Science, Faculty of
Computer Science, Department of
DSpace
UBCV
Domagoj, Babić
2008-08-26T16:40:45Z
2008
Doctor of Philosophy - PhD
University of British Columbia
Software bugs are expensive. Recent estimates by the US National Institute of Standards and Technology claim that the cost of software bugs to the US economy alone is approximately 60 billion USD annually. As society becomes increasingly software-dependent, bugs also reduce our productivity and threaten our safety and security. Decreasing these direct and indirect costs represents a significant research challenge as well as an opportunity for businesses.
Automatic software bug-finding and verification tools have a potential to completely revolutionize the software engineering industry by improving reliability and decreasing development costs. Since software analysis is in general undecidable, automatic tools have to use various abstractions to make the analysis computationally tractable. Abstraction is a double-edged sword: coarse abstractions, in general, yield easier verification, but also less precise results.
This thesis focuses on exploiting the structure of software for abstracting away irrelevant behavior. Programmers tend to organize code into objects and functions, which effectively represent natural abstraction boundaries. Humans use such structural abstractions to simplify their mental models of software and for constructing informal explanations of why a piece of code should work. A natural question to ask is: How can automatic bug-finding tools exploit the same natural abstractions? This thesis offers possible answers.
More specifically, I present three novel ways to exploit structure at three different steps of the software analysis process. First, I show how symbolic execution can preserve the data-flow dependencies of the original code while constructing compact symbolic representations of programs. Second, I propose structural abstraction, which exploits the structure preserved by the symbolic execution. Structural abstraction solves a long-standing open problem --- scalable interprocedural path- and context-sensitive program analysis. Finally, I present an automatic tuning approach that exploits the fine-grained structural properties of software (namely, data- and control-dependency) for faster property checking. This novel approach resulted in a 500-fold speedup over the best previous techniques. Automatic tuning not only redefined the limits of automatic software analysis tools, but also has already found its way into other domains (like model checking), demonstrating the generality and applicability of this idea.
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Exploiting Structure for Scalable Software Verification by Domagoj Babić Dipl.Ing., University of Zagreb, 2001 M.Sc., University of Zagreb, 2003 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Computer Science) The University Of British Columbia (Vancouver) August, 2008 c© Domagoj Babić 2008 Abstract Software bugs are expensive. Recent estimates by the US National Institute of Standards and Technology1 claim that the cost of software bugs to the US economy alone is approximately 60 billion USD annually. As society becomes increasingly software-dependent, bugs also reduce our productivity and threaten our safety and security. Decreasing these direct and indirect costs represents a significant research challenge as well as an opportunity for businesses. Automatic software bug-finding and verification tools have a potential to completely revolutionize the software engineering industry by improving relia- bility and decreasing development costs. Since software analysis is in general undecidable, automatic tools have to use various abstractions to make the anal- ysis computationally tractable. Abstraction is a double-edged sword: coarse abstractions, in general, yield easier verification, but also less precise results. This thesis focuses on exploiting the structure of software for abstracting away irrelevant behavior. Programmers tend to organize code into objects and functions, which effectively represent natural abstraction boundaries. Humans use such structural abstractions to simplify their mental models of software and for constructing informal explanations of why a piece of code should work. A natural question to ask is: How can automatic bug-finding tools exploit the same natural abstractions? This thesis offers possible answers. More specifically, I present three novel ways to exploit structure at three different steps of the software analysis process. First, I show how symbolic execution can preserve the data-flow dependencies of the original code while constructing compact symbolic representations of programs. Second, I propose 1For details, see [1]. ii Abstract structural abstraction, which exploits the structure preserved by the symbolic execution. Structural abstraction solves a long-standing open problem — scal- able interprocedural path- and context-sensitive program analysis. Finally, I present an automatic tuning approach that exploits the fine-grained structural properties of software (namely, data- and control-dependency) for faster prop- erty checking. This novel approach resulted in a 500-fold speedup over the best previous techniques. Automatic tuning not only redefined the limits of auto- matic software analysis tools, but also has already found its way into other domains (like model checking), demonstrating the generality and applicability of this idea. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Research Context . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1 Abstraction and Structure . . . . . . . . . . . . . . . . . 3 1.2.2 Model Checking and Static Checking . . . . . . . . . . . 4 1.2.3 Scalability and Precision . . . . . . . . . . . . . . . . . . 6 1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . 10 2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1 Notational Conventions . . . . . . . . . . . . . . . . . . . . . . . 12 iv Table of Contents 2.2 Verification Conditions . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Program Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . 16 2.3.2 Single Assignment Forms . . . . . . . . . . . . . . . . . . 21 2.3.3 Alias Analysis . . . . . . . . . . . . . . . . . . . . . . . . 28 2.3.4 Summarization for Interprocedural Analysis . . . . . . . 30 2.4 Low Level Virtual Machine . . . . . . . . . . . . . . . . . . . . . 31 3 Structure-Preserving Symbolic Execution . . . . . . . . . . . . 33 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.1.1 Sources of Complexity . . . . . . . . . . . . . . . . . