978 M.A., Washington S t a t e U n i v e r s i t y , 1981 M.A., The U n i v e r s i t y o-f B r i t i s h Columbia, 1983 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHILOSOPHY We accept t h i s t h e s i s as con-forming to the r e q u i r e d standard: THE UNIVERSITY OF BRITISH COLUMBIA January, 1989 C} @ Harold Campbell M o r r i s , 1989 By HAROLD CAMPBELL MORRIS DOCTOR OF PHILOSOPHY IN Permission has been granted t o the N a t i o n a l L i b r a r y of Canada t o m i c r o f i l m t h i s t h e s i s and to lend or s e l l c opies of the f i l m . The author (cop y r i g h t owner) h a s r e s e r v e d o t h e r p u b l i c a t i o n r i g h t s , and n e i t h e r t h e t h e s i s n o r extensive e x t r a c t s from i t may be p r i n t e d or otherwise reproduced without h i s / h e r w r i t t e n permission. L ' a u t o r i s a t i o n a ete accordee a l a B i b l i o t h e q u e n a t i o n a l e du Canada de m i c r o f i l m e r c e t t e these et de p r e t e r ou de vendre des exemplaires du f i l m . L'auteur ( t i t u l a i r e du d r o i t d ' a u t e u r ) se r e s e r v e l e s autres d r o i t s de p u b l i c a t i o n ; n i l a t h e s e n i de l o n g s e x t r a i t s de c e l l e - c i ne d o i v e n t e t r e imprimes ou autrement r e p r o d u i t s sans son a u t o r i s a t i o n e c r i t e . ISBN 0-315-50697-0 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date J ^ H 22s \1SCI Abstrac ABSTRACT T h i s t h e s i s d e a l s with a b s t r a c t models o-f propagation ( e s p e c i a l l y , s e l f — r e p l i c a t i o n ) . As some of these r e f l e c t or borrow from nature, a summary of b i o l o g y ' s c u r r e n t understanding of n a t u r a l r e p r o d u c t i o n ( m i t o s i s ) i s provided f o r background. However, the predominant concern i s with e n t i t i e s r e a l i z e d i n u n i n t e r p r e t e d symbolic systems, and a s s o c i a t e d p h i l o s o p h i c a l and design problems. Thus the comparison t h a t i s made between a r t i f i c i a l and n a t u r a l modes of propagation i s intended p r i m a r i l y t o enhance co n c e p t i o n of the former. Automata c o n s t i t u t e one type of formal model. With a simple T u r i n g Table the concept of a s e l f - r e p l i c a t i n g s t r i n g i s i l l u s t r a t e d . The i d e a of a l o g i c a l u n i v e r s e i n which propagating " v i r t u a l " e n t i t i e s emerge and i n t e r a c t i s explored with r e f e r e n c e t o c e l l u l a r automata. A formal system c a l l e d Typogenetics p r o v i d e s the c e n t e r p i e c e of t h i s t h e s i s . The system, f i r s t presented i n an incomplete form i n H o f s t a d t e r (1979), i s here f u l l y developed (augmented with a u s e f u l program f o r personal computers). A Typogenetics s t r i n g ("strand," i n analogy t o a DNA strand) codes f o r o p e r a t i o n s t h a t act t o A b s t r a c t i i i transform t h a t very s t r a n d i n t o descendant s t r a n d s . Typogenetics s t r a n d s e x h i b i t e d i n c l u d e , among o t h e r s , a p a l l i n d r o m i c s e l - f - r e p l i c a t o r coding -for o p e r a t i o n s su-f-ficient t o r e p l i c a t e i t s e l f ; a "sel-f-perpetuator" deforming and then reforming i t s e l f through f u l l y compensatory o p e r a t i o n s ; and an " i n f i n i t e l y f e r t i l e " s t r a n d b e a r i n g an i n f i n i t u d e of unique descendants. M e t a - l o g i c a l p r o o f s e s t a b l i s h c e r t a i n general p r o p o s i t i o n s about the Typogenetics system, e.g. t h a t f o r every s t r a n d t h e r e i s a mother s t r a n d . Redactio r e a s o n i n g , of p o t e n t i a l g e n e r a l i s a b i 1 i t y beyond Typogenetics, shows how a h y p o t h e t i c a l s t r a n d can be r u l e d out by e s t a b l i s h i n g the incommensurability of i t s two i d e n t i t i e s qua packet of o p e r a t i o n s and qua operand. A R u s s e l 1 i a n — t y p e p a r a d o x i c a l s t r a n d t h a t has a l l and o n l y the n o n — s e l f - r e p l i c a t i n g s t r a n d s f o r o f f s p r i n g i s considered ( i s i t a s e l f — r e p l i c a t o r ? ) , s p u r r i n g d i s c u s s i o n of the Theory of Types and H o f s t a d t e r ' s "strange l o o p s . " Table of Contents i v TABLE OF CONTENTS ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES x i LIST OF FIGURES x i i ACKNOWLEDGEMENTS x i i i PREFACE 1 Scope and L i m i t s of the I n q u i r y 1 Plan of the Work 5 I. PROPAGATION 9 Propagation i n Nature 9 Beyond Natural Propagation 13 I I . EARLY APPROACHES TO UNDERSTANDING PROPAGATION 17 The P r i n c i p l e s of the Seed 17 A r i s t o t l e 19 I I I . EMPIRICAL SCIENCE'S CONTRIBUTION TO THE UNDERSTANDING OF PROPAGATION 25 Preformationism vs. E p i g e n e s i s „ 26 M i t o s i s vs. M e i o s i s 31 Chromosomes 33 Table o-f Contents v N u c l e o t i d e s 34 N u c l e i c A c i d s as Genetic C a r r i e r s 35 The S t r u c t u r e o-f DNA 37 Sel-f-Rep 1 i c a t i o n of DNA 38 DNA as Info r m a t i o n a l V e h i c l e 39 T r a n s c r i p t i o n vs. T r a n s l a t i o n 41 The Role of DNA i n " I n d i r e c t i n g " I t s R e p l i c a t i o n 42 T r a n s c r i p t i o n and T r a n s l a t i o n i n DNA R e p l i c a t i o n 44 E p i g e n e s i s vs. Preformationism, R e p r i s e 44 Summing Up 46 IV. AUTOMATON MODELS OF REPLICATION 48 Models New t o B i o l o g i c a l S c i e n c e 48 Models Of vs. Models For 49 Automata 51 E f f e c t i v e Procedures and Algo r i t h m s 52 Something t o Aim For 55 S e l f — R e p 1 i c a t i o n with Automata 57 Turing Table Automata 59 A Self-Reproducing Program 65 A Guiding I n t u i t i o n 66 Approaches t o Problem S a l v i n g 70 S t a h l ' s T u r i n g Table Enzyme S i m u l a t i o n s 77 Von Neumann 81 The U n i v e r s a l Machine's S e l f - R e p l i c a t i o n 82 T a b l e o-f Contents v i The I n - f i n i t e Regress Paradox o-f Sel-f—Repl i c a t i o n 83 E v a l u a t i n g the Argument 85 A P a r a d o x i c a l Machine? 87 Rosen's Paradox 88 Von Neumann and C e l l u l a r Automata 90 C e l l u l a r Automata Capable o-f P a r a l l e l P r o c e s s i n g 93 P r o p e r t i e s o-f a C e l l u l a r Automaton 93 LIFE 95 V i r t u a l Automata 96 V i r t u a l Computers 99 C l a s s e s o-f C e l l u l a r Automata 100 R e s u l t s Obtained with C e l l u l a r Automata 102 Langton's S e l - f — R e p l i c a t i n g P a t t e r n 104 Other Computing and Propagating Automata 109 Langton's V i r t u a l T u r i n g Machine 111 On t o Typogenetics 112 V, TYPOGENETICS 113 Completing the System 114 Alphabet o-f Symbols 116 Recursive D e - f i n i t i o n o-f a Well—Formed Typogenetic Strand 116 No V a r i a b l e s 119 Units/Bases Level 119 H o r i z o n t a l and V e r t i c a l Unit R e l a t i o n s h i p s 120 Table of Contents v i i Enzyme Attachment 122 Operations 128 Quali-fi cat ions on Operations 130 Daughters 132 Procedure I l l u s t r a t e d 132 Trans-forms 134 Types o-f Strands Producible 137 Self-Replicators 139 VI. META—TYPOGENETICS 143 Reductio Strategy for Disproving a Hypothetical Strand 144 A Self-Erasing Strand? 144 A Hypothetical Self-Replicator Disproved 147 A Cascading Strand—Line 151 Computabi1ity 154 How Many Strands? 154 Possible Daughters 155 How Many Daughters 157 How Many Descendants? 159 Relative Frequency of Bases 160 Frequencies of Types of Strands 162 Asymmetry in Allocation of Binding Preferences 167 No Motherless Children in Typogenetics 170 A l l Strands Are Siblings 172 A Paradoxical Strand 173 Table o-f Contents v i i i What the Paradox Implies 175 V i c i o u s C i r c u l a r i t y or Strange Loop? 177 A Strange Loop of S e l f — R e p l i c a t i o n 179 The U r s t r a n d 181 The P h i l o s o p h e r ' s Strand 182 Relevance t o Natu r a l P h i l o s o p h y 1S4 VII . TYPOGENETICS AND NATURE'S LOGIC 188 A Log i c For DNA 189 Same Alphabet, D i f f e r e n t Codons 193 Coding Schemes 195 Operations 196 Genes 197 The Value of Typogenetics t o Natural Science 200 V I I I . LOGIC AND TYPOGENETICS 205 A M u l t i - L e v e l e d System 205 Complementarity 211 Typogenetics and the Regress Paradox of S e l f - R e p l i c a t i o n 214 Typogenetic vs. Standard L o g i c D e r i v a t i o n s 216 Typogenetic Strands vs. Programs 217 S e l f - P r o v i n g Formulas 219 S e l f - P r o v i n g Formulas and Typogenetic Strands 222 Proposed Lemma Proving Sentences 223 Ancestry 226 Table o-f Contents Least Ancestor A l g o r i t h m s 228 Typogenetics: D i r e c t Re-f 1 ex i v i t y 228 S e l f — C o n s t r u c t i n g Sentences 229 EVALUATING MODELS OF PROPAGATION 232 What's In a D e f i n i t i o n ? 233 T r i v i a l Instances 234 Keeping the S e l f i n S e l f - R e p l i c a t i o n 236 Keeping the P r o d u c t i o n i n Reproduction 238 S e l f - R e p l i c a t i o n as a R e l a t i o n s h i p 241 R e p l i c a t i o n i n a Vacuum 242 The Value of a S e l f - R e p l i c a t i n g S t r u c t u r e 244 I m p r o b a b i l i t y a C r i t e r i o n 248 The Problem of Dependent P r o b a b i l i t y of Occurrence 249 Why Improbable Is B e t t e r 253 Rosen's P a r a d o x — R e p r i s e 255 A Find Problem 256 Abduction and Deduction i n Design L o g i c 259 Deduction vs. Abduction i n Mathematics 260 Abduction vs. Deduction: Symbolic L o g i c 263 Rosen's Paradox: A F a i l u r e of Deduction 265 C o r r e l a t i v e Procedures 267 EXTENSIONS AND APPLICATIONS 268 An I n t e r p r e t a t i o n 269 Table of Contents x T e r t i a r y S t r u c t u r e z/i Adding the T h i r d Dimension 273 Typogenetic Graphing o-f Geometrical F i g u r e s 274 Sel-f-Repl i e a t i n g Geometrical F i g u r e s 277 Modeling the Descent o-f Geometrical Form 278 A Tool -for Thought 280 P a r a l l e l P r o c e s s i n g Strands 282 Automatization o-f Typogenetics 284 Automatic Generation o-f Random Strands 286 A P o s t e r i o r i I n v e s t i g a t i o n o-f F e c u n d i t y R a t i o 288 AL Through I n d i r e c t Design 289 A r t i f i c i a l Zygotes 291 A B i o l g e b r a 292 Beyond Models 294 A r t i f a c t s and A r t i f i c e r s of a New Kind 295 BIBLIOGRAPHY 299 APPENDIX I Comparing H o f s t a d t e r ' s Typogenetics t o the Ve r s i o n Presented In T h i s T h e s i s 306 APPENDIX II Gene I s o l a t i o n and Bindi n g P r e f e r e n c e Determining BASIC Program 311 APPENDIX III D e r i v a t i o n s f o r Strands E x h i b i t e d i n Text 321 APPENDIX IV "Mothei—maker" Algorithms 328 INDEX 338 L i s t o-f T a b l e s x i LIST OF TABLES Table 1 T u r i n g Table 63 Table Sample Run o-f T u r i n g Table Automaton 64 Table 3 S t r i n g R e p l i c a t i o n 67 Table 4 Typogenetics T r a n s l a t i o n Table 124 Table 5 T e r t i a r y S t r u c t u r e and B i n d i n g P r e f e r e n c e 125 L i s t of F i g u r e s x i i LIST OF FIGURES F i g u r e 1 LIFE, a c e l l u l a r automaton 97 F i g u r e 2 Enzyme t e r t i a r y s t r u c t u r e s 126 F i g u r e 3 A 64-sided t e r t i a r y s t r u c t u r e 275 Acknowledgements x i i i ACKNOWLEDGEMENTS S p e c i a l a p p r e c i a t i o n i s extended here t o my t h e s i s s u p e r v i s o r , P r o f e s s o r Thomas E. Pat t o n , who pro v i d e d sound ad v i c e and i n v a l u a b l e c r i t i c a l feedback at every stage of t h i s p r o j e c t , and without whom t h i s t h e s i s would not have been w r i t t e n . Thanks a l s o t o my other t h e s i s committee members: P r o f e s s o r Lawrence Ward, who in t r o d u c e d me t o the world of computers and automata theory, and t o P r o f e s s o r Daniel R. Brooks, f o r h i s guidance i n matters touching on b i o l o g y . Acknowledgement t o Douglas H o f s t a d t e r , and t o the l a t e Jacques Monod, whose s p l e n d i d books opened up t o me the realms, r e s p e c t i v e l y , of the formal s c i e n c e s , and of molecular b i o l o g y ; and t o C h r i s t o p h e r Langton and a l l the p a r t i c i p a n t s of the most i n s p i r i n g conference I have ever been p r i v i l e g e d t o a t t e n d , the f i r s t i n t e r n a t i o n a l Workshop on A r t i f i c i a l L i f e , h e l d September, 1987, at Los Alamos Na t i o n a l Laboratory, where my work i n the area of the l o g i c of propagation was f i r s t a i r e d . To J u n i u s M o r r i s , PhD, my l a t e f a t h e r and primary i n t e l l e c t u a l p r o g e n i t o r : L et t h i s t h e s i s serve t o express b e l a t e d thanks f o r the book on c r y s t a l s , quoted i n these pages, you gave me so long ago. F i n a l l y , t o my mother, Vera Campbell M o r r i s , s i n c e r e s t g r a t i t u d e f o r a l l her p a t i e n t support during my long s t r u g g l e t o produce t h i s document. P r e f a c e 1 PREFACE Nature and L i m i t s o-f the Inquiry T h i s t h e s i s i s a study o-f the l o g i c of propagation, e s p e c i a l l y (asexual) s e l f - r e p r o d u c t i o n . I t i s o f f e r e d as a c o n t r i b u t i o n t o philosophy, r a t h e r than t o b i o l o g y , which f a c t deserves emphasis at the o u t s e t . I t s o r i g i n a l i t y and main concern i s i n demonstrating phenomena of propagation by means of formal systems and automata models, although i t does c o n t a i n a summary account of p e r t i n e n t b i o l o g i c a l theory, and even devotes one chapter (VII) t o e x p l o r i n g the r e l e v a n c e t o n a t u r a l s c i e n c e of t h i s kind of r e s e a r c h . I should c a l l t h i s p hilosophy i n the t r a d i t i o n of Boole, R u s s e l l and Whitehead, and most immediately, H o f s t a d t e r (1979), a l l of whom employed symbolic systems i n a priori endeavors t o grasp the l o g i c a l s t r u c t u r e of the world. T h i s i s , i n other words, a l o g i c t h e s i s . The p h i l o s o p h e r and l o g i c i a n , S.K. Langer (1953), has termed l o g i c "the s c i e n c e of form." In the i n t r o d u c t i o n t o her l o g i c book one f i n d s a f a i r l y standard account of the d i s t i n c t i o n between the l o g i c i a n ' s task and the s c i e n t i s t ' s , and i n d i c a t i o n of how the former f u r t h e r s the general program of philosophy. On the f i r s t p o i n t , she observed that "the s c i e n c e s P r e f a c e 2 deal o n l y with forms which may be i n t e r p r e t e d f o r t h e i r s p e c i a l subject—matter. L o g i c d e a l s with any forms whatever without r e f e r e n c e t o content." (p43). In the present undertaking, i t w i l l be seen t h a t our concern i s with uninterpreted symbolisms. A b s t r a c t forms w i l l be e x h i b i t e d which exemplify v a r i e t i e s of propagation, with l i t t l e or no regard as t o whether or how these e n t i t i e s might be p h y s i c a l l y i n s t a n t i a t e d . In c o n t r a s t , n e i t h e r the s c i e n t i s t nor the engineer can be s a t i s f i e d with l e a v i n g t h i n g s purely at a symbolic l e v e l . As f o r ph i l o s o p h y , the aim of which Langer d e c l a r e d t o be "to see a l l t h i n g s i n the world i n p r o p o r t i o n t o each other, i n some or d e r , i . e . t o see r e a l i t y as a system, or at l e a s t any p a r t of i t as belonging t o some system," l o g i c i s , she h e l d , i n d i s p e n s a b l e , "being the s c i e n c e of order par excellence." (p40). S i n c e p h i l o s o p h e r s endeavor t o put together the b i g p i c t u r e l o g i c i s e s p e c i a l l y v a l u a b l e because i t enables c o n c e p t u a l i z a t i o n t o operate in abstracto, f r e e of the d i s t r a c t i o n s of the p a r t i c u l a r s one d e a l s with c o n s t a n t l y as a n a t u r a l s c i e n t i s t or i n everyday 1 i v i n g . T h i s t h e s i s p l a c e s at the d i s p o s a l of the reader t o o l s of thought—most notably, a formal system c a l l e d T y p o g e n e t i c s — d i f f e r e n t from, e.g. the t r a d i t i o n a l p r e d i c a t e l o g i c , but d e f i n i t e l y f a l l i n g under the r u b r i c of P r e f a c e the " s c i e n c e of form," with which a b s t r a c t i n v e s t i g a t i o n s i n t o propagation can be mounted heedless of m a t e r i a l or other c o n t i n g e n t c o n s t r a i n t s , so suggesting q u i t e general c o n c l u s i o n s and hypotheses. In keeping with the c h a r a c t e r of l o g i c a l i n v e s t i g a t i o n s , t h e r e w i l l be found i n the f o l l o w i n g pages a g r e a t emphasis on houi r e s u l t s are obtained. These c o n t r i b u t i o n s t o the theory of formal problem s o l v i n g , should not, however, be confused with c o g n i t i v e psychology. An excursus i n Chapter IV c o n t r a s t s i n some d e t a i l the l o g i c i a n ' s approach with those of the c o g n i t i v e p s y c h o l o g i s t and the other p r a c t i t i o n e r s of the formal s c i e n c e s : mathematicians and computer s c i e n t i s t s . One w i l l a l s o f i n d metalogical p r o o f s prominently f e a t u r e d h e r e i n . That i s , i n a d d i t i o n t o r e s u l t s obtained i n or with a system, p r o p o s i t i o n s w i l l be e s t a b l i s h e d about the system. Again, t h i s c o n t r a s t s with the usual e x c o g i t a t i o n s on formal models made by n o n - l o g i c i a n s ( i n c l u d i n g those c i t e d h e r e i n ) , which o n l y d e c l a r e (surface) parameters of a system, and suggest—but do not prove—other, more l a t e n t , p r o p e r t i e s or l i m i t a t i o n s of t h a t system. The a b i l i t y t o perform such s e r v i c e i s , I t h i n k , one of the most v a l u a b l e (and d i s t i n c t i v e ) a c quired i n the l o g i c i a n ' s t r a i n i n g . There i s a way i n which t h i s t h e s i s d i f f e r s from customary a priori researches. We are i n the h a b i t of t h i n k i n g of the formal s c i e n c e s as pursued i n a very "pure" P r e f a c e domain, a domain making no r e f e r e n c e t o "the way the world works." We can then be p l e a s a n t l y s u r p r i s e d when s c i e n t i s t s or engineers f i n d correspondences between our (formal) laws and p h y s i c a l r e a l i t y . However, I have no doubt t h a t , c o u l d we t r a c e a r i t h m e t i c and geometry back t o t h e i r r o o t s , we would f i n d men s t r u g g l i n g t o accomodate t h e i r r easoning t o nature's ways. In the present case, the systems we w i l l look at w i l l r e f l e c t what i s now known of b i o l o g i c a l s t r u c t u r e s and processes. T h i s p r a c t i c e p a r a l l e l s r e c e n t advances i n computer s c i e n c e (e.g. development of "neural nets") i n s p i r e d by o b s e r v a t i o n of naturally a r i s i n g computers (nervous systems). F i n a l l y , as a note r e g a r d i n g " p r e r e q u i s i t e s , " i t may be s a i d t h a t no s p e c i a l background i n b i o l o g y , p h i l o s o p h y and h i s t o r y of s c i e n c e , or mathematics, i s presupposed of the reader of t h i s t h e s i s . More p a r t i c u l a r l y , t he reader i s not expected t o have any p r e v i o u s exposure t o the automata or formal systems (e.g. Typogenetics) at the heart of t h i s t h e s i s . A background i n formal l o g i c i s necessary t o get the most out of some s e c t i o n s , however. Pre-f ace Plan o-f the Work Chapter I compares and r e l a t e s animate and inanimate propagation i n nature, as a p r e l i m i n a r y t o statement o-f a g u i d i n g assumption: Formal understanding of the l o g i c of propagation w i l l c o n t r i b u t e t o our a b i l i t y t o engender propagating e n t i t i e s , and forms of p r o p a g a t i o n , t h a t chance has not brought f o r t h i n nature. Mankind's e f f o r t s t o a t t a i n such an understanding are subsequently t r a c e d from e a r l i e s t times t o the most r e c e n t achievements, i n Chapters II-IV. The c r u c i a l accomplishment of people of N e o l i t h i c times was simply t o grasp the f a c t that a seed d i f f e r s from, e.g. a rock. In a n c i e n t Greece, A r i s t o t l e founded s c i e n t i f i c b i o l o g y , and h i s theory of the aitiai (the "four causes") and i n p a r t i c u l a r h i s i d e n t i f i c a t i o n of the "end" or development of a t h i n g as i n h e r e n t i n i t s "form" or o r g a n i z a t i o n , paved the way f o r a s c i e n t i f i c approach t o understanding propagati on. In modern times, the understanding of r e p r o d u c t i o n has progressed along two f r o n t s : the e m p i r i c a l , and the symbolic (logico—mathematical). Chapter II I o f f e r s a b r i e f and l a r g e l y n o n — t e c h n i c a l sketch of what e m p i r i c a l s c i e n c e has d i s c o v e r e d about m i t o s i s and the r o l e of DNA i n r e p r o d u c t i o n . T h i s chapter, which c o u l d be skipped b y the knowledgeable, a f f o r d s h e l p f u l background p r e p a r a t o r y t o p r e s e n t a t i o n of a r t i f i c i a l systems r e f l e c t i v e of what we have learned from P r e f a c e 6 b i o l o g i c a l s c i e n c e . Chapter IV d e a l s with automaton models of s e l f — r e p l i c a t i o n . T u r i n g T a b l e s and the s i m u l a t i o n models of S t a h l , von Neumann's proof of the s e l f — r e p r o d u c i n g c a p a c i t y of an hypothesized " u n i v e r s a l machine," and Langton's c e l l u l a r automaton—imbedded s e l f — r e p l i e a t i n g l o o p , a re covered i n p a r t i c u l a r d e t a i l . Chapter V develops one p a r t i c u l a r new formal system, Typogenetics, designed t o emulate DNA's mode of embodying i n f o r m a t i o n coding f o r o p e r a t i o n s t o be worked i n i t s s e l f -t r a n s f o r m a t i o n ( r e p l i c a t i o n ) . A "how t o " t u t o r i a l e x p l a i n s the system's workings thoroughly, and BASIC code f o r a u s e f u l computer program f o r personal computers has been i n c l u d e d f o r the r e a d e r ' s convenience as an appendix. F i n a l l y , the kinds of i n t e r e s t i n g propagating forms ("strands") Typogenetics harbors, i n c l u d i n g a s e l f - r e p l i c a t o r , are e x h i b i t e d with i nstances. Where Chapter V concentrated on f i n d i n g r e s u l t s in Typogenetics, Chapter VI proves t h i n g s about the system. It i s a showcase both of m e t a — l o g i c a l r e s u l t s , and of methods f o r o b t a i n i n g such r e s u l t s . Among the t h i n g s shown: (1) t h a t a p a r t i c u l a r s t r a n d can g i v e r i s e t o i n f i n i t e l y many d i s t i n c t descendants; (2) t h a t l o g i c a l reduct io s t r a t e g i e s e x i s t f o r proving the non-existence of h y p o t h e t i c a l s t r a n d s ; (3) t h a t the system (Typogenetics) possesses an i n h e r e n t , q u i t e P r e f a c e 7 unobvious asymmetry, which i s i t s e l f n e v e r t h e l e s s q u i t e r e g u l a r and mathematically e x p l i c a b l e ; (4) t h a t an a l g o r i t h m e x i s t s f o r d e r i v i n g a mother s t r a n d f o r any given s t r a n d ; and (5) t h a t a p a r a d o x i c a l Typogenetic s t r a n d isomorphic t o R u s s e l l ' s p a r a d o x i c a l s e t i s c o n c e i v a b l e . Chapter VII compares and c o n t r a s t s Typogenetics with the s o r t of modeling l o g i c nature (DNA) would r e q u i r e . An important s e c t i o n c o n s i d e r s the v a l u e of Typogenetics f o r n a t u r a l s c i e n c e and e n g i n e e r i n g , f i n d i n g t h i s s o r t of l o g i c i a n ' s endeavor u s e f u l : as f u r t h e r i n g development of new symbolic systems and methods; as s u p p l y i n g concepts b i o t e c h n o l o g i s t s may draw on; as e s t a b l i s h i n g l i m i t a t i o n s good a c r o s s a l l p o s s i b l e worlds; and as suggesting hypotheses about nature s c i e n t i s t s may wish t o i n v e s t i g a t e e m p i r i c a l l y . Chapter VIII analyzes Typogenetics as a d i s t i n c t i v e , i r r e d u c i b l y m u l t i - 1 e v e l e d system, then compares i t t o standard symbolic l o g i c s and automata, to s e l f - p r o v i n g formulas ( i n mathematical l o g i c ) , and t o s e l f — c o n s t r u c t i n g sentences. Chapter IX d i s c u s s i o n c o n s i d e r s c r i t e r i a on which one might judge the r e l a t i v e i n t e r e s t of proposed models of propagation. Propagating e n t i t i e s themselves have value i n p r o p o r t i o n t o how much procedural i n f o r m a t i o n ( e s p e c i a l l y , procedural i n f o r m a t i o n r e l a t i n g t o t h e i r own geneology or c o n s t r u c t i o n ) they i m p l i c i t l y c o n t a i n . Models of propagation P r e f a c e 8 have value i n p r o p o r t i o n t o what unexpected l o g i c a l t r u t h s they make conspicuous. To the extent "unexpected" equates with s t a t i s t i c a l deviance, improbable e n t i t i e s w i l l appear "more i n t e r e s t i n g . " Yet, i n the course of t h i s d i s c u s s i o n an important o b s e r v a t i o n i s made t h a t the p r o b a b i l i t y of an e n t i t y e o c c u r r i n g i n a system can be p a r t i a l l y dependent on the r e s p e c t i v e p r o b a b i l i t i e s of i t s ancestors having p r e v i o u s l y appeared i n the system, i m p l i c a t i n g computation of e's p r o b a b i l i t y i n a r e g r e s s which would i n some systems (such as Typogenetics) be i n f i n i t e . A f o l l o w i n g s e c t i o n of Chapter IX, a c o n t r i b u t i o n t o theory of formal problem s o l v i n g , p r o v i d e s an o r i g i n a l r e s o l u t i o n of a n o t o r i o u s , long unanswered "paradox" (Rosen, 1959) t h a t c h a r a c t e r i z e d the s e l f — r e p l i c a t i n g automaton as an u n a t t a i n a b l e design o b j e c t i v e . F i n a l l y , Chapter X p o i n t s t o ways t o improve or go beyond Typogenetics, mentioning p o s s i b i l i t i e s f o r i n t e r p r e t i n g t h i s symbolic system, f o r p r o v i d i n g i t with p a r a l l e l p r o c e s s i n g c a p a b i l i t i e s , f o r e x p l o i t i n g i t s p o t e n t i a l t h r e e — d i m e n s i o n a l i t y , and f o r automatizing i t t o l e s s e r or g r e a t e r degrees. A conclu d i n g note s p e c u l a t e s on the f u t u r e development of A r t i f i c i a l L i f e , and suggests the need f o r r e v i s i o n of t r a d i t i o n a l o n t o l o g i c a l c a t e g o r i e s . Chapter CHAPTER I: PROPAGATION Propagation In Nature Be-fore t h e r e was l i f e , t h e general p h y s i c a l p r i n c i p l e of " l e a s t a c t i o n " posed the main l i m i t a t i o n on the formation of t h i n g s . C r y s t a l s , of which a l l s o l i d s save the g l a s s e s are composed, farmed—we might say, i n a crude way propagated—because the most probable or " e a s i e s t " way f o r one atom t o d i s p o s e i t s e l f was ipso -facto " e a s i e s t " f o r o t h e r s of the same k i n d , r e s u l t i n g i n r e p e t i t i v e a c c r e t i o n and alignment, i . e . "growth." What i s " e a s i e s t " depends on circumstances, of course. Carbon atoms formed diamonds on l y under c o n d i t i o n s of high p r e s s u r e and temperature. Sometimes, even under a p p a r e n t l y e q u i v a l e n t e x t e r n a l c o n d i t i o n s atoms of a c e r t a i n kind can f a l l i n t o two or more d i f f e r e n t kinds of p a t t e r n s . In such cases, the i n i t i a l i n c l i n a t i o n can be the key circumstance l e a d i n g to one i n s t e a d of the other arrangement. Imagine a s e t of dominoes set on end, arranged i n a c i r c u l a r p a t t e r n . If the t a b l e i s shaken a b i t the f i r s t f a l l i n g domino's d i r e c t i o n of f a l l — c l o c k w i s e or c o u n t e r c l o c k w i s e — w i l l determine the p a t t e r n of f a l l f o r the e n t i r e c i r c l e . In c r y s t a l i z a t i o n an analogous phenomenon occurs. Sodium c h l o r a t e , f o r example, can c r y s t a l i z e i n e i t h e r a clockwise or a c o u n t e r c l o c k w i s e , Chapter d i r e c t i o n . The i n i t i a l i n c l i n a t i o n (o-f the "seed") imposes the d i r e c t i o n o-f growth the s t r u c t u r e w i l l take from t h e r e on. Which of d i f f e r e n t n a t u r a l s t r u c t u r e s becomes more common depends on (1) how common the p r e d i s p o s i n g circumstances (among which i n c l u d e the presence of a r e q u i s i t e seed) are; and (2) how durable t h a t s t r u c t u r e i s . B u b b l e s — g a s — f i l l e d s p h e r e s — a r e produced with g r e a t frequency on E a r t h but la c k d u r a b i l i t y , so we are not mountain deep i n foam. At the other extreme, diamonds are very durable but not so e a s i l y formed. Quartz c r y s t a l s are e a s i l y formed and f a i r l y d u r a b l e — h e n c e very p l e n t i f u l . The p o i n t i s t h a t even i n the preanimate e r a the world was being "populated" with s t r u c t u r e s a c c o r d i n g t o a kind of s e l e c t i o n Dawkins (1976) c a l l s a general p r i n c i p l e of " s u r v i v a l of the s t a b l e , " of which the famous " s u r v i v a l of the f i t t e s t " p r i n c i p l e of b i o l o g i c a l e v o l u t i o n i s r e a l l y a s p e c i a l case. The t r a n s i t i o n from an inanimate t o an animate world began about 3 b i l l i o n years ago, when molecules appeared t h a t were, though probably c r y s t a l l i n e , new and d i f f e r e n t , i n th a t they were aper iodic i n the sequencing of t h e i r c o n s t i t u e n t u n i t s , meaning t h e i r order was not as repetitive as c r y s t a l s u s u a l l y are. Monotony i m p l i e s i n f o r m a t i o n a l redundancy. The s t r u c t u r a l v a r i e t y of these new e n t i t i e s gave them the Chapter p o t e n t i a l t o embody i n f o r m a t i o n whole o r d e r s o-f magnitude g r e a t e r than t h a t contained i n r e g u l a r c r y s t a l s . That was not the only d i f f e r e n c e . These molecules possessed the c a p a c i t y t o c r e a t e autonomous c o p i e s of themselves—-copies t h a t c o u l d make f u r t h e r c o p i e s of the type. An o r d i n a r y c r y s t a l l i n e seed a s s i m i l a t e s adjacent f r e e atoms i n t o a p r e - e x i s t i n g p a t t e r n u n t i l t h e r e i s e i t h e r no f u r t h e r space or the raw m a t e r i a l s are exhausted, but i t does not produce autonomous descendants. A diamond may grow; and when cleaved i n two by e x t e r n a l f o r c e s , each of the r e s u l t a n t h a l v e s may grow. But the s p e c i a l new molecules spoken of deserve t o be c a l l e d replicators because t h e i r tendency t o s p l i t i n t o r e p l i c a s of the o r i g i n a l was as inherent as t h e i r tendency t o grow. Put another way, under " i d e a l " c o n d i t i o n s , a diamond w i l l grow a r b i t r a r i l y l a r g e , always maintaining i t s u n i t y , where a r e p l i c a t o r w i l l m u l t i p l y i n t o many d i s t i n c t r e p l i c a s . T h i s means t h a t where the c h a r a c t e r <e.g. l e f t asymmetric) of the seed can determine the e n t i r e subsequent progress of the growth of a s i n g l e c r y s t a l , the character of a replicator determines an entire subsequent line o-f descent. But t h a t l a s t sentence i s perhaps o v e r s t a t e d . Although these r e p l i c a t o r s were s a i d t o make c o p i e s of themselves, i t must be understand t h a t t h e i r descendants were not always p e r f e c t l y i d e n t i c a l t o t h e i r p r o g e n i t o r or to each other; copying f i d e l i t y i n other words was imperfect. Some Chapter d i f f e r e n c e s were u t t e r l y t r i v i a l , but some mattered; As compared t o t h e i r p r o g e n i t o r or t o t h e i r co-descendants, some descendants were more s t a b l e (enjoyed g r e a t e r l o n g e v i t y ) ; some were more fecund (producing more descendants "per l i t t e r " o r , simply produced l i t t e r s at a f a s t e r r a t e ) ; and f i n a l l y , some had g r e a t e r copying f i d e l i t y . I t i s i n t h i s l a s t r e s p e c t t h a t the advent of these r e p l i c a t o r s i n t r o d u c e d a new b a s i s of s e l e c t i o n i n the world ( f o r , as a l r e a d y s t a t e d , inanimate t h i n g s were a l r e a d y s e l e c t e d f o r a c c o r d i n g t o t h e i r d u r a b i l i t y and r a t e of p r o p a g a t i o n ) . Simple r e f l e c t i o n shows t h a t e n t i t i e s with g r e a t e r copying f i d e l i t y g i v e r i s e t o mare of their kind than do those with l e s s e r copying f i d e l i t y . Other t h i n g s being equal, they came to be more numerous, through time. Copying f i d e l i t y was s e l e c t e d f o r , though p e r f e c t i o n was never achieved and probably cannot be i n the p h y s i c a l world. The upshot of a l l t h i s was t h a t new v a r i a n t s kept (and keep) a r i s i n g among the descendants of even those e n t i t i e s with the best copying f i d e l i t y . C o n d i t i o n s on E a r t h were such as t o permit spread of r e p l i c a t o r s (we do not assume the e a r l i e s t ones were capable of locomotion). As happens with c r y s t a l s , competition f o r the n e c e s s a r i l y f i n i t e i n g r e d i e n t raw m a t e r i a l s and space occurred; but with a new wr i n k l e . If a v a r i a n t r e p l i c a t o r Chapter happened to decrease the l o n g e v i t y , - f e r t i l i t y , or c opying-f i d e l i t y of i t s " r i v a l s , " t h i s " o f f e n s i v e " tendency was preserved and was l i k e l y t o a u t o m a t i c a l l y become more p l e n t i f u l . Conversely, r e s i s t a n c e t o the d i s r u p t i v e a c t i v i t i e s of competitors was a l s o preserved. Species now competed with s p e c i e s i n a more a c t i v e f a s h i o n than had been c h a r a c t e r i s t i c of the inanimate world; the " f i t t e s t , " those with s u p e r i o r " o f f e n s i v e " and/or " d e f e n s i v e " a t t r i b u t e s , s u r v i v e d t o r e p l i c a t e themselves <and new v a r i a n t s ) , while the l e s s f i t went e x t i n c t . Bevon d Nat ur a1 Propaqat i on So went the process, o b v i o u s l y r e c o g n i z a b l e as b i o l o g i c a l e v o l u t i o n , f o r perhaps t h r e e thousand m i l l i o n years, u n t i l descendants of those f i r s t r e p l i c a t o r s , having evolved i n t e l l i g e n c e , became capable of reasoning about t h e i r o r i g i n s and the phenomenon of propagation. Before reviewing those i n t e l l i g e n t r e p l i c a t o r s ' e f f o r t s t o understand how n a t u r a l r e p l i c a t i o n works though an important hypothesis must be brought out. In t h i s connection a s t o r y recounted by the c r y s t a l l o g r a p h e r s Holden and Singer i s i n s t r u c t i v e . In the e a r l y 1950's an i n d u s t r i a l concern began growing l a r g e c r y s t a l s of ethylene diamine t a r t r a t e . But a year i n t o production c r y s t a l s i n the growing tanks began growing s t r a n g e l y . The d e s i r e d c r y s t a l l i n e m a t e r i a l had been Chapter anhydrous e t h y l e n e diamine t a r t r a t e , and the new c r y s t a l l i n e m a t e r i a l turned out t o be the uonohydr ate of t h a t substance. During s e v e r a l years of r e s e a r c h and development, and another year of manufacture, no seed of the monohydrate had f o r m e d — i t s e x i s t e n c e remained unsuspected. But then i t emerged— appa r e n t l y a product of sheer c h a n c e — a n d q u i c k l y e s t a b l i s h e d i t s e l f as an actual a l t e r n a t i v e way f o r the atoms t o a l i g n themselves. Holden and Singer (1960) hypothesize: "Perhaps i n our own world many other p o s s i b l e s o l i d s p e c i e s are s t i l l unknown, not because t h e i r i n g r e d i e n t s are l a c k i n g , but simply because s u i t a b l e seeds have not put i n an appearance." p81. Extending t h i s l i n e of reasoning i t i s s u r e l y the case t h a t t h e r e are propagating forms, inanimate and animate, and forms (ways) of propagation, t h a t could e x i s t i n t h i s world but which are u n l i k e l y , perhaps extremely u n l i k e l y , t o spontaneously develop. Though many, many l i f e forms today e x i s t , and even more have appeared i n the past o n l y t o be found wanting and disappear, i t i s s t i l l c e r t a i n t h a t nature has not exhausted all possibi1 ities f o r such forms. At every stage of the game c e r t a i n t h i n g s were p o s s i b l e i n p r i n c i p l e ; from among these, o n l y some were, by op e r a t i o n of chance, a c t u a l l y t r i e d . What was t r i e d opened up new o p p o r t u n i t i e s , new p o s s i b i l i t i e s , o f t e n reducing at the same Chapter time the p r o b a b i l i t y o-f occurrence of the u n t r i e d a l t e r n a t i v e s (without completely e l i m i n a t i n g them as p o s s i b i l i t i e s ) . The development of the biosphere has thus m u l t i p l i e d chance by chance, r e s u l t i n g i n a 1ow—probabi1ity s t r u c t u r e of m u l t i - l a y e r e d h i s t o r i c a l contingency (Brooks and Wiley, 1986), the development of which c o u l d not have been p r e d i c t e d from f i r s t p r i n c i p l e s (Monad pp. 42—43, 1971). Yet f o r l i f e forms as well as f o r c r y s t a l s , sometimes the o n l y missing " r e l e v a n t i n i t i a l circumstance" i s the presence of a seed; i f such a seed can be f a b r i c a t e d by man "again s t a l l odds," i t may propagate q u i t e " n a t u r a l l y . " That i s the promise of a r t i f i c i a l l i f e . What's more, e x i s t e n c e of the new a r t i f i c i a l form may open up p o s s i b i l i t i e s f o r new n a t u r a l f o r m s — a s the i n d u s t r i a l p r o d u c t i o n of anhydrous EDT provided the o p p o r t u n i t y f o r monohydrate EDT t o come i n t o being. That i s the -further promise, and perhaps a l s o the danger, of i n t r o d u c i n g a r t i f i c i a l l i f e i n t o the world at l a r g e . The b a s i s f o r c r e a t i n g a t r u l y novel propagating e n t i t y l i e s i n g a i n i n g an understanding of how b a s i c o r g a n i c p r o p e r t i e s and p r o c e s s e s — s u c h as propagation—work in principle. O bviously such an understanding may draw c o n s i d e r a b l e i n s p i r a t i o n from study of extant n a t u r a l forms, but i t cannot be l i m i t e d to t h a t i f the goal i s t o go beyond what nature has thus f a r manifested. Chapter In t h i s and the next c h a p t e r s of t h i s t h e s i s man's e f f o r t s t o a c q u i r e a theory of propagation are sketched. Then the present author's own c o n t r i b u t i o n s t o t h a t theory are p r e s e n t e d — w i t h a primary i n t e n t i o n of e x p l o r i n g propagation simply as a t o p i c i n pure l o g i c , but always l o o k i n g back t o nature f o r s t i m u l u s , and sometimes, ahead t o p o s s i b i l i t i e s of a p p l i c a t i o n . The e x p l o r a t i o n proceeds on, and s h a l l p r o v i d e evidence r e i n f o r c i n g , the assumption t h a t the l o g i c a l space of p o s s i b l e forms of propagation and propagating forms i s g r e a t e r than the n a t u r a l r e p e r t o i r e t h a t n a t u r a l s c i e n c e researches. Chapter CHAPTER I I : EARLY UNDERSTANDING OF PROPAGATION The P r i n c i p l e s of the Seed The mystery of h i s own kind's generation and descent l i k e l y commanded the f i r s t i n t e l l i g e n t s p e c i e s ' (man's) keenest i n t e r e s t , and the f r u i t s of h i s most elementary reasonings on those matters must have been among the f i r s t a r t i c l e s of conceptual thought. Yet the remoteness of that time, and those people, s c a r c e l y allows us t o r e c o n s t r u c t t h e i r fumblings f o r terms and concepts of r e p r o d u c t i o n and k i n s h i p r e l a t i o n s the modern d i s c u r s i v e i n t e l l e c t takes f o r granted. However, by the time of the N e o l i t h i c era (10-20 thousand years B.C.), a conceptual framework s u f f i c i e n t t o support the great undertaking of c u l t i v a t i o n was i n p l a c e . We might venture t o s p e c u l a t e as t o what the elements of that framework must have been by g i v i n g due c o n s i d e r a t i o n to what c u l t i v a t i o n i t s e l f r e q u i r e s . Husbandry r e q u i r e s , at the very l e a s t , t h a t the p l a n t e r be able t o d i s t i n g u i s h between such t h i n g s as rocks and seeds. One p l a n t s the banana, i n s t e a d of a stone Chapter spearhead, though more o-f each kind of t h i n g are d e s i r e d , because the banana i s fertile. From i t there may grow a t r e e that p r o v i d e s more f r u i t . And t h i s happens " n a t u r a l l y " — o f i t s own accord, l i k e a boulder r o l l s down a h i l l . By c o n t r a s t , a stone implement may be used to b r i n g more stone implements i n t o being, but the process i s f a r from automatic, r e q u i r i n g the e x e r t i o n s of the a r t i s a n . The f i r s t s e c r e t then t o be understood, f o r anyone t o have the i d e a t o p l a n t , i s t h i s : A seed will make more seeds, automatically. In a word, it propagates. A r t i f a c t s then known were r e p r o d u c i b l e , but not i n and of themselves reproducing, and i t i s u n l i k e l y t h a t c r y s t a l l i n e growth was known at t h a t time. A second, c l o s e l y r e l a t e d r e a l i z a t i o n i s induced by the o b s e r v a t i o n that a banana, and only a banana, g i v e s r i s e t o banana—bearing t r e e s . Neither s p e c i a l n u t r i e n t s , nor w i s h f u l t h i n k i n g , w i l l r e s u l t i n a banana t r e e a r i s i n g from the p l a n t i n g of a f i g . T h i s idea the a n c i e n t s encapsulated i n the p r e c e p t — a l r e a d y hoary with age by the time of the f i r s t p h i l o s o p h i c a l G r e e k s — t h a t "like gives rise to like." We say t h a t l i f e forms reproduce themse1ves. These two "obvious" r e a l i z a t i o n s put nature at man's d i s p o s a l as never before, and r e v o l u t i o n i z e d h i s e x i s t e n c e . But simple, b a s i c t r u t h s o f t e n have a depth of inner l a y e r s Chapter t o be r e v e a l e d . N e o l i t h i c man recognized merely that some t h i n g s are - f e r t i l e , and w i l l g i v e r i s e t o more o-f t h e i r kind; we a r e , thousands o-f years l a t e r , s t i l l endeavoring t o gain a r e a l l y f u l l understanding of hot* r e p r o d u c t i o n works i n n a t u r e — a n d how i t can be simulated a r t i f i c i a l l y . T h i s t h e s i s i s intended t o f u r t h e r t h a t understanding. A r i s t o t l e With the Greeks may be seen a refinement of the ways of understanding the w o r l d — p r a c t i c a l , s c i e n t i f i c , and m e t a p h y s i c a l — t h a t were u n d i f f e r e n t i a t e d i n p r e h i s t o r i c man. The e a r l i e s t H e l l e n i c t h i n k e r s , o f t e n t a k i n g as t h e i r p o i n t of departure n a t u r a l i s t i c o b s e r v a t i o n s , a t t a i n e d new h e i g h t s of a b s t r a c t i o n , pressed on t o confounding l o g i c a l extremes, and i n the process developed d i s t i n c t l y metaphysical p h i l o s o p h i e s . T h e i r i n t e l l e c t u a l h e i r , A r i s t o t l e , was founder of both s c i e n t i f i c b i o l o g y and formal l o g i c , and thus i s a p r i n c i p a l h i s t o r i c a l antecedent t o the present undertaking, which uses a new l o g i c t o t r e a t something t r a d i t i o n a l l y considered to be i n b i o l o g y ' s province. A r i s t o t l e ' s ontology p o s i t e d an u n d e r l y i n g matter, i t s e l f immutable as E l e a t i c Being, yet rearrangable by " e f f i c i e n t causes," i n a Becoming he f e l t was " f i n a l l y caused," i . e . l a w f u l l y c o n s t r a i n e d toward the attainment of Chapter a s p e c i f i c form. C o r r e l a t i v e t o h i s ontology A r i s t o t l e taught four completely general ways of understanding the world: the a i t i a i , f o u r "causes" or "becauses." (Moravcsik, 1975; M o r r i s , 1981). He c r e d i t e d h i s f o r e r u n n e r s f o r having f i r s t employed the " m a t e r i a l , " " e f f i c i e n t , " and "formal" becauses i n t h e i r attempts t o e x p l a i n the world; the " f i n a l because" was h i s s p e c i a l c o n t r i b u t i o n . ( I t w i l l be seen how h i s a i t i a i apply t o genera t i o n below). A r i s t o t l e however was as much h e i r t o the e m p i r i c a l t r a d i t i o n of the H i p p o c r a t i c medical men as t o the s p e c u l a t i v e t r a d i t i o n of the m e t a p h y s i c i a n s — h i s f a t h e r was court p h y s i c i a n t o P h i l i p of Macedon. His r a t i o n a l quest f o r s y s t e m e t i z a b l e t r u t h s , and h i s resp e c t f o r and assiduous o b s e r v a t i o n of l i v i n g nature came together i n h i s z o o l o g i c a l s t u d i e s that earned him the t i t l e of founder of bi o l o g y . He c l a s s i f i e d the animal kingdom i n a systematic and c r e d i b l e f a s h i o n , organized a wealth of e m p i r i c a l i n f o r m a t i o n , much of i t gathered through d i s s e c t i o n and of t e n accurate t o a s t a r t l i n g l e v e l of d e t a i l , and of course attempted t o e x p l a i n the z o o l o g i c a l sphere i n terms of h i s four becauses. He was e s p e c i a l l y emphatic i n urging the s u p e r i o r d e s i r a b i l i t y of knowing the forms nature assumes as opposed t o attempting t o e x p l a i n l i f e i n terms of some s p e c i a l v i t a l "plasm" or type of p a r t i c l e . ( T a y l o r , 1963). Chapter S e l f - r e p r o d u c t i o n was f o r A r i s t o t l e the d i s t i n g u i s h i n g f e a t u r e of l i f e , H i s t o r i a Animalium VIII I. (588b.-589a), yet nowhere i n h i s extant works does he t r y t o e x p l i c a t e that phenomenon i n i t s f u l l e s t g e n e r a l i t y , d e a l i n g o n l y with the narrower problem of how conception occurs i n p a r t i c u l a r s p e c i e s . But i t i s p o s s i b l e , and f a i r l y s t r a i g h t f o r w a r d , employing h i s theory of the a i t i a i , t o c o n s t r u c t an A r i s t o t e l i a n account of s e l f — r e p r o d u c t i o n . Take the t r a n s f o r m a t i o n of the acorn i n t o the acorn-bearing oak. In A r i s t o t e l i a n s c i e n c e , one i s t o : Understand the m a t e r i a l from which acorns and oaks are c o n s t i t u t e d ; understand the dynamic f o r c e s ( e f f i c i e n t c auses), laws of f l u i d t r a n s p o r t , e t c . , at work i n the growth of the seed i n t o a seed—bearing t r e e ; and understand the form or s t r u c t u r e of seed and t r e e . But one should a l s o a p p r e c i a t e the f a c t t h a t the acorn does not transform i t s e l f i n t o j u s t any o l d t h i n g , but i n t o the acorn-bearing oak, i n a most c e r t a i n and p r e d i c t a b l e , most iawfui-seeming process! T h i s l a s t element of understanding i s of course the " f i n a l because." I t l e a d s i n t o A r i s t o t l e ' s t e l e o l o g y , about which many misapprehensions e x i s t . Perhaps some of those misapprehensions can be disarmed by g i v i n g ear t o what h i s " f i n a l cause" was and was not. "EFor A r i s t o t l e ! l i v i n g t h i n g s c o n t a i n t h e i r own sources of motion and d i r e c t i v e n e s s . There i s no world-soul Chapter or d i v i n e providence o u t s i d e them. ...Again, w i t h i n nature i t s e l f t h ere i s no o v e r a l l design nor c o o r d i n a t i n g agency, no quasi-conscious purpose. ...He never speaks of "purpose" i n nature. H i s expression f o r the f i n a l cause i s "the end f o r the sake of which the development occurs," and t h i s end i s the i n d i v i d u a l ' s p e r f e c t i o n and r e p r o d u c t i o n . " (p234, Balme, 1973). The key t o understanding A r i s t o t l e ' s n o t i o n of " f i n a l c a u s a t i o n " i s contained i n h i s remark, a p p r o p r i a t e l y made i n De Sen An I. I. (715 a.) t h a t the f i n a l and formal a i t i a i "we may regard p r e t t y much as one and the same." T h i s equation, coupled with h i s i n s i s t e n c e t h a t f i n a l c a u s a t i o n works from w i t h i n , was a step i n the r i g h t d i r e c t i o n because i t i n v i t e s one to understand how compresent s p a t i a l r e l a t i o n s can d i c t a t e a temporal (t r a n s f o r m a t i o n a l ) sequence, and v i c e versa. The acorn has a c e r t a i n d i s c o v e r a b l e f o r m — i t i s an arrangement of matter i n space. I n t r i n s i c t o t h i s form i s the f a c t t h a t i t c o n s t r a i n s , under normal circumstances, m a t e r i a l and dynamic f o r c e s t o the e f f e c t t h a t something s p e c i f i c , a f u r t h e r s e t of acorns, i s produced. To understand the form i s t o understand what i t w i l l tend t o become. A round marble r o l l s down a moderate slope; a cube does not. The s p h e r e — a l m o s t without regard t o i t s c o n s t i t u e n t m a t e r i a l — i s a form with the p o t e n t i a l f o r Chapter r o l l i n g . To understand t h a t i s t o understand why the marble r o l l s . S i m i l a r l y , the acorn i s s e l f - r e p r o d u c i n g ; i t s form i s an " a c o r n — f e r t i 1 e form." If the acorn i s ground up i t w i l l l o s e i t s p o t e n t i a l t o s e l f — r e p r o d u c e . Why? The same m a t e r i a l i s present, but the form has been l o s t . C l e a r l y , knowing what m a t e r i a l i s present i s not enough t o know whether s e l f - r e p r o d u c t i o n i s p o s s i b l e . T h i s t h e s i s i s , i n l a r g e p a r t , a demonstration, on a symbolic plane, of how an arrangement of p a r t s of an e n t i t y t h a t is accounts f o r what i t becomes, and c o n v e r s e l y , of how a d e s i r e d t r a n s f o r m a t i o n a l sequence can d i c t a t e a compresent i n f o r m a t i o n a l sequence, a l o g i c a l form. The goal i s to understand generation without r e f e r e n c e t o the " m a t e r i a l " or " e f f i c i e n t " causes, and without invoking some e x t e r n a l g u i d i n g agency. It would t h e r e f o r e seem t o be much i n the s p i r i t of A r i s t o t e l i a n s c i e n c e . A f i n a l word on A r i s t o t l e qua " f a t h e r of l o g i c . " A r i s t o t e l i a n l o g i c was the f i r s t p r e c i s e , formal c l a s s l o g i c . I t was of such a nature as to lend i t s e l f well t o the i n f e r e n c e drawing of a taxonomist l i k e i t s i n v e n t o r . But of course, t h i s s y l l o g i s t i c l o g i c was only the b e g i n n i n g — i t s range of a p p l i c a b i l i t y being very r e s t r i c t e d . Modern symbolic l o g i c dates to Boole (1854). From that time i t has grown, and i n t e r a c t e d almost t o the Chapter p o i n t of merger with mathematics, i n t o the v a s t , h i g h l y a b s t r a c t "formal s c i e n c e s " of today, t h a t are more Pythagorean than A r i s t o t e l i a n . In sum, the legacy of the Greeks—most p a r t i c u l a r l y , A r i s t o t l e — i n c l u d e s both the e m p i r i c a l (natural s c i e n c e s ) and r a t i o n a l (formal sciences) approaches t o understanding generation. Subsequent s e c t i o n s of t h i s t h e s i s take up the f u r t h e r h i s t o r y of these two approaches i n t u r n . Chapter III CHAPTER I I I : EMPIRICAL SCIENCE'S CONTRIBUTION TO THE UNDERSTANDING OF PROPAGATION In the e a r l y 16th century, a new kind of e m p i r i c a l p u r s u i t commenced t h a t has evolved i n t o what we today know as sc i e n c e . The progress of that e n t e r p r i s e with r e s p e c t t o our understanding of propagation i s reported here, i n an h i s t o r i c a l format p r o v i d i n g a general overview. I t w i l l be d i s c o v e r e d t h a t as DNA e n t e r s the p i c t u r e the account becomes r a t h e r d e t a i l e d , t o the p o i n t t h a t the non-b i o l o g i s t may wonder whether i t i s not e x c e s s i v e (whereas the b i o l o g i s t might f e e l i t s l i g h t i n g l y p o t t e d ! ) . By way of j u s t i f i c a t i o n f o r t h i s s e c t i o n , then, the reader should be reminded (or forewarned, i f he skipped the Preface) that some of the formal methods about t o be c i t e d , notably S t a h l ' s automata, and Typogenetics, the formal system that i s e s p e c i a l l y prominent i n t h i s t h e s i s , are somewhat i m i t a t i v e or emulative of the workings of n a t u r a l (DNA—based) propagation. In Chapter VII that "somewhat" i s rendered a good deal more e x p l i c i t . But i n any case resemblances, and overlap of terminology, are as apt t o be deceptive as suggestive t o the unknowing. Hence the need, i n t h i s author's o p i n i o n , t o Chapter I I I 26 i n s u r e t h a t the reader i s in-formed s u f f i c i e n t l y t o a p p r e c i a t e , f i r s t the s i m i l a r i t i e s , and then, the d i f f e r e n c e s , between the ways of nature and these a r t i f i c i a l systems. Preformationism vs. E p i g e n e s i s Bacon was very keen on i n v e n t i o n of instruments t h a t would open up new v i s t a s f o r the mind as the mariner's compass had guided seagoing e x p l o r e r s t o new c o n t i n e n t s . P r e c i s e l y such a d e v i c e — t h e m i c r o s c o p e — a p p e a r e d i n the 1660s, r e v e a l i n g d e t a i l t h a t had been i n a c c e s s i b l e t o even the most assiduous A r i s t o t e l i a n r e s e a r c h e r , h e r a l d i n g a new er a f o r b i o l o g i c a l i n v e s t i g a t i o n of ge n e r a t i o n . But, as though t o i l l u s t r a t e Bacon's admonitions about the human p r o c l i v i t y f o r f i n d i n g i n ambiguous phenomena or pe r c e p t i o n s what i t needs or wants t o see, when e a r l y m i c r o s c o p i s t s i n s p e c t e d human spermatazoa, they r e p o r t e d with great enthusiasm having seen great numbers of t i n y h o m u n c u l i — m i c r o s c o p i c people. Some took pains t o make drawings of the l i t t l e f i g u r e s the wonderful d e v i c e had d i s c l o s e d . T h i s appeared t o be e m p i r i c a l c o r r o b o r a t i o n f o r the contemporary theory of ejRboitement—at l e a s t u n t i l b e t t e r microscopes and m i c r o s c o p i s t s came along. What was t h i s theory? "Leeuwenhoek's d i s c o v e r y of the spermatazoa suggested t o the 18th-century mind a much grea t e r refinement of a b s u r d i t y Chapter I than spontaneous g e n e r a t i o n . T h i s was "emboitement," or preformationism c a r r i e d t o i t s l o g i c a l l i m i t a c c o r d i n g to emboitement, the sperm (thus become the s o l e -factor) must c o n t a i n every d e t a i l of the mature c r e a t u r e ready formed, though on a s m a l l e r s c a l e . . . . i f i t (t h e r e f o r e ) a l r e a d y c o n t a i n s every d e t a i l of the mature c r e a t u r e , i t must, inter alia, c o n t a i n spermatazoa; and each of these, another complete c r e a t u r e i n l i t t l e . " (Pledge, pS7, 1959). Preformationism was an answer t o the anc i e n t metaphysical question of "from whence does a new l i v i n g organism, such as a new human being, come from?" The theory, as i t was espoused i n those days ("emboitement"), almost i n e v i t a b l y s t r i k e s the modern as l u d i c r o u s . To get any understanding of i t one must see what c o n s i d e r a t i o n s motivated i t . The b a s i c i d e a of preformationism i s t h a t a l i v i n g t h i n g does not r e a l l y come t o be from nothing, but was always t h e r e , o n l y i n v i s i b l y s m a l l . I t becomes evident by growing i n t o something n o t i c e a b l e . A f t e r a l l , as one t r a c e s the o r i g i n of the human being one sees at each e a r l i e r time a complete human being, only s m a l l e r . The baby has the f u l l s e t of eyes, e t c . Even i n t o the womb, the f o e t u s at l e a s t i s r e c o g n i z a b l y a human form. But as th i n g s get s m a l l e r , too small f o r the eye to see...what then? The microscope re v e a l e d t h a t there can be very tiny Chapter III 28 l i v i n g organisms indeed. Thousands i-f not m i l l i o n s could -fit into one drop o-f water. That staggering r e a l i z a t i o n lent p l a u s i b i l i t y to the idea that, e.g. the human form could have very small dimensions. A drop of human sperm then might l i t e r a l l y contain thousands of l i t t l e human beings, one of which would grow into the foetus, then into the infant, and f i n a l l y into the adult. So much for embryology; now to go further, the above must be taken in l i g h t of a further consideration. The Inquisition's persecution of G a l i l e o (1615—16)—who, among other things, had claimed to have seen through his telescopes worlds not spoken of in the Bible (e.g. the moons of Jupiter)—was an object lesson for s c i e n t i s t s of the time to interpret t h e i r data, and frame th e i r theories, within the bounds of Church doctrine. One such doctrine i n s i s t e d that the world was created a l l at once in a f a i r l y recent time—one divine, Bishop Usher, reckoned t h i s time of creation, on the basis of b i b l i c a l geneologies, to be 4004 B.C.—and would not be around for any very lengthy future period. Thus, the idea arose that a l i v i n g form at the time of Creation might have been endowed with s u f f i c i e n t — a n d s u f f i c i e n t l y tiny—preformed individuals to account for a l l i t s descendants, to the end of t h i s f a i r l y short-lived mundane sphere. The f i r s t acorn contained a number, n, of Chapter III 29 nested m i c r o s c o p i c acorns (enough -for a l l t i m e ) . I t s immediate s e t o-f descendants d i . . ,d M r e c e i v e d some apportionment of the n—x yet—to—emerge nested m i c r o s c o p i c acorns. And so f o r t h , u n t i l the l a s t , nth acorn grew out of the next t o l a s t g e n e r a t i o n of t r e e s . One c r e a t i o n , or C r e a t i o n , then s u f f i c e d ; t h e r e were no f u r t h e r c r e a t i o n s t o e x p l a i n , o n l y a growth of the i n v i s i b l y t i n y i n t o the n o t i c e a b l y l a r g e . Even i n f e r t i l i t y was e x p l a i n a b l e i n t h i s way: the i n f e r t i l e acorn simply had not been endowed with f u r t h e r t i n y acorns. Opposed to t h i s elegant theory was epigenesis. T h i s theory, more "tough—minded," mechanistic and m a t e r i a l i s t i c than i t s r i v a l , a s s e r t e d that the new t h i n g s t a r t s out as something l e s s than what i t w i l l grow i n t o . The germ i t s e l f i s nothing s p e c i a l , i n f a c t , i s e s s e n t i a l l y u n s t r u c t u r e d protoplasm, c l a y t o be molded. An a p p r o p r i a t e environment, e.g. the womb, making a v a i l a b l e the r i g h t raw m a t e r i a l s and dynamic f o r c e s , creates the new organism. A l i n e of descent i s a s e r i e s of new c r e a t i o n s . " J u x t a p o s i t i o n " and " c o n t i g u i t y " — f a c t o r s a p p a r e n t l y drawn from e x p l a n a t i o n of c r y s t a l growth—were considered s u f f i c i e n t t o s i m i l a r l y e x p l a i n the r e p r o d u c t i o n of l i f e forms. By the e a r l y 19th century, under attack from l e a d i n g e p i g e n e t i c i s t s l i k e Diderot and Erasmus Darwin, Chapter II preformationism i n i t s extreme forms had f a l l e n i n t o d i s r e p u t e ; by the mid—point o f the century, the e n t i r e debate was seen as a n t i q u a t e d . In r e t r o s p e c t , both t h e o r i e s were r e g r e s s i o n s t o p r e -A r i s t o t e l i a n u n s o p h i s t i c a t i o n . In ignorance of e v o l u t i o n i t i s reasonable t o say t h a t the banana form i s t r a c e a b l e back "to the beginning." But c a r e f u l d i s t i n c t i o n has to be made between the banana form and the n u m e r i c a l l y d i f f e r e n t g e n e r a t i o n s of bananas. It was the p r e f o r m a t i o n i s t s * s i l l i n e s s t o t h i n k t h a t each n u m e r i c a l l y d i s t i n c t banana i s contained i n i t s predecessors. On the other hand, e p i g e n e t i c i s t s ' " m a t e r i a l " and " e f f i c i e n t " causal e x p l a n a t i o n s l e f t much to be d e s i r e d . The womb happens t o be an anatomical f e a t u r e of the very type of t h i n g t o be shaped. At the very l e a s t , the s p e c i a l environment then i s making more s p e c i a l environments, which i m p l i e s a per petuation of form that cannot be ignored. True enough, growth does r e q u i r e a p p r o p r i a t i o n o f raw m a t e r i a l s as each n u m e r i c a l l y d i f f e r e n t banana comes t o be, and that i s a kind of c r e a t i o n . But the e p i g e n e t i c i s t s ' " j u x t a p o s i t i o n " and " c o n t i g u i t y " d i d not go very f a r towards accounting f o r the apparent lawfulness of t h i s becoming. A r i s t o t l e had s a i d : Look to the form, the o r g a n i z a t i o n , to e x p l a i n i t . The modern s c i e n t i f i c understanding of n a t u r a l r e p l i c a t i o n r e s u l t e d from a p p r e c i a t i n g how a body's form F Chapter I enables i t t o a p p r o p r i a t e m a t e r i a l s and impart t o them the new body po s s e s s i n g form F. And t h a t s e a r c h — c o n t r a r y t o one dogma of the e p i g e n e t i c i s t s — l e d d i r e c t l y t o a r e c o g n i t i o n of the special Cinformat ion all properties of the germ. M i t o s i s vs. M e i o s i s At the beginning of the 19th century b i o l o g y was s t i l l i n i t s i n f a n c y , f a r behind p h y s i c s as a developed s c i e n c e . Generation was s t i l l as much or more a metaphysical problem as a s c i e n t i f i c one. But the middle pa r t of t h a t century saw tremendous advances with promulgation of Darwin's theory of e v o l u t i o n , the d i s c o v e r y t h a t cells are the b u i l d i n g b l o c k s of a l l p l a n t s and animals, and the advent of biochemistry. L a t e r s t i l l i n the century the much more powerful microscopes then a v a i l a b l e allowed b i o l o g i s t s t o observe r e p r o d u c t i v e phenomena at the c e l l u l a r l e v e l . T h i s was one of the t r u l y great e m p i r i c a l advances i n our understanding of gene r a t i o n : For the f i r s t time, two kinds of r e p r o d u c t i o n c o u l d be d i s t i n g u i s h e d i n nature, mitosis and meiosis. In sexual r e p r o d u c t i o n , or meiosis ^ two c e l l s , each bearing o n l y h a l f as many chromosomes as normal f o r c e l l s of t h a t s p e c i e s , fuse t o form the zygote, which t h e r e a f t e r grows i n t o a m u l t i - c e l l e d e n t i t y through s u c c e s s i v e c e l l d i v i s i o n s . And t h a t d i v i s i o n of one c e l l i n t o two (each of Chapter II the two r e c e i v i n g f u l l a l l o t m e n t of the chromosomes of the o r i g i n a l , which had t e m p o r a r i l y doubled the number of same) i s mitosis. A l l higher p l a n t s and animals s e x u a l l y reproduce, while the s i m p l e s t e n t i t i e s s e l f - r e p l i c a t e o n l y through m i t o s i s . T h i s f a c t r a i s e s a p o i n t concerning terminology, and a f f o r d s an o p p o r t u n i t y t o narrow t h i s t h e s i s ' s f o c u s . Sexual r e p r o d u c t i o n i s not, a t the l e v e l of tokens, s e l f - r e p l i c a t i o n . That i s , my c h i l d i s not a r e p l i c a of me. I t i s , of course, s e l f — r e p r o d u c t i o n at the t y p a l (species) l e v e l , because my c h i l d i s a human being. But a simple o n e — c e l l e d amoeba does s p l i t i n t o two r e p l i c a s of the p r o g e n i t o r ( i g n o r i n g the p o s s i b i l i t y of mutation). In a r e a l sense, a l l amoeba are i d e n t i c a l twins or n-t u p l e t s . There are d i f f e r e n t kinds of r e p r o d u c t i o n i n nature, then. In t h i s t h e s i s I w i l l use the term " s e l f - r e p l i c a t i o n " to r e f e r only t o the t o k e n - l e v e l r e p r o d u c t i o n of m i t o s i s or analogous processes. As f o r sexual r e p r o d u c t i o n — i t i s a more complicated matter, and w i l l be d e a l t with i n these pages only b r i e f l y , i n the l a s t chapter. T h i s i s not to say though that t h i s t h e s i s i s of r e l e v a n c e only t o o n e — c e l l e d organisms or analogous e n t i t i e s , f o r r e c a l l t h a t even i n s e x u a l l y reproducing s p e c i e s l i k e man, apart from the m e i o t i c generation of the zygote ( o r i g i n a l c e l l ) , a l l c e l l s Chapter III 33 comprising the mature c r e a t u r e are m i t o t i c a l l y generated. In th a t sense we are mostly products o-f m i t o s i s ! Nor i s i t t o say t h a t self-replication w i l l be my e x c l u s i v e c o n c e r n — f o r v a r i e t i e s of propagation having no d i r e c t correspondence t o nature w i l l a l s o be i n evidence. Chromosomes To resume our h i s t o r y : The m i c r o s c o p i s t s who had l a i d bare the stages of r e p r o d u c t i o n at the c e l l u l a r l e v e l used new s t a i n i n g methods t h a t had r e v e a l e d chromatin f i b e r s e p a r a t i n g i n t o d i s t i n c t r o d s — t h e chromosomes—that v i s i b l y f i g u r e d , as ex p l a i n e d above, i n both m e i o s i s and m i t o s i s . (Chromosomes ("colored bodies") were so named because they were p e c u l i a r l y s u s c e p t i b l e t o d y i n g ) . Each c e l l (save the r e p r o d u c t i v e c e l l s — e g g and sperm) of a normal m u l t i - c e l l e d c r e a t u r e has the exact number of chromosomes c h a r a c t e r i s t i c of i t s s p e c i e s , e.g. man has 46, or 23 p a i r s . The s i g n i f i c a n c e of the chromosomes themselves was not immediately obvious, but by the mid 1890's Weismann had concluded t h a t the r e p r o d u c t i v e c e l l s ' chromosomes were s p e c i a l : the c a r r i e r s of h e r e d i t y (he mistakenly r e f u s e d t o a f f i r m t h i s of the non-reproductive c e l l s ' chromosomes). He captured the imagination of the time with the suggestion that the chromosomes were immortal. U n l i k e " o r d i n a r y " c e l l s t h a t l i v e d no longer than the i n d i v i d u a l organism they Chapter II helped c o n s t i t u t e , as -far as he co u l d t e l l the individual chromosome p e r s i s t e d u n a l t e r e d down the l i n e of descent, being passed from generation to g e n e r a t i o n , a l b e i t combining and recombining with other chromosomes through sexual mixing. (In t h i s he was onl y c l o s e t o the t r u t h : i f one s u b s t i t u t e s "gene" f o r "chromosome," h i s p o s i t i o n approaches t h a t of modern t h e o r i s t s l i k e Dawkins). Contemporaries c r i t i c i z e d Weismann as a new kind of p r e f o r m a t i o n i s t , and s c o f f e d at the i d e a t h a t some pa r t of an organism c o u l d c o n t a i n the whole i n m i n i a t u r e , even i f as a model or map ( c f . Chapter IV on "the r e g r e s s paradox of s e l f — r e p l i c a t i o n ) . Within another g e n e r a t i o n though the i d e a t h a t the chromosomes were the s i t e of the genes, or d i s c r e t e u n i t s of h e r e d i t y , was well e s t a b l i s h e d , by, f o r example, Morgan's s c h o o l ' s r e s e a r c h with the f r u i t f l y . N u c l e o t i d e s It began t o appear t h a t t o understand the chromosomes was to understand n a t u r a l r e p r o d u c t i o n . The f i r s t question t o be answered was: What are chromosomes made o f ? Chromosomes are n u c l e o p r o t e i n s ; that i s , p r o t e i n i n a s s o c i a t i o n with n u c l e i c a c i d s . The l a t t e r are s t r u c t u r e d i n terms of nucleotides. A n u c l e o t i d e i s a t h r e e p a r t u n i t : a sugar molecule, a phosphoric a c i d molecule, and a s i n g l e purine or pyramidine molecule. The pu r i n e s are adenine and guanine, known u s u a l l y as A and G. The p y r i m i d i n e s are Chapter I I I 35 c y t o s i n e , thymine, u r a c i l , or C,T, and U. N u c l e o t i d e s chain by connecting the phosphoric a c i d of one n u c l e o t i d e t o the sugar group of the neighboring n u c l e o t i d e . These c h a i n s of n u c l e o t i d e s come i n two types, d i s t i n g u i s h e d i n t h e i r composition p r i n c i p a l l y by t h e i r p y r i m i d i n e s : DNA (Deoxyribonucleic Acid) c o n s i s t s of A,C,S, and T (not U); RNA ( R i b o n u c l e i c Acid) of A,6,C, and U (not T ) . N u c l e i c A c i d s as Genetic C a r r i e r s The 19th century had pronounced p r o t e i n s the d i s t i n c t i v e b i o c h e m i c a l . U n t i l 1944 the n u c l e i c a c i d s were thought t o be of p e r i p h e r a l importance. T h i s changed i n t h a t year through the experiments of Avery's group, which suggested t h a t i n some cases n u c l e i c a c i d could be the g e n e t i c c a r r i e r . Avery's group was working with two types of pneumococci b a c t e r i a , the S ("Smooth") and R ("Rough"). The former had an outer capsule the former lacked. Now, an e x t r a c t of n u c l e i c a c i d from the S v a r i e t y when added t o the R enabled the R to grow the capsule (and become an S v a r i e t y ) . It looked very much as though the R s t r a i n ' s g e n e t i c i n h e r i t a n c e had been en r i c h e d by the i n j e c t i o n of the n u c l e i c a c i d e x t r a c t . Over the next dozen years the weight of many other experiments l e f t no doubts as to t h i s i n t e r p r e t a t i o n . Much of t h i s experimentation concerned v i r u s e s , which have a p r o t e i n s h e l l w i t h i n which i s a core of DNA or RNA. Chapter H I 36 S t u d i e s demonstrated t h a t the v i r u s ' s n u c l e i c a c i d minus i t s p r o t e i n s h e l l had some in-f e c t i v i t y ; the s h e l l , none. Moreover, new techniques allowed m i c r o b i o l o g i s t s t o observe bacteriophages ( v i r u s e s t h a t attack b a c t e r i a ) at work i n j e c t i n g t h e i r n u c l e i c a c i d i n s i d e the c e l l . Once i n s i d e t h i s n u c l e i c a c i d seemed t o take over the host c e l l ' s chemical machinery, s y n t h e s i z i n g s p e c i a l p r o t e i n s t o make r e p l i c a s o-f i t s e l f . With the v i r u s ' s n u c l e i c a c i d the host c e l l became a v i r u s reproducer i n s t e a d o-f reproducing i t s e l f . To t r y t o p l a c e t h i s s o r t of experimental f i n d i n g i n the p e r s p e c t i v e t h a t has a l r e a d y been gi v e n , one might r e t u r n t o the image of N e o l i t h i c man t r y i n g t o understand the banana. What i f one c o u l d r e p l a c e a s p e c i a l p a r t of the banana's core with an homologous core from a f i g , and from t h i s grow a f i g t r e e ? One would s u r e l y t h i n k : t h a t core i s i t as f a r as where the "essence of the t h i n g " inheres, t h i s was e x a c t l y what s c i e n c e had f o u n d — i n the core of every c e l l of every s p e c i e s , humble or g r e a t , was a n u c l e i c a c i d (apart from some v i r u s e s , t h i s i s always DNA), s e r v i n g as the c a r r i e r of h e r e d i t y , and, as w i l l be seen, the r e g u l a t o r y a u t h o r i t y . Replace i t , or a l t e r i t — a n d you have a new organism ( i f not an a b o r t i o n , as such changes are f r e q u e n t l y d e s t r u c t i v e ) — t h a t i s indeed the general i d e a behind "recombinant DNA" technology. Chapter III 37 The S t r u c t u r e o-f DNA Having determined the "m a t e r i a l because" o-f g e n e t i c s , s c i e n c e next turned t o the "formal because." The f i r s t c l u e t o DNA's s t r u c t u r e came as a r e s u l t of a n a l y s i s of the r e l a t i v e p r o p o r t i o n s of the f o u r bases (A,C,G and T) i n a s i n g l e c e l l ' s DNA. The r a t i o of the adenine—thymine p a i r s t o the guanine-cytosine p a i r s v a r i e s g r e a t l y between s p e c i e s (holding constant i n t r a — s p e c i e s so t h a t , given the r a t i o f o r some fragment of an u n i d e n t i f i e d i n d i v i d u a l bioform, i t s s p e c i e s c o u l d i n many cases be a s c e r t a i n e d d i r e c t l y or by process of e l i m i n a t i o n ) . But, across the gamut of l i f e forms, the amounts of A to T and C to G were found t o be equal. One could put i t : A=T and C=G. T h i s l a s t f a c t pointed t o the " s p e c i f i c p a i r i n g " hypothesis: There i s something about the s t r u c t u r e of DNA that p a i r s an A with every T, and a C with every G. Under the i n f l u e n c e of P a u l i n g ' s h e l i c a l model f o r the other great c l a s s of biomolecules, the p r o t e i n s , Cr ick and Watson i n 1953 h i t on the i d e a of a double helix s t r u c t u r e of DNA i n order t o account f o r s p e c i f i c p a i r i n g . The i d e a was t h a t , given two st r a n d s (I and I I ) , at every n u c l e o t i d e i n I where an A occurred, a complementary T occurred i n the adjacent n u c l e o t i d e belonging t o strand I I . And d i t t o f o r C and G. The f u r t h e r i m p l i c a t i o n of t h i s was tha t the e n t i r e strand I would be complementary end to end Chapter III 38 wi th s t r a n d 11. The Watson/Crick model co u l d not be immediately con-firmed simply by lo o k i n g through an o p t i c a l microscope. The f u l l complement o-f DNA of one c e l l of a human being would form a continuous s t r a n d 5 t o 6 f e e t long; but i n f a c t t h i s u l t r a — t h i n s t r a n d f o l d s back on i t s e l f l i k e a township-s i z e d t a n g l e of wire. It was a d i f f i c u l t t h i n g t o e y e b a l l . However, ingenious s t u d i e s of DNA's shadow through X—ray-c r y s t a l lography confirmed the double h e l i x model. Now, f o r the f i r s t time i t became p o s s i b l e t o see how the form of DNA could e x p l a i n : S e l f - r e p l i c a t i o n . S e l f - R e p l i c a t i o n of DNA S e l f - r e p l i c a t i o n i n m i t o s i s begins with the u n f o l d i n g of the double h e l i x i n t o two s t r a n d s , I and I I , each being the complement of the other. The n u c l e i c a c i d of each n u c l e o t i d e of a strand has, as we now understand f o r chemical reasons, an a f f i n i t y f o r i t s complement (A f o r T, C f o r G). I f , then, s u f f i c i e n t n u c l e o t i d e s are made a v a i l a b l e , they w i l l n a t u r a l l y p a i r up with the n u c l e o t i d e s of the s i n g l e s t r a n d s (base t o base u n i t e d i n e a s i l y formed noncovalent chemical bonds). Thus each s i n g l e strand w i l l , through t h i s s u c c e s s i v e process of " e l e c t i v e " p a i r i n g , have m a t e r i a l i z e d f o r i t s e l f a complementary s t r a n d , l e a v i n g u l t i m a t e l y two new double str a n d s . Chapter III 39 Monod emphasises that i t i s the s t e r e o s p e c i f i c i t y o-f b i o m o l e c u l e s — t h e i r e x t r a o r d i n a r i l y p r e c i s e s e l e c t i v i t y i n bonding or c a t a l y z i n g — t h a t makes the c e l l u l a r "machinery" work. That c e r t a i n l y i s the case with DNA's r e p l i c a t i o n . Given the r i g h t environment ( e s p e c i a l l y , -freely a v a i l a b l e bases, but a l s o a c a t a l y s t t o r e l e a s e a v a i l a b l e energy necessary f o r p u t t i n g together the strong covalent bonds between the sugar and phosphate molecules t h a t l i n k the nu c l e o t i d e s ) r e p l i c a t i o n occurs spontaneously. Of course, nature does not leave t h i s process t o spontaneous a s s e m b l y — a t l e a s t not at t h i s l a t e date i n e v o l u t i o n a r y time. The r e p l i c a t i o n process i s a c o n t r o l l e d one, from the s e p a r a t i o n of the strand s t o the l a s t n u c l e o t i d e l i n k a g e . And i t i s no co i n c i d e n c e t h a t the " r i g h t " bases and c a t a l y s t s are present. DNA i s the c o n t r o l l e r of i t s own d e s t i n y , and d i r e c t i n g i t s own s e l f — r e p l i c a t i o n i s a v i t a l p art of that d e s t i n y ; i t i s necessary t o see how t h i s i s so. DNA as Informational V e h i c l e DNA i s an in f o r m a t i o n a l v e h i c l e with two dimensions. In one d i r e c t i o n , as between the complementary bases of the two strands, there are no degrees of freedom: given that one of the bases i s C, the other can be nothing but G. T h i s i s p r e c i s e l y what would be wanted f o r s e l f — r e p l i c a t i o n : a Chapter III 40 s i n g l e h e l i c a l s t r a n d o-f DNA c o n t a i n s e x a c t l y and o n l y the i n f o r m a t i o n t o produce a complement with which i t may be pai red. But i n the other d i r e c t i o n , as f a r as the s t r u c t u r e ' s geometry d i c t a t e s , the sequence of bases can be i n any order, and t h e r e i s no i n h e r e n t length l i m i t a t i o n . In other words, lengthwise along a DNA s t r a n d , t h e r e i s no way to t e l l what w i l l come a f t e r the present C, or how many f u r t h e r bases t h e r e w i l l be. In t h i s dimension, t h e r e i s v i r t u a l l y u n l i m i t e d c a p a c i t y f o r v a r i e t y , and thus f o r information—DNA i s an example of the " a p e r i o d i c c r y s t a l " r e f e r r e d t o i n Chapter I. What s o r t of i n f o r m a t i o n ? Here e n t e r s the famous "genetic code." In the 1950s and 1960s i t became c l e a r t h a t a t r i p l e t (a contiguous t h r e e base s e r i e s , e.g. ATG) of n u c l e i c a c i d bases a c t u a l l y p r e s c r i b e s the formation of an amino a c i d . A segment of m u l t i p l e t r i p l e t s codes f o r c o n s t r u c t i o n of an e n t i r e chain of amino a c i d s . These amino a c i d chains ("polypeptides") i n t u r n comprise o p e r a t o r s and operands of the c e l l ' s b iochemistry. There are 20 d i f f e r e n t kinds of amino a c i d s , and 64 p o s s i b l e t r i p l e t s ; the f u n c t i o n , though q u i t e unambiguous, i s thus not 1 to 1 but many to 1; e.g. (JUL) and UDC both code f o r Phenyl a l a n y l . Moreover, a few t r i p l e t s do not s i g n i f y amino a c i d s but r a t h e r act as punctuators, analogous t o parentheses or p e r i o d s , demarcating Chapter I beginnings and endings o-f segments. T r a n s c r i p t i o n vs. T r a n s l a t i o n D e t a i l s o-f the code, or o-f the c e l l ' s o p e r a t i o n s , need not concern us. But the processes o-f transcr iption and translation remain t o be e x p l a i n e d . DNA r e s i d e s i n the nucleus o-f the c e l l , p r o t e c t e d by the nuclea r membrane (I am i g n o r i n g the DNA some o r g a n e l l e s have -for t h e i r own s e l -f - r egul at i on , and a l s o e x c l u d i n g some very p r i m i t i v e l i f e forms); i t i s not known t o have any "sensory" c a p a c i t y . Nor does i t possess any direct f a b r i c a t i v e c a p a c i t y . But i n the a p t l y named process of transcription "messenger" mRNA e n t e r s the nucleus and, through s t e r e o s p e c i f i c a f f i n i t y , r e c e i v e s an impress of the sequence of bases of one segment of DNA, complementary base f o r complementary base. T h i s mRNA molecule then r e t u r n s t o the cytoplasm and se r v e s as a template i n the process of translation. The r i b o s o m e — a t i n y but important o r g a n e l l e , p a r t l y made of RNA, and which i n c i d e n t a l l y possesses the marvelous power t o spontaneously assemble i t s e l f from the r i g h t b i t s — b u i l d s c h a i n s of amino a c i d s drawn from " t r a n s f e r " tRNA molecules, the l a t t e r having matched up complement f o r complement t o the mRNA's copy of the DNA sequence. The r e s u l t i n g amino a c i d c h a i n s — p r o t e i n s — s e r v e as the a c t i v e , r e g u l a t o r y agents (enzymes) and s u b s t r a t e s f o r biochemical processes of the c e l l , metabolic and Chapter III 42 c o n s t r u c t i v e . In s h o r t , the DNA molecule i s a long message on how to b u i l d amino a c i d s ; p a r t s of i t s message are "read" from time t o time ( t r a n s c r i b e d ) by mRNA; and the assembling of the amino a c i d s ( t r a n s l a t i o n ) i s done by the ribosome, working with the mRNA copy of the message. These amino a c i d s run the c e l l ; and t o a l a r g e extent, are the c e l l . The Role of DNA i n " I n d i r e c t i n g " I t s R e p l i c a t i o n At f i r s t impression, DNA's r o l e appears t o be pass i v e . There i s an appearance t h a t the ribosome and the RNA (taken c o l l e c t i v e l y ) are using DNA as one uses a book. How can t h i s be r e c o n c i l e d with the popular image of the DNA molecule as the " b r a i n " of the c e l l ? A c t u a l l y , i n t h i s author's o p i n i o n i t cannot be; the " b r a i n " analogy i s simply a poor one. Be th a t as i t may i t i s p o s s i b l e t o g i v e some i d e a of why DNA's "pas s i v e " r o l e i s i l l u s o r y : The r i g h t t h i n g s happen "to" DNA p r e c i s e l y i n s o f a r as DNA has p r e s c r i b e d the circumstances by i n i t i a t i n g the c r e a t i o n of the a c t i v e agents and raw m a t e r i a l s . Imagine some s p e c i a l c r y s t a l C t h a t , planted i n the r i g h t b r o t h , could c a t a l y z e formation of some other kinds of c r y s t a l s that i n turn c a t a l y z e formation of the raw m a t e r i a l s which then spontaneously combine i n t o , of a l l t h i n g s , another c r y s t a l C; and now m u l t i p l y the complexity of that r e a c t i v e c i r c l e by sev e r a l orders of magnitude, then one can get some i n k l i n g Chapter II of DNA's i n d i r e c t d i r e c t i o n of i t s s e l f — r e p l i c a t i o n . DNA's r e p l i c a t i o n r e q u i r e s s y n t h e s i s of the t h r e e p r i n c i p a l enzymes that a c t u a l l y do the job. DNA endonuclease i s the "unzipping" enzyme. E n t e r i n g the nucleus i t separates one segment of the double s t r a n d s . F o l l o w i n g i t up i s DNA polymerase which draws on f r e e l y a v a i l a b l e n u c l e o t i d e s to plug complement on to complement, doubling each s t r a n d . F i n a l l y , DNA l i g a s e plugs up any gaps r e s u l t i n g from the process. E v e n t u a l l y the whole double s t r a n d has been unraveled, and each o r i g i n a l s t r a n d now has a new complement mated t o i t . T h i s enzyme-directed s y n t h e s i s i s , by one estimate, 100,000,000 times more accurate than non—enzymatic assembly of DNA. Three stages of e r r o r - a v o i d a n c e combine t o reduce the e r r o r i n c i d e n c e i n r e p l i c a t i o n of the human genome to an estimated one base i n IO b i l l i o n . In general terms, DNA's high copying f i d e l i t y i s an example of the " s u r v i v a l of the s t a b l e " e n t a i l e d by the p r i n c i p l e of l e a s t a c t i o n , mentioned i n Chapter I. Noncomplementary n u c l e o t i d e s normally-f a i l t o bond t o the template strand because mismatches are e n e r g e t i c a l l y suboptimal; and when the wrong n u c l e o t i d e i s added, the c o n t i n u i n g s y n t h e s i z i n g a c t i v i t y of the polymerase enzyme i s i n h i b i t e d , which f a c t encourages a "proofreading" t h a t r e s u l t s i n c o r r e c t i v e a c t i o n . (For an a c c e s s i b l e , but d e t a i l e d account of t h i s see Radman and Wagner, 19S8). Chapter III 4 4 T r a n s c r i p t i o n and T r a n s l a t i o n i n DNA R e p l i c a t i o n DNA was termed e a r l i e r an " i n f o r m a t i o n a l v e h i c l e . " However, i t i s important t o p o i n t out that i n the copying process j u s t d e s c r i b e d the i n f o r m a t i o n i n the base sequence i s i r r e l e v a n t t o the enzymes r e p l i c a t i n g DNA; only the complementarity i n f o r m a t i o n i s taken i n t o account. I.e. the copying enzymes n o t i c e only " t h i s i s a C so I need t o match i t with a 6"; the f a c t t h a t t h i s C i s p a r t of Csome t r i p l e t ! coding f o r Csome s p e c i f i c amino a c i d ! i s t o t a l l y ignored. In the t e r m i n o l o g y — w h i c h i s worth remembering, s i n c e i t w i l l r e s u r f a c e t o r e f e r t o something analogous i n T y p o g e n e t i c s — t h i s i s pure transcription, not t r a n s l a t i o n . On the other hand, how i s i t t h a t the polymerase and helper enzymes were on hand t o e f f e c t the r e p l i c a t i o n ? They were of course s y n t h e s i z e d as a r e s u l t of translation. I t i s c u r i o u s t o conceive, but somewhere, then, along the length of the DNA str a n d i s a sequence of bases, the t r i p l e t s of which code f o r the timely s y n t h e s i s of the a c t i v e agents and raw m a t e r i a l s t h a t w i l l d u p l i c a t e the e n t i r e DNA s t r a n d , i n c l u d i n g of course that l i t t l e s u b s ection t h a t p r e s c r i b e d the s y n t h e s i s of the copying enzymes. E p i g e n e s i s vs. Preformationism, Reprise E a r l i e r , the t h e o r i e s of "preformationism" and Chapter II " e p i g e n e s i s " were presented, and both c r i t i c i z e d . But at t h i s p o i n t i t may be seen t h a t each theory a l s o had p a r t of the s t o r y r i g h t . The DNA molecule seems t o f i t the p r e f o r m a t i o n i s t ' s conception of the germ. From the i n c e p t i o n of the c r e a t u r e — i n d e e d , from the i n c e p t i o n of the s p e c i e s , i n the case of the h a p l o i d a l (one-chromosomed) c r e a t u r e — i t i s a complete, master plan f o r the organism. One could say, an amoeba's form was th e r e from the beginning; the f i r s t amoeba's DNA might as well be s a i d t o have l a t e n t w i t h i n i t a l l i t s descendants. Or c o n v e r s e l y , a p r e s e n t l y l i v i n g amoeba was " i n " i t s p r o g e n i t o r ' s DNA. E p i g e n e s i s , i n i t s way, was r i g h t too. Thanks t o DNA's "arrangements," the r i g h t environment ( s p e c i f i c a l l y , one with the a v a i l a b l e n u c l e o t i d e s , the r i g h t enzymes, etc.) recreates each new c r e a t u r e . Monod puts i t t h i s way: the modern s c i e n t i f i c a n a l y s i s " p l a i n l y reduces the o l d d i s p u t e between p r e f o r m a t i o n i s t s and e p i g e n e t i c i s t s t o a q u i b b l i n g over words. No preformed and complete s t r u c t u r e p r e e x i s t e d anywhere; but the a r c h i t e c t u r a l plan f o r i t was present i n i t s very c o n s t i t u e n t s . I t can t h e r e f o r e come i n t o being spontaneously and autonomously, without o u t s i d e help and without the i n j e c t i o n of a d d i t i o n a l i n f o r m a t i o n . The necessary i n f o r m a t i o n was present, but unexpressed, i n the Chapter II c o n s t i t u e n t s . The e p i g e n e t i c b u i l d i n g of a s t r u c t u r e i s not a c r e a t i o n ; i t i s a r e v e l a t i o n . " Monod, p 87, 1971. Summing Up B i o l o g y would have t o say now t h a t A r i s t o t l e was l a r g e l y r i g h t ; the formal cause and the f i n a l cause are p r a c t i c a l l y e q u i v a l e n t , with r e s p e c t t o n a t u r a l propagation. The m i t o t i c r e p l i c a t i o n of DNA i s e v i d e n t l y a lawf u l process. Not j u s t anything i s going t o come t o be, and not j u s t i n any o l d way, but something s p e c i f i c through a d e f i n i t e sequence. To understand f u l l y the form of the DNA—the o r d e r l y arrangement of the b a s e s — i s t o understand t o a great extent how that s p e c i f i c r e s u l t i s obtained i n i t s s p e c i f i c way. F r a n c i s Bacon, r e v i v e r of the s c i e n t i f i c t r a d i t i o n A r i s t o t l e s t a r t e d , accepted A r i s t o t l e ' s "four causes." And he too i n s i s t e d t h a t of a l l p a r t s of knowledge, knowledge of the laws of "the Forms" are "worthiest t o be sought." (Advancement of Lear n i n g , I I , v i i ) . But Bacon was of course f a r more application-minded than A r i s t o t l e ; as f a r as he was concerned the b e t t e r part of the j u s t i f i c a t i o n f o r a c q u i r i n g e m p i r i c a l l y based r a t i o n a l l y j u s t i f i e d knowledge was t h a t i t co u l d be used f o r making t h i s world over t o our s p e c i f i c a t i o n s . T h i s world of the a r t i f i c i a l he c a l l e d "Mundus A l t e r . " Among i t s i n v e n t i o n s he was sure would be v a r i e t i e s of a r t i f i c i a l l i f e . In h i s l a t e essay The New A t l a n t i s (1624) Bacon portrayed a s c i e n t i f i c s o c i e t y with Chapter III 47 z o o l o g i c a l gardens e x h i b i t i n g such novel a r t i f i c i a l organisms. For him, the r e a l question was not "whether," but "how." In h i 5 Novum Orqanum. I I , x x i x , (1620), he decides t h a t " . . . t o produce new s p e c i e s would be very d i f f i c u l t ; but t o vary known s p e c i e s , and thereby produce many r a r e and unusual r e s u l t s i s l e s s d i f f i c u l t . " The knowledge we have obtained of "the Forms" of propagation s i n c e Bacon's time has i n f a c t made p o s s i b l e a new technology, DNA recombination, t h a t permits us t o "vary known s p e c i e s " with p o t e n t i a l l y s p e c t a c u l a r r e s u l t s . A technology f o r producing DNA sequences t o order i s a l s o developing, which may enable us t o do the " d i f f i c u l t " t h i n g of producing e n t i r e l y new s p e c i e s from s c r a t c h . So f a r have we come on the path these two great i n t e l l e c t s marked out. However, i t seems n e i t h e r A r i s t o t l e nor Bacon r e a l l y a p p r e c i a t e d the p o t e n t i a l f o r advancing knowledge a f f o r d e d by the formal s c i e n c e s o p e r a t i n g at a high plane of symbolic a b s t r a c t i o n . Pythagoras d i d , and he and h i s f o l l o w e r s would probably be enthused with these a p r i o r i researches, researches paving the way f o r the development of a r t i f i c i a l l i f e beyond Bacon's experience or powers t o e n v i s i o n . Chapter IV 48 CHAPTER IV: AUTOMATON MODELS OF PROPAGATION Models are such c l o s e kin to diagrams, graphs, c a s t s , maps and i l l u s t r a t i o n s t h a t i t would be hard t o draw a l i n e s e p a r a t i n g any of the l a t t e r from the former. Leaving t h a t a s i d e , a model could be d e f i n e d as an artifact or logico-mathematical construct designed in such a Nay as to display in a conspicuous and i1luminating manner a logical relation or operation, or set of samer of interest to the scientific community. Models range from the a b s t r a c t , e.g. a symbolic system implementable as a computer program, t o the very t a n g i b l e , e.g. p l a s t i c and wire. S u f f i c e t o say here that t h i s chapter w i l l be e x c l u s i v e l y concerned with very a b s t r a c t models. Models New t o B i o l o g i c a l Science The Pythagoreans (6th—5th c e n t u r i e s B.C.) pioneered formal modeling of the u n i v e r s e , using f o r t h a t purpose mathematics and music. But A r i s t o t l e d i d not seem i n c l i n e d towards the use of m o d e l s — e i t h e r a b s t r a c t or c o n c r e t e — i n h i s b i o l o g y , and when m e d i c o / b i o l o g i c a l s c i e n c e was reborn i n the Renaissance i t s main e x t r a - l i n g u i s t i c medium was the analogue p i c t o r i a l i l l u s t r a t i o n ( V e s a l i u s / Leonardo's Anatomy). T h i s d i f f e r e n c e between the p h y s i c a l s c i e n c e s and the l i f e s c i e n c e s p e r s i s t e d r i g h t i n t o the 20th century: Chapter IV 49 P h y s i c s using very a b s t r a c t mathematical models; chemistry using diagrams and three—dimensional wire and paper c o n s t r u c t i o n s ; b i o l o g y using i l l u s t r a t i o n s . No doubt the d i f f e r e n c e s i n the s u b j e c t matter go f a r towards e x p l a i n i n g t h i s . P h y s i c s o f t e n d e a l s with t h i n g s t h a t cannot be seen or handled, such as atoms or microwaves, b i o l o g y with such t h i n g s as can be c a s t or drawn. Ab s t r a c t formal models of b i o l o g y began appearing i n connection with p o p u l a t i o n g e n e t i c s i n the f i r s t t h i r d of the 20th century. Geometrical modeling of biomolecules helped advance b i o c h e m i s t r y i n the e a r l y 1950s. Today, thanks e s p e c i a l l y t o the computer, a b s t r a c t model—making i s a c e n t r a l a c t i v i t y of a new breed of t h e o r e t i c a l b i o l o g i s t s . Some of t h i s modeling employs what are known as automata. T h i s chapter w i l l survey the use of automata with p a r t i c u l a r r e f e r e n c e t o the r e p r e s e n t a t i o n of propagation. And, j u s t as the l a s t chapter explained the workings of n a t u r a l r e p r o d u c t i o n at a l e v e l s u i t e d t o the n o n - b i o l o g i s t , some p r e l i m i n a r y background on automata theory w i l l be provided here f o r the b i o l o g i s t or philosopher l a c k i n g same. But even before launching i n t o that p r e p a r a t o r y m a t e r i a l , i t i s a good idea t o get s t r a i g h t on what kind of models we are d e a l i n g about. Models of vs Models f o r Lewontin (1973) makes an important d i s t i n c t i o n between Chapter models of and models f o r , a d i s t i n c t i o n he b e l i e v e s b i o l o g i s t s have sometimes f a i l e d t o make, r e s u l t i n g i n c o n f u s i on. The l a t t e r kind of model d i s p l a y s a l o g i c a l r e l a t i o n or o p e r a t i o n isomorphic t o one a l l e g e d l y present i n some designated n a t u r a l b i o l o g i c a l form or process. The Watson-C r i c k model f o r the s t r u c t u r e of DNA i s a p e r f e c t example. Such models are contingent; they are tantamount t o hypotheses, i n th a t they are s u b j e c t t o c o n f i r m a t i o n or d i s c o n f i r m a t i o n . To o f f e r such a model i s t o make an a s s e r t i o n , t r u e or f a l s e , e.g. about the geometrical r e l a t i o n s of the components of the DNA molecule. In t h i s sense they are the same as maps of geographical areas. To put one i s to make an a s s e r t i o n about what i s and what i s not the case i n the area, e.g. th a t Puget Sound opens onto the P a c i f i c . In c o n t r a s t , the model of i s i l l u s t r a t i v e of some possibi1 i c y or i d e a l case. Such models are analytic. They are not t e s t a b l e by comparing the a r t i f a c t with e m p i r i c a l phenomena. Such models can only be confirmed i n the sense that a l o g i c a l form i s judged v a l i d , a machine o p e r a t i o n a l , or a program i n t e r n a l l y c o n s i s t e n t . The a p p r o p r i a t e d i s t i n c t i o n having been made, i t w i l l now be understood t h a t , whereas the previous chapter might have a l l u d e d t o models for n a t u r a l r e p l i c a t i o n , the present Chapter w i l l be e n t i r e l y devoted to models or' r e p l i c a t i o n . Thus they are q u i t e u n l i k e the Watson-Crick model, which could be shown to be t r u e or -false. A l l t h a t w i l l be asked o-f the modeling l o g i c s we w i l l be l o o k i n g a t , l i k e Typogenetics (Chapters V and V I ) , or the automata presented below, i s t h a t they be well de-fined, c o n s i s t e n t , and a b l e t o demonstrate i n t e r e s t i n g propagative phenomena. Automata , The term "automaton" probably b r i n g s t o many a mind the image of a mechanical device; say, one of those clockwork r o b o t i c f i g u r e s t h a t emerges from c e r t a i n 17th century European clocktowers t o s t r i k e chimes on the hour. However, something more a b s t r a c t i s meant by the term here. As a f i r s t approach t o the t e c h n i c a l sense, an automaton can be considered t o be the schema f o r a machine. But even t h a t i s not saying enough; f o r a b l u e p r i n t too i s a schema f o r a machine, and an automaton i s not a b l u e p r i n t ( i . e . not a r e p r e s e n t a t i o n of the s p a t i a l s t r u c t u r e of a machine), but r a t h e r a schema f o r the l o g i c a l form of the operation of a mach in e . J u s t as t h e r e were b u i l d i n g s before there were b l u e p r i n t s , there were machines long before there was automata theory. In e f f e c t , the need f o r such a t h i n g as automata theory was only r e a l i z e d when machines—looms, c a l c u l a t o r s and cash r e g i s t e r s , etc.—became extremely Chapter IV 52 complicated, and could be more c o n v e n i e n t l y thought about symbolically than as p h y s i c a l hardware. Or again, i t should be s t r e s s e d — w h e n t h e i r opera t ions became so complicated (a d e v i c e with an e l a b o r a t e b l u e p r i n t but simple -function probably wouldn't i n t e r e s t an automata t h e o r i s t ) . More or l e s s i n c i d e n t a l l y t o the study and design o-f such d e v i c e s i t was r e a l i z e d t h a t any machine t h a t does anything can be represented by an automaton. Thus the concept of an automaton came t o be recognized as extremely g e n e r a l — s o general that i t i s no longer thought necessary that an automaton correspond t o an actual ( p r e e x i s t i n g ) or even possible—for-th is—mor Id p h y s i c a l d e v i c e . J u s t as Leonardo Da V i n c i drew b l u e p r i n t s f o r machines that cannot work on E a r t h (e.g. h i s corkscrew h e l i c o p t e r ) , automata are invented and considered today without regard t o t h e i r p h y s i c a l v i a b i l i t y , so long as they possess some i n t r i n s i c l o g i c a l i n t e r e s t . E f f e c t i v e Procedures and Algorithms Contemporaneous with the r i s e of a b s t r a c t machine (automata) theory (turn of the c e n t u r y ) , effective procedures became an o b j e c t of l o g i c a l i n t e r e s t . Some procedures seem t o be of a kind that they can be performed with absolute r e l i a b i l i t y — t o undertake them i s t o meet with c e r t a i n success ( b a r r i n g extraneous p r a c t i c a l d i f f i c u l t i e s , such as i n t e r r u p t i o n , e t c . ) . Sorting f o r example, rods by length so as to impose an order on them from s h o r t e s t to longest Chapter IV 53 (within some agreed upon range o-f e r r o r o-f measurement), can be done e f f e c t i v e l y ; s i m i l a r l y , numbers can be s o r t e d i n t o , e.g. odd and even. We take our own (human) a b i l i t y t o perform e f f e c t i v e procedures l i k e s o r t i n g r a t h e r f o r granted. But a p r e s c h o o l e r or feeble-minded person might have t o be told e x a c t l y how t o s o r t . The e x a c t l y s p e c i f i a b l e sequence of o p e r a t i o n s t h a t suffices t o capture an e f f e c t i v e procedure i s c a l l e d an algorithm. T y p i c a l l y , as i n the case with s o r t i n g , t h e r e i s more than one a l g o r i t h m c o n c e i v a b l e t o perform an e f f e c t i v e procedure. And now we reconnect t o automata theory, much of which c o n s i s t s of s p e c i f y i n g a l g o r i t h m s to achieve e f f e c t i v e procedures—machines being "dumber" than f e e b l e -minded persons, they must have a t a s k ' s performance l a i d out f o r them i n f u l l d e t a i l . Consider the design of a simple d e v i c e , a music box supposed t o p l a y " D i x i e . " The d e s i g n e r ' s task i s to arrange a s e r i e s of c o n s t r a i n t s t h a t w i l l i n e x o r a b l y b r i n g about the p l a y i n g of that a i r . In a t y p i c a l model, there i s a t u r n i n g c y l i n d e r with p i n s s t i c k i n g out of i t , p i n s p o s i t i o n e d i n j u s t such a way that each w i l l , without f a i l , pluck one among an immobile comb—like row of tuned bars the c y l i n d e r r e v o l v e s by. Of course, a designated pin must s t r i k e not j u s t any tuned bar, but t h a t one corresponding to the r i g h t p i t c h . And the p l u c k i n g must occur, not j u s t any time, but i n keeping Chapter with the c o r r e c t temporal sequence d e f i n i n g the musical p i e c e . Thus t h e r e must be a 1 t o 1 correspondence between the s p a t i a l r e l a t i o n s h i p s of the p a r t s of t h i s machine and the musical r e l a t i o n s h i p s d e f i n i n g the musical composition t o be played. Now, the s e r i e s of events t h a t c o n s t i t u t e s the p l a y i n g of the tune—-or, r e a l l y , the l o g i c a l s c h e m a t i z a t i o n of that s e r i e s of e v e n t s — i s an a l g o r i t h m f o r p l a y i n g " D i x i e . " In the case of a t y p i c a l one-tune music box the automaton a l l o w s no input (aside from the wind-up). But a pl a y e r piano (I am using examples a p p r o p r i a t e t o the era th a t gave b i r t h t o automata theory) accepts d i f f e r e n t r o l l s . It can p l a y many d i f f e r e n t tunes. Another way t o put t h i s i s t o say t h a t i t i s one automaton capable of executing many-alg o r i t h m s , when provided with the r i g h t i n p u t s . There w i l l be one a l g o r i t h m f o r performing " D i x i e , " and another f o r performing "Camptown Races." More advanced s t i l l i s a f u l l y programmable automaton, which can accept not only such p a s s i v e data as a p l a y e r piano r o l l as inp u t , but input i n the way of instructions. A computer i s such a d e v i c e . . I t i s t r u l y protean: New input can gi v e i t i n c r e d i b l y new ranges of behavior. The a r t and sc i e n c e of the computer programmer i s t o expand the r e p e r t o i r e of that v e r s a t i l e automaton's algorithms. Most computing i n v o l v e s combining of a v a r i e t y of Chapter IV 55 a l g o r i t h m s , which programmers have attempted to encode so as to r e s u l t i n the s m a l l e s t , most e f f i c i e n t and powerful program developable. T h e r e f o r e , f i n d i n g ways of packaging a l g o r i t h m s i n a s i m p l i f i e d way, would be a boon t o t h a t f i e l d . Something t o Aim For An application of the design l o g i c we are working towards i n t h i s t h e s i s would enable a computer programmer t o input a simple, i n f o r m a t i o n a l l y compact seed s of a program, where s i s capable of doing o n l y one s p e c i a l t h i n g , t h a t i s , of s eeing t o i t t h a t i t "grows" or transforms through r e f l e x i v e o p e r a t i o n s i n t o an e l a b o r a t e new program p capable of behaving i n a v a r i e t y of ways, which perhaps e v e n t u a l l y too metamorphizes i n t o yet a d i f f e r e n t descendant program p%, possessing f u t h e r c a p a b i l i t i e s . We w i l l i n l a t e r chapters see how strange a r t i f i c i a l e n t i t i e s , s o r t of h a l f DNA strand models, and h a l f computer programs, c a l l e d "Typogenetic s t r a n d s , " can pack i n t o one i n d i v i d u a l ' s s y n t a c t i c a l form an e n t i r e f a m i l y t r e e of descendant s t r a n d s t h a t can, along some branches, be i n f i n i t e . A Typogenetic s t r a n d i s l i k e a s i n g l e piano p l a y e r r o l l t h a t metamorphizes " m i r a c u l o u s l y " (by design!) through one tune a f t e r another. Some keep metamorphizing u n t i l t h e i r r e p e r t o i r e i s exhausted and the l i n e " d i e s . " Others keep Chapter IV 56 metamorphizing u n t i l the o r i g i n a l form, the " o r i g i n a l tune" i f we speak of piano r o l l s , recurs to s i g n a l r e b e g i n n i n g of a f i x e d c y c l e . Others beget metamorphizing l i n e s of descent without end. But we w i l l not on l y see t h a t such e n t i t i e s have been found. The d i s t i n c t i v e l y l o g i c a l bent of t h i s t h e s i s i s manifest i n the care taken t o g e n e r a l i z e and e x p l i c i t l y f ormulate usable general methods t h a t have proven t h e i r e f f e c t i v e n e s s . There i s , f o r example, a c a n o n i c a l p r o c e d u r e — a m o u l d — g i v e n f o r r o u t i n e manufacturing of s t r a n d s embodying i n f i n i t e "cascading" s t r a n d - l i n e s . An o u t l i n e f o r a program i n Appendix IV e x p l a i n s how t o generate an i n f i n i t e a n c e s t r a l l i n e f o r any given s t r a n d . There i s no more c h a r a c t e r i s t i c product of formal s c i e n c e , than the elegant u n i t y of a r e c u r s i v e formula or other general scheme that has encapsulated some variety. Perhaps i t i s to be expected then that (at l e a s t some) formal s c i e n t i s t s would e v e n t u a l l y have taken s e r i o u s l y the evident s i m i l a r i t y between a r e c u r s i v e generation of the d i g i t s of p i and a seed propagating an endless l i n e of descendant seeds, and have sought t o use something l i k e the former t o model or even i n s t a n c e something l i k e the l a t t e r . We next w i l l see how formal s c i e n t i s t s attempt t h i s with automata. Chapter S e l f - R e p l i c a t i o n with Automata For many persons, the term "automaton" n e c e s s a r i l y connotes inanimate. And automata theory began with study and design o-f conventional machines -few would h e s i t a t e t o c a t e g o r i s e as inanimate. Some such machines roughly i m i t a t e d organism!c behavior though. One can v i s u a l i z e a p l a y e r piano that i s equipped with a mechanical, man—shaped pseudo—"piayer" t h a t moves i t s hands about j u s t above the keys while, i n s i d e , the piano p l a y e r r o l l i s "read" by i n t e r i o r " t r a n s l a t i o n " machinery and executed as p h y s i c a l s t r i k i n g o-f piano s t r i n g s . Looks l i k e a human p l a y e r , and p l a y s l i k e a human p l a y e r ! Again, ingenious d e s i g n e r s had automatic looms weave p a t t e r n s l i k e s p i d e r s weave webs, and mechanical b i r d s " s i n g i n g " l i k e b i r d s It was apparently obvious -from very e a r l y on that automata theory might become an important c o n t r i b u t o r t o the modeling o-f l i - f e processes. The Greeks r e a l i z e d t h i s p o t e n t i a l back i n mythical times, i n the legendary r o b o t s of Daedalus. The stronger idea t h a t a l i v i n g organism " i s " a machine has been popular at l e a s t s i n c e the Enlightenment. One i m p l i c a t i o n of t h i s equation would seem to be that i t would be a p p r o p r i a t e t o c a l l upon automata theory t o f u l f i l l the dream of c r e a t i n g a r t i f i c i a l l i f e . The equation a d d i t i o n a l l y i m p l i e s the converse: that there i s no e s s e n t i a l difference Chapter between an organism and a machine, excepting a connotation that the l a t t e r i s much simpler and more i n f l e x i b l e than the former, a d i m i n i s h i n g connotation i n the age of computer technology. Descartes, f o r one, propounded the idea t h a t o r g a n i c bodies belong i n the same o n t o l o g i c a l category as mechanical e n t i t i e s (though he c a r e f u l l y avoided mechanizing away the human soul r e q u i r e d by Church dogma). While employed as a t u t o r of Queen C h r i s t i n a of Sweden, however, h i s p u p i l r a i s e d a question: How could organisms be machines, i n view of the f a c t t h a t they reproduce, which no machine was known t o be capable of doing? C h r i s t i n a ' s question might be r e s t a t e d as an i n v i t a t i o n f o r engineers t o s e t t l e an o n t o l o g i c a l question. Is s e l f -r e p l i c a t i o n a mechanizable process? A c t u a l l y , t h a t way of p u t t i n g i t i s vague. It could mean at l e a s t two t h i n g s : (1) Could s e l f - r e p l i c a t i o n worthy of the name occur within a r i g i d l y deterministic automaton—universe? (2) Could i t be the f u n c t i o n of an automaton t h a t i t could somehow replicate itself"? R e s u l t s suggesting a f f i r m a t i v e answers to both these questions w i l l be considered below. Yet another question t h a t probably corresponds most c l o s e l y of a l l to C h r i s t i n a ' s , would be "Could a p h y s i c a l machine reproduce i t s e l f ? " Because that v e r s i o n may well turn on what p h y s i c a l law a l l o w s , i t Chapter I w i l l not be pursued here (the p h y s i c i s t , E. Wigner, wrote a paper arguing, with some r e s e r v a t i o n s , t h a t on the b a s i s of quantum and s t a t i s t i c a l mechanics, s e l f - r e p r o d u c t i o n i s not p h y s i c a l l y p o s s i b l e — s e e Wigner, 1961). Our q u e s t i o n s have been posed, yet s t i l l more background must be provided before they can be d i r e c t l y addressed. Turing Table Automata The Turing Table i s a v e r s a t i l e l o g i c a l t o o l , well known to automata t h e o r i s t s . It o r i g i n a t e d i n connection with the Turing Machine (about which more w i l l be s a i d l a t e r ) — a n automaton d e s c r i b e d by Alan Turing (1937). The machine. The T u r i n g Table i s simply an automaton represented i n a t a b u l a r format. It presupposes a p r o c e s s i n g u n i t ("machine") capable of being i n f i n i t e l y many separate s t a t e s ; those s t a t e s ( q i , q=,...) are l i s t e d on the l e f t s i d e of the t a b l e . D i f f e r e n t t a b l e s might make d i f f e r e n t demands on the machine intended t o execute them; one might r e q u i r e a machine capable of 6 s t a t e s , another a machine capable of 6,OOO. A human mind, perhaps aided by p e n c i l and paper, can serve as the processor executing a simple Turing Table, but a complicated t a b l e can only p r a c t i c a l l y be executed by a p h y s i c a l computer. The tape. The machine i s to be thought of as r e c e i v i n g input on a program "tape." The tape i s a sequence of squares; Chapter again, depending on the Table i n v o l v e d , t h i s sequence may have to be s h o r t , long, or even i n - f i n i t e . A square can be empty, or i t can c o n t a i n a symbol. The machine has a "window th a t d i s p l a y s one square on the t a p e — t h e c u r r e n t one. The machine can read what, i f anything, i s i n that square. The machine executing the T u r i n g Table i s a l l o w e d — t h a t i s , e x p e c t e d — t o be very focused and a l s o u n t i r i n g l y p e r s i s t e n t . I t s span of apprehension i s no wider than the current square and i t s r e s i d e n t memory long enough on l y to r e c a l l the most r e c e n t l y given o p e r a t i o n or command. (The tape though, s u p p l i e s a p o t e n t i a l l y i n f i n i t e e x t e r n a l memory). The sequence of symbols i n the squares of the tape comprise the program the automaton i s going to run; the a l l o w a b l e vocabulary of symbols i s l i s t e d a c r o s s the top of the t a b l e . Oper at ions. The c u r r e n t s t a t e of the machine, and the c u r r e n t symbol i n the window ( i f any) together are the c o n d i t i o n s c o n j o i n t l y mandating what oper at ion the machine should perform. The o p e r a t i o n must s p e c i f y t h r e e t h i n g s : the next s t a t e the machine should be i n ; the next square on the tape that should be read; the change i f any to be made i n th« tape. An o p e r a t i o n then might be: Go to machine s t a t e q 2; move the window to the l e f t one square and read what i s i n that square; erase whatever i s i n that square. Reading the Table. A t a b l e can thus be made: As s t a t e d , Chapter I on the le-ft column are the p o s s i b l e s t a t e s the machine can be i n ; on top, the p o s s i b l e symbols t h a t could be i n the window being c u r r e n t l y read. At the i n t e r s e c t i o n of the machine s t a t e row and the symbol column, i s l i s t e d the o p e r a t i o n the machine should remember and perform. Table 1 i s a sample Turing Table. Table 2 i s a step by step t r a c i n g of a "run" of the automaton d e f i n e d by Table 1 . The input i s a s t r i n g of the allowed symbols, haObb. By convention #3 the f i r s t square of the tape t o be viewed by the "machine" i n i t s p r o c e s s i n g of t h i s input s t r i n g i s the square c o n t a i n i n g the l e f t m o s t punctuator symbol, h. By convention #2 the machine s t a r t s o f f i n s t a t e q x . Thus one w i l l f i n d the next o p e r a t i o n t o be executed by l o o k i n g at the i n t e r s e c t i o n of the t a b l e ' s h column and qx row. That o p e r a t i o n i s q^Rb, which means: "Switch t o machine s t a t e q^ ,; move the window over one square t o the Right, and, only i f th a t square i s empty (blank), i n s e r t the symbol b." The o p e r a t i o n i s performed (keep i n mind that the effect of the o p e r a t i o n l i s t e d i n l i n e n i s shown at l i n e n + 1 ) ; the next l i n e (2) i n our t r a c i n g of the run shows that the machine i s indeed i n s t a t e q,^ . The window's present p o s i t i o n over, the s t r i n g i s shown with an u n d e r l i n e , and i t i s thus c l e a r t h at the "move r i g h t " command has been c a r r i e d out. No b has been i n s e r t e d t h o u g h — s i n c e that square was not, as r e q u i r e d , empty. Again the machine looks up the next Chapter o p e r a t i o n , t h i s time at the i n t e r s e c t i o n o-f q<^  and a. The corresponding o p e r a t i o n i s found t o be q^Ra. A c c o r d i n g l y , the machine switches t o ( a c t u a l l y , remains at) s t a t e q^ ,, moves r i g h t one square on the tape, a n d — t h i s square being empty— e n t e r s the a symbol. The reader can h o p e f u l l y now f o l l o w the remainder of the run t o i t s end. Not t h a t one can always do t h a t . Many input s t r i n g s t r i g g e r a non-term in at in g run of t h i s p a r t i c u l a r automaton. A f t e r some p o i n t the automaton f a l l s i n t o , say, an "add b t o the r i g h t " c y c l e t h a t i t cannot escape from. Because such i n f i n i t e loops are r a t h e r commonplace i n the world of automata, Turin g machines are, u n l e s s otherwise e x p l i c i t l y s t a t e d , assumed t o have access t o i n f i n i t e tapes, on which they may enter symbols from here t o e t e r n i t y . Chapter TABLE 1. TURING TABLE h a b O q^ ,Rb qxL- q*La q^.Pa qxPO qiRa q 3 L a q 3Pb q=LO HALT q=Ra q 3Rh q* q i P - q 3P- q=La HALT q= qaRh q»RO HALT q 3Rh q* q=Ra q<-,Ra q-»Rb HALT Symbols. A tape square may be blank (symbolized with the O p l a c e h o l d e r where h e l p f u l t o r e a d e r ) ; or may c o n t a i n a, 6, or the punctuator symbol, h. Operat ions. P means window remains i n Pl a c e . R means move window to Right one square. L means move window t o L e f t one square. — means i n v e r t a t o 6 , or b to a; (leave h or 0 a l o n e ) . A window command (P, R or L) fo l l o w e d by a symbol (a, b, or h means i n s e r t t h a t symbol i n the square moved t o , but only i f that square i s empty. A window command (P, R, or L) followed by a 0 means erase or leave square moved to blank. Con vent ions. #1. A well—formed input s t r i n g c o n s i s t s of the allowed symbols and/or blanks, enclosed i n a p a i r of h punctuators ( p o s i t i o n e d at l e f t and r i g h t ends). #2. The machine always begins a run i n s t a t e q i . #3. The f i r s t square read of an input s t r i n g i s the leftm o s t h punctuator. # 4 . Upon t e r m i n a t i o n of a machine run, the tape i s parsed i n t o s u b s t r i n g s , with s e p a r a t i o n made at every h punctuator; r e s u l t a n t s u b s t r i n g s are enclosed i n h punctuators and then outputted. Contiguous rt's are t r e a t e d as one. Thus abOahahhh separates i n t o output s t r i n g s habOah and hah. Chapter IV 64 TABLE 2. SAMPLE RUN OF TURING TABLE AUTOMATON Run Current Tape; Current Next L i n e Machine Windowed Square Operation S t a t e Underlined 1 q* haObh q^Rb •—> haObh q&Ra 3 q*. haabh q^Ra 4 q*. haabh q&Rb 5 q*. haabh qoRa 6 q= haabha q=RO 7 q= haabhaO q 3Rh 8 q 3 haabhaOh q 2LO 9 q= haabhaOh q=sPb 10 q 3 haabhabh q^Ra 11 q=a haabhabh qxPO 12 qi haabhabO q-aPa 13 q* haabhaba q 3P-14 qs haabhabb q=Ra 15 q= haabhabba qiRa 16 q i haabhabbaa q*L-17 qi haabhabbba q*La 18 q* haabhabbba q=La 19 q= haabhabbba qsLa 20 q=s haabhabbba HALT By convention #4, the termi n a l tape content separates at the punctuator p o i n t (the h) -for o u t p u t t i n g as the two punctuator enclosed s t r i n g s haabh and habbbah. Chapter I A Self-Reproducing Program Now t h a t the reader has learned t o -follow a Turing Table's run, he may wish t o experiment with p r o c e s s i n g d i f f e r e n t input s t r i n g s . Then, t o t e s t the depth of h i s understanding, he might t a c k l e t h i s q u estion: Is there a " s e l f - e r a s i n g " input s t r i n g ? That would be an input s t r i n g which c o n s i s t s of something more than blanks and punctuators, but which t r i g g e r s a p r o c e s s i n g run t h a t e v e n t u a l l y terminates l e a v i n g nothing but blanks on the tape. <An e q u i v a l e n t question i s asked and answered f o r the formal system, Typogenetics, i n Chapter V I ) . Be t t e r s t i l l , the reader could t r y to f i n d a s t r i n g such t h a t , once i t has been i n p u t t e d i n t o t h i s T u r i n g Table automaton, the r e s u l t i n g run's output c o n s i s t s of two ident ical copies of the s t r i n g . The reader may a s c e r t a i n t h at i t i s not immediately obvious whether t h i s s t r i n g e x i s t s , nor what i t s i d e n t i t y i s , i f i t does. But i f such a s t r i n g could be found, i t s e x i s t e n c e might be thought t o answer the ( f i r s t ) question posed e a r l i e r as t o whether s e l f - r e p l i c a t i o n can be r e a l i z e d within an automaton "environment." In f a c t , t here i s s t r i n g answering to t h i s d e s c r i p t i o n : habbbah—which, s i g n i f i c a n t l y , happens t o be one of the o f f s p r i n g of the sample input s t r i n g we saw above, haObh ( a l e r t i n g us t o the p o s s i b i l i t y t h a t an e n t i t y might not Chapter I i t s e l f be a s e l - f - r e p l i c a t o r , but have a s e l - f - r e p l i c a t o r among i t s "descendants"). I t s complete run i s recorded i n Table 3. Is i t not c l e a r , on seeing t h i s program run, how c l o s e l y T uring Table programming resembles the programming o-f a music box or p l a y e r piano? A Guiding I n t u i t i o n A r e s u l t i s e x h i b i t e d i n Table 3. It i s not "the" p r i n c i p a l r e s u l t of t h i s t h e s i s , but i t does a f f o r d an e x c e l l e n t o p p o r t u n i t y t o r a i s e q u e s tions, and o c c a s i o n s some ponderings on how d i f f e r e n t d i s c i p l i n e s of the formal s c i e n c e s operate. What i s an interest in g and re levant result! The f i r s t q uestion might be: Should the kind of s t r i n g d u p l i c a t i o n we've j u s t seen count as an i n s t a n c e of s e l f — r e p l i c a t i o n ? Or i s t h i s r e s u l t , t o put i t c o l d l y , " t r i v i a l " ? It i s c e r t a i n l y a good time t o s t a r t t h i n k i n g about c r i t e r i a f o r e v a l u a t i n g the "genuineness" of propagative phenomena—something we may have taken f o r granted i n surveying what i s known of b i o l o g i c a l phenomena. While i t seems best to postpone d i s c u s s i o n of t h i s matter (see Chapter IX) u n t i l all our " r e s u l t s " are on the t a b l e , I w i l l i n v i t e the reader to share some i n t u i t i o n s t h a t have guided my t h i n k i n g and which may guide h i s through the next chapters. But f i r s t , a l i t t l e more about our r e s u l t . If one experiments f o r a while, i n p u t t i n g randomly Chapter TABLE 3, STRING REPLICATION Run Current Tape; Current Next L i n e Machine Windowed Square Operation S t a t e Underlined 1 qi habbbah q^Rb q*> habbbah q«»Ra 3 q& habbbah q^Rb 4 habbbah q<-,Rb c' q*. habbbah q^Rb 6 q*» habbbah q^Rb 7 q<=» habbbah q=»Ra 8 q= habbbaha q=sR0 9 q=» habbbahaO q 3Rh 10 q 3 habbbahaOh. q^LO 11 q= habbbahaOh q 3Pb 12 q 3 habbbahabh q 3Ra 13 q= habbbahabh qxPO 14 q i habbbahabO q^-Pa 15 q* habbbahaba q 3P-16 q 3 habbbahabb q^Ra 17 q= habbbahabba qiRa 18 q i habbbahabbaa qxL-19 habbbahabbba q*La 20 q-v habbbahabbba qzLa 21 qz habbbahabbba q 3 L a 22 q 3 habbbahabbba HALT By convention #4, the tape contents output as two punctuator enclosed s t r i n g s , habbbah and habbbah, that happen to be i d e n t i c a l t o each other and to the o r i g i n a l input s t r i n g . Chapter IV 68 chosen s t r i n g s i n t o the Table 1 Tu r i n g automaton, one w i l l v e r i f y t h a t the r e p l i c a t i o n we saw i n the case of habbbah i s exceptional—that i s , whether the reader t r i e s 10 such s t r i n g s , or 1,000 of them, I w i l l expect none (excepting habbbah i t s e l f ! ) t o induce a run r e s u l t i n g i n output of two c o p i e s of the input s t r i n g . Now, an i n t u i t i o n t h a t I t h i n k r e l e v a n t here, s t a t e d c r u d e l y , says: " I f any o l d s t r i n g gets r o u t i n e l y d u p l i c a t e d i n or by some system, you're t a l k i n g about production or replication, not reproduction or self—replication. Such an automaton (or environment) i s l i k e a photocopier t h a t d u p l i c a t e s any o l d document; and t h a t i s not an exemplary i n s t a n c e of r e p r o d u c t i o n i n the sense i n which we are i n t e r e s t e d ! But i f o n l y s p e c i a l , r e l a t i v e l y r a r e s t r i n g s get d u p l i c a t e d by a given automaton, then maybe you've got something." Well, on t h a t i n t u i t i o n , the r e s u l t i s n o n t r i v i a l — a s remarked not j u s t any o l d s t r i n g gets d u p l i c a t e d by t h i s automaton. But w a i t — t h e r e i s a second i n t u i t i o n I wish to share. Again s t a t e d c r u d e l y , i t says "I would think that s e i r - r e p l i c a t i o n means the reproduced item's own p a r t i c u l a r structure or s p e c i a l p r o p e r t i e s , somehow must be r e s p o n s i b l e f o r guiding or s u p p l y i n g the operators that perform the assembly or copying that r e s u l t s i n r e p r o d u c t i o n . " To understand t h i s , consider the f o l l o w i n g . When you Chapter IV f i r s t saw the run, you might have been impressed with the way the machine moved around an the tape, changing symbols, changing them back, a l l q u i t e u n p r e d i c t a b l y (to you), u n t i l , amazingly, with the job done, i t "knew" when t o h a l t (short of wrecking what i t had accomplished). And s i n c e the o p e r a t i o n s t h a t are c a r r i e d out depend on what the window reads o f f the s t r i n g , i t might have seemed l i k e habbbah's unique u n i t sequence was,responsible f o r the sequence of o p e r a t i o n s t h a t c r e a t e d i t s twin. Yet your i n d u c t i v e i n v e s t i g a t i o n s may have r e v e a l e d the f a c t t h at the s t r i n g habbbah i s a frequent product of t h i s automaton. In other words, th e r e are q u i t e a few s t r i n g s you can input i n t o t h i s t h i n g t h a t w i l l r e s u l t i n the o u t p u t t i n g of habbbah (among other, d i f f e r i n g s t r i n g s ) . And i f you t r a c e d these d i f f e r e n t runs, you saw t h a t the c r e a t i o n of habbbah i s done i n the same way each time. Does that not put matters i n a d i f f e r e n t l i g h t ? It now begins to appear that Table 1 d e s c r i b e s a v e r i t a b l e habbbah—generating automaton. If that i s so, perhaps i t i s l i t t l e more than a funny coincidence that habbbah, qua input s t r i n g , r e s u l t s i n the output of two habbbah^. In other words, i t h a r d l y seems l i k e the structure of habbbah has anything s p e c i a l about i t . On the c r i t e r i o n t h a t a self-replicating e n t i t y ' s s t r u c t u r e should o r c h e s t r a t e the a c t i v i t y r e s u l t i n g i n r e p l i c a t i o n , our " r e s u l t " does not hold up. Chapter What may be surmised -from a l l t h i s perhaps i s t h i s : A l l e g e d i n s t a n c e s o-f s e l - f - r e p l i c a t i on may be c a s t i n t o doubt by e s t a b l i s h i n g e i t h e r (1) that most any a r b i t r a r y e n t i t y gets r e p l i c a t e d i n the environment i n q u e s t i o n , or (2) t h a t the environment r o u t i n e l y produces c o p i e s o-f the a l l e g e d l y s e l T-repl i c a t i ng e n t i t y even i n the absence o-f "seed" c o p i e s o-f t h a t e n t i t y . It i s hoped the r a t h e r vague and t e n t a t i v e o b s e r v a t i o n s made above do possess some value , and w i l l be kept i n mind by the reader through the next c h a p t e r s , u n t i l they can be returned to (Chapter IX). Approaches t o Problem S o l v i n g I see t h i s t h e s i s as having two r a t h e r di-f-ferent " s p e c i a l i s t " audiences: One r e s u l t - o r i e n t e d , p r i m a r i l y i n t e r e s t e d i n understanding propagative phenomena. The other, t e c h n i q u e s - o r i e n t e d , i n t e r e s t e d i n how a formal s c i e n c e l i k e l o g i c can shed l i g h t on something l i k e propagation. T h i s l a t t e r group might well have been l e f t unmoved by the f o r e g o i n g d i s c u s s i o n on the "genuineness" of the r e s u l t , because they are not so i n t e r e s t e d i n the meaning of the r e s u l t as they are i n what made the r e s u l t —whatever i t " r e a l l y " i s — a c h i e v a b l e . They might have had t h e i r i n t e r e s t aroused by the very concept of a Turing Table, and perhaps are t r y i n g t o r e c o n s t r u c t the "abductive" reasoning (see Chapter Chapter IX) processes t h a t designed the automaton, or the problem s o l v i n g steps taken to -find the r e p l i c a t i n g s t r i n g habbbah-To both groups I w i l l now address some o b s e r v a t i o n s about how I see the r e s p e c t i v e d i s c i p l i n e s doing the "formal s c i e n c e s " — t h i s of course being a p r e l i m i n a r y t o f o c u s i n g i n on what a logician can be expected t o c o n t r i b u t e i n coming chapters. What would a mathematician do with the automaton d e f i n e d by Table 1? That i s , what very c h a r a c t e r i s t i c a l l y mathematical approach might he take t o understanding t h i s automaton? One t h i n g he might do would be to employ the technique of godel—number ing. H o f s t a d t e r (1979) does a b e a u t i f u l job e x p l a i n i n g how godel-numbering works. S u f f i c e t o say here t h a t a mapping i s invented that p l a c e s the symbols of our automaton i n 1 to 1 correspondence with numbers. Now, f o r every s t r i n g t r a n s f o r m a t i o n during the automaton's run, there i s a corresponding numerical t r a n s f o r m a t i o n . The (probably q u i t e complex) mathematical f u n c t i o n that holds between the set of numbers corresponding on the mapping t o the s e t of input s t r i n g s and the set of numbers corresponding on the mapping to the s e t of output s t r i n g s t h e r e f o r e r e p r e s e n t s the automaton i n question. Using t h i s r e p r e s e n t a t i o n the problemf e.g. of finding another string that self—replicates, becomes a truly mathematical Chapter one; A search for the right argument-However, given the mathematician's p r e f e r e n c e f o r g e n e r a l i t y , I would more expect him t o produce a theorem as to the f i n i t u d e (or i n f i n i t u d e ) of t h a t s et of arguments, than t o see him a c t u a l l y a r r i v e at a s p e c i f i c , i d e n t i f a b l e s e l f - r e p l i c a t i n g s t r i n g . And we can be sure t h a t he w i l l not make any s u r f e i t d i s c l o s u r e of the how of h i s modus operandi that would l e a d our minds t o easy emulation of h i s achievements, f o r t h i s proud s c i e n c e demures t o "set down all one's rea s o n i n g . " • f course, the computer science approach would i n v o l v e implementing the Table 1 T u r i n g automaton i n a computer program. That simple e x e r c i s e would then provide him with a t o o l , a way t o r u l e out hypotheses on a posteriori grounds. He can now, i n other words, see what runs. The "brute f o r c e " way to f i n d a d d i t i o n a l s e l f -r e p l i c a t i n g s t r i n g s f o r t h i s automaton i s to s y s t e m a t i c a l l y generate each next s t r i n g , run i t through the (computerized) automaton, and check the output t o see i f d u p l i c a t i o n has occurred. But the computer s c i e n t i s t would frown on such an arrangement; though i t might work i n p r i n c i p l e , i t s u r e l y could not work i n the r e a l world (too many s t r i n g s ) . He would seek the intelligent t h i n g t o do, which would be to f i n d ways to screen out unpromising s t r i n g s i n a maximally economical way, and t o otherwise Chapter prune the oroblam tree as much as p o s s i b l e . The computer s c i e n t i s t might seek al g o r i t h m s that s h o r t c u t the ( o r i g i n a l ) automaton's s t r i n g p r o c e s s i n g a l g o r i t h m s (generating the same output i n fewer s t e p s ) . In t h i s endeavor, one t h i n g he would s u r e l y n o t i c e i s the r e c u r r i n g p a t t e r n that c r e a t e s habbbah—the movement of the window over the tape that s t a r t s l e f t , goes to the r i g h t end, adds and i n v e r t s a few symbols, then comes back l e f t u n t i l i t reaches a H A L T . That p a t t e r n of o p e r a t i o n s might be c o n c e p t u a l l y " l i f t e d " out of the automaton, f o r study; understanding how to make a s t r e a m l i n e d v e r s i o n of i t might provide f u r t h e r c r i t e r i a f o r p r e - s c r e e n i n g candidate s t r i n g s to be run through the automaton f o r r e p l i c a t i o n . Cogn it ive psychologists have yet another agenda. Researchers ( i n the t r a d i t i o n , say, of Simon and Newell) might do something l i k e t urn t h i s automaton over to human s u b j e c t s with t h i s charge: "Now that you see how s e l f -r e p l i c a t i o n with t h i s automaton works, t r y to f i n d another s t r i n g that does the t r i c k . " From c o l l a t i o n of the s u b j e c t s ' w r i t t e n i n t r o s p e c t i o n s ( " p r o t o c o l s " ) , and through i n t e r r o g a t i o n , they would t r y to r e c o n s t r u c t how "the human mind" frames (and re-frames) t h i s problem space; how i t r e c o g n i s e s progress (or lack of same) towards i t s goal; what s o r t of operations (mental "moves") i t employs; a n d what s o r t of heuristics ( r u l e s of thumb) i t b r i n g s to the problem. Some Chapter IV 74 c o g n i t i v e p s y c h o l o g i s t s would a l s o be i n t e r e s t e d i n i d e n t i f y i n g commonly o c c u r r i n g i n s t a n c e s o-f fallacious i n f e r e n c e (e.g. c i r c u l a r reasoning) or l e v e l s c o n f u s i o n evidenced by t h e i r s u b j e c t s . At l e n g t h , a computer s i m u l a t i o n model might be produced that i s supposed t o " t h i n k " about the problem of f i n d i n g more s e l f — r e p l i c a t i n g s t r i n g s the same way people do. T h i s s i m u l a t i o n would be judged a success o n l y i f i t f a i l e d where people (the s u b j e c t s ) f a i l and succeeded where people succeed. And, f i n a l l y , what of the logician'? I see h i s approach as d i s t i n c t from a l l the f o r e g o i n g , and three—pronged i n nature. F i r s t , I expect him t o i n v e s t i g a t e s y n t a c t i c a l p r o p e r t i e s of the known s e l f - r e p l i c a t o r f o r c l u e s as to what makes i t special. He would n o t i c e , f o r example, t h a t habbbah i s a pallindrome (reads the same backwards or forwards). Thus the hypothesis that being p a l l i n d r o m i c i s necessary or s u f f i c i e n t f o r — o r at l e a s t i n some i n t e l l i g i b l e way related to—self—replication, would bear checking out. True, the Turing Table i t s e l f does not appear to be symmetrical; and the known s e l f - r e p l i c a t o r ' s z ig—zagging run does not make any obvious r e l i a n c e on i t s p a l l i n d r o m i c s t r u c t u r e . These f a c t s discourage, but do not e l i m i n a t e , the hypothesis. Symmetries are sometimes well hidden, and s t r u c t u r e / p r o c e s s dependencies well d i s g u i s e d . Therefore onne would be wise to run through Chapter some t e s t pal 1indromes. From such t e s t s ( t r y , e.g. hbbbaabbbh), i t w i l l be concluded t h a t the p a l l i n d r o m i c property i s not a s u f f i c i e n t one f o r s e l f — r e p l i c a t i o n . Of course no l o g i c i a n would mistake s u f f i c i e n t f o r necessary c o n d i t i o n s , and the p o s s i b i l i t y of the l a t t e r remains tenab l e So the l o g i c i a n attempts t o o b t a i n f u r t h e r r e s u l t s i n h i s own way. Yet that i s not a l l there i s to i t — p e r h a p s not even the main t h i n g . The l o g i c i a n i s i n t e r e s t e d i n methods of reasoning. Sometimes—though he may not admit i t — h e uses himself as a s u b j e c t i n somewhat the way that the c o g n i t i v e p s y c h o l o g i s t uses others. However, u n l i k e the p s y c h o l o g i s t , he i s not out to document h i s thought processes, such as they were (he would not, f o r example, thi n k memory c o n s t r a i n t s worth heeding), but t o i d e n t i f y what the course of reasoning ought t o be. From out of the fragments of consciousness, the e r r o r s , wanderings and g e s t a l t i n s i g h t s of h i s mind, he somehow ( r e — ) c o n s t r u c t s something s t r e a m l i n e d , sure, i d e a l i z e d , a g e n e r a l i z a b l e , valid l i n e of thought or technique o t h e r s can f o l l o w i n t h e i r p s y c h o l o g i c a l l y imperfect way, to a c o n c l u s i o n or s o l u t i o n . T h i s communicable product of h i s normative s c i e n c e he might embody i n a symbolic n o t a t i o n f o r c l a r i t y , p r e c i s i o n , and f u r t h e r manipulation or i n c o r p o r a t i o n i n t o a system. F i n a l l y , t o be a l o g i c i a n i s to be a meta - 1ogician. That means he proves t h i n g s about the systems (and l o g i c s ) he Chapter works with, i n a d d i t i o n t o or i n s t e a d of f i n d i n g r e s u l t s in those systems. A m e t a — l o g i c a l r e s u l t , f o r example, might e s t a b l i s h the f a c t t h at t h e r e i s a well—formed s t r i n g t h a t can never be outputted by t h i s automaton ( i . e . i s no s t r i n g ' s "descendant"). Obviously, such a p r o p o s i t i o n must be proved by some way other than comprehensively l i s t i n g the outputs y i e l d e d by the p r o c e s s i n g of the e n t i r e i n f i n i t u d e of w e l l -formed s t r i n g s ! I t might be proved (say, with a redactio ad absurdum argument) without ever i d e n t i f y i n g which s t r i n g i s the p e c u l i a r one, and t h a t would be f i n e with the l o g i c i a n , who i s s c a r c e l y l e s s general and a b s t r a c t than the mathematician. In t h i s t h e s i s , a logic t h e s i s , abundant examples of these t h r e e kinds of l o g i c i s i n g w i l l be found. But most of those r e s u l t s w i l l be obtained, not f o r our humble Turing Table, but f o r that r i c h e r , more i n t e r e s t i n g formal system, described p r e v i o u s l y , Typogenetics, a system which n e v e r t h e l e s s does have a l o t i n common with our Turing Table. As f o r mathematics, computer s c i e n c e or c o g n i t i v e p s y c h o l o g y — h a v i n g given them t h e i r due—we b i d them adieu. Chapter S t a h l ' s T u r i n g Table Enzyme Si m u l a t i o n s Thus f a r , t h i s chapter has been f a r removed from the world of e m p i r i c a l b i o l o g y p r e v i o u s l y examined. The present s e c t i o n , however, w i l l b r i n g automata theory back t o b i o l o g y . Though many had p r e v i o u s l y spoken of c e l l u l a r enzyme a c t i o n i n terms of automata metaphors, Walter R. Stahl (Stahl and Soheen, 1963; S t a h l , 1965a; S t a h l , 1965b; S t a h l , 1967) and c o l l e a g u e s were the f i r s t t o undertake s i m u l a t i o n of enzymatic processes o p e r a t i n g on biochemical s t r i n g s using the T u r i n g Table format. Again r e f e r r i n g t o the d i s t i n c t i o n between models of and models f o r , S t a h l ' s s i m u l a t i o n s d i d not a c t u a l l y t r y t o p r e c i s e l y mimic known biochemical processes, but were meant to be i d e a l i z e d models of same. H i s e n t e r p r i s e i n v o l v e d c r e a t i n g d i f f e r e n t Turing f u n c t i o n a l t a b l e s , one f o r each of a number of enzymes. One t a b l e , f o r example ( h i s Table 3, S t a h l and Goheen, 1963), i s an analogue t o a polymerase copying enzyme. In a d d i t i o n t o the Turing Tables or l o g i c a l o p e r a t o r s , h i s encompassing computer environment pro v i d e s r e s e r v o i r s f o r the s t r i n g s , conventions f o r r a t e c o n t r o l s and an o v e r a l l c o n t r o l executive which determines op e r a t i o n sequence. (Like a l l other e n t i t i e s i n t h i s and the next chapters, Stahl s "enzymes," e t c . , are p u r e l y a b s t r a c t , i . e . nonchemical). U n f o r t u n a t e l y , because h i s s i m u l a t i o n s were embodied i n Chapter I complex programs running on a main-frame computer, the present author was unable t o s c r u t i n i z e or e v a l u a t e h i s s i m u l a t i o n s -first—hand. S t a h l , o-f course, s u p p l i e d g e n e r a l , d e s c i p t i v e i n f o r m a t i o n i n h i s p u b l i s h e d works. In h i s words: "...a s t r i n g - p r o c e s s i n g c e l l model c o n s i s t s of a set (40 t o lOO) of s t r i n g - p r o c e s s i n g enzyme automata. ... With i t one can simulate general b i o - s y n t h e s i s , end-product i n h i b i t i o n , i n d u c t i o n and r e p r e s s i o n mechanisms, v a r i o u s kinds of chain copying, membrane b a r r i e r p r o p e r t i e s ( i n c l u d i n g a c t i v e t r a n s p o r t ) , and other t y p i c a l c e l l u l a r enzyme a c t i v i t i e s . "The main design philosophy has been t o have at l e a s t one enzyme automaton c o n t r o l each b a s i c d i f f e r e n t kind of general a c t i v i t y i n the c e l l — e . g . , n u c l e o t i d e s y n t h e s i s , l i p i d s y n t h e s i s , RNA polymerase a c t i o n , ribosome a c t i o n , energy pro d u c t i o n , e t c . Conformation or general o r g a n e l l e a c t i o n , as by ribosomes, i s a l s o modeled by the standard s t r i n g -p r o c e s s i n g automata, with or without use of energy, as a p p r o p r i a t e . " S t a h l , 1965a, pp 381-382. Reproduction. Reproduction i n S t a h l ' s model i s r a t h e r d i f f e r e n t from the e a r l i e r presented s t r i n g d u p l i c a t i o n example: more complex, and l e s s p r e c i s e . Here the idea i s f o r one " c e l l " t o manufacture enough i n g r e d i e n t s f o r another " c e l l . " The emphasis i s on c y b e r n e t i c s of production mechanics. "...each gene-enzyme automaton f u n c t i o n s i n t u r n and i s Chapter implemented by about 600 t o 1,000 Turing code i n s t r u c t i o n s . If i t s s p e c i f i c c o n t r o l c o n d i t i o n s are met, mRNA or enzyme i s generated. When produced, the enzymes themselves then a c t , with consumption of s u b s t r a t e s and energy. T h i s s o r t of pro c e s s i n g continues f o r hundreds of thousands of s y n t h e t i c s t e p s , u n t i l a l l the c e l l p r o t e i n s and s u b s t r a t e s have been b u i l t up t o a l e v e l c o n s i s t e n t with r e p r o d u c t i o n ( m i t o s i s ) , as determined by a complex i n t e r l i n k i n g of gene and enzyme c o n t r o l c o n d i t i o n s . When m i t o s i s i s p o s s i b l e , a s p e c i a l s e t of c o n t r o l conventions t u r n s o f f almost a l l the enzymes and i n i t i a t e s copying of DNA and s p l i t t i n g of r e s e r v o i r contents i n t o two approximately equal p a r t s ( i t must be kept i n mind t h a t small random f a c t o r s a re added t o a l l generation r a t e s ) . On completion of m i t o s i s , the genes are r e s t o r e d t o a " s t a r t i n g " c o n d i t i o n . The r e s u l t i n g daughter c e l l - s t r i n g i s found t o be i d e n t i c a l with i t s parent and capable of reproducing i t s e l f i n d e f i n i t e l y when placed i n a U n i v e r s a l Turing Machine ( s p e c i f i c a l l y , the TASP simulator) together with the automaton coding tape." S t a h l , 1965a, p383. Stahl views h i s achievements as having c o n t r i b u t e d t o the understanding of c e l l u l a r homeostasis. But he a l s o a n t i c i p a t e s h i s l i n e of research as l e a d i n g t o something l i k e the present t h e s i s , noting that " a n a l y s i s of a l g o r i t h m i c enzymes op e r a t i n g on biochemical s t r i n g s , as des c r i b e d above, r a i s e s a number of problems that are of c o n s i d e r a b l e i n t e r e s t Chapter t o students o-f mathematical l o g i c . It may even be the case that study of organismal l o g i c w i l l be p r o v o c a t i v e f o r the f u t u r e development of computation theory." (Stahl and Boheen, 1963, p284). St a h l (1965b) drew loose p a r a l l e l s between c o m p u t a t i o n a l / l o g i c a l problems (e.g. the " h a l t i n g problem") and those cropping up i n the s i m u l a t i o n of c e l l u l a r processes. Among the l o g i c a l problems he mentions i s R u s s e l l ' s Paradox ("Is the s e t of a l l s e t s t h a t are not members of themselves a member of i t s e l f ? " ) , but o f f e r s no c l o s e analogue t o th a t paradox i n the realm of "organismic l o g i c . " However, i f the reader proceeds t o the next s e c t i o n of t h i s chapter, he w i l l f i n d e x a c t l y such an analogue, i n the s e c t i o n on the "Paradoxical Machine"; another, more f u l l y developed analogue appears i n Chapter VI ("the Par a d o x i c a l S t r a n d " ) . Chapter Von Neumann An important pioneer worker i n the a b s t r a c t modeling o-f r e p l i c a t i o n was mathematician/automata t h e o r i s t J . von Neumann, About 1950 he became i n t e r e s t e d i n the second o-f the two b a s i c questions asked e a r l i e r : Can an automaton itself (as opposed t o some s t r i n g or p a r t o-f the automaton) make more of i t s own kind? (As w i l l be seen, he a c t u a l l y wound up co n v e r t i n g t h i s question i n t o a form of the f i r s t q u estion, by imbedding h i s reproducing automaton i n s i d e another automata complex). • f course i t was t r u e then, as i t was t r u e r e s p e c t i n g a r t i f a c t s back i n N e o l i t h i c times, and i s t r u e today, t h a t , though machines may make other machines, no machine s e l f -r e p l i c a t e s . But t h a t f a c t could have a p u r e l y p r a c t i c a l e x p l a n a t i o n , and von Neumann wanted t o know what was p o s s i b l e i n p r i n c i p l e . H i s e f f o r t s t o answer t h i s question s et the tone and d i r e c t i o n f o r r e s e a r c h e r s i n the f i e l d f o r a f u l l g e n e r a t i o n . In the beginning von Neumann labored f o r some time with the s c e n a r i o of a lake stocked with machine p a r t s , seeking t o d e s c r i b e a device capable of s e l e c t i n g needed f r e e — f 1 o a t i n g components that i t could put together i n t o a r e p l i c a of i t s e l f . The l o g i s t i c a l d e t a i l s , e.g. of how to a c t u a l l y r e c o g n i z e , pluck out and manipulate the p a r t s , were most vexatious and consumed more a t t e n t i o n than he Chapter -felt they deserved. The r e s u l t a n t " d e m o n s t r a t i o n " — t h a t such a machine was p o s s i b l e — w a s as u n s a t i s f y i n g t o von Neumann as to o t h e r s . F o r t u n a t e l y , he d i d not leave i t at t h a t . Von Neumann was a d i v e r s e genius, and h i s involvement i n other p r o j e c t s turned out t o d e c i s i v e l y shape h i s c o n t i n u i n g work on theory o-f r e p l i c a t i o n . As an automata t h e o r i s t he had been impressed by the work o-f Tu r i n g (1937), who had shown t h a t a c e r t a i n e n t i t y , now termed a " u n i v e r s a l T u r i n g machine," c o n c e p t u a l l y q u i t e simple, having access t o an i n e x h a u s t i b l e memory, or "tape," c o u l d compute anything t h a t c o u l d be computed i n p r i n c i p l e . Put a l t e r n a t i v e l y , as Tu r i n g T a b l e s have a l r e a d y been presented, the u n i v e r s a l T u r i n g machine i s an automaton provably capable o-f -formally i m i t a t i n g ("emulating") the behavior of any T u r i n g Table automaton. The U n i v e r s a l Machine's S e l f - R e p l i c a t i o n Armed with t h i s concept of the u n i v e r s a l T u r i n g machine, von Neumann's t h i n k i n g moved along the f o l l o w i n g l i n e s t o a r r i v e at an i n t e r e s t i n g c o n c l u s i o n (von Neumann a c t u a l l y d i e d b e f o r e f i n i s h i n g h i s proof; i t was completed and pub l i s h e d by Burks; c i t e d as von Neumann, 1966. A h i g h l y f o r m a l i z e d v e r s i o n of von Neumann's argument, with some exte n s i o n s , i s i n M y h i l l , 1964). It seems t h a t any machine's construction ought t o be p r e s c r i b a b l e i n terms of a well d e f i n e d sequence of Chapter o p e r a t i o n s . T h i s sequence of o p e r a t i o n s would be (at l e a s t , on Church's Thesis) l o g i c a l l y e q u i v a l e n t t o an a l g o r i t h m . Any al g o r i t h m "runs" on a u n i v e r s a l T u r i n g machine. Now suppose th a t a u n i v e r s a l T u r i n g machine were endowed with a un iver sal constructor t h a t could a c t u a l l y c a r r y out those o p e r a t i o n s . If the o p e r a t i o n s were s u f f i c i e n t l y p r i m i t i v e and st a n d a r d i z e d , such a u n i v e r s a l c o n s t r u c t o r should be f e a s i b l e , t h e o r e t i c a l l y (even though one might have t o s p e l l out the o p e r a t i o n s at a molecular, atomic, or even subatomic l e v e l of r e l a t i o n s ) . Then the r e s u l t a n t " u n i v e r s a l machine" UM ( u n i v e r s a l T u r i n g machine with u n i v e r s a l c o n s t r u c t o r ) could c o n s t r u c t any p o s s i b l e machine, because i t could compute and c a r r y out any machine—construction algorithm. Among other t h i n g s , t h a t means, i f the r e are any p o s s i b l e s e l f - r e p l i c a t i n g automata, the UM can b u i l d them. Now, i n the present context, the next move i s probably obvious: UM i t s e l f , qua machine, i s c o n s t r u c t i b l e ; an algo r i t h m would e x i s t s p e c i f y i n g the op e r a t i o n s needed t o b u i l d i t . UM then ought t o be abl e t o run and execute the alg o r i t h m corresponding to i t s own c o n s t r u c t i o n . The I n f i n i t e Regress Paradox of S e l f - R e p l i c a t i o n T h i s i s an ingenious diagonal argument, but there i s s t i l l more. The u n i v e r s a l machine can only output a machine i f i t i s p r o p e r l y programmed t o do so; i t i s not defined as knowing e v e r y t h i n g , but of being able to run and execute any Chapter a l g o r i t h m given proper i n p u t s (programming). So, to b u i l d machine m the u n i v e r s a l machine would need some inp u t . The input might be the a l r e a d y completely e x p l i c i t r e c i p e f o r b u i l d i n g the machine, or i t might be such as t o r e q u i r e a great deal of computation on the p a r t of the machine. But i t would have t o be, at bare minimum, s u f f i c i e n t input t o enable the u n i v e r s a l machine t o determine i n a f i n i t e number of st e p s the exact sequence of o p e r a t i o n s needed tD b u i l d the t a r g e t machine. Now, when i t was s a i d the u n i v e r s a l machine can b u i l d a copy of i t s e l f , i t was presupposed t h a t t h e r e was that minimal b i t of input s u p p l i e d t o the p r o g e n i t o r , p e r m i t t i n g i t t o determine the sequence of s t e p s i t f o l l o w e d i n c o n s t r u c t i n g the descendant. But what about the descendant? Without the same i n p u t , or program, i t i s ignorant of how to b u i l d a copy of i t s e l f — i s not, i n the f u l l e s t sense then, a s e l f — r e p l i c a t o r . S ince s e l f — r e p l i c a t i o n i s t r a n s i t i v e , ergo, the o r i g i n a l u n i v e r s a l machine too was not s e l f — r e p l i c a t i n g . C l e a r l y , the o r i g i n a l u n i v e r s a l machine must be s u p p l i e d with an input program t h a t e x p l a i n s how to b u i l d a universal machine already pre—programmed to build another universal machine. But, i s n ' t there something v i c i o u s about t h a t ? There seems t o be a program p t h a t d e s c r i b e s a machine m—wi th—program—p. A p w i t h i n a p? That opens up an e n t i r e i n f i n i t e r e g r e s s , s i n c e the p wi t h i n the p must too Chapter have a p w i t h i n i t , and so -forth. Von Neumann thought t o avoid t h i s r e g r e s s i n a manner analogous t o how nature works. R e c a l l the d i s t i n c t i o n between transcr iption and translation. The " t r i c k " t o a v o i d i n g the r e g r e s s i s t o handle the o r i g i n a l program p i n two ways. T h i s can be c o n c e p t u a l i z e d by t h i n k i n g o-f the sel-f-r e p l i c a t i n g machine as having separate subcomponents, one of which obeys (computes and c a r r i e s out the c o n s t r u c t i o n algorithm) the program, the other which copies the program p and a t t a c h e s t h i s copy t o the newly c o n s t r u c t e d machine. The c o p i e r subcomponent, once i t r e c e i v e s p's command to s t a r t copying, goes t o work copying p u t t e r l y o b l i v i o u s t o the i n s t r u c t i o n s i n p j u s t as the enzymes t h a t copy DNA during i t s r e p l i c a t i o n do so with complete d i s r e g a r d of what the DNA t r i p l e t s code f o r . Now i t i s unnecessary t o t h i n k of p as c o n t a i n i n g a p; r a t h e r , p simply has one step that says, "begin copying the tape," i t so happening that p i s the program on the tape. E v a l u a t i n g the Argument Von Neumann's i d e n t i f i c a t i o n of the i n f i n i t e r e g r e s s problem, and h i s demonstration of how i t can be avoided, has a me t a - t h e o r e t i c a l value i n t h a t i t shows that r e p r o d u c t i o n need not i n v o l v e i n f i n i t i e s , thus need hot be " i l l o g i c a l " or " p a r a d o x i c a l " i n that way. I t s t h e o r e t i c a l value i s that i t g i v e s an an a n a l y t i c a l account as t o how the job can be Chapter IV 36 done ( i . e . i n t e r p r e t p - f i r s t as a program to be c a r r i e d out and then as u n i n t e r p r e t e d date t o be copied) t h a t can a c t u a l l y guide those who set out to achieve sel-f-rep 1 i cat i o n . A number of authors have c r e d i t e d von Neumann with having proved t h a t r e p r o d u c t i o n -for an automaton i s p o s s i b l e . Poundstone (1985) d e c l a r e s t h a t "von Neumann showed t h a t t h e r e was no magic i n se l - f - r e p r o d u c t i o n , that the exact process could be s p e l l e d out and programmed i n t o a machine with a c e r t a i n minimum l e v e l of complexity. By i t s e l f , von Neumann's work made no s p e c i f i c a s s e r t i o n s about b i o l o g y . Reproduction of l i v i n g organisms might s t i l l i n v o l v e an imponderable l i f e f o r c e . Von Neumann d i d , however s t r i k e down the argument t h a t s e l f - r e p r o d u c t i o n must proceed by supernatural means." pl90 C e r t a i n l y , von Neumann drew out a deep i m p l i c a t i o n l a t e n t i n Church's T h e s i s and the concepts of the u n i v e r s a l Turing computer and the u n i v e r s a l c o n s t r u c t o r (which presupposes that d i g i t a l machine components can be i d e n t i f i e d out of which any c o n c e i v a b l e machine can be assembled). However, von Neumann's proof s t i l l lends no j u s t i f i c a t i o n t o the i d e a that physical machines can s e l f -reproduce (s i n c e he d i d not do the very c o n s i d e r a b l e work of e x p l i c a t i n g e x a c t l y how the u n i v e r s a l c o n s t r u c t o r works i n the p h y s i c a l world, how i t comes by i t s raw m a t e r i a l s , e t c . ) ; nor d i d i t shed much l i g h t on b i o l o g y . I t must be s a i d that Chapter IV 87 would-be de s i g n e r s of A r t i f i c i a l L i f e seeking t o invent new s e l f - r e p r o d u c i n g e n t i t i e s , and b i o l o g i s t s , s tudying e x i s t i n g , n a t u r a l s e l f - r e p r o d u c i n g e n t i t i e s , are not going t o f i n d anything of p r a c t i c a l value f o r t h e i r r e s p e c t i v e t a s k s , on studying von Neumann's proof. A P a r a d o x i c a l Machine? L o g i c i a n s , however, w i l l d e l i g h t i n the i n g e n u i t y of h i s argument. They might even be i n s p i r e d t o c o n s i d e r some p o s s i b i l i t i e s von Neumann overlooked. A machine t h a t can c o n s t r u c t any machine, i n c l u d i n g i t s e l f ? Then how about a machine t h a t only c o n s t r u c t s every machine that cannot c o n s t r u c t i t s e l f ? T h i s l a t t e r d e v i c e could produce automobiles, r a d i o s , o r d i n a r y computers, a l l the kinds of machines we now make, p l u s many more not yet conceived. Only, i t s d e v i s e r s d i d not want i t producing any of those p e c u l i a r machines that make more of t h e i r own kind (those tending t o exc e s s i v e p r o l i f e r a t i o n ) . Does t h i s seem any l e s s p l a u s i b l e than von Neumann's machine? Before answering, consider t h i s question: Can t h i s machine c o n s t r u c t i t s e l f ? If i t c o n s t r u c t s i t s e l f , then i t makes at l e a s t one s e l f — c o n s t r u c t o r (which i t was not, by d e f i n i t i o n , supposed t o do). If i t does not, i t f a i l s to produce "every" n o n — s e l f — c o n s t r u c t o r , as i t was defined t o do. Diagonal arguments have a d i s c o n c e r t i n g way of lea d i n g t o paradox! T h i s i s of course an analogue to R u s s e l l ' s Paradox. Yet another analogue t o i t w i l l be t r e a t e d Chapter IV 88 i n Chapter VI. Rosen's Paradox Rosen (1959) was bothered by what he thought o-f as a question—begging aspect of von Neumann's proof. He s t a t e d a paradox t h a t c a s t s doubt on the very p o s s i b i l i t y of s e l f — r e p l i c a t i n g automata of any s o r t . It i s a paradox which has never, i n t h i s author's o p i n i o n , been e f f e c t i v e l y countered. Si n c e Moore (1966) attempted to d i s m i s s the paradox with a remark made i n passing t o the e f f e c t t h a t Rosen's paradox f a i l e d t o d i f f e r e n t i a t e between the imbedded (see next s e c t i o n s ) automaton supposed t o s e l f — r e p l i c a t e and the c e l l u l a r automaton s e r v i n g as medium, i t i s reformulated below i n a c o n c i s e and general form, making no r e f e r e n c e t o whether i t i s imbedded i n another automaton, that precludes any such d i s m i s s a l . That Rosen c l e a r l y meant to s t a t e a problem of extremely general r e l e v a n c e i s apparent i n h i s remark: "This paradox i s apparently inherent i n any attempt to formulate the n o t i o n of s e l f - r e p r o d u c i n g automaton; hence the very notion of such automata appears to c o n t a i n an i n t e r n a l i n c o n s i s t e n c y . " pp 388-389. Any automaton i s a b s t r a c t l y e q u i v a l e n t to a mapping r : A—>B where A i s the set of a l l p o s s i b l e i n p u t s , B the set of a l l p o s s i b l e outputs. Now, a s e l f — r e p r o d u c i n g automaton outputs itself. So i t s s e t B would i n c l u d e r , s i n c e f i s none other than t h a t automaton. But we e v i d e n t l y can only d e f i n e B Chapter IV 89 when we know f , and at the same time can onl y d e f i n e f when we know B. Rosen concluded t h a t " i f t h i s paradox cannot be r e s o l v e d , then...the e x i s t e n c e of a s e l f — r e p r o d u c i n g automaton i s a l o g i c a l i m p o s s i b i l i t y . " Guttman's (1966) " r e s o l u t i o n " of Rosen's paradox was l i t t l e more than a s i d e s t e p . He i n e f f e c t d e c l a r e d t h a t p e r f e c t s e l f - r e p l i c a t i o n i s not p o s s i b l e i n n a t u r a l systems, because of the law of entropy, hence the paradox i s onl y a l o g i c i a n s ' d i s t r a c t i o n . Another kind of " r e s o l u t i o n " i s simply t o say t h a t the s e l f — r e p l i c a t o r r e q u i r e s a s p e c i a l kind of set theory which does not i n s i s t t h a t a mapping cannot be de f i n e d without i t s range. In t h i s author's o p i n i o n Rosen's "paradox" i s n e i t h e r a paradox nor a dilemma; nor does i t come down to how one does set theory. Rosen has simply misframed a problem space. T h i s i s d e a l t with i n depth i n Chapter IX. In any case, von Neumann's u n i v e r s a l c o n s t r u c t o r i s not p r a c t i c a l l y r e a l i z a b l e i n the present t e c h n o l o g i c a l era. F o r t u n a t e l y , s e l f — r e p l i c a t i n g e n t i t i e s i n nature do not have u n i v e r s a l c o n s t r u c t i n g nor u n i v e r s a l computing power, which encourages us to b e l i e v e that s e l f — r e p r o d u c i n g automata can get by with l e s s e r c a p a c i t i e s as w e l l . The next step a f t e r von Neumann had to be t o "come down to e a r t h " and i d e n t i f y Chapter the lower bounds f o r complexity of s e l f - r e p l i c a t i n g e n t i t i e s . We w i l l only o b t a i n nuts and b o l t s understanding of propagation by i n v e n t i n g and studying a c t u a l , demonstrable propagating e n t i t i e s . Von Neumann and C e l l u l a r Automata Von Neumann had, as mentioned, played a key r o l e i n des i g n i n g the new " l o g i c machines," computers, one of which was a v a i l a b l e t o workers at Los Alamos Laboratory. Among these workers was the mathematician Ulam, who experimented with the new machine's c a p a c i t y t o carry out huge numbers of c a l c u l a t i o n s i n s h o r t order, t o c r e a t e what he c a l l e d " r e c u r s i v e geometric s t r u c t u r e s . " A l a r g e number of i t e r a t i o n s of a few simple but well-chosen r u l e s generated complex s t r u c t u r e s , and these s t r u c t u r e s were o f t e n anything but what one would have expected i n advance of the act u a l running of the program. Ulam had di s c o v e r e d then a way to very simply d e f i n e a p o t e n t i a l l y very complex and u n p r e d i c t a b l e world. He suggested t h a t von Neumann t r y h i s luck i n c r e a t i n g a s e l f -r e p l i c a t o r w i t h i n such an a r t i f i c i a l world. The problem now would be p u r e l y l o g i c a l , not l o g i s t i c a l . Von Neumann accepted t h i s suggestion, and invented the f i r s t cellular automaton ( a l s o c a l l e d , but seldomly now, a " t e s s e l a t i o n s t r u c t u r e " ) . T h i s kind of automaton w i l l be e x p l i c a t e d f u l l y below. But i t i s perhaps well here t o f i n i s h Chapter IV 91 accounts with von Neumann. The s o l e motive -for t u r n i n g t o the c e l l u l a r automaton would seem t o be that i t o f f e r e d an o p p o r t u n i t y f o r one to engineer a s p e c i f i c s e l f — r e p l i c a t i n g e n t i t y without concern f o r " r e a l world" c o n s t r a i n t s , p a r t i c u l a r l y r e s p e c t i n g the u n i v e r s a l c o n s t r u c t o r . In t h i s l o g i c a l u n i v e r s e , computation is c r e a t i o n ; t h e r e i s no need f o r the a d d i t i o n a l step of p h y s i c a l p r o d u c t i o n . But von Neumann was s t i l l under the s p e l l of h i s " s e l f - r e p l i c a t i n g u n i v e r s a l machine." Rather than c o n s t r u c t a s p e c i f i c , demonstrably s e l f - r e p l i c a t i n g e n t i t y what he d i d was t o show that a l l the l o g i c a l p r i m i t i v e s f o r a u n i v e r s a l computing/constructing machine c o u l d be imbedded i n a c e l l u l a r automaton; t h i s imbedded u n i v e r s a l machine would then be capable of s e l f — r e p l i c a t i o n according t o the argument alre a d y covered above (von Neumann, 1966). A l i t t l e more w i l l be s a i d as t o what i s meant by an imbedded (or " v i r t u a l " ) computer below; but a good enough id e a of i t f o r present purposes can be gained from t h i s analogy: On my personal computer I can run a l i t t l e program which d i s p l a y s on my screen a picture of a c a l c u l a t o r . Using my "mouse" device I can use t h i s v i r t u a l c a l c u l a t o r j u s t as one would use a p h y s i c a l c a l c u l a t o r : I can press i t s buttons and o b t a i n readouts on the c a l c u l a t o r ' s l i t t l e d i s p l a y screen. T h i s c a l c u l a t o r i s imbedded w i t h i n a l a r g e r c a l c u l a t o r my personal computer. By the same token the Chapter IV 92 imbedded computer i s a con-figuration w i t h i n the c e l l u l a r automaton. Neither von Neumann nor h i s c o l l a b o r a t o r Burks, nor t h e i r subsequent i m i t a t o r s ever a c t u a l l y put together an imbedded computer i n a c e l l u l a r automaton; they showed t h a t i t s parts could be de-fined. The a c t u a l con-figuration would be too complex and would r e q u i r e too l a r g e a g r i d (see Poundstone's c a l c u l a t i o n s , pp 226-229, 1985). It i s i n t e r e s t i n g t o know t h a t a c e l l u l a r automaton can harbor a u n i v e r s a l computer. Among other t h i n g s t h i s seems t o imply t h a t a part of an automaton can compute anything the whole can compute. If th a t p a r t i s the " b r a i n " of some p s e u d o — l i v i n g organism " i n h a b i t i n g " t h a t l o g i c a l world, then i t shows that " A l " may be a c h i e v a b l e t h e r e i n . However, none of t h i s adds anything f u r t h e r t o our understanding of s e l f — r e p l i c a t i o n — s p e c i f i c a l l y , nothing i s added by imbedding the " s e l f - r e p l i c a t i n g computer" i n a c e l l u l a r automaton. Chapter IV 93 Cel l u l a r Automata Capable o-f P a r a l l e l Processing A cellular automaton ( h e r e a f t e r , c.a.; also known as a tesselation structure) i s a special type o-f automaton supporting parallel processing. A single c a . i s a system composed o-f many " c e l l s , " each one o-f which instances the same primitive automaton. These c e l l s collaborate in the productions o-f the system as a whole. It i s the capacity o-f a c.a. to be at once a unity and a m u l t i p l i c i t y that makes i t so interesting, unpredictable, and useful. Properties of a C e l l u l a r Automaton Two of the four properties defining a c.a. are, f i r s t , the number of states a c e l l may assume, and second, the rules that dictate change of a c e l l ' s state. An individual c e l l in a c.a. can be in one of a fixed number of states (in many computer implementations "states" are indicated by dif f e r e n t colors, e.g. a square of a grid may be green, or red, or blue; in paper and pencil implementations, a state i s t y p i c a l l y designated by a number used nominally, e.g. possible states are "O", "1", and "2"). There i s a temporal dimension along which the c.a. changes. This temporal dimension i s discrete, progressing time frame by time frame. A c e l l in a certain state at time frame 1 can either remain in that state or change to another state at Chapter IV 94 time -frame 2; whether i t remains the same or changes i s a f u n c t i o n of the c.a.'s t r a n s i t i o n r u l e s . These r u l e s are the c.a.'s "laws of nature." The remaining two p r o p e r t i e s d e f i n i n g a c.a. r e l a t e the c e l l s s p a t i a l l y . The tessellation (the ar r a y of c e l l s ) must have a geometry. Often a r e c t i l i n e a r l a t t i c e i s u s e d — o n e can p i c t u r e a checkerboard, with each square a c e l l — b u t t e s s e l l a t i o n s can be more e x o t i c : t h r e e dimensional, bounded back on themselves t o r u s — f a s h i o n , e t c . A molar u n i t i n the c.a. i s the neighborhood. A t y p i c a l two-dimensional neighborhood c o n s i s t s of the c e n t r a l c e l l and i t s n o r t h , e a s t , south and west neighbors. The f u t u r e of a c e l l i s dependent on the s t a t e s of i t s neighboring c e l l s ; a t r a n s i t i o n r u l e has as i t s antecedent c o n d i t i o n the co n j u n c t i o n of the c e l l ' s neighbors' s t a t e s . (Since every c e l l save those on the boundary of a f i n i t e g r i d i s at the center of a neighborhood, neighborhoods o v e r l a p ) . The s i z e of the neighborhood, and the number of d i f f e r e n t s t a t e s a c e l l may take, determines the maximum number of p o s s i b l e d i f f e r e n t neighborhood-sized permutations of c e l l s t a t e s there can be. S p e c i f i c a l l y , i f n i s the number of c e l l s i n a neighborhood, and s i s the number of s t a t e s each c e l l has a v a i l a b l e t o i t , there are s" d i s t i n c t neighborhood-sized permutations of c e l l s t a t e s . As s t a t e d above, each c e l l can be regarded as a d i s t i n c t Chapter automaton- The t r a n s i t i o n r u l e s of the c.a. are common to a l l c e l l s , but one c e l l ' s neighbors w i l l d i f f e r from another's, hence, under the i n f l u e n c e of i t s neighbors, i t may a r r i v e at a d i f f e r e n t computation, which w i l l be r e f l e c t e d i n i t s next s t a t e . As t h i s may go on i n d e f i n i t e l y , the c e l l s , though always i n temporal synchrony, have independent p r o c e s s i n g " c a r e e r s " — t h e r e s u l t i s p a r a l l e l p r o c e s s i n g . Once a c.a. i s d e f i n e d the "user" s p e c i f i e s an i n i t i a l configuration, i . e . s p e c i f i e s the s t a t e s of a l l c e l l s i n the a r r a y at time frame zero. From t h a t p o i n t on the automaton's t r a n s i t i o n r u l e s take over. For each i t h c e l l a d e c i s i o n must be made: what w i l l i ' s next s t a t e be? The d e c i s i o n i s i n each case made according to the one common set of t r a n s i t i o n r u l e s . However, s i n c e t h a t set of r u l e s takes as i t s "input" the present s t a t e s of the c u r r e n t c e l l i s neighbors, and i ' s neighbors are d i f f e r e n t than j ' s neighbors, i ' s next s t a t e may be d i f f e r e n t than j's. LIFE I l l u s t r a t i v e of the c.a. i s "LIFE," invented by Conway ((Berlekamp, Conway, and Guy, 1982), widely a v a i l a b l e on computer software f o r personal computers. I t uses the center— p l u s - e i g h t - a d j a c e n t c e l l s (N/S/E/W and diagonals) neighborhood. In LIFE a c e l l can take only one of two s t a t e s : "0" or "1". I t s t r a n s i t i o n r u l e s are extremely simple (which Chapter IV 96 f a c t accounts -for much o-f i t s ap p e a l ) : For any c e l l , i (1) If three of i ' s neighbors are p r e s e n t l y i n the 1 s t a t e , i w i l l be i n the 1 s t a t e next time frame; (2) If two of i 's neighbors are p r e s e n t l y i n the 1 s t a t e , i r e t a i n s i t s present s t a t e over the next time frame; (3) Otherwise i w i l l be i n the 0 s t a t e next time frame. These r u l e s are p i c t u r e d i n F i g u r e 1. V i r t u a l Automata Many commonly encountered p a t t e r n s i n LIFE have r e c e i v e d names, and t h e i r i n t e r a c t i o n s have been much st u d i e d (see Poundstone, 1985, f o r d e t a i l s and r e f e r e n c e s ) . A s o — c a l l e d " g l i d e r " (Figure 1) i s a p a t t e r n t h a t moves amoeba—like i n a diagonal d i r e c t i o n a c r o s s the LIFE g r i d , one of a c l a s s of locomoting p a t t e r n s g e n e r i c a l l y c a l l e d "spaceships." A more complex c o n f i g u r a t i o n , the " g l i d e r gun," p e r i o d i c a l l y emits g l i d e r s . When a g l i d e r c o l l i d e s with another a b j e c t new p a t t e r n s may r e s u l t , depending on the angle of i n t e r s e c t i o n , the nature of the obj e c t s t r u c k , and the phase the g l i d e r i s i n when contact occurs. I n t e r e s t i n g l y , a p r e c i s e l y choreographed c o l l i s i o n of 13 g l i d e r s can produce a g l i d e r gun—hence a f i n i t e number of g l i d e r s can engender a pa t t e r n that w i l l spawn an i n f i n i t e Chapter IV 97 FIGURE 1: LIFE, A CELLULAR AUTOMATON The i t h c e l l to which the demonstrated t r a n s i t i o n r u l e a p p l i e s i s u n d e r l i n e d . It i s shown i n the context o-f a c e r t a i n neighborhood at time 1, and, i n consequence, the s t a t e i has at time 2. Boundary c o n d i t i o n s have not been de-fined here, so the reader should r e s t r i c t a t t e n t i o n t o the t r a n s i t i o n s o-f the interior c e l l s of each g r i d fragment. Time 1 Time 2 0 0 o 0 0 o 0 0 0 o Rule 1: 0 1 1 1 0 0 1 0 1 0 0 I 1 0 o o 1 0 1 0 0 0 0 o 0 0 0 0 0 0 Rule 2: 0 o 0 o 0 o 0 0 0 0 O o 1 0 0 0 0 1 1 0 0 g 1 1 0 o g 1 1 o 0 0 0 o 0 o 0 0 0 0 Rule 3: 0 0 0 0 0 0 0 0 0 0 o 1 0 o 0 o o 0 0 0 0 0 0 g 1 0 0 0 g 1 0 0 1 l 1 0 o 1 I 1 The s t a t e of i i t s e l f , at time 1, i s ignored by r u l e 1 i n determining i ' s s t a t e at time 2. Rule 2, though, i s a s t a t e p r e s e r v i n g r u l e f o r i , so two d i f f e r e n t i c e l l s are i d e n t i f i e d at time 1, one i n each s t a t e ; at time 2 we see that each such c e l l has r e t a i n e d the s t a t e i t was i n before. Rule 3's two i d e n t i f i e d i c e l l s show that a s t a t e goes to zero i f i t s neighborhood i s underpopulated ( l e s s than 2 on neig h b o r s ) , or overpopulated (more than 3 on neighbors). A g l i d e r p a t t e r n propagating southeastwardly i l l u s t r a t e s a c o l l e c t i v e phenomenon t y p i c a l of c e l l u l a r automata. It w i l l creep southeastwardly u n t i l meeting a boundary, or incompatible new neighbors (perhaps p a r t s of another "spaceship"). On a high speed computer, the g l i d e r shoots d i a g o n a l l y a c r o s s the screen l i k e a d i s t a n t skyrocket; on a low speed, i t oozes through i t s c o n t o r t i o n s l i k e an amoeba. t l t2 t3 t4 t5 o o o o o 0 0 0 0 0 0 0 0 0 0 0 0 0 0 o o o 0 0 o 0 o o o 1 0 o o o 0 0 o o 0 0 0 0 0 o 0 0 o 0 0 o 0 0 0 0 o 0 0 1 0 o 1 0 1 o 0 0 0 1 0 o 0 1 0 0 0 0 o O 1 o o o I 1 1 o 0 0 1 1 o 0 1 0 1 o o o 0 1 1 o 0 0 0 0 1 o 0 0 0 0 o o o 1 o o 0 0 1 1 0 0 0 1 1 0 o o 0 1 1 1 0 0 0 0 0 0 0 0 O 0 o o 0 0 0 o o 0 o o o 0 0 0 0 o o 0 Chapter IV 98 number o-f g l i d e r s (shown i n Poundstone, 106-107) . It i s extremely d i f - f i c u l t t o d i s c o v e r new kinds o-f "spaceships" or other i n t e r e s t i n g l a r g e s c a l e p a t t e r n s i n LIFE, while r e s t r i c t i n g a t t e n t i o n e x c l u s i v e l y t o the l e v e l o-f s i n g l e c e l l s and t h e i r t r a n s i t i o n r u l e s . And i n t e r a c t i o n between such con-figurations (e.g. c o l l i s i o n ) , i s much more e a s i l y learned by experimental o b s e r v a t i o n than by c e l l -l e v e l -based deductions. For these reasons, the l a r g e r con-figurations deserve t o be c a l l e d "emergent" e n t i t i e s . T h i s i s not, o-f course, t o doubt t h a t t h e i r p r o p e r t i e s and i n t e r a c t i o n s can be understood or p r e d i c t e d , i n p r i n c i p l e , a t the c e l l u l a r l e v e l . Langton (1988) r e f e r s t o these "emergent" e n t i t i e s ( g l i d e r s , g l i d e r guns, etc.) as " v i r t u a l " automata, or " v i r t u a l s t a t e machines" (VSMs), and remarks on t h e i r s i g n i f i c a n c e : " C e l l u l a r automata provide us with s e v e r a l h i e r a r c h i c a l l e v e l s of " i n d i v i d u a l s " out of which aggregates may be composed. The f i r s t l e v e l c o n s i s t s of the i n d i v i d u a l c e l l s , each of which i s occupied by the same f i n i t e automaton. What makes CA so i n t e r e s t i n g i s that the g l o b a l behavior supported by a l a t t i c e of such automata i s much more complex and v a r i e d than the sum of the behaviors of the i n d i v i d u a l automata. Propagating s t r u c t u r e s Che uses the term "propagate" i n the p h y s i c a l , r a t h e r than b i o l o g i c a l , s e n s e ] — V S M ' s — c o n s t i t u t e Chapter IV 99 another l e v e l o-f i n d i v i d u a l s . Although they are d i r e c t l y supported by the r i g i d l y -fixed, homogeneous c e l l s o-f the l a t t i c e , VSM's are -free t o migrate around i n the l a t t i c e , c o n s t a n t l y changing t h e i r s e t of neighboring v i r t u a l automata. Furthermore, a homogeneous l a t t i c e of automata can support a heterogeneous p o p u l a t i o n of VSM's, and t h i s p o p u l a t i o n can vary i n s i z e and composition with time, as VSM's are c r e a t e d , modified, and destroyed by processes o c c u r r i n g i n the l a t t i c e . Thus, a r i g i d l y - f i x e d , uniform p o p u l a t i o n can support a polymorphic society of r e l a t i v e l y f r e e — r a n g i n g i n d i v i d u a l s . Higher l e v e l s i n the h i e r a r c h y of i n d i v i d u a l s are due t o the i n t e r a c t i o n of the processes t h a t are c o n s t i t u t e d of i n d i v i d u a l s of lower l e v e l s , i n much the manner of the b i o l o g i c a l h i e r a r c h y of m a l e c u l e s - c e l l s -t i s s u e s — o r g a n s — o r g a n i s m s — s o c i e t i e s and so f o r t h . " pp 14—15. V i r t u a l Computers If there can be " v i r t u a l automata," why not v i r t u a l "computers"? That was of course von Neumann's ide a . Conway has found i n LIFE a l l the c o n f i g u r a t i o n s corresponding t o the necessary p a r t s of a von Neumann imbedded u n i v e r s a l machine. A key t o the computer component of the u n i v e r s a l machine i s the g l i d e r gun, which a c t s as the computer's c l o c k . The emitted g l i d e r s i n t e r a c t with other p a t t e r n s — i n c l u d i n g products of c o l l i s i o n s i n v o l v i n g p r e v i o u s l y emitted g l i d e r s — Chapter IV 100 to c r e a t e new p a t t e r n s . T h i s a c t i v i t y has an interpretat ion i n terms o-f data s t r u c t u r e s and programs i . e . computation. In oversimpl i-f i e d terms, the s e l f - r e p l i c a t i n g u n i v e r s a l LIFE computer emits, at r e g u l a r i n t e r v a l s e s t a b l i s h e d by the p e r i o d i c i t y of g l i d e r guns, a s u c c e s s i o n of spaceships. These spaceships c o l l i d e at some d i s t a n c e from the o r i g i n a l p a t t e r n , producing new p a t t e r n s ("raw m a t e r i a l s " ) that subsequent spaceships c o l l i d e with, s t i m u l a t i n g f u r t h e r c o n s t r u c t i o n . . . E v e n t u a l l y , i f a l l the myriad d e t a i l s have been taken i n t o account, there c o a l e s c e s elsewhere i n the g r i d space a d u p l i c a t e of the o r i g i n a l g l i d e r e m i t t i n g c o n f i g u r a t i o n (again, see Poundstone f o r a more g e n e r o u s l y — o r , depending on your outlook, more t o r t u r o u s l y — d e t a i l e d account). No one has ever a c t u a l l y designed e i t h e r a computing or a s e l f — r e p r o d u c i n g LIFE p a t t e r n i n f u l l d e t a i l . Poundstone c o n s i d e r s i t " b a r e l y c o n c e i v a b l e that someone could someday go to the t r o u b l e " of doing t h a t (p229, 1985). C l a s s e s of C.A. The developments o c c u r r i n g i n a c.a. depend of course on i t s systemic a t t r i b u t e s . For one i n t e r e s t e d i n c r e a t i n g a r t i f i c i a l organisms w i t h i n a c.a. the question i s : which of these c.a. "worlds" t o choose? Wolfram (1984a, 1984b) i d e n t i f i e d four general types of c e l l u l a r automata: Chapter IV 101 C l a s s one automata i n v a r i a b l y evolve t o a homogeneous e q u i l i b r i u m s t a t e ; c h o i c e o-f the s t a r t i n g con-figuration o n l y can make a d i f f e r e n c e as t o how long i t w i l l take t o a t t a i n the end s t a t e . For example, the present author wrote a BASIC program f o r a c e l l u l a r automaton where c e l l s t a t e changes t r i g g e r e d t o n a l p r o d u c t i o n . T h i s "melody-maker" always r e s o l v e d e v e n t u a l l y t o an e q u i l i b r i u m s t a t e — t h e t o n i c. C l a s s two automata i n v a r i a b l y evolve t o a m u l t i p l i c i t y of simple s t r u c t u r e s that may be s t a t i c or p e r i o d i c but c o l l e c t i v e l y comprise what might be c a l l e d a heterogeneous e q u i l i b r i u m s t a t e . C l a s s three automata generate non—random but c h a o t i c developments. These automata "can be a s s o c i a t e d with the more i n t e r e s t i n g e n t i t i e s c a l l e d strange a t t r a c t o r s , which are c h a r a c t e r i s t i c of p h y s i c a l phenomena such as the onset of t u r b u l e n t flow. In a system governed by a strange a t t r a c t o r e v o l u t i o n proceeds toward a subset of a l l the p o s s i b l e c o n f i g u i r a t i o n s , but the subset can have an exceedingly i n t r i c a t e s t r u c t u r e . When the set i s v i s u a l i z e d as an array of p o i n t s i n space, i t i s i n many cases a f r a c t a l , a geometric f i g u r e with a f r a c t i o n a l number of dimensions." (Hayes, 1984). F i n a l l y , c l a s s f o u r automata balance o f f change and e q u i l i b r i u m , a l l o w i n g f o r a v a r i e t y of s t r u c t u r e and Chapter IV 102 growth. L I F E — t h e most s t u d i e d o-f a l l c . a . s — i s among those - f a l l i n g i n t h i s c l a s s . Langton (1986) has been as s e s s i n g the p o t e n t i a l -for occurrence o-f propagating e n t i t i e s i n automata. H i s search has l e d him i n t o the c l a s s -four c.a., "a re g i o n c h a r a c t e r i z e d by the e x i s t e n c e o-f r e l a t i v e l y s t a b l e , -fixed and propagating p e r i o d i c s t r u c t u r e s together with t h e i r i n t e r a c t i o n s , which can support an ongoing dynamic t h a t i s -far from e q u i 1 i b r i u m . . . i n c l u d i n g propagating p e r i o d i c s t r u c t u r e s as molecular o p e r a t o r s with the c a p a c i t y t o operate on themselves as well as on -fixed p e r i o d i c s t r u c t u r e s . " p 130. R e s u l t s Obtained with C e l l u l a r Automata As mentioned e a r l i e r von Neumann's id e a -for a s e l f — r e p l i c a t i n g imbedded u n i v e r s a l machine was taken up by others working with c.a.s (e.g. Codd, 1968; Berlekamp, Conway, and Guy, 1982). At the same time, doubts were r a i s e d as t o one the assumptions u n d r l y i n g the whole proof. Gar den—of-Eden configurations. The f l i p s i d e of a proof of the e x i s t e n c e of s e l f — r e p l i c a t i n g e n t i t i e s i s a proof of the e x i s t e n c e of n o n - s e l f — r e p l i c a t i n g e n t i t i e s . Moore (1964, 1966) proved t h a t there must be some p a t t e r n s — termed "Garden-of-Eden" c o n f i g u r a t i o n s — t h a t cannot occur i n a c.a. except as an i n i t i a l c o n f i g u r a t i o n . Banks worked out a Garden-of-Eden p a t t e r n f o r LIFE (Poundstone, 1985, Chapter I p 50). Since such a p a t t e r n cannot evolve from any p r e v i o u s p a t t e r n or combination of events i n the c.a., i t cannot be produced by any propagating s t r u c t u r e . As a c o r o l l a r y t o t h i s i t f o l l o w s that such p a t t e r n s cannot be s e l f - r e p l i c a t i n g . At f i r s t t h i s may seem almost s i l l y : who ever thought there wouldn't be n o n — s e l f - r e p l i e a t i n g p a t t e r n s i n any i n t e r e s t i n g system? But another c o r o l l a r y of t h i s r e s u l t i s that there can be no universal constructor imbedded in a c.a,. It w i l l be remembered t h a t a von Neumann-type proof of a s e l f -r e p l i e a t i n g u n i v e r s a l machine assumed such a component. To get around t h i s problem l i y h i l l (1964) proposed, i n essence, r e d e f i n i n g the u n i v e r s a l c o n s t r u c t o r as capable of c o n s t r u c t i n g any e v o l v a b l e c o n f i g u r a t i o n , i . e . any c o n f i g u r a t i o n t h a t can a r i s e a f t e r time frame zero. Such a c o n s t r u c t o r i s a l l one a c t u a l l y needs to handle s e l f — r e p l i c a t i o n , s i n c e of course a s e l f — r e p l i c a t o r cannot be a Garden—of-Eden p a t t e r n . Second generat ion results. Where von Neumann and f o l l o w e r s labored to e s t a b l i s h i n - p r i n c i p l e p o s s i b i l i t y of s e l f - r e p l i e a t i n g u n i v e r s a l machines i n c e l l u l a r automata, c u r r e n t , "second generation" r e s e a r c h e r s have concentrated on c o n s t r u c t i n g a c t u a l , working e n t i t i e s . In 1984 Langton, working with a m o d i f i c a t i o n of a c.a. developed by Codd (1968), e x h i b i t e d a s e l f - r e p l i c a t i n g e n t i t y and charted i t s r e s u l t a n t c o l o n y - l i k e p r o l i f e r a t i o n (Langton 1984, 1986). It Chapter IV 104 i s not of the von Neumann type, i . e . does not a s p i r e t o e i t h e r u n i v e r s a l computational or u n i v e r s a l c o n s t r u c t i v e c a p a b i l i t y . T h i s r e s u l t i s worth f u r t h e r examination. Langton's S e l f - R e p l i c a t i n g P a t t e r n A c e l l i n Langton's c.a. has e i g h t s t a t e s , denominated with the numbers 0—7, and the f i v e c e l l neighborhood. That means there are 8=, or 32,768 p o s s i b l e neighborhood c o n f i g u r a t i o n s . Since t h e r e c o u l d be a d i s t i n c t t r a n s i t i o n r u l e t r i g g e r e d by each d i s t i n c t neighborhood c o n f i g u r a t i o n Langton could have had 32,767 d i s t i n c t t r a n s i t i o n r u l e s f o r t h i s system (no r u l e would be t r i g g e r e d by the "quiescent" a l l — 0 s t a t e neighborhood)! By using combinations, i n s t e a d of permutations, he reduced the number of r e q u i r e d r u l e s . However, the t r a n s i t i o n f u n c t i o n r u l e s he has l i s t e d i n h i s publis h e d paper (1984), numbering 219, do not s u f f i c e t o cover a l l combinations. For example, i f a l l the c e l l s i n i ' s neighborhood are i n the 4 s t a t e , there i s no way to t e l l , from the 219 l i s t e d r u l e s , what i ' s s t a t e should be, next time frame. T h i s seems u n s a t i s f a c t o r y ; i n a very r e a l sense h i s c e l l u l a r automaton, as presented t o the world, i s not w e l l - d e f i n e d . The l i s t e d r u l e s do s u f f i c e t o handle a l l neighborhoods a r i s i n g i n the propagation of h i s reproducing patt e r n through an otherwise empty universe. Propagation. Given a row of c e l l s i n s t a t e 1, Chapter IV 105 sandwiched between two rows o-f c e l l s i n s t a t e 2, a c e l l i n the 1 s t a t e " i m i t a t e s " an adjacent neighbor t h a t i s i n a s t a t e >2. Thus a row o-f c e l l s i n s t a t e 1 w i l l be l i k e a row o-f dominos, each one i n t u r n " f a l l i n g " i n t o the same s t a t e as i t s predecessor; the s t a t e 2 c e l l s are l e s s in-f 1 uencable by neighbors, thus p l a y the p a r t of good " i n s u l a t o r s . " Now, i f a s t a t e 7 c e l l i s put at the end of the row of s t a t e i c e l l s , the 7 w i l l propagate down t h a t "data-path". T h i s i s shown below; the s i g n a l i s moving t o the r i g h t . One i s to imagine a checkerboard—1ike g r i d ; c e l l s i n the O s t a t e o u t s i d e the i l l u s t r a t i v e s t r u c t u r e are not shown. 2 2 2 2 2 2 2 2 Time 1 1 1 1 1 0 7 1 1 2 2 2 2 2 2 2 2 Time 2 1 1 1 1 1 0 7 1 2 2 2 2 2 2 2 2 To make the data—path r e u s a b l e (set the dominos back up again) one f o l l o w s the 7 s i g n a l with a 0 s i g n a l (as above). The t r a n s i t i o n r u l e s are such that the row of c e l l s w i l l r e v e r t t o t h e i r o r i g i n a l 1 s t a t e , which makes them amenable t o t r a n s m i t t i n g the next " s i g n a l . " Path extension. A data—path l i k e the one above can be extended i n t o the unoccupied 0 s t a t e t e r r i t o r y beyond the o r i g i n a l s t r u c t u r e . ( I t can a l s o be r e t r a c t e d ) . For example, when a 7 0 s i g n a l h i t s an i s o l a t e d c e l l i n the 2 s t a t e i t converts i t i n t o a 1 s t a t e : Chapter IV 106 Be-fore 2 2 2 2 2 2 A-fter 2 2 2 2 2 2 1 1 1 1 0 7 2 1 1 1 1 1 1 1 Now, when a 6 O s i g n a l a r r i v e s at that newly crea t e d 1 i t extends the data-path: Be-fore 2 2 2 2 2 2 A-fter 2 2 2 2 2 2 2 1 1 1 1 0 6 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 Obviously, a su c c e s s i o n of 7 0 and 6 0 s i g n a l s would p r o g r e s s i v e l y extend a data-path. The loop. The shell of Langton's s e l f - r e p l i c a t i n g p a t t e r n c o n s i s t s of t h i s c i r c u l a r s t r u c t u r e of "core" 1 s t a t e c e l l s sheathed i n s t a t e 2 c e l l s : 2 2 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 2 2 1 2 2 2 2 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2 1 JL JL xL JL xL 1 JL JL xL x- J-2 1 1 1 1 1 1 1 1 1 1 1 1 1 2 Now, by changing one of the i n t e r n a l 1 s t a t e c e l l s i n t o a >2 s t a t e , one w i l l s et up a "chain r e a c t i o n " and a l l the 1 s t a t e s w i l l t u r n to t h a t >2 s t a t e (one c e l l per t r a n s i t i o n frame). A 7 0 s i g n a l would a c t u a l l y loop about the c i r c l e : the 7 i n the le a d c o n verts each next c e l l t o a 7, but the f o l l o w i n g 0 r e s e t s the c e l l s t o 1. However, a c l o s e look at the s h e l l shows that i t i s not a c l o s e d c i r c l e . In the r i g h t lower corner there i s a T j u n c t i o n . The t r a n s i t i o n r u l e s of t h i s system are such that Chapter IV 107 a s i g n a l such as 7 0 (moving counterclockwise) goes o-ff- i n both d i r e c t i o n s at such a j u n c t i o n , i . e . one 7 0 s i g n a l moves v e r t i c a l l y back up i n t o the loop while the other 7 0 s i g n a l moves on t o the r i g h t . T h i s i s a very important f e a t u r e . Every time a s i g n a l comes around t h i s loop i t can send another "copy" o-f i t s e l - f o-f-f t o the r i g h t down that data-path. Suppose t h a t s i g n a l i s a path extender, as our 7 0 — 6 0 s e r i e s was i n the example above? Then every time the loopi n g prototype s i g n a l comes round t o the j u n c t i o n at the bottom r i g h t corner o-f the s h e l l a new extension "command" can be dispatched t o -further some c o n s t r u c t i o n p r o j e c t . Extension can thus go on as long as the prototype keeps l o o p i n g w i t h i n the s h e l l — p o t e n t i a l l y -forever. In -fact, i-f we put a 7 0 - 6 0 s i g n a l i n t h a t s h e l l the r e s u l t would be a never-ending sequence of extensions of that "arm" coming out of the r i g h t bottom corner of the s h e l 1 . The reproducing loop. Langton though had a d i f f e r e n t c o n s t r u c t i o n p r o j e c t i n mind. He h i t on a sequence to p l a c e i n t o h i s s h e l l t h a t would, a f t e r enough (151) loops, b u i l d another s h e l l equipped with the same prototype program! The sequence: 7 0 - 7 0 - 7 0 - 7 0 - 7 0 - 7 0 - 4 0 - 4 0. Thus t h i s s t r u c t u r e s e l f - r e p l i c a t e s : Chapter IV 108 7 ^. 'Tr '•"1 i 7 0 i 4 o 1 4 2 o JL. 0 -'—< 2 7 2 1 2 1 -~i 1 2 2 0 2 1 2 7 ^_ 1 2 1 2 2 2 2 2 1 2 2 '—1 _^ 2 2 2 0 7 i O 7 1 0 7 1 1 1 1 1 2 2 2 o 2 2 2 2 2 2 2 2 The s i g n a l loops around i n the s h e l l i n a counterclockwise -Fashion. B a s i c a l l y , the s i g n a l c o n t a i n s enough in-formation t o make a left—turn ing corner; with enough i t e r a t i o n s , s e v e r a l such c o r n e r s are made—and a new s h e l l has been c o n s t r u c t e d . A number of stages i n t h i s process are shown i n Langton (1984). Separation of offspring. When the second loop has been made a s o r t of f i g u r e 8 r e s u l t s , with c o l l i s i o n of s i g n a l s at the i n t e r s e c t i o n . Langton has arranged t h i n g s so that t h i s c o l l i s i o n has the e f f e c t of both s e a l i n g o f f the loops from each other and of i n i t i a t i n g c o n s t r u c t i o n of new c o n s t r u c t i o n arms ( l i k e the bottom lower r i g h t of the o r i g i n a l ) i n both parent and o f f s p r i n g . Proliferation. E v e n t u a l l y , two r e p l i c a s of the o r i g i n a l loop r e s u l t , d i f f e r e n t l y o r i e n t a t e d but i d e n t i c a l . Both of these generate o f f s p r i n g . But p r o l i f e r a t i o n of the sp e c i e s uses up a v a i l a b l e space. As the colony grows those i n d i v i d u a l s c e n t r a l l y l o c a t e d have no room t o extend t h e i r arms; t h e i r s i g n a l s come up against u n y i e l d i n g " w a l l s " of 2 s t a t e c e l l s on a l l s i d e s , and cease l o o p i n g . Only a dead Chapter IV 109 " s h e l l " remains i n p l a c e of that i n d i v i d u a l — t h o u g h the colony c o n t i n u e s t o p r o l i f e r a t e . Langton compares t h i s s i t u a t i o n of a " r e p r o d u c t i v e f r i n g e surrounding a growing core of empty loops" t o the growth of a c o r a l r e e f . Again, i l l u s t r a t i o n s of t h i s c o l o n i a l growth are i n Langton (1984 and 1986). Evaluation. Rosen's "Paradox" notwithstanding, Langton's c o n s t r u c t i o n seems t o be a v a l i d example of an automaton d i r e c t i n g i t s own r e p r o d u c t i o n , w i t h i n an automata complex. His achievement w i l l be evaluated i n the l i g h t of other r e s u l t s , i n Chapter IX. Other Computing and Propagating Automata Langton's loop needs t o be provided with i t s s e l f — c o n s t r u c t i n g program, j u s t as von Neumann's u n i v e r s a l machine needed the program g i v i n g the al g o r i t h m f o r i t s own c o n s t r u c t i o n . Laing (1975, 1977) though has devised schemata f o r what he c a l l s an " a r t i f i c i a l molecular machine" t h a t can analyze i t s own s t r u c t u r e t o ob t a i n a complete d e s c r i p t i o n f o r use i n r e p l i c a t i o n . In h i s model, a computing/constructing automaton c o n s i s t s of a pair of s t r i n g s , i n contact with one another at some p o i n t , and capable of s l i d i n g back and f o r t h t o change the p o i n t of connection (because of t h i s s l i d i n g c h a r a c t e r i s t i c , he c a l l s these s t r i n g s "kinematic"). At the same time, h i s s t r i n g s are l i k e l i n e a r c e l l u l a r automata: Chapter IV 110 Each i s segmented i n t o c e l l s , c e l l s t h a t can take on d i f f e r e n t s t a t e s . There are p a s s i v e s t a t e s (where a c e l l c o n t a i n s data to be r e a d ) , a c t i v e s t a t e s (where a c e l l c o n t a i n s a command and the ef-ficacy t o implement the command), and n u l l s t a t e s (the e q u i v a l e n t o-f blanks, a v a i l a b l e i n en d l e s s s u p p l y ) . Commands are rendered i n a Turing computer n o t a t i o n f a i r l y s i m i l a r t o the one e a r l i e r presented i n t h i s chapter. At the p o i n t where the p a i r of s t r i n g s are i n c o n t a c t , i t i s p o s s i b l e f o r an a c t i v e - s t a t e c e l l t o a l t e r the s t a t e of a p a s s i v e c e l l . Any p a s s i v e c e l l can be converted step by step t o any other i n the r e p e r t o i r e of p o s s i b l e s t a t e s . T h i s f a c t i s the key t o the " s e l f — i n s p e c t i o n " the p a i r of s t r i n g s needs t o c a r r y out t o r e p l i c a t e i t s e l f . The idea i s to s u b j e c t the unknown c e l l t o a s e r i e s of conversions u n t i l i t a t t a i n s a c e r t a i n basal s t a t e ; the number of conversions r e q u i r e d , counted o f f i n terms of the number of a c t i v e c e l l s employed, i n d i c a t e s what the o r i g i n a l s t a t e was. By r e v e r s i n g the procedure subsequently (working on a blank c e l l ) the o r i g i n a l s t a t e i s r e c r e a t e d as a c o n s t i t u e n t of the new automaton. Of course, the procedure i s repeated u n t i l the i d e n t i t y of each c e l l of the s t r i n g to be copied i s a s c e r t a i ned. Laing (1977) maps out two v a r i a n t s t r a t e g i e s f o r s e l f -r eproduction with t h i s kind of d o u b l e - s t r i n g automaton. His e x p o s i t i o n makes i t r e l a t i v e l y c l e a r how t h i s process would Chapter IV H I go, but he does not a c t u a l l y p r o v i d e a -fully s p e c i - f i e d , programmed machine and c h a r t i n complete d e t a i l i t s progress through the sel-f-reproductory process. In very general (condensed) terms, the o r i g i n a l machine has t h r e e e s s e n t i a l p a r t s : (1) a s p e c i a l - p u r p o s e c o n s t r u c t o r ; (2) a d e s t r o y e r (reduces c e l l s t o n u l l p r i m i t i v e s ) ; and (3) a general-purpose c o n s t r u c t o r . The preprogrammed s p e c i a l -purpose c o n s t r u c t o r begins the process by b u i l d i n g "analyzer" and " i n f e r r e r " s u b s t r i n g s . The analyzer i d e n t i f i e s the c o n s t i t u e n t n u l l p r i m i t i v e s of the ( e n t i r e ) o r i g i n a l machine by the method of s u c c e s s i v e conversion given above. The i n f e r r e r (bearing r e c o r d of how many conversions were needed by the analyzer to convert each c e l l t o n u l l ) now a l l i e s with the general-purpose c o n s t r u c t o r t o b u i l d a complete copy of the o r i g i n a l machine. The d e s t r o y e r u n i t s f i n a l l y e l i m i n a t e the t o o l s of p r o d u c t i o n — n o t a b l y the a n a l y z e r , and the i n f e r r e r l e a v i n g behind only the o r i g i n a l machine and i t s new r e p l i c a . Langton's V i r t u a l Turing Machine Langton (1988) has d e s c r i b e d an automaton s i m i l a r to Laing's, with two eel1-segmented p a r a l l e l s t r i n g s , one s e r v i n g as r u l e tape, the other as data tape. These are intended to be imbedded i n a c e l l u l a r automaton. The a b i l i t y f o r v i r t u a l o b j e c t s l i k e the g l i d e r s of LIFE t o migrate between s t r i n g s i s e x p l o i t e d i n h i s scheme ( i . e . s i g n a l s can Chapter I be dispatched -from the r u l e tape to c a r r y out o p e r a t i o n s on the data t a p e ) . On t o Typogenetics The -foregoing review, though sometimes q u i t e c u r s o r y , should have provided the reader with a sense of the nature and uses of automata i n modeling propagation, and brought l e a d i n g r e f e r e n c e s i n the f i e l d to l i g h t . It a l s o a f f o r d e d an oppor t u n i t y f o r i n t r o d u c i n g some b a s i c concepts (such as Gardens of Eden; s e l f - e r a s i n g and s e l f — r e p l i c a t i n g s t r i n g s ; u n i v e r s a l c o n s t r u c t o r s ; the p a r a d o x i c a l machine; etc.) and i s s u e s (e.g. the p u z z l e r a i s e d by Rosen's "Paradox"), t h a t w i l l be r e s u r f a c i n g , Yet, i n a sense, a l l t h a t has appeared thus f a r has been preparatory f o r what i s now to come, as we enter the world of Typogenetics. CHAPTER V : TYPOGENETICS Chapter Typogenetics was f i r s t i ntroduced by Douglas Hofsta d t e r (Chapter XVI of Godel. Escher, Bach, 1979), i n connection with h i s d i s c u s s i o n of the "tangled h i e r a r c h y " of DNA's r e p l i c a t i v e processes. A DNA s t r a n d , i t w i l l be r e c a l l e d from our Chapter I I I , c o n t a i n s , among other t h i n g s , i n s t r u c t i o n s p r e s c r i b i n g the production of enzymes capable of o p e r a t i n g on (de s t r o y i n g , r e p a i r i n g , copying, c o n v e r t i n g , purging, manufacturing, etc.) any of the contents of the c e l l — i n c l u d i n g the DNA s t r a n d . In f a c t , a p a r t of the information i n the DNA base sequence p r e s c r i b e s the s y n t h e s i s of the enzymes that make a copy of the DNA strand i t s e l f , i n pr e p a r a t i o n f o r m i t o t i c s e l f — r e p l i c a t i o n . H o f s t a d t e r saw an analogy to be f u l f i l l e d here. DNA i n s t i g a t e s the formation of operators that w i l l , i n the na t u r a l course, t r e a t that very DNA source as an operand. Why not have a formal system with axioms t h a t generate theorems that can come back a n d — d e v i a t i n g from standard p r a c t i c e i n use of formal s y s t e m s — o p e r a t e on the axioms? Thus, Typogenetics. With i t , one was supposed t o be ab l e to invent a r t i f i c i a l propagative e n t i t i e s reminiscent of DNA str a n d s , though obedient to t h e i r own "natural laws" and possessing t h e i r own s p e c i a l p o t e n t i a l i t i e s . Thus Typogenetics was i n s p i r e d by nature; e s p e c i a l l y as regard t o Chapter V 114 how i t encodes opera t o r s i n a s t a t i c s t r i n g -form. But the " b i o l o g i c a l " o r i g i n o-f Typogenetics, as a model of propagation, i s only hal-f the s t o r y . A l s o i n the context o-f Ho-f s t a d t e r ' s i n t r o d u c t i o n o-f Typogenetics i s h i s d i s c u s s i o n o-f s e l f - r e p r o d u c i n g computer programs, and h i s t h i n k i n g about the l a t t e r has c l e a r l y shaped the former. Thus, as e a r l i e r s a i d , the e n t i t i e s i n t h i s a r t i f i c i a l world are h y b r i d s , s i m i l a r on one hand to DNA, on the other, to s t r i n g programs l i k e those we looked at i n connection with the Turing Table automaton <pp 6 3 f f ) , and a l s o t o s t r i n g automata along the l i n e s of those we have seen of Laing (1977). It happens t h e r e i s no mention of Laing or Stahl i n Godel , Escher, Bach ( f a c t observed c r i t i c a l l y by Webb, 1983.). Yet there i s some very general s i m i l a r i t y between the systems these men devised. That men from d i f f e r e n t backgrounds (Laing an automata t h e o r i s t , Stahl a p h y s i c i a n and biomodeler, Hofstadter a m a t h e m a t i c i a n / p h y s i c i s t / c o g n i t i v e s c i e n t i s t ) , should be thus independently converging on a s i n g l e nexus, I f i n d i n d i c a t i v e of the r i p e n e s s of t h i s area f o r development. Completing the System The idea was great. The problem was that Typogenetics as presented by H o f s t a d t e r , was " i l l - f o r m e d " — n e c e s s a r y r u l e s and d e f i n i t i o n s were l a c k i n g . The essence of a formal system Chapter i s t h a t i t makes one use s e l e c t e d symbols according t o s t r i c t r u l e s , t o b r i n g about some unambiguous r e s u l t . So the system l i t e r a l l y c o uld not be used, so long as i t lacked those r u l e s and c l e a r e r d e f i n i t i o n of qivens. Then too, having given us t h i s system with such e x c i t i n g p o t e n t i a l , H o f s t a d t e r does not have a s i n g l e i n t e r e s t i n g r e s u l t t o share with us. I t i s as though the inventor never got around t o t r y i n g out h i s i n v e n t i o n (one gets the same impression of L a i n g ) i Which might account f o r why he d i d not n o t i c e t h a t t h e r e are many s i t u a t i o n s one inevitably gets i n t o i n Typogenetics f o r which there i s no appl y i n g r u l e . The f i r s t order of business f o r t h i s chapter, then, i s to o f f e r a completed v e r s i o n of Typogenetics, with a step by step how—to t u t o r i a l . Then t h e r e w i l l be an e x h i b i t i o n of some of the r e s u l t s t h a t can be obtained with i t : a g a l l e r y of a r t i f i c i a l , symbolic e n t i t i e s possessing d i s t i n c t propagative or, i f you w i l l , s e l f — t r a n s f o r m a t i v e , powers. Appendix 1 summarizes the m o d i f i c a t i o n s and extensions made on H o f s t a d t e r ' s proto—system. To hazard an estimate t h a t the c u r i o u s reader might weigh, perhaps the schema f o r Typogenetics, as given i n Godel, Escher, Bach, c o n s t i t u t e s t hree q u a r t e r s of the system as given here. H o f s t a d t e r , of course, deserves e x c l u s i v e c r e d i t as o r i g i n a t o r of the concept of such a formal system. Chapter V 116 Alphabet of Symbols The alphabet o-F symbols from which a s t r i n g w i l l be cons t r u c t e d c o n s i s t s o-f the l e t t e r s A,C,S or T. As explained below, the lower case o-f these l e t t e r s w i l l sometimes be used t o i n d i c a t e e l e c t i o n o-f a b i n d i n g s i t e . There are no -functors nor q u a n t i f i e r s i n t h i s system. A p e c u l i a r i t y of the system's n o t a t i o n allows a vertical dimension t o our s t r i n g s . During the course of a d e r i v a t i o n a strand may temporarily add an upper l e v e l , e.g.: CCTATG There can be no more than two l e v e l s (measured v e r t i c a l l y ) t o a s t r i n g . An a d d i t i o n a l graphic n o t a t i o n f o r rendering s o — c a l l e d t e r t i a r y structures i s employed; i t w i l l be explained i n i t s pl a c e . Recursive D e f i n i t i o n of a Wei1—Formed Typogenetic Strand Ordinary languages l i k e E n g l i s h do not have simple s y n t a c t i c a l r u l e s f o r s o r t i n g out which permutations of the alphabet are words. Among other t h i n g s , that means you cannot dis c o v e r or f o r e s e e or compute some h e r e t o f o r e unknown E n g l i s h word. Because i t i s not predetermined what a l p h a b e t i c a l combinations are going to be added to the E n g l i s h language. On the whole i t i s not a lawful business. Chapter V 117 The s i t u a t i o n i s r a d i c a l l y d i f f e r e n t i n an a r t i f i c i a l "language" l i k e Typogenetics. The s t a t u s o-f every permutation of our alphabet, as to well-formedness, i s predetermined by a s i n g l e formula. T h i s formula d e s c r i b e s the c r i t e r i a and the c o n d i t i o n s f o r d e c i d i n g whether a p u t a t i v e s t r i n g is a well formed s t r a n d or not; i t promises a f i n i t e number of s t e p s w i l l s u f f i c e t o make the dete r m i n a t i o n , f o r any f i n i t e — l e n g t h s t r i n g of the approved symbols. Perhaps more i m p r e s s i v e l y , t h i s formula can be used t o r e c u r s i v e l y enumerate each next s t r a n d , ad infinitum (see Chapter IX's s e c t i o n on generation of s t r a n d s ) . One modest set of r u l e s and i n i t i a l c o n d i t i o n s — r e p r e s e n t i n g a c e r t a i n , r e l a t i v e l y small amount of i n f o r m a t i o n — c a n t h e r e f o r e g i v e r i s e t o an i n f i n i t u d e of i n f o r m a t i o n . L a t e r , i n Chapter VI, we w i l l see how numerical computation can be done of the number of strand s there are of a given length. We w i l l see that the numbers grow so b i g so f a s t t h a t there i s l e s s practical value t o the c l a i m , that we can enumerate a l l the well—formed s t r i n g s using our r e c u r s i v e formula, than one might t h i n k . The e n t i r e c h a l l e n g e of Typogenetics s h a l l then be r e a l i z e d t o l i e i n d i s c o v e r i n g interesting new strands from among the s t u p i f y i n g l y l a r g e realm of u n i n t e r e s t i n g strands i n h a b i t i n g t h i s r e c u r s i v e space. F o r t u n a t e l y , these strands can be conceived or found or molded or o u t p u t . . . i n a word, f o u n d . w i t h o u t having to generate a l l t h e i r Chapter V 118 r e c u r s i v e l y antecedent well—formed st r a n d s ! F i r s t , an in-formal statement, which w i l l g i v e the reader the g i s t o-f the idea r i g h t away. A s t r i n g , or strand as i t i s known, i s any permutation of the base denoted u n i t s , with the r e s t r i c t i o n t h a t , to be well—formed, i t must have at l e a s t one duplet (defined below) other than AA. Now, -formally, l e t * or # represent any s i n g l e symbol of our alphabet. Then, (1) *# i s a s t r a n d , except f o r AA; (2) If * i s a s t r a n d , then $* i s a s t r a n d ; (3) If $ i s a s t r a n d , then AA$ i s a s t r a n d . Examples: CG and TT are s t r a n d s , by (1). Since CG i s a strand by ( 1 ) , CGT i s a strand by (2); and s i nee CGT i s a s t r a n d , CGTT i s a s t r a n d , a l s o by (2). Since TT i s a strand by (1), AATT i s a s t r a n d , by (3); and s i n c e AATT i s a s t r a n d , AAAATT i s a s t r a n d , a l s o by (3) . Any strand must meet c r i t e r i o n (1), or (1) i n c o n j u n c t i o n with e i t h e r or both (2) or (3). From (1) i t can be seen t h a t a well—formed strand w i l l always be at l e a s t two l e t t e r s long, and cannot c o n s i s t only of the l e t t e r A. Chapter V 119 No V a r i a b l e s There are no v a r i a b l e s at any l e v e l o-f t h i s system. ( V a r i a b l e s w i l l be employed i n the me t a — l o g i c a l p r o o f s o-f Chapter VI, however). Units/Bases Level A strand i s a well—formed s t r i n g (see previous s e c t i o n ) o-f two or more o-f the l e t t e r s A, C, G and T. Those l e t t e r s r epresent bases—attributes o-f the system's o b j e c t language l e v e l atoms, c a l l e d units. A d i f f e r e n c e here between a standard p r e d i c a t e symbolic l o g i c ' s n o t a t i o n and Typogenetics must be made c l e a r . The former g i v e s an i n d i v i d u a l at the o b j e c t l e v e l d i s t i n c t symbolic r e p r e s e n t a t i o n . To a t t r i b u t e t o the i n d i v i d u a l o b j e c t a the p r e d i c a t e H one might w r i t e Ha. But i n Typogenetics the u n i t i s not symbolized apart from i n d i c a t i o n of i t s base p r e d i c a t e . Rather than Ga ("unit a i s of the G base type") i n Typogenetics one simply w r i t e s G. For convenience, the u n i t ' s base G, i s of t e n spoken of as though i t named the i n d i v i d u a l , as when I say something l i k e "the next op e r a t i o n chops o f f that G on the r i g h t end." Where there i s more than one u n i t , as i s always the case when t a l k i n g about a s t r a n d , one u n i t i s d i s t i n g u i s h e d from another by i t s typographical p o s i t i o n . E.g. the duplet Chapter CC c o n s i s t s of two d i s t i n c t u n i t s , though both are o-f the same base type. A u n i t must be o-f one and only one of the f o u r base types. A u n i t has no other a t t r i b u t e than i t s base type, but may e x i s t i n v a r i o u s r e l a t i o n s h i p s with other u n i t s . The bases themselves have f i x e d higher order p r e d i c a t e s : A and G are pur inesi C and T are pyrimidines. The bases a l s o have a f i x e d r e l a t i o n s h i p of complementarity: A being complementary t o T, C t o G. H o r i z o n t a l and V e r t i c a l Unit R e l a t i o n s h i p s P a i r i n g from the l e f t of the s t r i n g , a u n i t i n a h o r i z o n t a l t y p o g r a p h i c a l r e l a t i o n s h i p with one other u n i t i s i n a duplet r e l a t i o n s h i p . E.g. ACTAT i s a strand c o n s i s t i n g of the d u p l e t s AC, TA, and the l e f t over b i t T — p a r t of the strand but not i n a duplet r e l a t i o n s h i p . In double s t r a n d s , which are always temporary e n t i t i e s a r i s i n g during d e r i v a t i o n of a s t r a n d ' s daughters, a u n i t may be i n a v e r t i c a l t y p o g r a p h i c a l r e l a t i o n s h i p with another u n i t . U n i t s i n a v e r t i c a l r e l a t i o n s h i p must be complementary to one another. E.g. an A base u n i t can be i n a v e r t i c a l r e l a t i o n s h i p only with a T base u n i t . A u n i t t h a t stands alone, i . e . i s not i n any r e l a t i o n s h i p to any other u n i t , i s c a l l e d a " b i t . " Chapter V 121 The l e f t - r i g h t order o-f the d u p l e t s i n the s t r a n d corresponds t o the order i n which o p e r a t i o n s are executed; the o p e r a t i o n corresponding to the le-ftmost duplet i s the first to be executed. The f i r s t "gene" of a s t r a n d begins with the f i r s t l e f t m o s t non-AA duplet and extends u n t i l the next AA d u p l e t or the s t r a n d ' s end i s reached. If there i s more than one gene then each next gene begins with the next non-AA duplet and continues t o the next AA duplet or s t r a n d ' s end. AA, then, which does not code f o r an o p e r a t i o n and i s not p a r t of a gene, serve s as the punctuator, or boundary, between the genes of a multi-gene s t r a n d . E.g. AACTGTAATCG c o n s i s t s of two genes: CTGT and TCG. TAAT though i s a s i n g l e gene strand; d u p l e t s are p a i r e d from the l e f t , so TAAT's adjacent AA u n i t s are not read as a duplet. A s i n g l e duplet s u f f i c e s as a gene; a l e f t o v e r b i t , as the T at the end of GGAAT, i s not a gene. Since each duplet codes f o r an amino a c i d embodying a s p e c i f i c o p e r a t i o n there corresponds to a gene a s e r i e s of operations. T h i s s e r i e s of o p e r a t i o n s c o n s t i t u t e s an "enzyme" program. An enzyme can be v i s u a l i z e d as a robot arm operating on the s t r a n d , c a r r y i n g out the commands i t s corresponding gene codes f o r . In a d e r i v a t i o n of a strand's descendants the enzyme's present p o s i t i o n (point of attachment to the strand) i s recorded t y p o g r a p h i c a l l y by reducing that u n i t ' s base to lower case, e.g. CTTaT. Order of enzyme a c t i v a t i o n f o l l o w s Chapter V 122 the l e f t - r i g h t order of genes; the enzyme corresponding to the leftmost gene i s a c t i v a t e d f i r s t . Enzyme Attachment P r o c e d u r a l l y , once a st r a n d i s posed, e.g. GCCACT, i t i s analyzed i n t o i t s g e n e s — h e r e t h e r e i s only one, c o n s i s t i n g of the du p l e t s GC, CA, and C T — a n d the d u p l e t s t r a n s l a t e i n t o amino a c i d o p e r a t i o n s that may transform the st r a n d . The f i r s t d uplet of GCCACT codes f o r the command t o i n s e r t a C base u n i t . But i t now should be apparent that the point of attachment of the enzyme i s a key c o n s i d e r a t i o n , because i t w i l l i n s e r t the u n i t at p r e c i s e l y that p o i n t (that i s , to the r i g h t of the u n i t t o which i t i s bound). Each enzyme has a " t e r t i a r y s t r u c t u r e " which p o i n t s t o a binding preference, f o r a s s i g n i n g that enzyme an i n i t i a l b i n d i n g s i t e , a place t o begin work, i n d i c a t e d t y p o g r a p h i c a l l y as a lower case l e t t e r . Binding preference i s determined by the r e l a t i v e o r i e n t a t i o n (in the t e r t i a r y s t r u c t u r e ) of the first and last amino a c i d s of the enzyme (which would be the same, i n a one duplet gene). T h i s determination i s r e a l l y not as d i f f i c u l t to do as to e x p l a i n , and the understanding to be gained by going through the procedure on one's own i s i n v a l u a b l e . Nevertheless, the convenience of automation f o r longer genes or multiple—gene strands i s overwhelming. Code f o r a BASIC program f o r personal computers which accepts any input strand Chapter V 12 (to length 255 characters) , i s o l a t e s the s t rand 's genes, and for each determines and displays the corresponding enzyme's binding preference, i s printed in Appendix II . On the r ight of a d u p l e t ' s entry in Table 4 i s displayed the folding i n c l i n a t i o n for that d u p l e t ' s amino a c i d , indicated with a l e t t e r ("1" for l e f t - t u r n i n g , " r " for r i g h t , or "s" for s t r a i g h t ) . Once a l l f o l d i n g i n c l i n a t i o n s are known, the t e r t i a r y s tructure can be g r a p h i c a l l y represented as in Figure 2; a more spectacular example i s Figure 3 (p275). Notice that the f i r s t arrow in t h i s graphing of the t e r t i a r y s tructure i s always pointed r i g h t by convention. This does not mean that the f i r s t amino a c i d ' s f o l d i n g i n c l i n a t i o n i s ignored, however. The f i r s t amino a c i d ' s i n c l i n a t i o n determines which subtable of Table 5 i s used. In any case, once started to the r ight the t e r t i a r y structure may turn l e f t or r i g h t , f o l d i n g back and over i t s e l f in a complex fashion. Ult imately the las t arrow points in one of the four d i r e c t i o n s . To determine binding preference one then goes to Table 5, f i n d s the r ight subtable on the basis of the f i r s t amino a c i d ' s f o l d i n g i n c l i n a t i o n , then locates in that subtable an arrow whose d i r e c t i o n corresponds to that o f the f i n a l arrow of that enzyme's t e r t i a r y s t ructure . That tabular entry w i l l have associated with i t a base—which i s the preferred base for the enzyme. Chapter V 124 Pup1et Command AC s cut TABLE 4: TYPOGENETICS TRANSLATION TABLE Operation Described AG s AT r CA s CC s CG r del swi mvr mvl cop Cut the s t r a n d t o the r i g h t of the present u n i t ; through both l e v e l s i f double strand. Delete t h i s u n i t , then move one u n i t r i g h t . Switch enzyme to u n i t ( i f any) i n v e r t i c a l r e l a t i o n s h i p with present u n i t . Enzyme moves one u n i t r i g h t . Enzyme moves one u n i t l e f t . Turn on copy mode. U n t i l turned o f f or detached, enzyme produces complementary u n i t s v e r t i c a l t o a l l u n i t s i t touches or i n s e r t s . CT 1 o f f Turn o f f copy mode. GA s i na Inser t A t o r i g h t of t h i s u n i t . GC r i n c I n s e r t C to r i g h t of t h i s u n i t . GG r ing I n s e r t G t o r i g h t of t h i s u n i t . 6T 1 i n t I n s e r t T t o r i g h t of t h i s u n i t . TA r rpy Fi n d nearest p y r i m i d i n e (C or T) to r i g h t TC 1 rpu Find nearest purine (A or G) t o r i g h t . TG 1 lpy F i n d nearest p y r i m i d i n e (C or T) to l e f t . TT 1 1 pu Find nearest purine (A or G) to l e f t . Chapter V 125 TABLE 5: TERTIARY STRUCTURE AND BINDING PREFERENCE (1). LAST fffllHQ flHP BINDING PREFERENCE FIRST flniNO ACID fl INCLINATION: STRAIGHT FIRST AMINO ACID INCLINATION: LEFT i C G T ( 2 ) . IflST flKHO ACID BINDING PREFERENCE i t G T O ) . , LAST nHNO ACIS BINDING PREFERENCE FIRST AFIINO ACID 1f * INCLINATION: RIGHT < = = C M P G T I ] Chapter V 126 (1) cop==Kut =#mvl IS) cop =4swi t mvr t lpu=7>lpu FIGURE 2: Enzyme tertiary structures of strand CGflCCCflftCGflTTTTTCflT's two genes! (1) CGflCCC; (2) CGfiTTTTTCftT Chapter V 127 Election of binding site. The b i n d i n g preference i s s t a t e d as a base. Where more than one u n i t i n the strand i s of that base type, an e l e c t i o n must be made as to where f i r s t t o a t t a c h the enzyme. For example, i n the s t r a n d ACAAACAA both enzymes have a bind i n g p r e f e r e n c e f o r the base A — a n d there are no l e s s than s i x d i f f e r e n t u n i t s of that base type. An enzyme must a t t a c h to o n e — a n d only o n e — o f those u n i t s to e f f e c t i t s cut command. Since d i f f e r e n t c h o i c e s of b i n d i n g s i t e w i l l very f r e q u e n t l y lead t o d i f f e r e n t r e s u l t s , much of the "gamesmanship" of Typogenetics i s i n making a f r u i t f u l s e l e c t i o n ( i n the present case, the r i g h t c h o i c e s w i l l r e s u l t i n ACAAACAA c u t t i n g i t s e l f i n t o i d e n t i c a l twin daughters). Such e l e c t i o n s of i n i t i a l enzyme bi n d i n g s i t e are comparable t o the c o n d i t i o n a l assumptions one makes i n n a t u r a l deduction l o g i c s : The f a c t t h a t a c e r t a i n descendant i s produced, i s c o n d i t i o n e d on the c h o i c e s of i n t i t i a l b i n d i n g s i t e that were made i n i t s d e r i v a t i o n . Another way to look at t h i s i s t o acknowledge that the e l e c t i o n of i n i t i a l enzyme binding s i t e i s a way f o r the user to choose one p a r t i c u l a r d e r i v a t i o n from among the many p o s s i b l e f o r a s t r a n d t h a t o f f e r s a l o t of a l t e r n a t i v e b i n d i n g s i t e s f o r i t s enzymes. Once the enzyme i s attached, i t remains there unless and u n t i l r e l o c a t i o n i s e x p l i c i t l y d i c t a t e d by a command. Chapter V 128 Operations There are 15 o p e r a t i o n s , summarily s t a t e d i n Table 4. <i) Relocation o p e r a t i o n s are o-f three s o r t s : (a) . Searches send the enzyme down the str a n d r i g h t or le-ft t o -find a u n i t o-f the s p e c i f i e d base type (pyrimidine or p u r i n e ) . There i s no l i m i t t o the d i s t a n c e i t can t r a v e l i n t h i s search. (b) . Moves have the enzyme move h o r i z o n t a l l y e x a c t l y one u n i t . (c) . Switching i s the only mode of v e r t i c a l r e l o c a t i o n ; a switch attaches the enzyme t o the other s t r a n d of a double s t r a n d , hence two cons e c u t i v e switches would r e t u r n the enzyme t o i t s s t a r t i n g p l a c e . ( i i ) Effective o p e r a t i o n s d i m i n i s h or augment the st r a n d : (a) . The insertion commands have the enzyme i n s e r t e x a c t l y one new u n i t of a s p e c i f i e d base i n t o the strand immediately t o the r i g h t of the enzyme's point of attachment. The enzyme "makes room" even i n a double st r a n d . In the f o l l o w i n g example a T w i l l be i n s e r t e d : 13W990H 13W9 93fc> AGTcCGT AGTcTCGT (b) . Deletion commands are the r e c i p r o c a l of i n s e r t i o n , except that the one u n i t d e l e t e d need not be of any C h a p t e r V 129 p a r t i c u l a r b a s e t y p e , a n d i s t h e v e r y u n i t t h e e n z y m e i s c u r r e n t l y a t t a c h e d t o . A - f t e r d e l e t i n g t h e u n i t , t h e e n z y m e a t t a c h e s i t s e l f t o t h e u n i t to t h e r i g h t . T h i s i s i n t e r p r e t e d a s c r e a t i n g a g a p i n o n e l e v e l o f a d o u b l e l a y e r s t r a n d . B u t a s i n g l e l a y e r s t r a n d r e j o i n s i t s e l f , " s q u i n c h i n g t o g e t h e r " t o m a i n t a i n i t s s p a t i a l a n d s y n t a c t i c a l c o n t i g u i t y . < c ) . Cutting s e p a r a t e s t h e s t r a n d i n t o t w o p a r t s . T h e p a r t " c u t o f f " i s t h a t l y i n g i m m e d i a t e l y r i g h t o f t h e e n z y m e ' s p o i n t o f a t t a c h m e n t ; i t i s f r o m t h a t m o m e n t a d a u g h t e r o f t h e o r i g i n a l s t r a n d a n d n o t s u b j e c t t o a n y f u r t h e r o p e r a t i o n s o f t h i s o r o f a n y s u b s e q u e n t e n z y m e ' s o p e r a t i o n s . T h e p a r t o f t h e s t r a n d t h e e n z y m e h a s r e m a i n e d a t t a c h e d t o ( w h i c h m i g h t c o n c e i v a b l y b e o n l y o n e u n i t l o n g , i f t h e e n z y m e p e r f o r m e d i t s c u t w h i l e a t t a c h e d t o t h e l e f t m o s t u n i t ) i s n o w t h e o p e r a n d f o r a n y a n d a l l f u r t h e r o p e r a t i o n s . ( d ) . Copying. W h e n c o p y m o d e i s t u r n e d o n t h e i m m e d i a t e r e s u l t i s t h e v e r t i c a l i n s e r t i o n o f a complement o f t h e b a s e o f t h e u n i t t h e e n z y m e i s p r e s e n t l y a t t a c h e d t o . C o p y m o d e r e m a i n s o n u n t i l t h e e n z y m e d e t a c h e s , e x h a u s t s i t s r u n o f o p e r a t i o n s , o r t h e c o p y o f f c o m m a n d i s g i v e n ( w h i c h e v e r c o m e s f i r s t ) . W h e n i t i s o n t h e e n z y m e d o u b l e s t h e s t r a n d ( i n t h a t p e c u l i a r f a s h i o n o f m a n u f a c t u r i n g complements) w h e n e v e r a n d w h e r e v e r i t m o v e s , w i t h t h i s e x c e p t i o n : i f c o m m a n d e d t o c o n d u c t a s e a r c h t h a t w i l l t a k e i t c l e a r o f f t h e s t r a n d t h e e n z y m e s u c c e e d s m e r e l y i n d e t a c h i n g i t s e l f ( s e e n e x t s e c t i o n ) Chapter V 130 i f i t cannot f i n d the p y r i m i d i n e or purine t h a t i s the o b j e c t of i t s s e a r c h — n o copying having r e s u l t e d . Q u a l i f i c a t i o n s on Operations Detachment. In some circumstances o p e r a t i o n s w i l l r e s u l t i n detachment of the enzyme; subsequent commands supposed to be c a r r i e d out by t h a t enzyme thus cannot be executed and the next enzyme i s a c t i v a t e d . T h i s happens when the enzyme i s commanded t o move or search o f f the end of the stra n d ; when i t i s t o switch t o a nonexistent second l e v e l s t r a n d ; and when i t moves o f f the end of a strand pursuant t o the second p a r t of a d e l e t e command. Arranging f o r t i m e l y detachment of an enzyme i s yet another important aspect of Typogenetics design s t r a t e g y . Benign impossibility. Benign i m p o s s i b i l i t y occurs when the enzyme i s commanded t o c u t , copy, or turn o f f copy mode when i t cannot. (If the enzyme i s alre a d y on the f a r r i g h t hand end base i t can't cut the strand to the r i g h t of the present u n i t ; i f the copy mode i s al r e a d y on, i t can't be turned on again; and when o f f , can't be turned o f f a g a i n ) . In these circumstances the enzyme simply ignores the command and proceeds to execute the next operation i n the amino a c i d sequence. Operations on double strands. U n i t s of an upper l e v e l of a double strand are recorded upside—down (see Chapter V 131 i l l u s t r a t i v e d e r i v a t i o n below). When the enzyme switches to t h a t upper l e v e l , i t s sense o-f r i g h t and l e f t w i l l be in r e verse t o r i g h t and l e f t on the lower l e v e l . Formally: For a l l u n i t s w,x,y,and z, i f x and y are v e r t i c a l l y r e l a t e d , and w i s l e f t of x and z i s v e r t i c a l t o w, then z i s r i g h t of y. An enzyme sw i t c h i n g from x t o y and then moving r i g h t w i l l a t t a c h t o z. Gaps in strands. If str a n d segments remain separated by a gap a f t e r a l l enzymes t r a n s l a t e d out of the o r i g i n a l s t rand are exhausted, then the segments are permanently separated. Double s t r a n d s may have, e i t h e r through d e l e t i o n or i n t e r m i t t e n t copying, a gap at one but not both l e v e l s ; such a gap a f f o r d s no bases onto which an enzyme may be bound, hence presents an o b s t a c l e t o sw i t c h i n g and moving (unless and u n t i l plugged). However such a gap can be tr a v e r s e d by a search i n g enzyme (the enzyme r e q u i r e s only something t o " s l i d e along" when se a r c h i n g , and the i n t a c t l e v e l of the double s t r a n d provides t h i s ) . J u s t as an enzyme can t r a v e r s e an i n d e f i n i t e number of u n i t s i n a search across an i n t a c t s t r a n d , i t can span an i n d e f i n i t e l y wide gap so long as the complementary strand upon which i t i s s l i d i n g i s conti n u o u s l y i n t a c t . An enzyme t r a v e r s i n g a gap while i n copy mode cop i e s the bases of the i n t a c t l e v e l of the double strand i t passes over (doubles the strand as i t moves h o r i z o n t a l l y , Chapter V f i l l i n g the gap i t passes through with the complements of the bases above or below the gap). Daughters One or more s t r a n d s (or, at bare minimum, a s i n g l e b i t ) w i l l be l e f t over a f t e r a l l o p e r a t i o n s have been c a r r i e d out by a l l enzymes. If a double s t r a n d has r e s u l t e d , i t must separate at t h i s p o i n t so t h a t the upper and lower l e v e l s are d i s t i n c t s t r a n d s . Strands l e f t a f t e r a l l t r a n s f o r m a t i o n s have occurred, i n c l u d i n g segments cut o f f by an enzyme, are the daughters of the o r i g i n a l s t r a n d . Each daughter can be t r a n s l a t e d and transformed t o produce daughters of i t s own (granddaughters t o the o r i g i n a l s t r a n d ) . Typogenetics can be c a r r i e d on l i t e r a l l y f o r generations! Procedure I l l u s t r a t e d D e r i v a t i o n of the descendants of a Typogenetics strand i s given below i n a format resembling a d e r i v a t i o n i n a standard l o g i c a l system. T h i s s t r a n d was designed to show a r e p r e s e n t a t i v e s e l e c t i o n of the amino a c i d o p e r a t i o n s i n a c t i o n , but i s of no s p e c i a l i n t e r e s t beyond that capaci t y . Notice t h a t the p o i n t of enzyme attachment on the c u r r e n t transform i s always i n d i c a t e d by reducing that Chapter u n i t ' s base t o lower case. The s t r a n d : CGACCCAACGATTTTTCAT T r a n s l a t i o n : 1st Gene Duplets: CG AC CC Amino Ac i d Operations: cop cut mvl F o l d i n g I n c l i n a t i o n s : Binding Preference: 2nd Gene Duplets: Amino Ac i d Operations: F o l d i n g I n c l i n a t i o n s : Binding F'reference: s (See F i g u r e 2) G Stage CG AT TT TT CA T cop swi l p u l p u mvr r r 1 1 5 (See F i g u r e A DERIVATION OF A SAMPLE STRAND'S DESCENDANTS Strand Transform Operati on CGACCCAACgATTTTTCAT 1st enzyme attached CGACCCAACgATTTTTCAT Copy on CGACCCAACg ATTTTTCAT Cut 00 CGACCCAAcG Move l e f t 93 CGaCCCAACG 2nd enzyme attached Chapter V 134 6 1 03 CGaCCCAACG Copy on 7 } 93 CGACCCAACG Switch 8 199911&3 CGACCCAACG Find f i r s t Purine Left 9 19991193 CGACCCAACG Find f i r s t Purine Left At step 9 of our example the enzyme moved o f f the strand; no f u r t h e r commands can be c a r r i e d out. There being no f u r t h e r enzymes t o be a c t i v a t e d , s t r a n d t r a n s f o r m a t i o n i s completed. The double strand separates i n t o the two st r a n d s : C6TTGGGT and CGACCCAACG. The fragment of double s t r a n d ATTTTTCAT cut o f f from the r e s t of the strand at stage 3 i s a l s o a daughter. There are thus t h r e e daughters of the s t r a n d , given the b i n d i n g s i t e c h o i c e s made above; as they are w e l l -farmed s t r a n d s , f o r each of them too descendants could be d e r i ved. Transforms It should be made c l e a r here t h a t the s y n t a c t i c a l e n t i t i e s o c c u r r i n g intermediate t o a d e r i v a t i o n of descendants are "ephemera," i . e . not considered t o be strands, but r a t h e r snapshots of stages i n a process. At each l i n e of a d e r i v a t i o n we are " f r e e z i n g " the a c t i o n t o see what Chapter V 135 has r e s u l t e d -from the l a t e s t enzyme o p e r a t i o n . But the t r a n s f o r m a t i o n continues, and only the s t r i n g t h a t i s l e f t a f t e r a l l enzymes are f i n i s h e d i s viewed as the descendant strand of the process. "Double strands" are not strands . What I have l o o s e l y c a l l e d "double s t r a n d s " are a c t u a l l y always transforms, because they i n v a r i a b l y separate i n t o a p a i r of s i n g l e s t r a n d s at the end of a d e r i v a t i o n . J u s t the opposite of nature, i n Typogenetics the double s t r i n g form i s temporary ("unstable"), the s i n g l e form, normal ( " s t a b l e " ) . Strands , not trans-forms are translated. The strand transform appearing on some l i n e i n a d e r i v a t i o n of descendancy, i s not t r a n s l a t e d , because i t i s a temporary t h i n g , o n l y an operand i n one of i t s stages of t r a n s f o r m a t i o n . T h i s i s very important to avoid confusion on. In the beginning, one has a given s t r a n d . One t r a n s l a t e s i t , a l l at once and completely, t o l e a r n what op e r a t i o n s i t s h a l l c a r r y out on i t s e l f . Once the o p e r a t i o n s begin, we get t h i s s u c c e s s i o n of transforms, u n t i l the o p e r a t i o n s have a l l been c a r r i e d out. What i s l e f t are one or more strand s that can be t r a n s l a t e d . The Typogenetic strand can be viewed as a " c r e a t u r e " with a c t i v e and p a s s i v e phases. In i t s a c t i v e phase i t t r a n s l a t e s out (releases) "amino a c i d s , " i n sequence, capable of doing s p e c i a l jobs, i n sequence, that could add up to an Chapter V 136 e n t i r e c o n s t r u c t i o n p r o j e c t . Then, the s t r a n d goes i n t o i t s p a s s i v e stage. I t l i e s prone while the amino a c i d s go to work on i t , lengthening i t here, shortening i t t h e r e , p o s s i b l y adding a second l a y e r t o i t , and p o s s i b l y c u t t i n g i t i n t o many p i e c e s . L i n e by l i n e , the transforms y i e l d e d by enzyme a c t i o n are l i s t e d , i n a d e r i v a t i o n o-f descendancy. Now, i n l i g h t of the f o r e g o i n g , one should r e a l i z e why, though i t i s tempting, i t i s very mistaken t o l a p s e i n t o b e l i e v i n g t h a t , say, the w h i t t l i n g down of one's operand i n t o s h o r t e r and s h o r t e r transforms through the course of a d e r i v a t i o n i s somehow a l t e r i n g the number or type of the o p e r a t i o n s t h a t w i l l yet be performed i n t h i s d e r i v a t i o n . One must keep i n mind that the number and type of the o p e r a t i o n s t o be c a r r i e d out i n t h i s d e r i v a t i o n was f u l l y and f i n a l l y s p e c i f i e d by d u p l e t s i n the t r a n s l a t e d s t r a n d . A transform l i k e the one at l i n e 4 of the sample d e r i v a t i o n given above, shows the cumulative (to that point) e f f e c t s of those o p e r a t i o n s t h a t have been performed. But we do not bother what i t s d u p l e t s code f o r , as long as i t i s a transform. We complete the d e r i v a t i o n , and only then, i f a s t r i n g has r e s u l t e d , do we c a l l t h a t a s t r a n d . However, i t i s t r u e that enzymes must have something to work on. So t h e r e are ways i n which one can arrange i t so that the nth transform of a designed strand's d e r i v a t i o n w i l l be such as to defy any f u r t h e r enzyme tra n s f o r m a t i o n . In that Chapter V 137 q u i t e d i f f e r e n t sense, by being " i n o p e r a b l e , " a transform can br i n g a d e r i v a t i o n t o a h a l t (and being t h a t l a s t transform i t would t h e r e f o r e be, by d e f i n i t i o n , a "daughter," the descendant s t r a n d d e r i v e d ) . Intra— vs. inter-generational succession. C o r o l l a r y t o a l l of the above i s the f a c t t h a t t h e r e are two kinds of succ e s s i o n i n Typogenetic world, i n t r a - g e n e r a t i o n a l and i n t e r - g e n e r a t i o n a l . The temporary being of the strand transform i s a b i t l i k e the stage of l i f e ( i n t r a -g e n e r a t i o n a l ) a metamorphizing c r e a t u r e such as a f r o g passes through (e.g. as t a d p o l e ) . Such l i f e stages a l l belong t o one i n d i v i d u a l ' s " l i f e t i m e . " But the i n t e r - g e n e r a t i o n a l r e l a t i o n s h i p of an c e s t r y , that strands proper can enjoy, holds between d i s t i n c t i n d i v i d u a l s . These two kinds of succ e s s i o n must be c l e a r l y d i f f e r e n t i a t e d i n the reader's mind, t o understand the f o l l o w i n g c l a s s i f i c a t o r y scheme, or to make any use of t h i s formal system. Types of Strands P r o d u c i b l e The f o l l o w i n g r e p r e s e n t s a c l a s s i f i c a t i o n , with examples u s u a l l y chosen f o r t h e i r s i m p l i c i t y , of strands p r o d u c i b l e i n Typogenetics. Since, f o r many of these strands, a p a r t i c u l a r binding s i t e of the se v e r a l a v a i l a b l e must be e l e c t e d f o r the strand t o demonstrate i t s p e c u l i a r c a p a b i l i t i e s , the lower—case l e t t e r Chapter V 138 convention has been employed t o designate which i s the a p p r o p r i a t e u n i t t o be e l e c t e d . D e r i v a t i o n s o-f these s t r a n d s are presented i n Appendix I I I . Two types of sterile s t r a n d s are incapable of any r e p r o d u c t i v e a c t i v i t y : ( i ) Duds do not o f f e r t h e i r enzyme(s) any b i n d i n g s i t e t o begin work, hence do not undergo any t r a n s f o r m a t i o n . An example: GGTT (Binding p r e f e r e n c e : A). ( i i ) Enzymes of trivial s t r a n d s s h u t t l e about moving, se a r c h i n g , or s w i t c h i n g , without ever s e t t l i n g down t o do any r e a l work (copy, i n s e r t , d e l e t e , or c u t ) . E.g.: TACCAT (Binding preference: T; i t i s t r i v i a l on e i t h e r b i n d i n g s i t e e l e c t i o n ) . Monstrous s t r a n d s beget descendants n o n i d e n t i c a l t o themselves, hence are not s e l f - r e p l i c a t i n g . E.g. our strand i l l u s t r a t i n g Typogenetic d e r i v a t i o n above, with i t s three daughters. These are the most commonly encountered strands. A seIf—perpetuator 's enzyme a c t i v i t y a l t e r s i t s operand at some stage(s) i n the game, only to undo whatever was done, e.g. u n i t s are dele t e d only t o be r e p l a c e d . Example: TCAGCGATGTCtAGAGCT. To design such a strand i s not p a r t i c u l a r l y d i f f i c u l t , and provides a good e x e r c i s e f o r the " T y p o g e n e t i c i s t . " Recurrors come i n two types: Chapter V 139 (i) A self-recurror i s l i k e a s e l -f -perpetuator, but r e t u r n s t o i t s e l - f inter-generational 1 y r a t h e r than intra-g e n e r a t i o n a l l y . ( i i ) An incidental recurror i s produced at r e g u l a r i n t e r v a l s as the byproduct o-f a r e c u r r o r 's r e p r o d u c t i v e a c t i v i t y . Example: The strand cGATGTGGCG i s a s e l - f - r e c u r r o r ; i t y i e l d s as daughters the s t r a n d s GGT and AcCGATGTGGCG. The l a t t e r y i e l d s i n tu r n (another) GGT, AACC, and (the sel-f-r e c u r r o r ) CGATGTGGCG. Thus CGATGTGGCG r e c u r s along i t s own l i n e o-f descent, every other g e n e r a t i o n . A l s o r e c u r r i n g along that l i n e at every generation i s the strand GGT— pu r e l y as an i n c i d e n t a l byproduct (GGT does not recur i n i t s own l i n e o-f descent, and thus cannot be considered a se It-recurror i, i t s only daughter i s the dud GGTT) . Sel-f—Repl i c a t o r s There appear t o be two modes t h a t s e l f — r e p l i c a t i o n might take i n Typogenetics: One mode would i n v o l v e a strand extending i t s e l f h o r i z o n t a l l y along one l e v e l (perhaps through s e v e r a l generations) and then c u t t i n g i t s e l f i n t o two pi e c e s which are e i t h e r already r e p l i c a s of t h e i r p r o g e n i t o r or w i l l beget same. No such strand i s known to have been c r e a t e d — t h e reader i s challenged t o c r e a t e one. The other type of s e l f - r e p l i c a t o r , an example of Chapter V 140 which has been found and which w i l l now be d i s p l a y e d , uses the copy f u n c t i o n t o c r e a t e a double strand that w i l l separate i n t o two daughters that are e i t h e r already r e p l i c a s of t h e i r parent or w i l l grow i n t o same. A simple, n c m - s e l f - r e p l i c a t i n g self-copier i l l u s t r a t e s an important c o n s i d e r a t i o n t o be borne i n c r e a t i n g t h i s l a t t e r type of s t r a n d . CGTTCCA succeeds i n copying i t s e l f i n i t s e n t i r e t y (see Appendix I I I ) . However, because copying i n Typogenetics i n v o l v e s making a "negative," or complement of what i s co p i e d , the r e s u l t a n t double strand separates i n t o these two s t r a n d s : CGTTCCA and T6GAACG. These stran d s are i n the Typogenetical sense c o p i e s (complements) of one another, but c l e a r l y TGGAACG i s n o n i d e n t i c a l t o i t s parent. S e l f -copying does not n e c e s s a r i l y mean s e l f - r e p l i c a t i n g . There i s reason then t o make the o r i g i n a l s trand self—complementary! i t s copy i s both complementary t o and a r e p l i c a of i t s e l f . Four of the 16 p o s s i b l e d u p l e t s of Typogenetics are self—complementary: AT, CG, TA, and AT. F o r t u n a t e l y , i t i s not necessary t o r e s t r i c t one's choice of d u p l e t s t o j u s t these s p e c i a l ones i n assembling a s e l f - r e p l i c a t o r (the four commands those d u p l e t s code f o r may not provide an adequate r e p e r t o i r e ) . Rather, one induces (in the A r i s t o t e l i a n sense: grasping the u n i v e r s a l from the wel1-apprehended p a r t i c u l a r ) from such examples the g e n e r a l i z a t i o n that any strand which i s l e f t / r i g h t Chapter V 141 JLV complementary i s self-comp1ementary (e.g. AT). Once t h i s i s seen i t becomes p o s s i b l e t o c o n c e p t u a l i z e one's t a r g e t strand as c o n s i s t i n g of two p a r t s , r i g h t and l e f t . To one of these p a r t s i s assigned the duty of copying the entire s t r a n d . As f o r the other p a r t i t need on l y be (1) the p r e c i s e complement of the copying h a l f , and (2) h a r m l e s s — i . e . does not code f o r any commands d i s r u p t i v e of the o v e r a l l o b j e c t i v e . A strand meeting these requirements i s given below (bottom, next page). The s t r a n d , S-R, has 64 u n i t s and 4 genes. Let the p o s i t i o n s of the u n i t s i n the s t r a n d , from the l e f t , be numbered 1 t o 64. Now, r e c a l l i n g what i t i s f o r two bases t o be complementary (C i s complementary to B, A to T ) , n o t i c e S-R's n i c e Pythagorean p r o p o r t i o n : I t s 1st u n i t i s complementary t o the 64th, as i s the 2nd with the 63rd, the 3rd with the 62nd, e t c . , on i n t o the complementing, innermost p a i r of the s t r a n d , an A (32nd) and T (33rd). Formally: Let n be the length i n u n i t s of S-R (64), and l e t u be a u n i t at p o s i t i o n #x i n S-R. Then t h a t u n i t c which i s at the p o s i t i o n #y, such that x+y—n+1, i s n e c e s s a r i l y complementary to u, i n the Typogenetic sense. The s t r u c t u r e i s e q u i v a l e n t to an a r i t h m e t i c a l magic square (an equivalence formulated more e x a c t l y infra, at p261). A magic square c o n s i s t s of an a d d i t i o n t a b l e such that the same sum i s obtained at every row and column i n t e r s e c t i o n . The p r e c i s e design o b j e c t i v e of the present c o n s t r u c t i o n Chapter V 142 was to obt a i n a sequence such t h a t , when read le-ft t o r i g h t , i t i s complementary to i t s e l - f read r i g h t t o l e - f t . It can be c o n c e p t u a l i z e d t h i s way: If one reads S—R -from le-ft t o r i g h t , c o n v e r t i n g each base one scans t o i t s complement (A and T interchange; C and 6 i n t e r c h a n g e ) , by the time one has -finished, one has run through a p e r f e c t preview of what i t would be l i k e t o scan S—R from r i g h t t o l e f t . T h i s means that the stra n d has a remarkable, u n d e r l y i n g pal1indromic s t r u c t u r e (see infra, pp 2 1 I f f , on complementarity and pal1indromes). The p o i n t of a l l t h i s i s t h a t a copy of a strand S i s (1) a complement of S and (2) i s l e f t - r i g h t reversed from S. Yet S—R i s b u i l t t o be i n v a r i a n t under a combined complementarity and l e f t / r i g h t order r e v e r s a l . So i t s copy i n the Typogenetic sense i s a l s o a perfect replica of the p r o g e n i t o r . For d e s i g n i n g S—R, h a l f the c h a l l e n g e was i n producing the self-complementary operand. The other h a l f was c o n t r i v i n g t o make a v a i l a b l e the r e q u i s i t e o p e r a t i o n s t o e f f e c t s e l f -r e p l i c a t i o n . To accomplish the l a t t e r , i t i s of course necessary t o i n c l u d e d u p l e t s coding f o r those o p e r a t i o n s , yet the d u p l e t s must f i t i n t o the h i g h l y r e s t r i c t i v e complementarity framework. Here i s the product, S—R CGCGCGCGTAATATAACGATCGCGCGTATTAATTAATACGCGCGATC6TTATATTACGCGCGCG Appendix III g i v e s a d e t a i l e d account of the d e r i v a t i o n of t h i s s t r a n d , and o f f e r s f u r t h e r understanding of i t s form. Chapter CHAPTER VI: META-TYPOGENETICS In the preceding chapter a l l r e s u l t s were obtained within the formal system of Typogenetics. That i s , the a s s e r t i o n s made (e.g. "there i s a s e l f - r e p l i c a t i n g strand") were proved by adducing a s t r a n d or s t r a n d - l i n e and e x h i b i t ! i i t s p r o p a g a t i v e / s e l f — t r a n s f o r m a t i o n a l p r o p e r t i e s i n the c h a r a c t e r i s t i c Typogenetical d e r i v a t i o n format. However, many important t h i n g s cannot be demonstrated within the system. For example, i f i t i s claimed that a designated s t r a n d I has i n f i n i t e l y many distinct descendants how could i t s s t r a n d - l i n e be e x h i b i t e d in the system? That would r e q u i r e an i n f i n i t e l i s t i n g of descendants. Again, i f negative a s s e r t i o n i s made that such and such a w e l l - d e f i n e d strand (e.g."a s t r a n d coding f o r s u f f i c i e n t o p e r a t i o n s t o completely erase i t s e l f " ) does not e x i s t , how can that be demonstrated i n the system shor t of l i s t i n g every p o s s i b l e strand and showing that each i s not the strand claimed not t< e x i s t ? Meta—Typogenetics c o n s i s t s of r e s u l t s obtained f o r and about T y p o g e n e t i c s — e i t h e r of the system as a whole or of part s of i t . These analyses and demonstrations ("informal," Chapter VI 144 i.e. to be c o n t r a s t e d with d e r i v a t i o n s ) will be argued and/or formulated with o r d i n a r y language, formal l o g i c , and simple a l g e b r a . Variables, which it w i l l be r e c a l l e d Typogenetics l a c k e d , w i l l be much used. Algorithms with p r o b a t i v e value w i l l a l s o be used t o e s t a b l i s h p r o p o s i t i o n s . Reductio S t r a t e g y f o r D i s p r o v i n g a Hy p o t h e t i c a l Strand The f o l l o w i n g two pr o o f s i l l u s t r a t e how a reductio s t r a t e g y can take one from an i n i t i a l l y p l a u s i b l e conception of a s t r a n d with c e r t a i n d e s i r e d s t r u c t u r a l and f u n c t i o n a l c h a r a c t e r i s t i c s t o the c o n c l u s i o n that the conception i s s e l f — c o n t r a d i c t o r y . The i n i t i a l conception of the s t r a n d must be very d e t a i l e d . The usual c o n t r a d i c t i o n t o be obtained c o n t r a s t s the s t r u c t u r e of the strand as h y p o t h e t i c a l operand with i t s s t r u c t u r e as impl i e d by the operators i t s propagative f u n c t i o n r e q u i r e s . A S e l f - E r a s i n g Strand? H o f s t a d t e r ' s Godei, Escher. Bach <1979) f e a t u r e s such t h i n g s as s e l f - s n u f f i n g p i p e s , s e l f - f a l s i f y i n g p r o p o s i t i o n s , s e l f - d e s t r o y i n g phonographs and the like. In a s i m i l a r s p i r i t , I ask: Is i t p o s s i b l e to devise a Typogenetic strand that will succeed i n completely a n n i h i l a t i n g i t s e l f ? F i r s t , l e t the nation of self-erasure Chapter VI 145 be de-fined (we encountered the concept e a r l i e r , i n connection with the Turing Table automaton, Chapter IV). A sel - f—erasing s t r a n d i s a n o n - t r i v i a l s t r a n d that leaves no daughters nor b i t s . In other words, a f t e r some s u b s t a n t i v e o p e r a t i o n s have been c a r r i e d out, i t s d e r i v a t i o n winds up with no t r a c e of a remaining transform. The f o l l o w i n g argument demonstrates t h a t , although a stra n d can break i t s e l f up, and p a r t i a l l y erase i t s e l f , t o t a l s e l f — e r a s u r e i s impossible. There are only two o p e r a t i o n s having the e f f e c t of d i m i n i s h i n g a s t r a n d . The cut op e r a t i o n though always leaves a b i t or d a u g h t e r — t h a t p a r t which was cut o f f , which i s beyond the reach of the enzyme t o f u r t h e r reduce. So the h y p o t h e t i c a l s e l f - e r a s i n g strand must not i n v o l v e c u t t i n g . That leaves the d e l e t e o p e r a t i o n . The s e l f - e r a s e r must d e l e t e i t s e l f i n t o o b l i v i o n . AG i s the duplet coding f o r d e l e t i o n . Since the h y p o t h e t i c a l s e l f — e r a s i n g strand cannot d i m i n i s h i t s e l f by c u t t i n g , there must be at l e a s t as many d e l e t i o n s c a r r i e d out as there are u n i t s i n the st r a n d , and that r e q u i r e s as many AG du p l e t s as there are u n i t s i n the strand. But t h i s i s c l e a r l y p r o b l e m a t i c a l . A duplet i s two u n i t s . We have s a i d that the strand of r» base length must have n AS dup l e t s , i . e . must be at l e a s t 2n u n i t s long. A strand s a i d to be both TI bases long and 2n u n i t s long i s s e l f — C h a p t e r V I 146 c o n t r a d i c t o r y . I n f a c t , t h e s i n g l e d u p l e t strand a 8 i s t h e c l o s e s t t h i n g t o a s e l f - e r a s e r p o s s i b l e , b e c a u s e i t c a n a t l e a s t l e a v e o n l y a s i n g l e b i t , G , b e h i n d . I t i s a n o n — t r i v i a l s t r a n d w i t h n o d a u g h t e r s . T h e o u t c o m e o f t h e a b o v e p r o o f i m p l i e s t h a t , o n o u r d e f i n i t i o n o f " s e l f - e r a s u r e " w i t h i t s r e q u i r e m e n t o f l e a v i n g n o t o n e b i t b e h i n d , a " s e l f — e r a s e r " c a n n o t e x i s t i n o u r s y n t a c t i c a l w o r l d . T o p u t i t a n o t h e r w a y , n e w r u l e s e x p e d i t i n g s e l f - e r a s u r e w o u l d h a v e t o b e a d d e d t o T y p o g e n e t i c s t o a l l o w f o r s u c h . T h e n e w r u l e w o u l d h a v e t o h a v e t h e e f f e c t o f p a c k i n g m o r e " s e l f — e l i m i n a t i v e " p o w e r i n t o a m o r e c o m p a c t c o d e , s o t h a t w e e s c a p e t h e f r u i t l e s s s e a r c h t o a d d m o r e d e l e t i o n p o w e r b y a d d i n g m o r e c o d e t h a t i n c r e a s e s t h e l e n g t h o f o u r o p e r a n d g i v i n g u s m o r e t o e r a s e ! We h a v e d i s c o v e r e d a logical truth a b o u t o u r p r e s e n t r u l e s , s o m e t h i n g t h e maker o f t h e s y s t e m d i d n o t r e a l i z e s i m p l y i n v i r t u e o f h a v i n g d e f i n e d t h e s y s t e m . A n d , b e c a u s e a l o g i c i a n d i s c o v e r e d t h i s f a c t , the reasoning h a s b e e n m a d e e x p l i c i t e n o u g h t h a t a n o t h e r m i n d c a n r e t r a c e t h e i n f e r e n c e . I t i s t h u s a p e r f e c t e x a m p l e o f w h a t l o g i c i a n s a d e p t a t m e t a — l o g i c a l a r g u m e n t a t i o n c a n a d d t o s y s t e m s s c i e n c e : We p r o v e t h i n g s about s y s t e m s . S o m e t i m e s , p r o o f l e a d s t o p r o o f . T h e r e m a y , f o r e x a m p l e , b e o t h e r i n t e r e s t i n g g e n e r a l p r o p o s i t i o n s w e c u r r e n t l y w o n d e r a b o u t o r a s s u m e Chapter VI 147 u n j u s t i f i e d l y t a be t r u e , t h a t e n t a i l the e x i s t e n c e of s e l f -erasure. They are negated by the c o n c l u s i o n reached above. The f a c t t h a t strand AG does e l i m i n a t e i t s e l f as a member of the o n t o l a g i c a l c l a s s " s t r a n d , " even though i t s r e d u c t i o n leaves a s i n g l e bit ( i s o l a t e d i n d i v i d u a l u n i t ) , has s i g n i f i c a n c e of i t s own. I t demonstrates that two d i f f e r e n t kinds of " s e l f - e r a s u r e " are c o n c e i v a b l e i n t h i s purely-s y n t a c t i c a l domain: a str a n d can reduce i t s e l f down to non-s t r a n d , but cannot reduce i t s e l f down t o nothingness. S u i c i d e yes, v a n i s h i n g a c t , no. A H y p o t h e t i c a l S e l f - R e p l i c a t o r Disproved The above proof e s t a b l i s h e d that there can be no member of an e n t i r e h y p o t h e t i c a l c l a s s of s t r a n d s , the s e l f - e r a s e r s . The present proof shows how one can r u l e out a spec if ic in stance of a general c l a s s that may yet have e x i s t e n t members. The author has produced many such d i s p r o o f s i n his.endeavors t o f i n d i n t e r e s t i n g strands. E a r l i e r i t was suggested t h a t there might be a type at s e l f — r e p l i c a t o r which extends i t s e l f h o r i z o n t a l l y , then cuts i t s e l f i n t o two daughters Dl and D2 which e i t h e r already are i d e n t i c a l t o the pr o g e n i t o r or w i l l grow i n t o same. The strand ACAaACAa does succeed i n c u t t i n g i t s e l f i n ha l f with two i d e n t i c a l daughters r e s u l t i n g ; but the Chapter VI 148 daughters are not i d e n t i c a l t o t h e i r parent, nor do they grow i n t o same, so t h i s strand i s not a s e l - f - r e p l i c a t o r , But one can c o n c e p t u a l i z e how such a strand could be. As one's c o n c e p t u a l i z a t i o n grows p r e c i s e , consequences are e n t a i l e d . If the consequences u l t i m a t e l y c o n f l i c t , then the imagined s t r a n d , i n s o f a r as i t has been s p e c i f i c a l l y d e s c r i b e d , has been proved untenable. The f o l l o w i n g i l l u s t r a t e s t h i s . $, the h y p o t h e t i c a l s t r a n d , has i d e n t i c a l halves *1 and *2, The f i r s t o p e r a t i o n executed by the f i r s t enzyme of the s t r a n d c u t s $ i n t o i t s two halves, *1 and *2. The second o p e r a t i o n executed detaches the enzyme. Remaining enzymes e i t h e r cannot f i n d p l a c e s of attachment on the remaining transform *1 or are t r i v i a l . No subsequent o p e r a t i o n s having been c a r r i e d out, *1 and *2 are daughters of $. C a l l these daughter strands, r e s p e c t i v e l y , Si and #2. As a l r e a d y s t a t e d , * propagated $1 and * propagated £2= But now add: --£1 propagates *3 and $3 i s i d e n t i c a l to * *2 propagates *4 and *4 i s i d e n t i c a l t o -t Thus * has two granddaughters i d e n t i c a l t o i t s e l f ; t h i s i s a s e l f - r e p l i c a t i n g strand. Further s p e c i f i c a t i o n : Among the duplets of $1 (or we Chapter VI 149 could be speaking of $2, s i n c e they are twins) i s , by n e c e s s i t y , AC, coding f o r the cut that separated *1 and *2. There are a l s o i n $1 r< du p l e t s coding f o r extension of the strand through insertion o p e r a t i o n s . Now, the propagative f u n c t i o n of t h i s strand $1 i s of course t o double i t s l e n g t h , so that i t s descendant i s i d e n t i c a l t o the o r i g i n a l p r o g e n i t o r , *. Thus th e r e must be s u f f i c i e n t i n s e r t i o n commands to double #1. $-1 had n d u p l e t s coding f o r i n s e r t i o n , and the one duplet coding f o r the cut. A t o t a l of 2(n+1) u n i t s then must be produced by the i n s e r t i o n o p e r a t i o n s to achieve doubling of SI. But to produce 2(n+1) bases through i n s e r t i o n , 2(n+1) i n s e r t i o n duplets are needed. That many du p l e t s adds up to 2(2(n+D) u n i t s — o r over twice as many i n s e r t i o n d u p l e t s as our strand was def i n e d t o have. (If we def i n e d our strand t o have more duple t s coding f o r more i n s e r t i o n o p e r a t i o n s , then i t becomes l o n g e r — t w i c e as l o n g — a n d twice as many i n s e r t i o n o p erations are needed, so there can be no ca t c h i n g up). It must be concluded that the exact kind of strand c o n c e p t u a l i z e d i n i t i a l l y cannot e x i s t . However, t h i s r e s u l t must not be o v e r g e n e r a l i z e d . S p e c i f i c assumptions were made, and i t i s t h e i r combination that produce the c o n f l i c t i n g consequences. By studying what these assumptions were and why they l e d to c o n f l i c t , i t becomes p a s s i b l e to conceive of a l t e r n a t i v e p o s s i b i l i t i e s that C h a p t e r VI 150 r e m a i n open. F o r example, i f t h e o r i g i n a l s t r a n d $ e x t e n d s i t s e l f q u i t e a b i t before c u t t i n g i t s e l f i n t o i d e n t i c a l t w i n s , t h e n t h e d a u g h t e r s would n o t have t o doable t h e m s e l v e s , and might even have enough d u p l e t s c o d i n g f c r i n s e r t i o n t o g e t t h e j o b done. A l s o , i t i s p o s s i b l e t h a t a s t r a n d $ might be a b l e t o a c h i e v e s e l f - r e p l i c a t i o n w i t h g e n e r a t i o n s f u r t h e r removed t h a n i t s g r a n d d a u g h t e r s . I t h a s been shown t h a t $1 and $2 c a n n o t p r o p a g a t e $ r e p l i c a s ' . But p e r h a p s d a u g h t e r s *1 can *2 can e a c h p r o p a g a t e longer g r a n d d a u g h t e r s £3 and $4 t h a t f i n a l l y c o m p l e t e t h e j o b , i . e . p r o p a g a t e * r e p l i c a g r e a t g r a n d d a u g h t e r s . C o n s i d e r t h i s p o s s i b i l i t y . We know t h a t $ (our g o a l s t r a n d ) i s 4n+4 u n i t s i n l e n g t h ( r e m i n d e r : Ti i s t h e number o f i n s e r t i o n duplets each i d e n t i c a l d a u g h t e r of $• p o s s e s s e s ) . We know t h a t an $1 o r $2 d a u g h t e r can a t b e s t p r o d u c e a g r a n d d a u g h t e r of l e n g t h 3n+2 u n i t s . I f t h i s g r a n d d a u g h t e r c o n s i s t s e n t i r e l y o f i n s e r t i o n commands, t h e n i t can i n s e r t (3n+2)/2 new u n i t s . The g r e a t g r a n d d a u g h t e r t h e n c o u l d be up t o (3n+2) •+( (3n+2) /2) u n i t s l o n g . S i m p l y i n g t h a t e x p r e s s i o n a l g e b r a i c a l l y we have a s t r a n d o f up t o 4n+3+(n/2) u n i t s — l o n g enough t o c o n s t i t u t e an . Chapter VI 151 Example: Let n=12 Progenitor * was 4n+4 or 52 u n i t s Dciaughters are 2n+2 or 26 u n i t s Granddaughters are at most 3n+2 or 38 u n i t s Great granddaughters are at most 4n +(n/2 +3) or 57 u n i t s Of course there may s t i l l be reasons why a strand of t h i s type, r e p l i c a t i n g at the great granddaughter stage, i s not p o s s i b l e . But the motives that l e d us to gi v e up on the r e p l i c a t o r at the granddaughter stage at l e a s t have been shown to be no n a p p l i c a b l e here. The next step o b v i o u s l y i s to c o n c e p t u a l i z e a str a n d that one hopes can surmount the i d e n t i f i e d d i f f i c u l t i e s and see i f i t s s p e c i f i c a t i o n s imply c o n t r a d i c t i o n . By t h i s course of reductio ad absurdum reasoning the p o s s i b l e routes t o the s o l u t i o n strand are s t e a d i l y e l i m i n a t e d u n t i l the r i g h t one i s found...or the answer set i s proved t o be empty. A Cascading S t r a n d - l i n e A strand e x i s t s which may be c a l l e d i n f i n i t e l y f e r t i l e . Of course, a s e l f - r e p l i c a t o r produces an i n f i n i t y of r e p l i c a s of i t s e l f , and a s e l f - r e c u r r o r produces an i n f i n i t y of both (1) r e p l i c a s of i t s e l f and (2) i n c i d e n t a l r e c u r r o r s . But the i n f i n i t e l y f e r t i l e s trand produces a descendant d i f f e r e n t than i t s e l f which i n turn produces a descendant d i f f e r e n t Chapter VI than i t s e l f , e t c e t e r a , ad infinitum. That i n f i n i t e l y f e r t i l e s trand thus packs i n t o one f i n i t e s t r i n g a cascading ( i n the l o g i c a l sense) s t r a n d - l i n e . The simplest such strand i s SAGA. GAGA adds by i n s e r t i o n a p a i r of A bases t o i t s r i g h t s i d e . That descendant, GAGAAa adds another p a i r of A's to become GAGAAAAa. So f o r each descendant, so long as the A chosen as binding s i t e i s rightmost of the s t r a n d . The AA duplet does not a f f e c t b i n d i n g p r e f e r e n c e , so every descendant of GAGA w i l l have the same bindin g preference as the o r i g i n a l p r o g e n i t o r . That b i n d i n g preference i s f o r A, the type of base that keeps being i n s e r t e d onto the r i g h t s i d e of each descendant down the s t r a n d — l i n e . These added AA du p l e t s do not of course code f o r o p e r a t i o n s . So the two op e r a t i o n s , the A i n s e r t e r s , r e t a i n e d from generation t o generation, are s o l e l y r e s p o n s i b l e f o r making each descendant a duplet longer; no descendant of GAGA has any other kind of operation (again, so long as the rightmost A i s s e l e c t e d at every generation as the enzyme's i n i t i a l binding s i t e ) . A kind of mould, or schemata, can be s t a t e d f o r making strands l i k e GAGA (these are s u f f i c i e n t , not necessary-c o n d i t i o n s , some of them being o v e r l y s t r i c t ) ; Let * be the o r i g i n a l one gene strand of length L u n i t s <L>3 and an even number) with i t s rightmost base, b. The enzyme of $ has a binding preference f o r b. The n i n s e r t i o n Chapter VI 15 commands coded f o r i n $ lengthen the strand at (to the r i g h t of) b e x a c t l y n bases (r< i s an even number) . There are two r e s t r i c t i o n s on what the n/2 dupl e t s added on can be. F i r s t , t h e i r corresponding amino a c i d f o l d i n g i n c l i n a t i o n s are, cum u l a t i v e l y , t o have a straight t e r t i a r y s t r u c t u r e ; second, any i n s e r t e d d u p l e t s must code e i t h e r f o r the AA duplet or f o r a command to detach the enzyme (e.g. s w i t c h ) . Now, f o r any such strand the gth generation descendant w i l l be L+gn u n i t s long; w i l l have a binding preference f o r £>; w i l l have the same i n s e r t i o n o p e r a t i o n s as $ possessed and new d u p l e t s e x c l u s i v e l y coding e i t h e r f o r AA or f o r a command to detach the enzyme. Because the o r i g i n a l i n s e r t i o n d u p l e t s are l e f t m o s t i n every descendant, they are always executed f i r s t . They grow a descendant that has no a d d i t i o n a l d u p l e t s coding f o r o p e r a t i o n s that would d i s r u p t the ba s i c p a t t e r n . S c h e m a t i c a l l y : Progenitor $ EUnits 1 to L - i l d Descendant #1 " " bn #2 13 t l brtTt # it b ri rt n II I I bTlTlTl Tl . . . Chapter VI 154 Computabi1i t y For some s e c t i o n s i s s u e s of computabi1ity, e.g. of str a n d s as comb i n a t o r i a l e n t i t i e s , w i l l now be considered. Where matters (e.g. formulas) are worked out i n d e t a i l , the reader may peruse s w i f l y , paying a t t e n t i o n mainly t o j u s t seeing what is available, with the assurance t h a t the d i s c o u r s e t u r n s more " p h i l o s o p h i c a l " l a t e r i n t h i s chapter, where some of the most important ideas and p u z z l e s of t h i s t h e s i s are e l a b o r a t e d . How Many Strands? For a st r a n d of n u n i t s length there are 4" permutations of the bases p o s s i b l e . However, a sequence of u n i t s having only the A base i s not a well—formed s t r a n d . Thus, f o r an (even) n, the number of strand s must be 4"—!. For example there are 16,777,216 d i s t i n c t permutations of bases f o r 12 u n i t length s t r a n d s , minus the one AAAAAAAAAAAA " r i n g e r . " There are, g i v e or take many t r i l l i o n s , 3.402 x 1 0 3 0 d i s t i n c t strands equal i n length t o the (64 u n i t ) s e l f — r e p l i c a t o r presented i n the previous chapter! For s t r a n d s of odd numbered length, permutations whose dup l e t s c o n s i s t e x c l u s i v e l y of AAs are not well—formed r e g a r d l e s s of what base the f i n a l odd b i t i s . Thus where n i s odd, the number of d i s t i n c t strands i s 4"-4. For example, Chapter VI 155 i f r<=3, AAA, AAC, AAG, and AAT are not well-formed and thus the t o t a l number of d i s t i n c t t hree u n i t strands i s 64-4 or 60. P o s s i b l e Daughters As e a r l i e r e x plained (pl27), election of a str a n d ' s i n i t i a l b i n d i n g s i t e s must f r e q u e n t l y be made by the user, i n the system as now def i n e d (but see Chapter X's d i s c u s s i o n of complete automatization of Typogenetics). T h i s must happen whenever there e x i s t s , i n the cu r r e n t transform being operated on, more than one in s t a n c e of the base the cur r e n t enzyme p r e f e r s t o i n i t i a l l y bind i t s e l f t o . Having no mind of i t s own, and with no s t o c h a s t i c d e c i s i o n b a s i s , the enzyme must be i n i t i a l l y attached by the a c t i v e c h o i c e of the user to one of those e l i g i b l e s i t e s . And t h i s c h o ice can make a l l the d i f f e r e n c e i n the world. The s e l f — r e p l i c a t o r S-R (given on pl42 above) d i s p l a y e d before only propagates i n the de s i r e d way i f the c o r r e c t c h o i c e s of i n i t i a l b inding s i t e f o r each of the enzymes are made (I have made what those c h o i c e s are c l e a r i n my d e s c r i p t i o n of the strand's s e l f -r e p l i c a t i o n ) . A completely d i f f e r e n t d e r i v a t i o n would almost c e r t a i n l y r e s u l t from d i f f e r e n t c hoices ( e s p e c i a l l y of the f i r s t or second enzymes). The m a j o r i t y of strands capable of doing anything i n t e r e s t i n g then can be expected to r e q u i r e t h i s choice of the user. Put another way, most strands encode m u l t i p l e Chapter VI 156 d i s t i n c t d e r i v a t i o n s , and the user has an important r o l e i n t h i s system d e c i d i n g on one route through t h i s branching network of p o s s i b l e d e r i v a t i o n s , r a t h e r than another. In a sense then, a strand has many p o s s i b l e daughters, and the user, having d e r i v e d some p a r t i c u l a r l y d e s i r e d daughters from a p r o g e n i t o r , may well ignore the other p o s s i b l e daughters d i f f e r e n t d e r i v a t i o n s would produce. The a l t e r n a t i v e d e r i v a t i o n s p o s s i b l e f o r the strand S-R have not been i n v e s t i g a t e d , though there are bound t o be many " p o s s i b l e s i b l i n g s " f o r that f o u r enzyme st r a n d . The f a c t i s that we are able t o achieve i n t e r e s t i n g or us e f u l propagative behavior out of our strands without exhausting any l a r g e f r a c t i o n of the information the strand r e p r e s e n t s . S—R has many t h i n g s i t can "do" (strands or b i t s i t can produce) besides s e l f — r e p l i c a t e . Our present concerns are narrow enough t o ignore S—R's f u l l c a p a b i l i t i e s . In Chapter X, I w i l l r a i s e the prospect of f i n d i n g some i n t e r p r e t a t i o n of Typogenetics that e x p l o i t s the information the strand has a d d i t i o n a l to i t s s e l f -r e p l i c a t i v e powers. Among other p o s s i b i l i t i e s , the t e r t i a r y -s t r u c t u r e of the strand might have, as a geometrical f i g u r e , some independent value t o make i t worth r e p l i c a t i n g . Perhaps i t i s a design of something. Perhaps i t i s a " b e a u t i f u l " design. In any case, i t a marvelous design, because within the laws of Typogenetics, i t "knows" how to make a copy of Chapter VI 157 i t s e l f ! How Many Daughters Though i t was remarked i n the preceding s u b s e c t i o n that there are many p o s s i b l e daughters f o r a t y p i c a l s t r a n d , the reader might be s u r p r i s e d at j u s t how many there are. The number o-f d i s t i n c t d e r i v a t i o n s f o r a strand i s c a l c u l a b l e . Keep i n mind that we are speaking s p e c i f i c a l l y of only the f i r s t generation descendants of a p r o g e n i t o r . For each enzyme t r a n s l a t e d out of the s t r a n d , a c h o i c e might have t o be made. In the branching t r e e of p o s s i b l e d e r i v a t i o n s , a new f o r k occurs each time i t becomes necessary to a t t a c h an enzyme t o one among s e v e r a l a v a i l a b l e binding s i t e s . Thus the q u a n t i t y e, that i s the number of enzymes i n the strand (equaling the number of genes i n the s t r a n d , of course) i s a key one. How many branches are there at a node? That depends on the q u a n t i t y £>/, which i s the number of u n i t s i n the transform having the p r e f e r r e d base. T h i s q u a n t i t y must be c a l c u l a t e d anew f o r each ith enzyme, because the operand i s always changing as one transform succeeds another. The q u a n t i t y Q of unique der ivations a given strand can g i v e r i s e t o then i s defined by t h i s r e c u r s i v e formula, w r i t t e n i n a manner f a c i l i t a t i v e of implementation i n BASIC: For 1=1 to e 1 1 Chapter VI 158 ; Let Q = Q x o/ Next I For example, a c e r t a i n strand may have three genes, hence three enzymes, hence three attachment ch o i c e s t o make, hence e=Z. Suppose the F i r s t enzyme's (1=1) b i s 5 (5 u n i t s had the base p r e f e r r e d by t h i s enzyme). Q i s then set at 5. The second enzyme's o=3, S O Q i s now 15. F i n a l l y , the t h i r d ; enzyme's 0=2, so Q becomes 30. A l l of the I=e=3 loops having been given, we have a f i g u r e of 30 p o s s i b l e d e r i v a t i o n s f o r t h i s s t r a n d . T h i s g i v e s us the systematic c a l c u l a t i o n of a unique d e r i v a t i o n viewed as a p o s s i b l e permutation of operations, i That i s not the same as saying how many d i f f e r e n t daughters can be de r i v e d from t h i s s t r a n d . T h i s author has been unable to a r r i v e at a general method f o r computing number of d i s t i n c t daughters that a strand can have. The problem i s | that d i f f e r e n t d e r i v a t i o n s can r e s u l t i n the same outcome. 1 To again r e f e r t o . s t r a n d S—R, i t s l a s t two enzymes are trivial. So i t does not matter where you choose t o i n i t i a l l y bind those enzymes, as f a r as what change i s wrought i n the transform as i t was handed down from the second-activated | enzyme. Yet those d i f f e r e n t e l e c t i o n s do r e s u l t i n d i f f e r e n t | movement of the enzymes around the transform; they are | recorded as d i s t i n c t d e r i v a t i o n s , even though they a l l lead t o the same descendants (the p e r f e c t t w i n s ) . •i Chapter VI 159 There are many reasons why two d i s t i n c t d e r i v a t i o n s o-f the same strand r e s u l t i n the same descendants. We thus have an i r r e g u l a r r e c u r s i v e space where branches reconverge down the road, -for reasons at l e a s t p a r t i a l l y independent o-f what made the branches d i v e r g e (which was the a v a i l a b i l i t y o-f a l t e n a t i v e b i n d i n g s i t e s ) . In any event, the number o-f d i s t i n c t d e r i v a t i o n s f o r a s i n g l e s t r a n d can be stupendous. For a 10 gene strand designed such t h a t each next ith transform o f f e r s at l e a s t 12 p o s s i b l e i n i t i a l b i n d i n g s i t e s f o r the next Ith enzyme, there are at l e a s t 1 2 l ° d i s t i n c t d e r i v a t i o n s . With the r i g h t design taqic, perhaps a strand could be made such that each of i t s d e r i v a t i o n s produces a unique daughter (or set of daughters). In t h a t case, the p r o g e n i t o r would have 12*° p o s s i b l e daughters. How Many Descendants? Because we cannot always compute the number of possible daughters a s t r a n d may have (only the number of d i s t i n c t der ivations of daughters), the most obvious way t o r e c u r s i v e l y c a l c u l a t e the number of possible descendants a strand may g i v e r i s e t o i s s i m i l a r l y l i m i t e d . If we d i d possess the c a l c u l a b l e quantity d* f o r p o s s i b l e daughters of the i t h generation descendant of a p r o g e n i t o r , then the r e c u r s i v e enumeration of that p r o g e n i t o r ' s p o s s i b l e Chapter VI 160 Where G equals the number o-f generations c a l c u l a t i o n i s to extend through: For 1=1 t o G 7 = 7 x D* Next I 7 would increment the t o t a l number of branches of descent the p r o g e n i t o r could engender. For example, over 3 generations, i f a pr o g e n i t o r has 10 daughters, t h a t each have IO daughters, that each have 10 daughters, then there are 10 3 or 1000 branches at the great granddaughter l e v e l . However, there i s a problem of reconverging branches of t h i s f a m i l y t r e e , s i m i l a r t o the problem we had with branches of p o s s i b l e daughters. Two d i f f e r e n t s t r a n d s can have the same o f f s p r i n g . Thus we cannot be c e r t a i n t h at a l l of the branches of the s t r a n d — l i n e are separate. Some may reconverge. The value 7 s e t s only a maximum on the number of descendants. 7 minus the number of reconvergences of the fa m i l y t r e e (two d i f f e r e n t s t r a n d s having i d e n t i c a l o f f s p r i n g ) would give the c o r r e c t e d ( l e s s e r ) number of d i s t i n c t descendants. R e l a t i v e Frequency of Bases The c r i t e r i a set f o r d e f i n i n g well-formed strands e n t a i l that the base A w i l l not occur with as great a frequency i n Chapter VI 161 s t r a n d s as w i l l the other bases. I n t u i t i v e l y one can understand t h a t by r e a l i z i n g t h a t GGGG i s a strand whereas AAAA i s not a s t r a n d . A i s -further d i s c r i m i n a t e d a g a i n s t i n the case of odd numbered length s t r a n d s , e.g. GGT i s a s t r a n d but AAT i s not. That f a c t , taken i n c o n j u n c t i o n with the e a r l i e r s e c t i o n on the asymmetry of t e r t i a r y s t r u c t u r e p r e f e r e n c e s , p o i n t s to a f a i n t tendency f o r dud s t r a n d s : the type of base p r e f e r r e d above a l l others as a b i n d i n g s i t e i s a l s o the l e a s t f r e q u e n t l y a v a i l a b l e base i n strands. In f a c t the frequency of occurrence of duds happens to be t r e a t e d i n the next s e c t i o n . F i r s t though, some p r e c i s e statement of the r e l a t i v e frequency of the bases must be made. For the s d i s t i n c t (see above) strands of length n u n i t s : Let 7 = sn T then i s the t o t a l number of c o n s t i t u e n t u n i t s i n a l l those s s t r a n d s of length n. Now, i f n i s even, l e t a = (7 - 3n)/4 a g i v e s , f o r s t r a n d s of length n, the number of u n i t s having the A base; (T-a)/3 equals the number of u n i t s having the C, or G, or T base. The percentage of u n i t s having a given base can be e a s i l y computed knowing these component t o t a l s . Example: Let r/=6. There are 4,095 6 u n i t length strands. T, the grand t o t a l of a l l those strands* Chapter VI 162 c o n s t i t u e n t u n i t s , comes to 24,570. ((24,570-(3x6))/4 = 6,138. Thus o-f those 4,095 6 u n i t s t r a n d s , 6,138 o-f t h e i r c o n s t i t u e n t bases are of base type A, 6,144 are Cs, 6,144 8s, and 6,144 Ts. 6,138/24,570 expresses the f r a c t i o n of u n i t s of the A base i n 6 u n i t s t r a n d s . If n i s odd, then l e t o = 4n Now a= ((T-o)/4)-3 The t o t a l of any of the non-A base types i s (T-a)/3. Example: Let r>=3. There are 60 3 u n i t length strands. 7 equals 180. o equals 12. a then i s 39. So t h e r e are 39 A base u n i t s , and 47 u n i t s of each of the other bases. Frequencies of Types of Strands In the l a s t chapter s t r a n d s were c l a s s i f i e d as duds, trivials, monstrous, self-perpetuators, self—recurrors, and self—replicators. Examples of each of these "natural kinds" were given, and i t might seem worth asking what percentage of strands f a l l i n t o each category. Working with the system f o r a time leads one t o the i n t u i t i v e judgment that the l a r g e m a j o r i t y of strands are monstrous. Could that be proved? In p r i n c i p l e — y e s . Yet the d i f f i c u l t i e s t o be overcome i n determining the f r e q u e n c i e s of the types of s t r a n d s — s o e a s i l y d e f i n e d — t e s t i f i e s t o the complexity of t h i s formal system. Some of those d i f f i c u l t i e s are i d e n t i f i e d below, beginning with the e a s i e s t case. Chapter V Duds. To be c l a s s i f i e d a dud a Typogenetics strand must be (1) of a c e r t a i n t e r t i a r y s t r u c t u r e d i c t a t i n g a s p e c i f i c base o; AND (2) l a c k i n g t hat s p e c i f i c base b. Consider the second c o n d i t i o n . Some st r a n d s lack one base only; some lack two, or even three bases. Take the f i r s t case. If b i s a given base, and r» i s the number of u n i t s i n the s t r a n d , then the number of fa-l e s s permutations w i l l be 3". E.g. f o r s i x u n i t lengths there w i l l be 4,096 p o s s i b l e permutations of the four bases, and 3*> or 729 of those permutations l a c k i n g , say, the S base. S l i g h t adjustments must be made to these f i g u r e s t o e l i m i n a t e the AAAAAA non-strand: there are 4,095 s i x u n i t strands^ and 728 of them lack the G base. Of those 728 strands, those whose t e r t i a r y s t r u c t u r e s p r e f e r G are duds. Now, i t i s tempting t o say that one f o u r t h of those 728 s t r a n d s w i l l have t e r t i a r y s t r u c t u r e s p r e f e r r i n g G and w i l l thus (meeting c o n d i t i o n 1 above i n our d e f i n i t i o n ) be duds. That may indeed roughly approximate the t r u t h . But as we have seen e a r l i e r , t e r t i a r y combinations are not p r e c i s e l y equiprobable. Even more confounding i s the f a c t t h a t a l l the o - l e s s s i x u n i t strands w i l l e n t i r e l y lack d u p l e t s having the b base. For example the G-less strands w i l l lack the AG, CG, GA, GC, GG, GT and TG d u p l e t s . Those dupl e t s of course have a s s o c i a t e d f o l d i n g i n c l i n a t i o n s : three are r i g h t t u r n i n g , two l e f t , and two s t r a i g h t , so the Chapter VI 164 728 G—less s t r a n d s w i l l evidence r e l a t i v e l y reduced r i g h t -t u r n i n g . The absence of those r i g h t t u r n s then w i l l s u b t l y skew the p r o b a b i l i t y d i s t r i b u t i o n f o r the t e r t i a r y s t r u c t u r e s , thus a f f e c t i n g the necessary second c o n d i t i o n to be a,dud, namely that i t have a preference f o r the very base i t l a c k s . T h i s skewing would be troublesome though not impossible t o c a l c u l a t e ; i t i s p a r t i c u l a r l y aggravating where odd numbered length s t r a n d s are i n v o l v e d . However, i t seems most u s e f u l here simply t o draw a t t e n t i o n to the non-independence of the probabilities of the two conditions defining a dud. The one case where we can a c t u a l l y go t o the t r o u b l e of computing e x a c t l y the frequency of a type of strand i s i n the case where the strand l a c k s three of the four bases. E.g. CCCCCC. The s t o r y i s the same as f o r the A—only non-strands. In other words, of the n"*—1 strands of length n, only three w i l l be the s i n g l e base type: the strand of n Cs, the strand of n Gs, and the strand of n Ts. C—only stran d s of any length w i l l always be duds, f o r the CC duplet has a s t r a i g h t f o l d i n g i n c l i n a t i o n and a l l C-only strand t e r t i a r y s t r u c t u r e s w i l l p r e f e r A. Three out of four G-only and T-only strands are duds. Recursive formulas i d e n t i f y e x a c t l y which are and which are not. For G—only strands: If i t s length n i s 2 or 3 then i t i s a non-dud. And i f Chapter V the strand o-f n-8 u n i t s length was a non-dud then the strands o-f n and n + 1 length are non-duds. A l l others are duds. For T-only s t r a n d s : I-f i t s length n i s 4 or 5 then i t i s a non-dud. And i f the s t r a n d of n-8 u n i t s length was a non-dud then the st r a n d s of n and n+1 length are non-duds. A l l others are duds. Multipie gene duds. The presence of m u l t i p l e genes i n a strand m u l t i p l i e s the c o n t i n g e n c i e s i n v o l v e d i n determining the p r o b a b i l i t y of i t s being a dud, easy as i t i s to invent such a s t r a n d . If t h e r e are n genes, there w i l l be n t e r t i a r y s t r u c t u r e s , and n enzymes. A multi—gened dud must be such that no one of i t s enzymes can attach to a base belonging t o any of the n genes. Among other t h i n g s that means that a Concatenation of dud one—gened strands i n t o a multi-gene strand w i l l u s u a l l y not make a dud. Suppose we have a one gene strand dud l a c k i n g only the base G, which i t s enzyme p r e f e r s , and concatenate that gene with another one gene strand dud, l a c k i n g only the base C, which i t s enzyme p r e f e r s . The r e s u l t i s not a dud, f o r the former enzyme can attach t o the G's of the l a t t e r gene, and the l a t t e r enzyme to the C's of the former gene. Only a small f r a c t i o n of dud gene concatenations r e s u l t i n dud multi—gene strands. Trivial strands. C a l c u l a t i o n of the frequency of the Chapter VI 166 dud proved complicated. Would one -fare b e t t e r with the t r i v i a l s trand? It i s a strand that does not c a r r y out any ope r a t i o n s -for r e a l a c t i o n s (copy on, cut, d e l e t e , i n s e r t ) . At - f i r s t one i s tempted t o b e l i e v e t h i s merely r e q u i r e s computation o-f the number o-f permutations of non-action coding d u p l e t s s e t over the t o t a l number of permutations of du p l e t s . But i n f a c t a st r a n d l i k e CAACCGGCGCAGGT, though coding f o r the whole gamut of a c t i o n o p e r a t i o n s , has a bin d i n g preference c h o i c e ( f o r the one and only T) and f i r s t o p e r a t i o n (move r i g h t ) that together f o r c e detachment of the (only) enzyme p r i o r t o the execution of the a c t i o n o p e r a t i o n s . It i s a bona f i d e t r i v i a l s t r a n d . It i s seen then that a strand may be t r i v i a l e i t h e r because i t e n t i r e l y l a c k s commands f o r such o p e r a t i o n s , or because i t s enzyme i s detached before such commands can be executed. But i n r e a l i z i n g t h i s l a t t e r case i t becomes c l e a r t h a t the i n t r i c a c i e s of bindin g preference and operation execution order are e s s e n t i a l t o the d e f i n i t i o n of t h i s type of s t r a n d , making computation of the p r o b a b i l i t y of i t s occurrence as hard or harder than f o r the dud. Conclusion. On the whole, i t appears that even f o r the humblest types of strands, the duds and t r i v i a l s , i t i s very d i f f i c u l t to determine a priori the r e l a t i v e f r e q u e n c i e s of t h e i r o c c u r r e n c e — t h o u g h no one doubts i t can be done i n p r i n c i p l e . The prospects f o r determining f r e q u e n c i e s of the Chapter VI 167 more i n t e r e s t i n g s e l f - c o p i e r s , s e l f - p e r p e t u a t o r s , s e l f -r e p l i c a t o r s and s e l f — r e c u r r o r s i s f a r more daunting. An a l t e r n a t i v e , pseudo-a poster ior i method would i n v o l v e generating s t r a n d s at random and t e s t i n g them f o r type. Without a very l a r g e sample, however, i n t u i t i o n suggests i t i s u n l i k e l y t h a t such a method would i d e n t i f y ar/y i n s t a n c e s of such r a r e strand types as s e l f - r e p l i cat or s or s e l f - r e c u r r o r s . How many, say, of the 5.725 b i l l i o n s t r a n d s of 8 d u p l e t s length or shorter can be expected to be capable of doing something as s o p h i s t i c a t e d as copying themselves? Asymmetry i n A l l o c a t i o n of Binding Preferences We have thus f a r d e a l t with the f r e q u e n c i e s of s t r a n d s , bases, types of s t r a n d s , e t c . We t u r n now to c o n s i d e r a t i o n of tertiary structures. The present s e c t i o n w i l l probe a deeply l a t e n t property of Typogenetics, answering the question: Are t e r t i a r y s t r u c t u r e s d i c t a t i n g a binding preference f o r one base as numerous as those d i c t a t i n g b i n d i n g preference f o r another base? Or i n other words, are enzyme binding preferences e q u a l l y probable? S t r i c t l y speaking the answer i s no. T h i s seems odd, almost u n b e l i e v a b l e , u n t i l one sees the reasons f o r i t . On the s u r f a c e , the binding preference t a b l e (p125 infra) i s p e r f e c t l y symmetrical, and the procedure f o r determining binding preference seems to completely r o t a t e the bases. Chapter VI 168 Let n be any s p e c i f i e d number of amino a c i d s . Each amino a c i d w i l l have one of the th r e e f o l d i n g i n c l i n a t i o n s . Thus a sequence of n amino a c i d s i s a permutation of n f o l d i n g i n c l i n a t i o n s . For each permutation of f o l d i n g i n c l i n a t i o n s t h e r e i s a unique enzyme t e r t i a r y s t r u c t u r e implying a s p e c i f i c b i n d i n g preference. Now, f o r each n t h e r e w i l l be one more enzyme p r e f e r r i n g A than T. T h i s discrepancy begins with the one duplet gene case. There i s no one-amino a c i d enzyme t e r t i a r y s t r u c t u r e that has a binding preference f o r T, though t h e r e i s one f o r A. t h i s : # of amino a c i d s 1 # of unique t e r t i a r y s t r u c t u r e s : 3 Base Type P r e f e r r e d : A 1 C 1 8 1 T 0 T e r t i a r y s t r u c t u r e s of course are permutations not of d u p l e t s as p a i r s of bases but of those d u p l e t s ' corresponding amino a c i d f o l d i n g i n c l i n a t i o n s . The permutation s, r , 1 ( s t r a i g h t , r i g h t , l e f t ) i s one t e r t i a r y s t r u c t u r e whether or The general p a t t e r n looks l i k e 17 81 7 21 7 20 7 20 6 20 Chapter VI 169 not the d u p l e t s i n v o l v e d are AC-CG-CT or AG-AT-TC. Note that the sequence s, r , 1 r e s u l t s i n a d i f f e r e n t t e r t i a r y s t r u c t u r e than the e q u i v a l e n t combination i n the sequence 1, r , s, even though e q u i v a l e n t f o l d i n g i n c l i n a t i o n combinations w i l l have the same bindin g p r e f e r e n c e — a f a c t on which the BASIC computer program i n Appendix II r e l i e s . The number of p o s s i b l e t e r t i a r y permutations i n c r e a s e s of course with the number of d u p l e t s (excluding the AA d u p l e t , which does not have a f o l d i n g i n c l i n a t i o n and so does not f i g u r e i n t e r t i a r y s t r u c t u r e ) . To be exact, i f n i s the number of d u p l e t s c o n s t i t u t i n g the gene, there are 3" unique permutations of those d u p l e t s ' corresponding amino a c i d s ' f o l d i n g i n c l i n a t i o n s . Nevertheless, a l l of these combinations w i l l always be c a t e g o r i s a b l e i n terms of the four p o s s i b l e b i n d i n g p r e f e r e n c e s . It i s i n f a c t the modulo play between the m u l t i p l e s of 4 and 3 that b r i n g s about the mysterious noted asymmetry. Suppose one had 3" cards t o deal out to four p l a y e r s . There would never be an even d i s t r i b u t i o n of hands. 9, 27, 31, 243, e t c . , powers of 3, d e a l t out four ways w i l l always r e s u l t i n unevenness. The modulo a r i t h m e t i c works out i n such a way that there are always 1 or 3 cards l e f t over to d i s t r i b u t e . As i t happens that the cards are d e a l t out always i n the same order, s t a r t i n g with A, whether there are one or three cards l e f t over A w i l l r e c e i v e one card more than the Chapter VI 170 -fourth and l a s t dealee, T; but when there are three cards le-ft over only T w i l l be l e f t short-handed, so T i s p r e f e r r e d l e a s t of any of the bases. Mo Motherless C h i l d r e n i n Typogenetics As recognized i n our c l a s s i f i c a t i o n of types, there are s t e r i l e Typogenetic s t r a n d s , i . e . s t r a n d s that have no propagative c a p a c i t y . Such stran d s are the dead ends of t h e i r s t r a n d - l i n e s . But what about the converse? Are there "motherless" s t r a n d s ? The answer i s no. Algorithms can be made (two are d e t a i l e d i n Appendix IV) which w i l l take any input strand and c r e a t e f o r i t a mother s t r a n d . One such al g o r i t h m scans the t a r g e t daughter st r a n d ; as each next u n i t ' s base i s determined a duplet coding f o r the i n s e r t i o n of that base i s added t o the growing mother s t r a n d . The mother strand then possesses the i n s e r t i o n d u p l e t s s u f f i c i n g t o i n s e r t the sequence of bases c o n s t i t u t i n g the daughter. No doubt a s i m i l a r a l g o r i t h m could do the t r i c k using copying. No Garden-of-Eden strands. The e x i s t e n c e of any such alg o r i t h m proves there are no motherless c h i l d r e n i n T y p o g e n e t i c s — n o "Garden of Eden" strands, t o borrow from the l e x i c o n of c e l l u l a r automata. In t h i s respect Typogenetics i s l i k e n a t u r e — a l l l i v i n g e n t i t i e s have p r o g e n i t o r s . The exi s t e n c e of a l t e r n a t i v e such algorithms shows t h a t , u n l i k e Chapter VI 171 n a t u r a l e n t i t i e s , any Typogenetics strand has more than one mother. For example, the s e l f — c o p y i n g , n a n — s e l f — r e p l i c a t i n g s t r a n d a l r e a d y i n t r o d u c e d , CGTTCCA, has as i t s daughter T6GAACG. Now, one would see that i f t h i s daughter were run through the mother—maker alg o r i t h m the mother outputted f o r i t would not be CGTTCCA. And d i f f e r e n t mother—maker algorithms (such as the two given i n the appendix) make d i f f e r e n t mothers. If someone goes t o the t r o u b l e of d e f i n i n g a completely g e n e r a l i z a b l e way of making a l t e r n a t i v e mother-maker algorit h m s (a mother—maker mother), then any str a n d could t h e n c e f o r t h be s a i d t o have infinitely many different mothers (at l e a s t as many as th e r e are such d i s t i n c t a l g o r i t h m s ) ! Recursive geneology. I t should be noted t h a t when looped back on i t s e l f (previous output strand becomes c u r r e n t input strand) such "mother—maker" algorithms w i l l produce an a r b i t r a r i l y long l i n e of ancestry f o r the o r i g i n a l s t r a n d . In such a scheme, the product of the nth loop of the alg o r i t h m i s p r o g e n i t o r t o a l l p r e v i o u s l y outputted strands; the o r i g i n a l input strand i s i t s the nth generation descendant. Comparison to cascading strand—lines. T h i s r e c u r s i v e generation c o n t r a s t s i n t e r e s t i n g l y with the cascading s t r a n d -l i n e . The i n f i n i t e l y f e r t i l e strand GAGA i s a known progenitor t o an endless l i n e of descendants d i f f e r e n t from each other and from GAGA. But with our looping algorithm, any Chapter VI 172 strand ( i n c l u d i n g GAGA) i s a known descendant of an endless l i n e of ancestors d i f f e r e n t from each other and from the input s t r a n d . Both these cases i n turn c o n t r a s t i n t e r e s t i n g l y with a r e s u l t obtained by Myhi 11 (1964), who proved v i a diagonal argument that t h e r e could be an automaton A capable of ou t p u t t i n g a s e t of theorems and a successor automaton A± i t s e l f capable of o u t p u t t i n g a l l the theorems A could plus an a d d i t i o n a l theorem plus another successor automaton Azt that i n t u r n outproduces Ax. . .and so f o r t h , ad infinitum. L i k e the i n f i n i t e l y f e r t i l e s t r a n d , the o r i g i n a l or input automaton here i s a known pr o g e n i t o r t o an endless l i n e of descendants. But i n terms of theorem-proving c a p a c i t y , the nth descendant of t h i s l i n e of automata can produce everything i t s predecessors c o u l d , c o l l e c t i v e l y , produce, and some. S i m i l a r l y , the nth ancestor produced by our looping mother-making alg o r i t h m can produce everything i t s descendants c o u l d , c o l l e c t i v e l y , produce, and some (where "produce" means "bear s t r a n d s " ) . A l l Strands Are S i b l i n g s Strangely enough, i t seems that any strand i s a s i b l i n g t o any other s t r a n d . T h i s i s e s t a b l i s h e d by an algorithm that w i l l accept as input any f i n i t e s et of strands and cr e a t e s f o r the e n t i r e membership of that s et one mother common to Chapter VI 173 all. The most s t r a i g h t f o r w a r d such "sibling-maker" a l g o r i t h m concatenates the input strands (each strand imbedded as a gene) and i n c l u d e s cut commands s u f f i c i e n t t o dismember t h i s parent i n t o the in p u t t e d s t r a n d s , i t s "daughters" (see Appendix IV) . A Pa r a d o x i c a l Strand Notwithstanding the aforementioned algorithms, t h e r e are l i m i t s on how many l i t t l e monsters one mother strand can bear, as we now see. F i r s t , l e t us lump together as " s e l f -reppers" the types of strands p r e v i o u s l y d i s t i n g u i s h e d as s e l f - p e r p e t u a t o r s , s e l f — r e p l i c a t o r s , and s e l f - r e c u r r o r s , s i n c e a l l these n o n - t r i v i a l s t r a n d s share the a t t r i b u t e of having a descendant i d e n t i c a l t o themselves. Now, consider the s e t of a l l str a n d s that are no n - s e l f - r e p p e r s (that being the l a r g e m a j o r i t y of a l l s t r a n d s ) . Could there be made or found f o r t h i s s et a s i n g l e mother common to a l l the s e t ' s members, such t h a t on the intended e l e c t i o n of binding s i t e s f o r i t s many enzymes, t h i s strand ( c a l l i t p) would have as i t s progeny a l l and only n o n — s e l f — r e p p i n g strands? It seems not. The conception of such a strand i s p a r a d o x i c a l , as can be seen by asking of i t whether or not i t i s a s e l f - r e p p e r . Suppose i t i s a s e l f - r e p p e r , which i m p l i e s that i t has a descendant i d e n t i c a l t o i t s e l f , p,. But p,, being i d e n t i c a l t o p, too would be a s e l f - r e p p e r . T h i s Chapter VI 174 descendant v i o l a t e s the p r o v i s o t h a t p has only n o n - s e l f -reppers -For descendants. It may then be assumed that p i s a non-sel-F-repper. According t o the de-Finition of p, p has all the non-sel-F-reppers as i t s progeny. So p, here hypothesized t o be a non-s e l -f-repper, must be among i t s own progeny. Yet that makes i t a s e l f — r e p p e r . Let us see i t i n a symbolic l o g i c r e p r e s e n t a t i o n , f o r c l a r i t y . We s h a l l presume that the r e l a t i o n of "strand x i s a n o n - t r i v i a l descendant of strand y," Dxy, has alre a d y been def i n e d i n more p r i m i t i v e terms, r e l a t i n g t o whether a d e r i v a t i o n r e c o r d i n g the a l l e g e d descendancy r e v e a l s any a c t u a l amino a c i d work done. Now we d e f i n e the a t t r i b u t e R of being a s e l f - r e p p e r . Our q u a n t i f i c a t i o n domain i s of Typogenetic strands, so i n a standard symbolism we w i l l express t h i s d e f i n i t i o n : (la) For a strand t o be a s e l f - r e p p e r (Rx) means that i t has a descendant i d e n t i c a l t o i t s e l f (Dxx). (x) (Rx —> Dxx) AND (lb) If a strand x has a descendant i d e n t i c a l to i t s e l f (Dxx), then i t i s a s e l f - r e p p e r , Rx. (x) (Dxx —> Rx) Now, the paradoxical strand p i s defined by b o t h of these c o n d i t i o n s : Chapter VI (2a) Every non-repper i s i t s descendant. (x) (-Rx — Dxp) AND (2b) I t has only non-reppers -for descendants. (x) (Dxp — -Rx) Now, the question i s : what i s p, the paradoxical s t r a n d , Rp or -Rp? If we assume p i s Rp, then by (la) Dpp, but by (2b), -Dpp. I-f we assume p i s —Rp, then by (lb) -Dpp, but by (2a) Dpp. What the Paradox Implies We seem t o be up against the law o-f excluded middle! Apparently, our assumptions have put us on the path o-f a reductio ad absurdum. But l e t us pursue t h i s l i n e a b i t f u r t h e r s t i l l . R e c a l l that " i n f i n i t e l y f e r t i l e " strands, l i k e GAGA, are capable of having i n f i n i t e l y many different descendants (proof, i n c i d e n t a l l y , that there i s an i n f i n i t u d e of n o n - s e l f - r e p p e r s ) . Now, could one of these s t r a n d - l i n e s i n c l u d e every non-self-repper? And would not the ulti m a t e progenitor of that l i n e be the paradoxical strand? GAGA, and the s e l f - r e p p i n g strands, are pr o g e n i t o r s t o i n f i n i t e l i n e s of descendants, but very predictable l i n e s that c l e a r l y would not include a l l non-self-reppers. However, almost any long strand generates very complex, m u l t i f a r i o u s s t r a n d - l i n e s , branching i n unpredictable ways, leading t o descendants both longer and shorter that the o r i g i n a l Chapter VI 176 pr o g e n i t o r . It seems q u i t e p l a u s i b l e that some o-f these s t r a n d - l i n e s are never—ending. Thus, we have a v a i l a b l e the p o t e n t i a l i n f i n i t u d e of v a r i a t i o n known to be possessed by some Typogentic s t r a n d s , to p l a c e i n t o 1:1 correspondence with the i n f i n i t u d e of n o n - s e l f - r e p p e r s t h a t the para d o x i c a l strand needs t o engender. Yet, again, i f reductio ad absurdum reasoning i s v a l i d , then i f an i n f i n i t e l y f e r t i l e s t rand can gi v e r i s e to all the n o n - s e l f - r e p p e r s , i t must a l s o bear at l e a s t one s e l f - r e p p e r (what a burden on t h a t one s e l f - r e p p e r , keeping h i s i n f i n i t u d e of monstrous brethren out of l o g i c a l t r o u b l e ! ) . Put another way, i f a l l n o n - s e l f - r e p l i c a t i n g s t r a n d s are branches of some p's f a m i l y t r e e , at l e a s t one s e l f - r e p p e r i s a l s o on that t r e e . A l l of t h i s suggests a c e r t a i n absolute selt-incom patibi1ity of the c l a s s of no n - s e l f - r e p p e r s . It remains t r u e that f o r any f i n i t e s et of them, a mother can be made, so any one i s a s i b l i n g t o any other one cares t o name. That was what our sibling-maker algorithm (Appendix IV) e s t a b l i s h e d . Yet the non- s e l f - r e p p i n g strands cannot p o s s i b l y " f i t " together as one b i g , happy, exclusive-and-comprehensive i n f i n i t e f a m i l y of monsters. T h i s i s s t r o n g l y reminiscent of what i n mathematical l o g i c i s c a l l e d omega inconsistency, which i s an i n c o n s i s t e n c y between a cascading s e r i e s of s i n g u l a r p r o p o s i t i o n s and the summarizing u n i v e r s a l Chapter VI 1 7 7 p r o p o s i t i o n -for that s e r i e s . V i c i o u s C i r c u l a r i t y or Strange Loop? The p a r a d o x i c a l strand above i s , of course, yet another analogue t o R u s s e l l ' s paradox ( c f . the "paradoxical machine," pS7, supra). R u s s e l l ' s i n t u i t i o n was that " v i c i o u s c i r c u l a r i t y " was i n v o l v e d i n the conception of a stra n d l i k e p. To develop h i s l i n e of t h i n k i n g , suppose I have at my bi d d i n g a genie, omnipotent i n the Typogenetics world, t o whom I make these wishes, i n order: ( 1 ) "Enumerate a l l the n o n - s e l f - r e p p e r s . " (2) "Find or make a mother f o r a l l the n o n - s e l f - r e p p e r s . " To carry out the second wish the genie i s i n i t i a l l y ready to run through every strand c o l l e c t e d i n (1) to see i f any one of them happens t o have a l l the no n - s e l f - r e p p e r s f o r i t s o f f s p r i n g . But he q u i c k l y sees, meta - 1ogical 1y, that t h i s i s precluded. Every strand here i s c e r t a i n l y not going to have itself f o r o f f s p r i n g (since these were screened t o be non-se l f-reppers) , so every strand there has " f e r t i l i t y incompleteness" r e s p e c t i n g i t s e l f . He has no recourse to the strand s not enumerated i n ( 1 ) , namely the s e l f -reppers, because i t i s s t i p u l a t e d p has only the n o n - s e l f -reppers f o r progeny, and a s e l f - r e p p e r by d e f i n i t i o n has a s e l f - r e p p i n g o f f s p r i n g . A l l the genie can do at t h i s point i s f a b r i c a t e some Chapter VI 178 "new" kind of s t r a n d to be p. Cantor managed to come up with h i s diagonal number to prove t h a t the set of the i n t e g e r s i s not " i n f i n i t e enough" to index the set of r e a l s (well presented i n H o f s t a d t e r , 1979, at p421). T h i s i s harder; we must not only f i n d a "new" s t r a n d , but p has very s p e c i a l propagative r e s p o n s i b i l i t i e s i t s s t r u c t u r e must support. We could l i t e r a l l y c r e a t e a diagonal Typogenetic strand by c l o s e l y f o l l o w i n g Cantor's method. But, (1) i t would r e s u l t i n an i n f i n i t e s t r a n d ("too long" to be a t y p o g r a p h i c a l e n t i t y ) , and (2) i n any case, what would lead us to t h i n k t h a t the diagonal s t r a n d would have a l l the n o n - s e l f - r e p p e r s f o r o f f s p r i n g ? Why not feed each next n o n — s e l f - r e p p i n g strand i n t o the "sibling-maker" algorithm of Appendix IV? The number of n o n — s e l f — r e p p i n g s t r a n d s i s i n f i n i t e , and the a l g o r i t h m i n question's mode of o p e r a t i o n i n v o l v e s imbedding every input (daughter) strand i n the parent, which would lead to an i n f i n i t e l y long strand i n t h i s case. But, could t h i s genie have a marvelous " l e a s t ancestor" a l g o r i thm that f i n d s , f o r any inputted s t r a n d , f i n i t e or i n f i n i t e , the s h o r t e s t ancestor p o s s i b l e , even one f a r removed? T h i s a l g o r i t h m could supposedly accept the input of our sibling-maker algorithm's output and r e t u r n a f i n i t e length strand bearing a l l the non-self-reppers. It could take a l l of GAGA's descendants and f i n d t h e i r l e a s t p r o g e nitor Chapter VI 179 ( i . e . 6AGA i t s e l f ) , f o r example. The p o i n t i s , i s the strand p something you could somehow f i n d or make, or i s i t a l o g i c a l mirage? Bertrand R u s s e l l viewed such n o t i o n s as the par a d o x i c a l s t r a n d as the products of reason gone a s t r a y . A somewhat R u s s e l l i a n a n a l y s i s of p i s as f o l l o w s : The d e f i n i t i o n of the a t t r i b u t e of being a s e l f - r e p p e r , Rx, i s dependently d e f i n e d i n terms of Dxx. But Dxx i s de f i n e d independently of Rx; i t i s a lower order p r e d i c a t e than Rx. The extension of Dxx i s t h e r e f o r e p r i o r t o that of Rx. In our example, what t h i s means e x a c t l y i s that i f the mother strand can be made, then the " a l l " of my second wish to the genie should be l i m i t e d t o the " a l l " of the f i r s t wish. If that i s held t o , then the mother's d e f i n i t i o n a l d e s c r i p t i o n as bearing " a l l the n o n - s e l f - r e p p e r s , " ought not i n c l u d e the strand p i n i t s scope. The c o n t r a d i c t i o n d i s s o l v e s when the s e l f — r e f e r e n c e f a i l s . R u s s e l l of course devised h i s Theory of L o g i c a l Types t o t r y to e l i m i n a t e s i m i l a r " f a l l a c i o u s " t h i n k i n g i n mathematical l o g i c , v i a s y n t a c t i c a l r e s t r i c t i o n s . On the f a s c i n a t i n g t o p i c of Theory of L o g i c a l Types see Copi (1971); Copi a l s o g i v e s a b r i e f e r but s t i l l i n f o r m a t i v e account i n h i s textbook (1979). A Strange Loop of S e l f - R e p l i c a t i o n What R u s s e l l regarded as " v i c i o u s c i r c u l a r i t y " Chapter VI ISO Ho-f s t a d t e r (1979) devoted much of Sodel , Escher. Bach t o documenting, as those mysterious but deeply s i g n i f i c a n t "strange loops." Of p a r t i c u l a r i n t e r e s t t o our present concerns, Hofs t a d t e r sought t o i d e n t i f y a strange loop i n the l o g i c of propagation. However, h i s candidate, being "a str a n d of DNA which, i f i n j e c t e d i n t o a c e l l , would, upon being t r a n s c r i b e d , cause such p r o t e i n s t o be manufactured as would dest r o y the c e l l (or the DNA), and thus r e s u l t s i n the non-reproduction of t h a t DNA" (p53&) seems r a t h e r lame. As Webb (19S3) noted, i n h i s review of Sodel, Escher. Bach, t h i s analogue t o the Sodel "undecidable" sentence, i s an unfortunate c h o i c e i n s o f a r as Hofstadter g i v e s us no c l u e as t o how the strand i n question i s t o be taken as ex p r e s s i n g i t s n o n - r e p l i c a b i 1 i t y . (If anything, H o f s t a d t e r ' s strand sounds l i k e an analogue t o the h y p o t h e t i c a l self—erasing s t r a n d , which i s a c u r i o s i t y t o be sure, but non-paradoxical). I t i s submitted that the paradoxical strand i d e n t i f i e d above (having a l l and only n o n — s e l f - r e p p i n g strands f o r i t s progeny) q u a l i f i e s as i n s t a n c i n g the strange loop of s e l f -r e p l i c a t i o n Hofstadter sought f o r what Webb c a l l s the "centerpiece of h i s b o o k " — t h e chapter comparing s e l f -r e f e r e n c e i n mathematical l o g i c with s e l f - r e f e r e n c e and s e l f -r e p l i c a t i o n i n molecular b i o l o g y , and which chapter Chapter VI 181 introduced Typogenetics t o the world. The Urstrand Now consid e r the obverse s i d e o-f the c o i n . There i s nothing ( i n t h i s author's perception) p a t e n t l y p a r a d o x i c a l about a s t r a n d — c a l l i t the urstrand, a f t e r Goethe's ur p-flanze, the primal p l a n t -from which a l l other p l a n t s stem — w h i c h manages t o have a l l (and perhaps only) the s e l f — r e p l i c a t o r s f o r i t s progeny. I t s e x i s t e n c e i s as yet n e i t h e r proved nor disproved. Nor i s i t known whether i t would have to have i n f i n i t e descendants (that depending on whether there are i n f i n i t e l y many s e l f - r e p l i c a t o r s ) . But even i f there are i n f i n i t e l y many, the e x i s t e n c e of i n f i n i t e l y f e r t i l e s t r a n d s allow us t o preserve hopes that i t could e x i s t . A c t u a l l y , t h e r e may at f i r s t appear to the reader t o be a problem. A strand cannot have i n f i n i t e l y many daughters, s i n c e t hat would r e q u i r e i n f i n i t e l y many du p l e t s coding f o r the i n f i n i t e l y many operations t o b u i l d a l l those s t r a n d s , and i n f i n i t e l y many dupl e t s cannot be accomodated i n a typo g r a p h i c a l e n t i t y . While i f i t has a finite number of s e l f - r e p l i c a t i n g daughters, and each s e l f - r e p l i c a t i n g daughter produces only i t s own kind, how do we get the infinitely many s e l f - r e p l i c a t o r s ? But i n f a c t there i s nothing i n the d e f i n i t i o n of a s e l f - r e p l i c a t o r which says i t cannot have descendants different from i t s e l f , so long as i t Chapter VI 182 has some that are i d e n t i c a l . The u r s t r a n d then would have t o have among i t s descendants some i n f i n i t e l y f e r t i l e descendants bearing s e l f - r e p l i c a t o r s , so that a l l the i n f i n i t u d e of s e l f -r e p l i c a t o r s can be unpacked out of the f i n i t u d e of daughters over i n f i n i t e l y many generations. T h i s assumes, then, than the great f a m i l y of s e l f - r e p l i c a t o r s i s a cascading s t r a n d — 1 i ne. Does the ur s t r a n d e x i s t ? It i s a pregnant question, f o r i f i t does, its -form captures the entire domain of self-replication in Typogenetics. Again, i f we represented Typogenetics with a standard symbolic l o g i c , the formula corresponding t o the d e s c r i p t i o n of t h i s s i n g l e s t r a n d would possess axiomatic f e c u n d i t y . The P h i l o s o p h e r s ' Strand In Medieval alchemy the "Philosopher's stone" was symbolic of "u n l i m i t e d t r a n s m u t a b i 1 i t y . " One can conceive of a p e c u l i a r kind of s t r a n d , the Philosopher's s t r a n d , which metamorphizes, i n the course of a s i n g l e generation d e r i v a t i o n , through a number of transforms i d e n t i c a l to members of some d e s i r e d c l a s s of strands. The reader h o p e f u l l y r e c a l l s that i n a Typogenetic d e r i v a t i o n , the p r o g e n i t o r , having t r a n s l a t e d i t s e l f i n t o a set of enzymes with programs, becomes an operand f o r those enzymes. Step by step i t i s transformed, the product of each Chapter VI 183 opera t i o n being what we c a l l here the "trans-form" present at one stage o-f the d e r i v a t i o n . F i n a l l y , when a l l enzymes have been exhausted, the " l a s t " trans-form is the descendant r e s u l t i n g from the process, a f i n i s h e d strand, i f well formed. The i d e a here i s t h a t you have a l i s t of s t r a n d s , i through j, that i n t e r e s t you f o r some reason, and some c a r e f u l l y designed P h i l o s o p h e r s ' strand that can, i n the course of i t s s e l f - d i r e c t e d t r a n s f o r m a t i o n i n t o descendant s t r a n d s , go through a sequence of transforms tj through tj such that f o r every s t r a n d x there i s some transform tx, and x i s s y n t a c t i c a l l y i d e n t i c a l t o t x . For example, could there be a strand that t r a n s l a t e s i n t o one s e t of enzymes capable, over the course of one long d e r i v a t i o n , of transforming the pr o g e n i t o r operation by op e r a t i o n , enzyme by enzyme, u n t i l every s i n g l e s e l f -r e p l i c a t o r has showed up as a transform at some stage of the d e r i v a t i o n ? That would be analogous t o an E a r t h l y animal passing metamorphical1y through i t s i n d i v i d u a l l i f e from stage t o stage, and seeming t o be i d e n t i c a l , at each given stage, t o some other mature animal. If one wished t o take an e n t i r e zoo along i n a small s p a c e c r a f t , t h i s c r e a t u r e would be very handy as an educational t o o l . Just as a f r o g i s a 3-i n - i animal < f i r s t egg, then tadpole, then f r o g ) , t h i s e n t i t y changes from beast t o beast i n the course of i t s one l i f e . Chapter VI 184 One week i t i s a p o l a r bear, the next a walrus, t i l l the whole l i s t has been exhausted. In the world o-f Typogenetics, such t h i n g s are c o n c e i v a b l e . But an argument proves that the number of transforms i t must become cannot be i n f i n i t e . A d e r i v a t i o n can continue o n l y so long as there are o p e r a t i o n s to c a r r y out, o p e r a t i o n s t r a n s l a t e d out of the o r i g i n a l p r o g e n i t o r . There must be as many du p l e t s i n the p r o g e n i t o r as there are o p e r a t i o n s t o c a r r y out. If t h i s e n t i t y i s presupposed to be a standard case -finite—length s y n t a c t i c a l e n t i t y , then i t can have only f i n i t e d uplet c o n s t i t u e n t s , thus code f o r only a f i n i t e number of o p e r a t i o n s , which e n t a i l s a f i n i t e d e r i v a t i o n . There are no i n f i n i t e l y transmutable s t r a n d s , i n c o n t r a s t t o the s i t u a t i o n with s t r a n d — 1 i n e s , which can be i nf i n i t e . Relevance t o Natural Philosophy A l i t t l e experience with the system informs one that i n the Typogenetic world, " l i k e g i v e s r i s e only to l i k e " ( s e l f -r e p l i c a t i o n ) i s the s p e c i a l case. A strand t y p i c a l l y has descendants d i f f e r e n t from i t s e l f . Does nature honor a " l i k e g i v e s r i s e only to l i k e " law such that a n a t u r a l kind only bears descendants of i t s own v a r i e t y ? And does that mean that such notions as the paradoxical strand and the urstrand have no relevance to nature, because Chapter VI 185 those a r t i f i c i a l e n t i l e s are not bounded by t h i s law? In the f a c e of the evidence proving branching descent of today's v a r i e t y of E a r t h l y s p e c i e s from common, simpler a n c e s t o r s , one can b e l i e v e i n a " l i k e g i v e s r i s e t o l i k e " law of nature only as an apparent r e g u l a r i t y , not as r e f l e c t i n g some i r o n c l a d general purpose n e c e s s i t y . Sexual recombination i s not token l e v e l s e l f - r e p l i c a t i o n , and s i g n i f i c a n t mutations are exceptions to t y p a l l e v e l s e l f -r e p l i c a t i o n , so l i k e does not always g i v e r i s e t o l i k e at e i ther 1evel. The great f a m i l y t r e e of Earth image can lead one i n a Goethean d i r e c t i o n t o a conception that Earth's very f i r s t s e l f — r e p l i c a t o r ( i f there was only one) was indeed a s o r t of natural urstrand, because every subsequent l i f e form i s i t s descendant. However, modern s c i e n t i f i c e v o l u t i o n a r y theory most d e f i n i t e l y does not t h i n k of a DNA strand as p r e s c r i b i n g i t s descendants the way a strand p r e s c r i b e s i t s s t r a n d - l i n e . The o r i g i n a l s e l f — r e p l i c a t o r spoken of i n Chapter II r e a l l y was not an "urstrand" because i t s t r e t c h e s c r e d u l i t y to b e l i e v e i t possessed the "formal and f i n a l cause" of a l l l i f e forms. A t h e o r i s t l i k e Monad (1971) f i n d s i t e a s i e r to b e l i e v e , on the evidence, that mutations and sexual recombination account f o r the v a r i a t i o n of s p e c i e s — t h e branching of the f a m i l y t r e e . Thus i n nature, the l i n e of descent i s shaped by events Chapter VI 186 e x t e r n a l t o the g e n e t i c i n-f ormati on, and does not have the "-final c a u s a t i o n " t h a t i s so evident i n embryology. Our a r t i f i c i a l e n t i t i e s are different in kind from n a t u r a l ones, because a strand does " c o n t a i n " ( i n f o r m a t i o n a l l y ) i t s descendants' forms; the whole s t r a n d -l i n e grows out of the p r o g e n i t o r l i k e one t r e e grows out of an acorn. The relevance of t h i s depends on whether one i s l o o k i n g at a Typogenetic s t r a n d as a model for e x p l i c a t i n g nature, or as a model of some i n t e r e s t i n g , new propagating e n t i t y , p r e c i s e l y formulated enough to i n s p i r e a design o b j e c t i v e . Breater designer c o n t r o l over descent i s o b v i o u s l y p o s s i b l e when v a r i a t i o n i s preprogrammed r a t h e r than the outcome of e x t e r n a l , random f a i l u r e s t o r e p l i c a t e with 100% f i d e l i t y . Along these l i n e s , M y h i l l <1964), i n h i s paper r e p o r t i n g an h i s s u c c e s s i v e l y more p r o d u c t i v e automata mentioned above, remarked " . . . i t seems t o me t h a t the p o s s i b i l i t y of producing an i n f i n i t e sequence of v a r i e t i e s of descendants from a s i n g l e program i s methodologically s i g n i f i c a n t i n a manner which might i n t e r e s t b i o l o g i s t s more than a r t i f i c i a l i n t e l l i g e n c e r s . It suggests the p o s s i b i l i t y of encoding a p o t e n t i a l l y i n f i n i t e number of d i r e c t i o n s to p o s t e r i t y on a f i n i t e l y long chromosomal tape, a p o s s i b i l i t y which seems h i t h e r t o to have escaped the notice- of b i o l o g i s t s . " p218. So l e t us wonder f o r a moment about a p a s s i b l e Chapter VI 18 convergence of our r e s u l t s and the p o s s i b i l i t i e s opened up by modern biotechnology. It seems very l i k e l y a DNA strand could be endowed with the c a p a c i t y to d i r e c t the manufacture of DNA s t r a n d s n o n - i d e n t i c a l to i t s e l f and to each other (I s t r e s s : through design, not random d e v i a t i o n as i n mutation). Somewhat l e s s c e r t a i n l y , t h e r e might some day be developed means of g e n e r a t i n g , f o r any set of DNA s t r a n d s D, another DNA s t r a n d P capable of manufacturing every member of D. F o l l o w i n g out t h i s p a r a l l e l t o our meta-Typogenetics r e s u l t s , a p r e f o r m a t i o n i s t dream, i t must be asked: Could there be a single "infinitely fertile" DNA ur strand able to synthesize all possible DNA—based life forms? I t s d i s c o v e r y would s u r e l y c o n s t i t u t e the u l t i m a t e achievement of the wetware s i d e of the A r t i f i c i a l L i f e e n t e r p r i s e ! (But would we have to know i n advance what a l l the p o s s i b i l i t i e s are f o r l i f e , i n order to put together such a progenitor?) These are of course f a n c i f u l , i f mind-expanding s p e c u l a t i o n s . But l e t us r e c a l l t h a t great minds, such as von Neumann (who conceived of the u n i v e r s a l machine-building machine we looked at i n Chapter IV), Soethe, who conceived of the u r p f l a n t z , Bertrand R u s s e l l , who discovered and then d i s a n n u l l e d h i s paradoxical s e t , and Hofstadter, who saw "strange loops" at the essence of both l i f e and i n t e l l i g e n c e , c e r t a i n l y thought on t h i s same plane. Chapter VII 188 CHAPTER VII: TYPOGENETICS AND NATURE'S LOGIC Typogenetics, the -formal system we have been d e a l i n g with the past two chapters, employed terms i t s inventor (Hofstadter) borrowed from molecular b i o l o g y : terms such as "enzyme," "gene," and so f o r t h . There are advantages but a l s o dangers, i n adopting p r e e x i s t i n g terms. The advantages are p l a i n . The terminology makes q u i t e transparent the extended analogy Hofstadter invented the system to f u l f i l l . It might enable the molecular b i o l o g i s t t o p i c k up Typogenetics more q u i c k l y , or g i v e the l o g i c i a n f a m i l i a r with Typogenetics a " l e a d — i n " i n understanding nature's " l o g i c , " should he decide to take up molecular b i o l o g y . The dangers are a l i t t l e l e s s patent, but not i n s i g n i f i c a n t . For one t h i n g , importing i n t o a formal system terms r i c h i n t h e i r own connotations and h i s t o r y can tempt the user i n t o v i o l a t i o n s of the requirements of f o r m a l i t y . It i s widely thought that e a r l y geometers sometimes strayed from r i g o r because the terms they used such as " l i n e " and "space" c o n s c i o u s l y or unconsciously i n f l u e n c e d them, b l i n d i n g them to some p o s s i b i l i t i e s (such as the p o s s i b i l i t y Chapter VI that two p a r a l l e l l i n e s might meet), and l e a d i n g them to i n f e r e n c e s not supported by t h e i r system's axioms and r u l e s . S i m i l a r l y , i t i s p o s s i b l e t h a t the knowledge one has of nature's enzymes, amino a c i d s , and so f o r t h , might c r e a t e preconceptions that would l i m i t or d i s t o r t one's use and understanding of Typogenetics, or that one's f a m i l i a r i t y with Typogenetic c o n s t r u c t s would g i v e one s e r i o u s mi simpressione about t h e i r namesakes i n nature. The other danger, r e c a l l i n g the d i s t i n c t i o n e a r l i e r made, would be to confuse Typogenetics with a model for nature. H o p e f u l l y t h i s danger was defused by the express d e c l a r a t i o n i n the f o u r t h chapter. However, i t i s worth examining i n some depth j u s t how a model Tor DNA would compare to Typogenetics, now t h a t the l a t t e r has been presented. A Logic For DNA A modeling l o g i c for DNA, i f and when one i s developed, would have a d i f f e r e n t raison d'etre than Typogenetics. It would e x i s t i n order to serve the end of p r e d i c t i n g nature's o p e r a t i o n s . One co u l d , f o r example, use t h i s l o g i c to deduce whether or not d e l e t i o n of a given n u c l e o t i d e sequence from a c e l l ' s DNA would e l i m i n a t e that DNA's c a p a c i t y to e f f e c t i t s s e l f - r e p l i c a t i o n . Such a model for DNA would be open t o em p i r i c a l r e f u t a t i o n , s i n c e i t s deductions must conform to the r e s u l t s of experimentation. Typogenetics, a model of Chapter VII 190 propagating e n t i t i e s , does not make any e m p i r i c a l c l a i m s , and has no use i n making p r e d i c t i o n s about nature. The b i o l o g i s t s who would c o n s t r u c t and use the l o g i c for DNA are i n a very d i f f e r e n t p o s i t i o n than the "Typogenetic!st." The T y p o g e n e t i c i s t i s provided the complete r u l e s of h i s system at the ou t s e t , and s t r i v e s t o i d e n t i f y i n t e r e s t i n g er>t it ies w i t h i n the system. The b i o l o g i s t does not s t a r t o f f i n possession of the " r u l e s " of h i s system. His task i s t o take the e n t i t i e s t h a t are given (the DNA of extant l i f e forms) and t r y t o f i g u r e out what nature's " r u l e s " are. A c r i t i c a l d i s t i n c t i o n the b i o l o g i s t i s t r y i n g t o make i s between a r u l e or c h a r a c t e r i s t i c t h a t i s p h y s i c a l l y necessary and the r u l e t h a t i s simply r e f l e c t i n g the i n e r t i a of h i s t o r y , a "frozen a c c i d e n t . " Is the ge n e t i c code a r b i t r a r y ? Must the RNA codon CAU code f o r the amino a c i d h i s t i d i n e , or could a d i f f e r e n t codon serve? (Hofstadter (1985a) has explored that p a r t i c u l a r q u e s t i o n ) . Now, a s l a v i s h r e d u c t i o n i s t b e l i e v e s that from a p e r f e c t p h y s i c s one could i n p r i n c i p l e deduce e x a c t l y what b i o l o g i c a l s t r u c t u r e s and processes there can be i n t h i s p h y s i c a l world. But a p e r f e c t p h y s i c s can only be developed a f t e r a l l observ a t i o n s of any p o s s i b l e relevance have been made. Of course, most of those observations have not been made, and i t i s not c e r t a i n whether they can be (some of the needed Chapter VII 191 obs e r v a t i o n s may l i e i n the p a s t — t h o s e i n i t i a l c o n d i t i o n s spoken o-f i n the - f i r s t c h a p t e r ) . And even i f the p e r f e c t p h y s i c s i s given to us, i t would r e q u i r e p e r f e c t d e r i v a t i o n a l p o w e r s — p o s s i b l y i n v o l v i n g trans-computable c a l c u l a t i o n s — t o make the necessary i n f e r e n c e s . At present, our ph y s i c s i s not well enough developed t o enable one, f o r example ( r e c u r r i n g again t o t h a t f i r s t c h a p t e r ) , t o know i n advance what c r y s t a l l i z a t i o n p a t t e r n s are p o s s i b l e , which not (though many a l t e r n a t i v e s can be r u l e d o u t ) , even f o r common substances i n mundane environments. A c l a s s i c i l l u s t r a t i o n of what we are t a l k i n g about i n v o l v e s the p e c u l i a r f a c t t h at living forms are not ambidextrous. That i s , a l l E a r t h ' s l i v i n g forms share some "handed-ness" b i a s e s , e.g. i n r e c o g n i t i o n and s y n t h e s i s of levogyrous r a t h e r than dextrogyrous compounds. It has been suggested t h a t t h i s handedness i s evidence of the f a c t that a l l l i f e forms evolved from a common ancestor. The idea i s that some o r i g i n a l l i f e form (that f i r s t r e p l i c a t o r ) , by sheer f 1 i p - o f — t h e — c o i n chance ( i t j u s t as e a s i l y could have been r i g h t handed) had t h i s l e f t handedness preserved i n i t s genetic m a t e r i a l and then passed i t on to a l l i t s descendants. However, i n the 1950's i t was proved that e l e c t r o n s p i n i s a l s o mirror asymmetric, meaning that the p h y s i c a l universe i s not ambidextrous. The p o s s i b i l i t y e x i s t s that t h i s mirror asymmetry at the subatomic l e v e l Chapter VII 192 n e c e s s i t a t e s the handedness o-f l i - f e , though t h i s has not yet been proved (Gardner, 1978). The p o i n t i s the DNA modeling l o g i c would not only need t o have t h i s handedness b i a s t o be tr u e t o i t s s u b j e c t matter, i t would, t o s a t i s f y b i o l o g i s t s , have t o r e f l e c t , perhaps through modal q u a n t i f i e r s , the d i s t i n c t i o n between the p o s s i b l e , the a c t u a l , and the necessary laws, tendencies or p r o p e r t i e s t h a t u n d e r l i e t h a t handedness. In Typogenetics h i s t o r i c i t y i s not recognized, so there can be no " i n i t i a l c o n d i t i o n s " t o f e r r e t out. It i s not meaningful t o say th a t one str a n d o r i g i n a t e d before another, even though there i s such a t h i n g as descent. A l l strands are "contemporaries" i n t h e i r p l a t o n i s t i c space. The l o g i c a l "laws of thought" provide the only kind of n e c e s s i t y i n Typogenetics s t r a n d s . Whether strand A i s an ancestor of strand B i s s t r i c t l y an a priori, not a posteriori question. Thus f a r not n e a r l y enough i s known about DNA to put together anything l i k e a complete modeling l o g i c f o r i t . And of those t h i n g s t h a t are known, i t ' s not s e t t l e d which are p h y s i c a l l y necessary and which a c c i d e n t a l / h i s t a r i c a l . S t i l l , some p o i n t s of comparison between how such a l o g i c for nature can be made. The next s e c t i o n s make some of those compari sons. Chapter VTT Same Alphabet, D i f f e r e n t Codons A Typogenetics strand c o n s i s t s o-f the l e t t e r s A,C,6, and T. B i o l o g i s t s u s u a l l y represent a DNA base sequence with those same l e t t e r s . In other words, the same alphabet i s drawn upon. T h i s p r o v i d e s a s u p e r f i c i a l s i m i l a r i t y , and a deep s i m i l a r i t y . The deep s i m i l a r i t y i s that the same law of combinatorics works f o r each. Whether t a l k i n g about n u c l e i c a c i d sequences or Typogenetic u n i t sequences, there are 4" unique sequences of a strand of length n. Duplets vs. triplets. One d i f f e r e n c e between TG and DNA str a n d s that would be most q u i c k l y n o t i c e d by a b i o l o g i s t would be the f a c t t h a t the codon f o r the TG strand i s 2 u n i t s , t h at f o r DNA, three. In other words, f o r the former, the duplet codes f o r an "amino a c i d , " whereas i n DNA the triplet codes f o r an amino a c i d . Reading frames of reference. In Typogenetics, the dup l e t s are read s t a r t i n g from the l e f t . That means there i s a f i x e d reading frame of r e f e r e n c e . T h i s matters q u i t e a b i t i n t r a n s l a t i n g a strand . Read i n the o r d i n a r y manner, the strand ACTTAAGGC has two genes, ACTT and BSC. If the rightmost u n i t f i x e d the frame of reading r e f e r e n c e , though, that same sequence would have a s i n g l e gene, and no d u p l e t s i n common with the ord i n a r y from-the - 1eft readi ng! S i m i l a r l y , i n DNA, the t r i p l e t s are read i n a c e r t a i n Chapter VII 194 d i r e c t i o n . And i n Typogenetics or DNA, t o d i s p l a c e the beginning of the frame of reading by one u n i t would u t t e r l y change the reading. Among other t h i n g s , t h i s means that two co n c e i v a b l y q u i t e lengthy sequences d i f f e r i n g by only one s i n g l e base, can be i n f o r m a t i o n a l l y totally u n l i k e . For example the two Typogenetic s t r a n d s , ACCCTTATATTTACGGT and CCCTTATATTTACGGT, are the same except f o r one l i t t l e A at the beginning of the former. Yet they have r a d i c a l l y d i f f e r e n t t r a n s l a t i o n s . Among other t h i n g s , f o r students of e i t h e r DNA or Typogenetics, the sheer f a c t t h a t one has found i d e n t i c a l s u b s t r i n g s w i t h i n strands being compared leads to no c e r t a i n equation of t h e i r "meanings." Put another way, extremely s i m i l a r operands may code f o r extremely d i f f e r e n t o p e rations. The importance of reading frame of r e f e r e n c e holds f o r both the l o g i c f o r DNA and Typogenetics. But the s i t u a t i o n with regard t o the former i s much l e s s simple than t h a t f o r the l a t t e r . When s c i e n t i s t s l a i d bare the e n t i r e genome (base sequence) of a n a t u r a l organism (a v i r u s ) f o r the f i r s t time, they discovered that i n nature the same base sequence i s sometimes "read" twice over according t o d i f f e r e n t frames of reading r e f e r e n c e (see Hofstadter, pp524-525). For example, the sequence CCATTGCTATTT might be read from the leftmost poin t the f i r s t time, as the sequence of t r i p l e t s CCA, TTG, CTA and TTT; then read a second time using the second—from-t h e - l e f t C as the s t a r t i n g p o i n t , y i e l d i n g the t r i p l e t s CAT, Chapter VII TGC, and TAT. One gene was contained e n t i r e l y i n s i d e another. One can imagine how s e t t i n g a s i d e some codon to d e f i n e the s t a r t i n g p o i n t f o r the t r a n s l a t i o n reading frame of r e f e r e n c e would complicate t h i n g s , and a l s o open up new p o t e n t i a l i t i e s , i n Typogenetics. As Hofstadter remarked, "The d e n s i t y of i n f o r m a t i o n packing i n such a scheme i s i n c r e d i b l e . pp524—525. Brooks and Wiley <1983) propose a formula f o r e s t i m a t i n g how much in f o r m a t i o n can be embodied at m u l t i p l e l e v e l s i n DNA, which helps e x p l a i n how there could be i n f o r m a t i o n enough to account f o r e p i g e n e s i s (growth). Coding Schemes In Typogenetics the coding scheme i s 1 to 1, i . e . each duplet codes f o r one and only one o p e r a t i o n . In the genetic code, the mapping i s sometimes many to one. That i s , there are 64 codons coding f o r only 20 amino a c i d s . Hence as e a r l i e r noted, there are cases where 2 or more t r i p l e t s code f o r the same amino a c i d . T h i s makes f o r a very s i g n i f i c a n t d i f f e r e n c e t h a t may not be immediately apparent. Far i t means that d i s t i n c t DNA strands can each code f o r the same amino a c i d s using different t r i p l e t s . In other words, d i f f e r e n t operands can code f o r the same operations. T h i s leads to i n t e r e s t i n g problems i n b i o l o g y (see Kauffmann, 1973). For t r a c i n g the branches of the e v o l u t i o n a r y t r e e , and f o r other purposes, the s i m i l a r i t y of base seqeuences i n genomes i s important evidence. But as painted out above, a Chapter VII 196 -finding o f i d e n t i c a l s u b s t r i n g s of two c r e a t u r e s ' DNA sequences cannot be assumed t o i n d i c a t e an i n f o r m a t i o n a l equivalence i n the absence o f knowledge of the reading frames of r e f e r e n c e . Now we a l s o see that two n o n i d e n t i c a l base sequences can a c t u a l l y t r a n s l a t e out as i d e n t i c a l . Which s i m i l a r i t y should count!? It w i l l be noted t h a t a many t o 1 coding scheme would p r e v a i l i n Typogenetics i f we adopted t r i p l e t codons i n p l a c e of the d u p l e t s , without much expanding the r e p e r t o i r e of ope r a t i o n s t o be coded f o r . Operati ons DNA t r i p l e t s code f o r p h y s i c a l amino a c i d s , Typogenetic d u p l e t s f o r a b s t r a c t "amino a c i d o p e r a t i o n s . " T h i s terminology conceals a disanalogy. In nature a s e r i e s of p h y s i c a l amino a c i d s i s assembled i n t o a "polypeptide," or p r o t e i n . It i s the p r o t e i n that has a very s p e c i f i c o peration to perform. By c o n t r a s t , i n Typogenetics, each "amino a c i d " has, by i t s e l f , a s p e c i f i c o p e r a t i o n t o discharge. (This i s noted by Ho f s t a d t e r , 1979, pp520-521). One might t h i n k of the d i f f e r e n c e between general purpose components l i k e nuts and b o l t s and complete, autonomous, s p e c i a l purpose machines l i k e p e n c i l sharpeners. In nature, the amino a c i d s are l i k e , nuts and b o l t s that go together t o make some machine; i t would be almost impossible Chapter VI to guess i n advance what machine they w i l l be made i n t o . But i n Typogenetics, each "amino a c i d " i s already autonomous, l i k e the p e n c i l sharpener, not a component -for some l a r g e r machine, though they may be used i n some sequence that adds up t o a g l o b a l program. Genes In molecular b i o l o g y , the sequence o-f bases l y i n g between two punctuator t r i p l e t s , coding -for one polypeptide c h a i n , i s c a l l e d a cistron. Obviously t h i s corresponds c l o s e l y t o the "gene" o-f a Typogenetics s t r a n d . But how does the b i o l o g i c a l term gene - f i t i n here? Molecular b i o l o g i s t s sometimes use the word "gene" interchangeably with " c i s t r o n . " T h i s i s f i n e f o r some purposes. A c i s t r o n codes f o r production of a p r o t e i n . There are innumerable p r o t e i n s ( r e c a l l that there were only 20 amino a c i d s i n the genetic code, but there are n e a r l y i n e x h a u s t i b l e numbers of ways these amino a c i d s can be put t o g e t h e r ) . Each d i f f e r e n t l i f e form (and d i f f e r e n t s p e c i a l i z e d c e l l s w i t h i n a l i f e from) s y n t h e s i z e s p r o t e i n s unique to i t s type. C l e a r l y , there i s a sense i n which "you are what your c i s t r o n s are." C i s t r o n s , and the p r o t e i n s y n t h e s i s a s s o c i a t e d with them, are i n h e r i t e d . Yet the concept of the gene as- an hereditary unit o r i g i n a t e d with Mendel, not with biochemistry, and i t i s not Chapter VII 1 9 8 always p o s s i b l e t o c o r r e l a t e an i n h e r i t e d c h a r a c t e r i s t i c ( e s p e c i a l l y those i d e n t i f i e d other than through biochemistry, e.g. "chreods" (embryological p a t t e r n s o-f change), or s t r u c t u r a l form i n morphology) with a s i n g l e c i s t r o n , or even a w e l l - d e f i n e d set of c i s t r o n s . For that reason e v o l u t i o n a r y b i o l o g i s t s (e.g. Dawkins) and " c l a s s i c a l g e n e t i c i s t s , " e.g. i n a g r i c u l t u r a l s c i e n c e , p r e f e r t o d e f i n e a gene as a DNA sequence of s u f f i c i e n t s i z e and i n t e g r i t y t o p o t e n t i a l l y l a s t through enough generations t o serve as a u n i t of n a t u r a l s e l e c t i o n . Dawkins (1976) g i v e s a good example, i n v o l v i n g the phenomenon of minickry, t o show why the cistron=gene equation i s not a good one f o r h i s purposes. Two caucasion human beings might d i f f e r as to eye c o l o r . But members of a c e r t a i n s p e c i e s S of b u t t e r f l y d i f f e r f a r more d r a s t i c a l l y . Some members of t h i s s p e c i e s S mimic one kind of " n a s t y - t a s t i n g " b u t t e r f l y , and other members of S mimic an e n t i r e l y d i f f e r e n t kind of " n a s t y — t a s t i n g " b u t t e r f l y , r e s u l t i n g i n the remarkable happenstance that one b u t t e r f l y may mimic s p e c i e s A while h i s brother mimics s p e c i e s B. "...But how can a s i n g l e gene determine a l l the m u l t i f a r i o u s aspects of m i m i c r y — c o l o u r , shape, spot p a t t e r n , rhythm of f l i g h t ? The answer i s that one gene i n the sense of a c i s t r o n probably cannot. But by the unconscious and automatic ' e d i t i n g ' achieved by i n v e r s i o n s and other Chapter VII 199 a c c i d e n t a l rearrangements of g e n e t i c m a t e r i a l , a l a r g e c l u s t e r of f o r m e r l y separate genes has come together i n a t i g h t l i n k a g e group on a chromosome. The whole c l u s t e r behaves l i k e a s i n g l e g e n e — i n d e e d , by our d e f i n i t i o n i t now is a s i n g l e gene..." pp 33-34. For Dawkins a gene may be a whole set of c i s t r o n s . Or i t could be a mere p o r t i o n of a c i s t r o n . The r e l a t i o n s h i p — a n d d i f f e r e n c e — b e t w e e n c i s t r o n s and genes leads t o two ways of doing b i o l o g y . On the one hand, b i o l o g i s t s i d e n t i f y c i s t r o n s and c i s t r o n complexes as s y n t a c t i c a l e n t i t i e s — t h e n t r y t o determine what they do f o r the organism, how they are r e a l i z e d p h e n o t y p i c a l l y . At the other end s c i e n t i s t s i d e n t i f y i n h e r i t e d s t r u c t u r e s , q u a l i t i e s (e.g. c o l o r ) and processes ( i n c l u d i n g behaviors) and encourage i n v e s t i g a t i o n of what chromosomal material i s a s s o c i a t e d with them. Again, t h i s f o r c e s us to consider the very d i f f e r e n t p o s i t i o n s of the b i o l o g i s t and the T y p o g e n e t i c i s t . The l a t t e r has p r a c t i c a l l y nothing but syntax to work with; h i s e n t i t i e s ' "genes" are not expressed phenotypical1y. The T y p o g e n e t i c i s t cannot, f o r example, i d e n t i f y a c e r t a i n c o l o r s t r a n d , or a substance l i k e i n s u l i n s ecreted by a s t r a n d , or an i n s t i n c t u a l p a t t e r n manifested by a strand under c e r t a i n t r i g g e r i n g circumstances, and then set to work t r y i n g to f i n d what duplet sequence d i c t a t e s i t . Chapter VII 2 0 0 The concerns t h a t motivate a Dawkins to di-f-ferentiate gene -from c i s t r o n do not apply t o Typogenetics. One could well say: Typogenetics has c i s t r o n s , but not genes. The Value of Typogenetics t o Natural Science To the b i o l o g i s t , the l o g i c i a n ' s job must almost seem too easy. He c r e a t e s h i s own world with h i s own r u l e s , s p i n n i n g out t a u t o l o g i e s without ever having to mount specimens or buy expensive equipment... So, what i s "pure r e s e a r c h " ( l i k e t h i s t h e s i s ) worth, anyway, t o n a t u r a l s c i e n c e ? F i r s t of a l l , t h i s kind of endeavor pioneers the development of symbolic methods. George Boole and h i s s u c c e s s o r s pioneered the symbolic r e p r e s e n t a t i o n of l o g i c before any r e a l e l e c t r i c a l engineering was being done. But when the p h y s i c s was s u f f i c i e n t l y understood, and some needs f o r e l e c t r i c a l d e v i c e s had been i d e n t i f i e d by i n v e n t o r s , the symbolic methods were ready to hand, t o be modified i n t o the switching c i r c u i t schematization that has proved so u s e f u l f o r e l e c t r i c a l , e l e c t r o n i c , and computer technology. Biology may not ready f o r a DNA modeling l o g i c yet. But we are g e t t i n g c l o s e r a l l the time. When we are ready, the modelers w i l l look around at symbolic systems l i k e Typogenetics f o r i n s i g h t s i n t o how t o approach the task of symbolic modeling of nature. Chapter VII 201 Second, and r e l a t e d l y , Typogenetics and the l i k e supply new concepts t h a t b i o t e c h n o l o g i s t s may drawn on, j u s t as they draw on those of nature now. Take the concept of the s e l f -r e c u r r o r . T h i s s t r a n d , i t w i l l be r e c a l l e d , has as i t s daughter one or more strand s n o n i d e n t i c a l to the p r o g e n i t o r , but has granddaughters (or more remote descendants) i d e n t i c a l to i t s e l f — i t r e c u r s i n t e r g e n e r a t i o n a l l y . Now, nature has something somewhat analogous: the metamorphizing c r e a t u r e s such as i n s e c t s or amphibians. The f r o g l i v e s as egg, then as water—breathing t a d p o l e , then r e c u r s to a i r — b r e a t h i n g f r o g . But the metamorphizing c r e a t u r e r e t a i n s the same DNA throughout i t s phase changes ( d i f f e r e n t genes being suppressed at d i f f e r e n t s t a g e s ) . Yet perhaps i t i s p o s s i b l e f o r one DNA s t r a n d , i n the manner of the s e l f — r e c u r r i n g Typogenetics s t r a n d , t o s y n t h e s i z e another DNA strand which becomes c r e a t u r e X, with i t s own d i s t i n c t DNA. Then X has (among i t s ) o f f s p r i n g c r e a t u r e Y, with i t s own d i s t i n c t D N A , that i s i d e n t i c a l t o the p r o g e n i t o r . There might be good reasons f o r making an a r t i f i c i a l e n t i t y l i k e t h i s . For example, the environment i n which t h i s c y c l e of beings l i v e s may f l u c t u a t e d r a s t i c a l l y , so that only "warm-blooded" X can l i v e i n winter, "cool-blooded" Y i n summer. And look at the i n c i d e n t a l r e c u r r o r . T h i s c r e a t u r e cannot s e l f - r e p l i c a t e , but i t i s produced by an e n t i t y that Chapter VII 202 s e l f - r e c u r s , so i t does get r e p l i c a t e d so long as the s e l f -r e c u r r o r keeps r e p l i c a t i n g . T h i s r e l a t i o n s h i p could be e x p l o i t e d . Suppose the i n c i d e n t a l r e c u r r o r s e c r e t e s a u s e f u l substance. There are at l e a s t two good reasons f o r not wanting t h i s e n t i t y t o s e l f - r e p l i c a t e . One i s , i t might become too p l e n t i f u l . It might be a good i d e a to keep the s e l f - r e c u r r o r t h a t i s the source of a l l the i n c i d e n t a l r e c u r r o r s i n c o n d i t i o n s of s t r i c t containment, and r e l e a s e f o r use o n l y the i n c i d e n t a l r e c u r r o r s which are s t e r i l e and thus not l i a b l e t o overpopulate. Then again, biotechnology i s a p o t e n t i a l l y commercial e n t e r p r i s e . It may be that the designer/owners of the s e l f — r e c u r r o r would wish to keep i t under lock and key as a trade s e c r e t . They might choose to only s e l l the i n c i d e n t a l r e c u r r o r , which does not r e p l i c a t e but does provide the u s e f u l product. The advantage t h i s has over the metamorphizing type c r e a t u r e i s t h a t , i n theory, you can take a tadpole, i n s p e c t i t s DNA, and the how-to-bui 1d-a-f r o g i n f o r m a t i o n i s there. But the information f o r the s e l f -r e c u r r o r simply i s not i n the i n c i d e n t a l r e c u r r o r . Logic a l s o provides some absolute l i m i t a t i o n s on what can and cannot be. At the end of the l a s t chapter the meta-Typogenetical r e s u l t s progressed to more and more expansive c o n c e p t s — t h e mother-maker algorithm, the u r s t r a n d , e t c . — u n t i l an absolute l i m i t a t i o n was discovered, one that probably had never been thought of before: The l o g i c a l Chapter VII i m p o s s i b i l i t y of an e n t i t y that could have a l l and only non-s e l f - r e p l i c a t i n g e n t i t i e s f o r i t s o f f s p r i n g . T h i s l i m i t a t i o n could not have been discovered e m p i r i c a l l y , of c o u r s e — o n e cannot f i n d what cannot e x i s t ! But i t i s a l i m i t a t i o n t h at would apparently cover a l l p o s s i b l e worlds, i n c l u d i n g t h i s p h y s i c a l one. L o g i c a l s o suggests hypotheses r e l e v a n t to n a t u r a l s c i e n c e t h a t might not have otherwise occurred t o the s c i e n t i s t s . Maybe there i s a scheme or procedure that nature uses i n i t s DNA " l o g i c " t h a t b i o l o g i s t s have not yet o b s e r v e d — b u t which l o g i c i a n s are using. Perhaps, f o r example, there are s e l f - r e c u r r o r s i n n a t u r e — a n d we simply have not i d e n t i f i e d them yet. Perhaps there i s a s t r a n d - l i n e o f , say, v i r u s e s corresponding to the c l a s s of i n f i n i t e l y f e r t i l e s t r a n d s : each descendant i s d i f f e r e n t from i t s p r o g e n i t o r , not through mutation, but by design. It i s s a f e t o say that most of the e m p i r i c a l f a c t s of b i o l o g y are as yet unknown. Our tendency i s always t o take t h i n g s we do know and g e n e r a l i z e them. Often such g e n e r a l i z a t i o n s are f a l s e , but we teach them and most people, even i n c l u d i n g workers i n the f i e l d , accept them unquestioningly. The very " f r e e l i c e n s e " of the l o g i c i a n encourages him to think up p o s s i b i l i t i e s t h a t seem to be f o r e c l o s e d by what i s "known" about b i o l o g y . If a venturesome b i o l o g i s t can be induced to look f o r these in s t a n c e s of the p o s s i b l e , even where he i s not supposed to Chapter VII 204 f i n d them, he sometimes w i l l f i n d them. C.S P e i r c e makes the v a l u a b l e p o i n t that our l o g i c a l i n t u i t i o n l i g h t s on t r u t h s that hold f o r nature much of t e n e r than chance would seem to a l l o w — b e c a u s e our l o g i c a l i n t u i t i o n emanates from a b r a i n that i s i t s e l f a product of and process w i t h i n that same nature (Fann, 1970). F i n a l l y , i t may come to pass that a l t e r n a t i v e models e x p l a i n some aspect of b i o l o g y about e q u a l l y w e l l . In that case, the c r i t e r i a f o r choosing between the r i v a l s w i l l be 1 o g i c o — a e s t h e t i c ones: Which i s more general? Which i s more elegant? Chapter VIII 205 CHAPTER V I I I : TYPOGENETICS AS A LOGICAL SYSTEM In the l a s t chapter, an e f f o r t was made t o compare Typogenetics with a h y p o t h e t i c a l DNA modeling l o g i c . Returning t o the domain o-f the "pure" formal s c i e n c e s , t h i s chapter analyzes Typogenetics as a system; compares i t to a standard symbolic l o g i c , and then t o s e l f - p r o v i n g c o n s t r u c t i o n s i n mathematical l o g i c . A M u l t i - L e v e l e d System Multi—leveled systems, and complementarity (of f i g u r e and ground; or form and content, e t c . ) , are main t o p i c s of i n t e r e s t i n the book which introduced Typogenetics to the world (Hofstadter, 1979). Not s u r p r i s i n g l y , then, h i s formal system Typogenetics provides a symbolic medium f o r studying the problems and p o s s i b i l i t i e s r e l a t i n g to m u l t i - l e v e l e d systems and complementarity. Let us r e c a l l that working with Typogenetics r e q u i r e d one t o deal with s e v e r a l i n t e r p e n e t r a t i v e l e v e l s of order. A strand embodies: Chapter VII (1) the order of i t s well-formed u n i t s ; (2) the order of i t s amino a c i d e n f o l d i n g ( t e r t i a r y s t r u c t u r e ) ; (3) the order of i t s duplets; (4) the order of i t s gene/enzyme a c t i v a t i o n ; (5) the order of i t s enzyme movements/operations; (6) the order of e l e c t e d i n i t i a l attachment s i t e s f o r enzymes; (7) and, f i n a l l y , the strand too may f i t w i t h i n a p a r t i c u l a r i z e d l i n e of descent, f o r i t has v a r i o u s p o s s i b l e progeni t o r s . IrreducibiIity. M u l t i - 1 e v e l e d systems r a i s e questions of r e d u c i b i 1 i t y . In Typogenetics, as we go up each l e v e l , new "laws" superimpose emergent order on the orders already imposed by preceding l e v e l s ; f o r that reason I c a l l i t an " i r r e d u c i b l e " system. Of course, there i s a great debate over whether nature i s such a system. Rosenberg (1985) r a t h e r narrowly d e f i n e s philosophy of b i o l o g y as e x c l u s i v e l y concerned with t h i s i s s u e (or more p r e c i s e l y , with the two questions; "Is b i o l o g y a p h y s i c a l s c i e n c e ? " and "Is the p o s t p o s i t i v i s t account of s c i e n c e , obtained from study of h i s t o r y of p h y s i c s , adequate f o r b i o l o g y ? " ) . Hofstadter h i m s e l f , a p h y s i c i s t by t r a i n i n g , f e e l s that nature i s , i n p r i n c i p l e , completely- descr ibable at physico-chemical levels well "below" the level of or gari isms, just as Chapter V I I I a program i n any higher l e v e l computer language (e.g. BASIC) can be f u l l y understood, w r i t t e n , e t c . , at the machine language l e v e l . However, i t has at l e a s t been suggested t h a t there might be i r r e d u c i b l e "molar" laws and emergent l e v e l s of order i n nature (Rosenberg c a l l s those who fav o r t h i s view "autonomists"; those who b e l i e v e t h a t b i o l o g y i s r e d u c i b l e t o p h y s i c a l s c i e n c e he c a l l s " p r o v i n c i a l i s t s " ) . Typogenetics pr o v i d e s a c l e a r example of what an i r r e d u c i b l e m u l t i - l e v e l system would be l i k e . T h i s can be seen by c h a r a c t e r i z i n g the limits of one's knowledge at each level of the system. (1) At the most b a s i c l e v e l one w i l l possess the r e c u r s i v e formula of well-formedness ( i n c l u d i n g , which symbols are allowable) f o r Typogenetics s t r a n d s . Then, s p e c i f i c a l l y , one can know a p a r t i c u l a r s t r a n d as a s t r i n g of a c e r t a i n length of u n i t s with i d e n t i f i a b l e bases. Yet, there i s no way to i n f e r from t h i s l e v e l , what the strand does, what i t s molar s t r u c t u r e i s , or where i t comes from. (2) At the next l e v e l , one would be aware of (possess the diagrams f o r ) a st r a n d ' s enzymes' t e r t i a r y s t r u c t u r e s . Such diagrams then r e v e a l how many genes are i n the s t r a n d , and how many amino a c i d s (.qua t h e i r f o l d i n g i n c l i n a t i o n s ) are i n each gene. But even i f one i s granted, at t h i s l e v e l , the knowledge that an amino a c i d i s coded f o r by a duplet, ambiguity would remain as t o e x a c t l y what duplets make up each gene, s i n c e , several d u p l e t s code f o r an amino a c i d Chapter VIII 208 having a given -folding i n c l i n a t i o n . (3) At the next l e v e l , one would be apprised o-f the order o-f duplets comprising the s t r a n d , and the from-the-1 e-ft reading frame r u l e . One could here i d e n t i f y the AA boundaries between genes. However, without more i n f o r m a t i o n one s t i l l cannot know t h i s sequence's a c t i o n p o t e n t i a l . ( 4 ) At t h i s l e v e l , one would be aware of the reading frame r u l e t h a t reads genes from the l e f t , which order corresponds e x a c t l y with the a c t i v a t i o n of the enzymes those genes code f o r . Notice t h i s i s d i f f e r e n t than reading duplets from the l e f t ; a read-genes—from—the—right r u l e would not be i n c o n s i s t e n t with the read—duplets—from—the—1 e f t r u l e ; i t i s only c o i n c i d e n t a l t hat they both s t a r t at the l e f t . In any case, at t h i s l e v e l , one knows which enzyme w i l l precede which, and what bases w i l l be p r e f e r r e d , but not yet what they w i l l da, nor where they w i l l i n i t i a l l y a t t a c h t o the strand when more than one i n s t a n c e of the p r e f e r r e d base type i s a v a i l a b l e . (5) At t h i s l e v e l , one has a v a i l a b l e the t r a n s l a t i o n t a b l e that allows i n t e r p r e t a t i o n of a duplet as an o p e r a t i o n . Now one knows what the strand's enzymes can do. But because v a r i o u s a l t e r n a t i v e i n i t i a l enzyme binding s i t e e l e c t i o n s can, i n most cases, be made, one does not know which d e r i v a t i o n of t h i s strand might be e n t e r t a i n e d (and so, which descendants i t w i l l have, and thus which s t r a n d - l i n e i s being Chapter VIII developed), though one can run through a l l the p o s s i b i l i t i e s and know i t must be one of these. ( 6 ) Here one has as given the s p e c i f i c c h o i c e s of the enzymes' i n i t i a l b i n d i n g s i t e s , p u t t i n g one i n e f f e c t i n possession of the d e r i v a t i o n of the str a n d ' s descendants. However, i t i s s t i l l not p o s s i b l e t o deduce what the progenitor was to the s t r a n d , s i n c e every Typogenetics strand has i n d e f i n i t e l y many p o s s i b l e "mothers." Thus the s p e c i f i c s t r a n d — l i n e e n t e r t a i n e d remains ambiguous. (7) At t h i s f i n a l , s t r a n d — l i n e l e v e l , the exact ancestry of the strand i s s p e c i f i e d . At t h i s l e v e l our c l a s s i f i c a t i o n of kinds of st r a n d s can be ap p r e c i a t e d . It i s thus evident that Typogenetics i s defined across i t s s e v e r a l l e v e l s , meaning i t has new laws, r e l a t i o n s and a t t r i b u t e s introduced at each l e v e l . Put a l t e r n a t i v e l y , t h e r e i s no way that higher l e v e l phenomena can be i n f e r r e d even i n p r i n c i p l e from lower level—based understanding. T h i s c o n t r a s t s s h a r p l y with c e l l u l a r automata. There may be "higher order" laws d e s c r i b i n g what r e s u l t s from c e r t a i n c o l l i s i o n s of LIFE o b j e c t s , but those laws are completely r e d u c i b l e t o the t r a n s i t i o n f u n c t i o n s o p e r a t i v e at the c e l l l e v e l . Given the information as t o the s t a t e s of the c e l l s of a g r i d , I can i d e n t i f y a LIFE "spaceship"; but there i s no way of knowing that a Typogenetics strand i s , e.g. a s e l f -r e c u r r o r , based only on knowing what i t s t e r t i a r y s t r u c t u r e Chapter VII i s . The way to have something e q u i v a l e n t i n the way of a c e l l u l a r automaton would be t o have higher l e v e l r u l e s t h a t are t r i g g e r e d by c o n f i g u r a t i o n s a r i s i n g through the lower l e v e l r u l e s . For example: i f a "spaceship" a r r i v e s at the edge of a f i n i t e LIFE g r i d , a "higher order" t r a n s i t i o n r u l e bounces the spaceship back t o whence i t came (rather than l e t t i n g i t d i s i n t e g r a t e at the edge). That would be an i r r e d u c i b l e r u l e , i n s o f a r as t h a t bouncing back could not be e f f e c t e d nor explained i n terms of LIFE'S normal c e l l - l e v e l r u l e s . It i s beyond the scope of t h i s t h e s i s t o t r y to address the question of whether nature i s a r e d u c i b l e system or not. As an e x e r c i s e i n p h i l o s o p h i c a l l o g i c , the main p o i n t i s that our system permits us to f o r m a l i z e c o h e r e n t l y the no t i o n . From the p o i n t of view of A r t i f i c i a l L i f e , the main question i s whether i r r e d u c i b l e systems o f f e r new p o s s i b i l i t i e s , or at l e a s t economies, f o r a r t i f i c i a l environments or e n t i t i e s . In f a c t , design work i n a m u l t i -l e v e l e d system l i k e Typogenetics does pose i t s own d i s t i n c t i v e c h a l l e n g e s . Interpenetrative levels. Design o b j e c t i v e s would normally be posed at the 6th or 7th l e v e l s of order of Typogenetics. And what I have, c a l l e d the interpenetrat ion of the l e v e l s becomes very apparent as soon as one goes to work Chapter VII to achieve those design o b j e c t i v e s . A change e f f e c t e d at one l e v e l u s u a l l y b r i n g s about change at one or more other l e v e l s . I f , f o r example, i t i s r e a l i z e d t h a t an a d d i t i o n a l command i s r e q u i r e d f o r a strand t o achieve a r e p r o d u c t i v e g o a l , then another duplet coding f o r that command must be added. But that changes the operand, and may a l t e r an enzyme's bi n d i n g preference. And of course, working at the l e v e l of the s t r a n d - l i n e s e t s exacting c o n s t r a i n t s : Any change wrought i n a strand under design that i s supposed t o be a n c e s t r a l t o a known strand can wind up n e c e s s i t a t i n g change of the known s t r a n d , which of course might amount t o undoing one's previous l a b o r s . Complementari t y Godel, Escher, Bach (Hofstadter, 197?) f e a t u r e s some i l l u m i n a t i n g examples of complementarity (see e s p e c i a l l y h i s Chapter I I I ) . Here i s one of h i s mathematical examples: Consider t h i s i n d u b i t a b l y r e c u r s i v e l y enumerable sequence: 1, 3, 7, 12, IS, 26, 35, 45, 56, 69... What i s i t s p a t t e r n ? How would i t be extended? We observe that from the domain of natur a l numbers, some numbers have been i n c l u d e d , and some excluded, from t h i s sequence. The excluded numbers sequence would be: 2, 4, 5, 6, 8, 9, 10, 11, 13... It turns out there i s a c u r i o u s r e l a t i o n s h i p between the included and the excluded. The gap between the f i r s t two Chapter VII numbers i n the sequence o-f i n c l u d e d s i s 2. 2 happens to be the - f i r s t number i n the sequence o-f the excludeds. Then the next gap between the i n c l u d e d numbers i s 4, which i s the next number i n the sequence o-f the excludeds. And so -forth. Each next gap between in c l u d e d numbers d e f i n e s the next number among the excludeds. Knowing t h i s , we can extend the sequence of i n c l u d e d s : 83, 98, 114, 131, 150..., knowing we are at the same time i m p l i c i t l y extending the sequence of the excludeds. A s t r i k i n g i n s t a n c e of complementarity, e x h i b i t i n g c o n s i d e r a b l e i n g e n u i t y on the p a r t of i t s a r t i s t (S. Kim), i s Figure-Figure (Hofstadter's F i g u r e 17, p 69), a drawing i n which the (white) "foreground" i s simply i t e r a t i o n of the word " f i g u r e , " while the (black) "background" i s also simply i t e r a t i o n of that word. T h i s " t i l i n g of the plane" i l l u s t r a t e s how f i g u r e and ground can contain the same i nformati on. (If you cut out the whi t e , foreground from the sheet of paper on which t h i s f i g u r e i s p r i n t e d , the cut out p o r t i o n (foreground) could be f i t t e d e x a c t l y onto the remaining black background). T h i s kind of complementarity i s somewhat the r e v e r s e of the mathematical example, inasmuch as Figure-Figure's " p o s i t i v e " and "negative" spaces were the same ( c o n s i s t i n g of i t e r a t i o n of the word " f i g u r e " ) , while the complementary number sequences were mutually e x c l u s i v e . The essence of complementarity i s t h a t , from e i t h e r of the two t h i n g s s a i d t o be complementary, the information Chapter VIII necessary to c o n s t r u c t both i s a v a i l a b l e i n j u s t one. For example, -from the sequence o-f " i n c l u d e d " numbers given, the s e t o-f "excluded" numbers could be deduced, and v i c e versa; they are, i n t e c h n i c a l terms, recursive. Now, t h i s should remind the reader of the complementarity of the two s t r a n d s making up DNA, e i t h e r of which possesses i n f o r m a t i o n s u f f i c i e n t t o r e c o n s t i t u t e the p a i r . S i m i l a r l y (by design) i t i s t r u e t h a t a Typogenetics s t r a n d such as the s e l f -r e p l i c a t o r produces i t s complement. But the s e l f - r e p l i c a t i n g Typogenetics s t r a n d was more than j u s t complementary with i t s copy, i t was i n a s p e c i f i c sense, sel/-complementary. Pal 1indromes manifest a kind of self—complementarity, s i n c e they read the same backward and forward, and Hofstadter managed to c o l l e c t some remarkable pallindromes f o r h i s book; In one b r i e f s e c t i o n he r e p r i n t s M.C. Escher's "Crab Canon" (a t i l i n g of the plane with crabs e q u i v a l e n t t o Kim's Figure-Figure)^ the music f o r J.S. Bach's "Crab Canon" (a musical composition which reads the same backwards or forwards); and presents a dialogue ( f e a t u r i n g a character named the Crab) which, amazingly enough, reads the same backwards and forwards. Now, the reader w i l l perhaps remember back t o the Turing Table automaton i n Chapter IV and the p a l l i n d r o m i c " s e l f -r e p l i c a t i n g " s t r i n g . In that case i t seemed the s e l f -complementarity of the s t r i n g was a "red h e r r i n g " having Chapter VIII l i t t l e to do with i t s r e p l i c a t i o n . But the Typogenetics s e l f -r e p l i c a t o r was, not c o i n c i d e n t a l 1 y , a pallindrome (not a l l self-complementary strands are p a l l i n d r o m i c , i n c i d e n t a l l y ; e.g. the one duplet strand CG i s self—complementary but not a pa l l i n d r o m e ) ; that property had eve r y t h i n g t o do with design c o n s i d e r a t i o n s r e v o l v i n g about the copy f u n c t i o n t h a t was employed t o e f f e c t i t s r e p l i c a t i o n . Comparing the Typogenetical s e l f — r e p l i c a t o r to the two kinds of complementarity ("same," as i n Figure—Figure, and "opposite" as i n the r e c u r s i v e number sequences), which i s i t ? Two strand s (such as ACTGGT and ACCAGT) that are "complementary" i n Typogenetics' own sense (complementary as to bases, with l e f t and r i g h t reversed) are complementary i n the "opposite" mode. But the p a l l i n d r o m i c s e l f - r e p l i c a t o r i s self—complementary i n the "same" way: The same information i s contained i n the l e f t h a l f ( f i r s t 16 duplets) of the strand as i n the r i g h t , j u s t as the Figure—Figure foreground matches i t s background. Typogenetics and the Regress Paradox of S e l f - R e p l i c a t i o n There i s yet another sense of "complementarity" that i s very important i n Typogenetics. T h i s can be analyzed i n connection with the "regress paradox of sel f - p a r a d o x " ( r e l a t e d i n Chapter IV). The r e g r e s s paradox arose from the pres u p p o s i t i o n that s e l f — r e p l i c a t i n g e n t i t y must possess a Chapter VIII d e s c r i p t i o n o-f i t s e l f — w h i c h d e s c r i p t i o n would, t o be complete, seemingly possess i n tu r n an imbedded d e s c r i p t i o n of i t s e l f , and so on, ad infinitum. But of course there i s no d e s c r i p t i o n of the Typogenetics strand in the strand because the commands the strand codes f o r are addresssed not at the c o n s t r u c t i o n of a design contained i n a b l u e p r i n t , but ra t h e r at the making of a copy of the strand i t s e l f . The s e l f -r e p l i c a t i n g strand i s , f i r s t , a sequence of o p e r a t i o n s (a program or r e c i p e ) d i r e c t i n g the c o n s t r u c t i o n of i t s copy from a model; i t i s second, u n i n t e r p r e t e d data forming a pass i v e operand, the model t o be copied. Now, what Typogenetics demonstrates p a r t i c u l a r l y well i s that i t i s r e a l l y no mean t r i c k t o arrange i t so that the uninterpreted—information—as—model s i d e of t h i n g s squares precisely with ( i s , i n t h i s f u r t h e r sense of the term, "complementary" to) the interpreted—information—as-commands-to-copy-the—model s i d e . In f a c t , the e n t i r e method of reductio ad absurdum d i s p r o o f of h y p o t h e t i c a l strands (covered i n Chapter VI) c o n s i s t e d i n the d e t e c t i o n of noncomplementarity of (or i n c o n s i s t e n c y between) the hy p o t h e t i c a l strand as (1) data and (2) as program. T h i s then i s another major design c o n s i d e r a t i o n : Where before i t was mentioned that m o d i f i c a t i o n s at one l e v e l of work could have adverse e f f e c t s at other l e v e l s (e.g. change in duplets might a l t e r t e r t i a r y s t r u c t u r e ) , here we have Chapter VIII the p o s s i b i l i t y of m o d i f i c a t i o n of one " h a l f " of a complement impacting adversely on i t s other h a l f . To tamper w i t i h the "foregound" of Figure-Figure i s to simultaneously a f f e c t the "background." And to modify a Typogenetic s e l f - r e p l i c a t o r l i k e the one given i n Chapter V i s t o run the r i s k of des t r o y i n g i t s several embodied complementarities. An engineer p r a c t i c e d i n t h i s system w i l l of n e c e s s i t y develop a f i n e a p p r e c i a t i o n f o r r e f l e x i v i t y ! Typogenetic vs. Standard L o g i c D e r i v a t i o n s How does Typogenetics resemble and c o n t r a s t with standard l o g i c ? A f a i r l y obvious mapping would equate Typogenetics' "amino a c i d o p e r a t i o n s " with the r u l e s of i n f e r e n c e and s u b s t i t u t i o n found i n a standard l o g i c a l system, and the Typogenetics strand t o be transformed as the equiva l e n t af the premises af the standard l o g i c a l form, the daughters of that strand the c o n c l u s i o n s . In the s p e c i a l case of the s e l f - r e p l i c a t o r , the premiss happens t o embody i t s own d e r i v a t i o n . That t h i s sounds almost v i c i o u s l y c i r c u l a r perhaps p a i n t s t a d i f f e r e n t motives f o r c o n s t r u c t i n g a semantics based l o g i s t i c on the one hand, and a syntax-only formal system l i k e Typogenetics on the other. A standard l o g i s t i c e x i s t s to b r i n g f o r t h , by t r u t h p r eserving t r a n s f o r m a t i o n s , the t r u t h s l a t e n t i n a formula or set of axioms. To go through the "work" of d e r i v i n g the "same Chapter VIII a i d t r u t h " — w h i c h i s what c i r c u l a r reasoning amounts t o — i s to commit a s i n of s u p e r f l u i t y . That c o n s i d e r a t i o n does not enter i n t o Typogenetics, where we are t r y i n g t o model processes of propagation, not i d e n t i f y formulas i n t e r p r e t a b l e as proven t r u t h s , and symbols are used because they are e a s i l y manipulated, r e g a r d l e s s (for now) of t h e i r p o t e n t i a l t o be given semantic i n t e r p r e t a t i o n s . We value the s e l f -r e p l i c a t o r S-R because i t s d e r i v a t i o n i l l u s t r a t e s well d e f i n e d r u l e s of t r a n s f o r m a t i o n a c h i e v i n g t h i s i n t e r e s t i n g o b j e c t i v e , step by step, i n the symbol domain. The t r a d i t i o n a l v i r t u e of a standard l o g i s t i c was that some info r m a t i o n ( l i k e the axioms and r u l e s of PM) could y i e l d a great deal more in f o r m a t i o n . The Typogenetic strand does have t h a t v i r t u e . One strand can embody an e n t i r e s t r a n d - l i n e . If i t s own form f a l l s w i t h i n the domain of what i t engenders, as i n s e l f - r e p l i c a t i o n or s e l f - r e c u r r e n c e , that should not reduce our i n t e r e s t , but r a t h e r enhance i t , f o r reasons that w i l l be more f u l l y developed i n the next chapter. Typogenetics Strands vs. Programs A Typogenetical " d e r i v a t i o n " d i f f e r s markedly from the standard s o r t . If a Typogenetics strand i s taken to be the formula c o n s t i t u t i n g the premiss, then i t i s at l e a s t a very s p e c i a l s o r t of premiss: It s p e c i f i e s q u i t e e x a c t l y what Chapter VIII r u l e s o-f i n f e r e n c e and s u b s t i t u t i o n are a p p l i e d , thus d i c t a t i n g the d e r i v a t i o n of the daughters or " c o n c l u s i o n s . " In a standard l o g i c , t here are no control r u l e s p r e s c r i b i n g when a r u l e of t r a n s f o r m a t i o n must be a p p l i e d i n a d e r i v a t i o n — t h a t i s l e f t up t o the d e r i v e r . Since, i n such a system, there are u s u a l l y i n d e f i n i t e l y many d i s t i n c t v a l i d p r o o f s of a theorem, the premises cannot be thought to co n t a i n a p a r t i c u l a r d e r i v a t i o n . T h i s c o n t r a s t s s h a r p l y with computer theorem—proving r o u t i n e s , which must have such c o n t r o l r u l e s s t r i c t l y g u i d i n g the computer from premises to c o n c l u s i o n (see Cohen & Feigenbaum (1982) f o r an overview of automatic theorem—proving), and with programming of automata g e n e r a l l y . Working with Typogenetics might well then come more n a t u r a l l y t o an i n d i v i d u a l accustomed to w r i t i n g computer programs than an i n d i v i d u a l used only t o standard l o g i c , s i n c e , i n designing a Typogenetics s t r a n d , one i s i m p l i c i t l y p r e s c r i b i n g a d e r i v a t i o n t o go along with i t , j u s t as one's input s t r i n g f o r the Turing Table of Chapter IV completely determined the processing i t underwent by that automaton. There i s , though, one important c h a r a c t e r i s t i c of Typogenetics that p l a c e s i t i n the company of standard formal systems. There are times when the user must choose between vari o u s l e g a l o p tions, namely-when more than one u n i t bearing the p r e f e r r e d base to which an enzyme can make i t s i n i t i a l Chapter VIII attachment i s a v a i l a b l e . The next chapter c o n s i d e r s the p o s s i b i l i t y o-f r e p l a c i n g t h i s " f r e e c h o i c e " with an automatic d e c i s i o n procedure (which could be p r o b a b i l i s t i c ) . S e l f — P r o v i n g Formulas Typogenetics can be looked at as a system designed f o r producing formulas that p r e s c r i b e d e r i v a t i o n s (sometimes t h e i r own). T h i s idea of " s e l f - p r o v i n g " or " s e l f -c o n s t r u c t i n g " symbolic e n t i t i e s has become a popular one r e c e n t l y . The concept has i t s r o o t s i n Godel's incompleteness proof, but more immediately i n Sodel, Escher, Bach. (Hofstadter, 1979). It i s not p o s s i b l e here t o r e c a p i t u l a t e Godel's Proof (recounted i n many p l a c e s , and e s p e c i a l l y p e r s p i c a c i o u s l y i n Nagel and Newman, 1958). Let us r e c a l l though the s p e c i a l framework the proof r e q u i r e d : It c o n s i s t e d of (1) a p r e d i c a t e l o g i c l i k e R u s s e l l and Whitehead's P r i n c i p i a M a t h e m a t i c a — c a l l i t P l i — d e s i g n e d t o represent the t r u t h s of number theory; and (2) a mapping a s s i g n i n g to each formula i n PM a unique "godel number" (often on a numeral-to-PM-symbol b a s i s ) , and to each r u l e of transformation i n PM, an exact a r i t h m e t i c a l e q u i v a l e n t . In the context of such a system and mapping, Godel showed that an a r i t h m e t i c a l f u n c t i o n could be defined f o r "proof p a i r s , " n a t u r a l numbers m and n, such that m i s the Chapter VIII godel number o-f the corresponding c o n d i t i o n a l t o a PM d e r i v a t i o n t h a t has the -formula with godel number n as i t s c o n c l u s i o n . Each and every v a l i d d e r i v a t i o n i n PM had i t s own unique corresponding H» godel number, and each and every theorem ( v a l i d l y d e r i ved -formula) had i t s own unique corresponding n godel number. Now, s i n c e PM was designed t o represent a r i t h m e t i c a l f u n c t i o n s , i t could represent that "proof p a i r " f u n c t i o n . T h i s can be i n t e r p r e t e d as meaning PM can represent the d e r i v a b i l i t y of i t s own theorems. That i s , s i n c e PM was r e p r e s e n t i n g mathematics which was " m i r r o r i n g " PM, PM was i n e f f e c t — q u i t e i n c i d e n t a l l y t o i t s o r i g i n a l rBison d'etre—representing itself ( i n d i r e c t l y , by re p r e s e n t i n g i t s "mirror" image, the r e l a t i o n s h i p s h o lding between godel numbers). Bodel himself used t h i s framework to e s t a b l i s h h i s famous Theorems by o f f e r i n g a s e l f - r e f e r e n t i a l PM formula that could be i n t e r p r e t e d as saying "There i s no number m such that » i s i n the proof p a i r r e l a t i o n s h i p Cas the godel number of the corresponding c o n d i t i o n a l t o n 's d e r i v a t i o n ! with the godel number n that corresponds to t h i s formula," which i s tantamount t o saying "There i s no d e r i v a t i o n f o r t h i s formula i n PM." If what t h i s formula a s s e r t s about i t s e l f i s an a r i t h m e t i c a l truth, there i s apparently one a r i t h m e t i c a l t r u t h PM f a i l s to represent, hence i t would be incomplete (unless inconsistent^ i f Chapter VIII i n c o n s i s t e n t , every -formula would be d e r i v a b l e i n a PM l o g i s t i c . Inconsistency, of course, i s j u s t what most l o g i c i a n s b u i l d t h e i r formal systems t o prevent). A f t e r Godel, the l o g i c i a n Leon Henkin used the same kind of framework t o c o n s t r u c t PM formulas i n t e r p r e t a b l e as a s s e r t i n g t h e i r p r o v a b i l i t y . Hofstadter mentioned these, and suggested t a k i n g the idea f u r t h e r : "...a Henkin sentence t e l l s nothing about i t s own d e r i v a t i o n ; i t j u s t a s s e r t s t h a t one e x i s t s . Now i t i s p o s s i b l e to invent a v a r i a t i o n on the theme of Henkin sentences—namely sentences which explicitly describe t h e i r own d e r i v a t i o n s . Such a sentence's h i g h - l e v e l i n t e r p r e t a t i o n would not be "Some Sequence of S t r i n g s E x i s t s Which i s a D e r i v a t i o n of Me", but r a t h e r , "The Herein-described Sequence of S t r i n g s . . . . . Is a D e r i v a t i o n of Me". Let us c a l l the f i r s t type of sentence an implicit Henkin sentence. The new sentences w i l l be c a l l e d explicit Henkin sentences, s i n c e they e x p l i c i t l y d e s c r i b e t h e i r own d e r i v a t i o n s . " (Hofstadter, 1979, p 5 4 2 i . Solovay (1985) was i n s p i r e d t o f o r m a l i z e t h i s i d e a . He constructed e x p l i c i t Henkin sentences and showed that some are t r u e . Chapter VIII Sel-f-Proving Formulas and Typogenetic Strands There i s some s i m i l a r i t y between a s e l f - p r o v i n g formula and a Typogenetics formula. In each case, the symbolic e n t i t y ' s s t r u c t u r e can be read as a map d e t a i l i n g i t s position, i n p o s s i b l e d e r i v a t i o n s , t o other determinable symbolic e n t i t i e s i n i t s r e s p e c t i v e system. A new dimension t o l o g i c a l syntax i s opened up when an i n t e r p r e t a t i o n i s made a v a i l a b l e t hat enables formulas i n a system t o r e f e r t o one another. S y n t a c t i c a l e n t i t i e s can r e f e r t o t h e i r descendancy, t h e i r o r i g i n s , or both. The advent of new ways of t h i n k i n g about t r a d i t i o n a l systems (as one can c h a r a c t e r i z e Godel's c o n t r i b u t i o n ) , and of new systems l i k e Typogenetics, o f f e r the l o g i c i a n the t o o l s t o access t h i s new dimension. Now one moves from c o n s i d e r i n g only i n d i v i d u a l d e r i v a t i o n s t o c o n s i d e r i n g e n t i r e " f a m i l y l i n e s " of theorems. Instead of t h i n k i n g only of a d e r i v a t i o n as a showing of how formula F got here, from the axioms, i t becomes p o s s i b l e t o f i n d i n F's form, necessary information as to what i t s predecessors were, and to what successors i t s very e x i s t e n c e p o i n t s t o . Chapter VIII 223 Proposed Lemma Proving Sentences We have i n t h i s t h e s i s seen much a t t e n t i o n devoted to showing that the -formulas of the Typogenetics system embody e n t i r e s t r a n d — l i n e s of successors. The s e l f — p r o v i n g Henkin sentence does not have comparable information about i t s successor theorems. An i n f i n i t u d e of d i s t i n c t theorems can succeed any lemma, i n a standard l o g i s t i c , and the Henkin sentence's s t r u c t u r e only records where i t has come from, not where other p o s s i b l e d e r i v a t i o n s are l e a d i n g . T h i s suggests the p o s s i b i l i t y of a c o n s t r u c t i o n we could c a l l a lemma proving sentence. T h i s formula, L would r e f e r t o the other theorems o c c u r r i n g i n a determinable d e r i v a t i o n t h a t continues on beyond t h i s lemma formula, t o a terminal formula 7. T would have other v a l u a b l e i n t e r p r e t a t i o n , not to do with d e s c e n t — p e r h a p s T i s i n t e r p r e t a b l e as some s u b s t a n t i v e t r u t h of number theory. Now, the value of t h i s lemma sentence i s that i t s s t r u c t u r e records how to get to t h i s v a l u a b l e formula 7. Our i n v e s t i g a t i o n s today probe ever more r e c o n d i t e reaches of the a priori, so our d e r i v a t i o n s (aided by computer or not) tend t o become more d e t a i l e d . Where the d e r i v a t i o n i s a lengthy one, i t has never been common to p u b l i s h i t i n i t s e n t i r e t y . Frequently, formulas o c c u r r i n g at s u b s i d i a r y l i n e s of the d e r i v a t i o n , lemmas, are p r i n t e d , as e x h i b i t s f o r an accompanying verbal explanation of how the proof proceeds. Chapter VIII Such lemmas are normally chosen -for t h e i r semantic value. Some mathematical thought has been completed., i n -furtherance of the o v e r a l l proof, and the lemma -formula, on the strandard i n t e r p r e t a t i o n , encapsulates that thought. The lemma proving sentence that has been p i c t u r e d , or some con j u n c t i o n o-f lemma proving sentences, would have a very di-f-ferent i n t e r p r e t a t i o n and value f o r the l o g i c i a n . It would s y n t a c t i c a l l y capture the p r o v a b i l i t y of the independently valued theorem T. c's very form would vouch f o r T's d e r i v a b i 1 i t y . T h i s seems a good example of c o l l a p s i n g a p o t e n t i a l l y long process (the d e r i v a t i o n as a procedure) i n t o an immediately presentable s t a t i c form, f o r i n f o r m a t i o n a l economy. The s t a r t l i n g t h i n g about t h i s i s that the "immediately p r e s e n t a b l e s t a t i c form" £ i s a formula j u s t l i k e 7, a formula which l i k e l y means something r a t h e r t r i v i a l i f i n t e r p r e t e d " l i t e r a l l y " as a number t h e o r e t i c a l t r u t h , as 7 i s p r i m a r i l y i n t e r p r e t e d . N a t u r a l l y , t h i s makes us at l e a s t wonder i f there could be any system wherein one could c o n s t r u c t a s y n t a c t i c a l formula that i s both 7 and L at once. That i s , has important content, and i s s e l f - p r o v i n g . In a way what we are doing i s applying two i n t e r p r e t a t i o n s to the same typographical domain, making sure that the r u l e s of syntax capture both semantics. Both 7 and L are WFFs of the same s y n t a c t i c a l world, yet under one i n t e r p r e t a t i o n L has an interesting-meaning: the whence of 7. Chapter VII 7 i t s e l f we can imagine to have an i n t e r e s t i n g t r u t h t o t e l l under a d i f f e r e n t i n t e r p r e t a t i o n — t h e number t h e o r e t i c a l (maybe 7 i d e n t i f i e s an even number t h a t i s not the sum of two primes, which would be a d i s p r o o f of Goldbach's Conjecture one of the great unsolved problems of number t h e o r y ) . But L would exemplify a new kind of l o g i c a l e n t i t y i t would be worth mathematical l o g i c i a n s ' e f f o r t s to invent. Our own system Typogenetics, a l r e a d y has the "descendancy i n t e r p r e t a t i o n " b u i l t i n t o i t . A Typogenetic s t r a n d ' s S's l i n e s of descent are inherent i n i t s form, and each generation i s determinable by a f i n i t e search. As argued e a r l i e r ( p 1 8 1 ) , a strand cannot have i n f i n i t e l y many daughters. T h i s assures us t h a t enumeration of a s t r a n d - l i n e cannot become hung up on in t e r m i n a b l e d e r i v a t i o n of any strand's i n f i n i t y of daughters. At each l i n k i n the l i n e , a f i n i t e number of daughters i s c a l c u l a b l e , and that s trand's d e r i v a t i o n ends, so the d e r i v a t i o n of a l l the next generation's daughters can be done and the s t r a n d - l i n e i s f u r t h e r enumerated. In a standard l o g i s t i c , t h e r e i s an i n f i n i t y of p o s s i b l e d e r i v a t i o n s any given formula F can appear on a s u b s i d i a r y (non—terminal) l i n e i n . For one t h i n g , i f "conjunction" i s a r u l e (allowing me to take any other theorem and, c o n j o i n i n g i t with F, .have thereby created the content of the next l i n e of a d e r i v a t i o n to which F was a Chapter VIII 226 mere lemma), then t h e r e are as many d e r i v a t i o n s going "beyond" the l i n e t h a t F i s on as the r e are theorems t o conjoin with F. Which means, i n t r a c i n g the l i n e a g e of theorems i n a standard l o g i s t i c , we are confronted at each branch of our t r e e of theorems with an i n f i n i t e f o r k . In Typogenetics, one never reaches a str a n d of whom i t can be s a i d " i t s daughters are i n f i n i t e . " C o n t r a r i l y , i n a standard l o g i s t i c , one never reaches a theorem of whom i t can be s a i d " i t s successor theorems i n p o s s i b l e d e r i v a t i o n s are limited in number." Typogenetics has nothing l i k e " c o n j u n c t i o n " p o s s i b l e between i t s WFFs. Strands r e l a t e t o other s t r a n d s only along l i n e s of descent. And descent i s governed by a " f i n i t e -daughters per generation only" law th a t l e t s the f a m i l y t r e e grow i n f i n i t e l y long ( i n some cases) without ever posing an i n f i n i t e fork f o r a search or enumeration program. Of course, i n Typogenetics the strands are themselves the "enumeration programs." Ancestry The determination of descent, r a t h e r than ancestry, was emphasized i n Chapter V. But the meta — l o g i c a l mother-maker algorithms of Appendix IV, discus s e d i n Chapter VI (ppl71-172), show t h a t any Typogenetic strand can serve as a seed f o r r e c u r s i v e generation of ancestors. Thus, the Chapter VIII 227 in f o r m a t i o n a l value of a str a n d i s s u f f i c i e n t t o allow f o r r o u t i n e determination of some v a l i d a n c e s t r a l l i n e , and i n that sense only, a strand can be s a i d t o c o n t a i n information about i t s ancestry. C e r t a i n l y , the mother-maker algorit h m s do take i n t o account the unique form of a strand t o make an ap p r o p r i a t e an a n c e s t r a l l i n e f o r i t . Determining descent i s not l i k e determining ancestry i n t h i s system, however. I t can be s a i d t h a t a t y p i c a l Typogenetic s t r a n d has many p o s s i b l e daughters (see pl55 supra), t hat some st r a n d s have no daughters, and that some have i n f i n i t e l y many descendants; yet the number of daughters a strand can bear i s always determinate (never i n f i n i t e : argued at p l S l supra). By c o n t r a s t , i f i n d e f i n i t e l y many d i f f e r e n t algorithms produce i n d e f i n i t e l y many d i f f e r e n t unending l i n e s of p o s s i b l e ancestry f o r a s i n g l e i nputted s t r a n d s e r v i n g as template, then s t r a n d — l i n e s can never be completely, f i n a l l y enumerated f o r any s t r a n d , i n the d i r e c t i o n of ancestry. In f a c t , exhaustive determination of p o s s i b l e parents cannot even be made. T h i s a l s o goes t o a major d i f f e r e n c e i n the t r a c i n g back of a Typogenetic strand t o i t s ancestor on the one hand, and the t r a c i n g back of a theorem t o i t s axioms. "Back" i s always a f i n i t e number of l i n e s i n the l a t t e r case, never i n the former. Chapter VII Least Ancestor Algorithms A problem with the algorithms given i n Appendix IV i s that they output longer s t r a n d s than were i n p u t t e d . Every strand f e d i n t o the mother-maker i s s h o r t e r than the mother str a n d generated by t h a t a l g o r i t h m , so we get longer and longer s y n t a c t i c a l e n t i t i e s as t h i s p o s s i b l e a n c e s t r y i s enumerated back. In the name of i n f o r m a t i o n a l e f f i c i e n c y , i t would be worth f i n d i n g an a l g o r i t h m t h a t generates f o r the input s t r a n d G a s e r i e s of l e g i t i m a t e a n c e s t o r s , each of which i s a shorter s t r a n d than G, u n t i l a l i m i t i s reached: the s h o r t e s t p o s s i b l e ancestor of G along t h i s s t r a n d -l i n e . I d e a l l y , t h a t would be the s h o r t e s t p o s s i b l e ancestor of <S, p e r i o d ; but I would be impressed by any algorithm that generates some ever-shortening ancestry f o r a given input s t r a n d . In e i t h e r case, there i s no danger of inte r m i n a b l e s e a r c h i n g , s i n c e the number of strands s h o r t e r than a given strand i s always f i n i t e . •nee i n possession of such an algor i t h m , the Typogenetics strand w i l l be i n t e r p r e t a b l e as e l e g a n t l y encoding p r o g r e s s i o n s i n e i t h e r or both d i r e c t i o n s of generation, f r e e of i n f i n i t e f o r k s . Typogenetics: D i r e c t R e f l e x i v i t y F i n a l l y , there i s t h i s fundamental d i s t i n c t i o n between these kinds of " s e l f — d e r i v i n g " e n t i t i e s : Those o c c u r r i n g i n Chapter VIII 229 the context of the Godeli an framework are indirect i n t h e i r s e l f - r e f e r e n c e . The s e l f — p r o v i n g sentence i s , at the shallowest l e v e l of i n t e r p r e t a t i o n , a l o g i c a l formula r e p r e s e n t i n g a f a c t of number theory. Only when one has gone deeper i n t o i n t e p r e t a t i o n — r e a l l y , gone round what Hofstadter c a l l s a "strange l o o p " — h a s one come t o the r e a l i z a t i o n that t h i s formula i s t a l k i n g about i t s e l f . Typogenetics is a formal system deliberately invented to avoid such circuitry, E n t i t i e s i n t h i s system embody i n s t r u c t i o n s f o r s e l f -t r a n s f o r m a t i o n by d e f i n i t i o n and design, not by a c c i d e n t , so as models they provide more conspicuous d i s p l a y s of t h e i r r e f l e x i v e r e l a t i o n s h i p s and t h e i r r e l a t i o n s h i p t o o t h e r s of t h e i r s t r a n d — l i n e . S e l f - C o n s t r u c t i n g Sentences There i s another l i n e of l i t e r a t u r e i n l o g i c a l l i n g u i s t i c s r e l e v a n t t o our i n t e r e s t . In a famous essay, "The Ways of Paradox," Quine (1966) introduced what i s now known as "qui ni ng," or "quine sentences." The idea i s to c o n s t r u c t a s i n g l e sentence which both mentions ( i n quotes) and uses the same c l a u s e . For example: I could get tired of saying " I could get tired of saying". Quine put a quine sentence s t r o n g l y reminiscent of the Chapter VIII 230 L i a r ' s Paradox (and t o Sodel's undecidable -formula, which, i n the course of a s s e r t i n g i t s n o n - d e r i v a b i 1 i t y , "mentions" i t s e l - f by r e f e r r i n g t o i t s godel number); "Yields falsehood when appended to its quotation" yields falsehood when appended to its quotation, Quine was i n t e r e s t e d i n s e l f — r e f e r e n c e as a route to paradox, not i n s e l f - c o n s t r u c t i o n . But Hofstadter noted t h a t , i n a d d i t i o n t o i t s p a r a d o x i c a l q u a l i t y , "Quine's sentence i n e f f e c t t e l l s the reader how t o c o n s t r u c t a r e p l i c a of the sentence being read..." H o f s t a d t e r , 1985b, p28. In s p i r e d with the p o t e n t i a l f o r t h i s i d e a , Hofstadter challenged readers of Scientific American to "Create a comprehensible and not unreasonably long self—documenting sentence that not only l i s t s i t s p a r t s (at the word l e v e l or, b e t t e r yet, the l e t t e r l e v e l ) but a l s o t e l l s how to put them together so that the sentence r e c o n s t i t u t e s i t s e l f . " p28. A winning e n t r y , by one Martin Weichert went as f o l l o w s : Alphabetize and append, copied in quotes, these words: "these append, in Alphabetize and words: quotes, copied" (in Ho f s t a d t e r , 1985b, p64). What at f i r s t appears to be g i b b e r i s h i n f a c t analyzes i n t o i n s t r u c t i o n s and raw m a t e r i a l s p r e s c r i b i n g i t s r e c o n s t i t u t i o n . The raw m a t e r i a l s are d n the quotes. The i n s t r u c t i o n s are, f i r s t , to a l p h a b e t i z e the raw m a t e r i a l s ; then to append a second copy of those raw m a t e r i a l s ( i n Chapter VI quotes) t o the c l a u s e that was rendered i n a l p h a b e t i c a l order. I t i s p r i m i t i v e , but works a f t e r a f a s h i o n ; we w i l l examine i t s s i g n i f i c a n c e i n the next chapter. For now, i t i s adduced as something a c t u a l l y c l o s e r i n s p i r i t t o Typogenetics than the s e l f — p r o v i n g formulas i n t h e i r g o d e l i a n framework. However, there i s always a b i g d i f f e r e n c e between working w i t h i n the c o n f i n e s of a formal system, and an informal system l i k e o r d i n a r y language. If we get s t r i c t , we might deny that Weichert's "sentence" i s well—formed. It probably would not pass muster i n an E n g l i s h c l a s s ! And the essence of a formal system i s that kind of " s t r i c t n e s s . " Chapter I CHAPTER IX EVALUATING MODELS OF PROPAGATION In Chapter I, some broad s t r o k e s were devoted t o d i f f e r e n t i a t i n g c r y s t a l growth from the propagation of l i f e forms. In t h a t context, there was l i t t l e m otivation t o argue over "marginal" cases (e.g. Is a virus a self—replicator or not ? ) . But, questions were r a i s e d , f i r s t i n Chapter IV, i n the context of demonstrating s t r i n g r e p l i c a t i o n i n a Turing Table automaton, as t o when a "genuine" i n s t a n c e of some kind of propagation i s on d i s p l a y . Since then, other models of propagating e n t i t i e s have been presented, i n c l u d i n g Langton's s e l f - r e p l i e a t i n g loop, and, of course, the strands produced i n Typogenetics. We should thus be able now t o d i s c u s s the question of "genuineness" with a stock of examples i n mind. F i r s t of a l l , because a l l of these models are models of r a t h e r than for ( i n Lewontin's sense defined p49 supra) I s h a l l begin by co n v e r t i n g the question of what makes a model of a propagating e n t i t y "genuine" i n t o a question of what i s " i n t e r e s t i n g . " We are t h e r e f o r e seeking some kind of s t i p u l a t i v e , h o p e f u l l y formal, grounds f o r e v a l u a t i n g models, grounds r e l a t i v e l y i n d i f f e r e n t to the f i d e l i t y of the model's r e p r e s e n t a t i o n of exte r n a l processes or e n t i t i e s . Chapter IX What's In a D e f i n i t i o n ? There w i l l be some non—phi 1osophers who f e e l r e l u c t a n c e at t h i s p o i n t . They w i l l be sympathetic t o these words of Dawkins (1976): "Should we then c a l l the o r i g i n a l r e p l i c a t o r molecules " l i v i n g " ? Who cares? I might say t o you "Darwin was the g r e a t e s t man who has ever l i v e d " , and you might say "No, Newton was", but I hope we would not prolong the argument. The p o i n t i s t h a t no c o n c l u s i o n of substance would be a f f e c t e d whichever way our argument was r e s o l v e d . The f a c t s of the l i v e s and achievements of Newton and Darwin remain t o t a l l y unchanged whether we l a b e l them "great" or not." pp 19-20. My r e p l y i s , f i r s t , t o admit t o being no stranger t o f e e l i n g s of f r u s t r a t i o n and tedium produced by encounters with d e f i n i t i o n a l d i s p u t e s , and second, t o i n s i s t that there i s n e v e r t h e l e s s good reason to engage i n them. Even i f we take Dawkins' example, though that d i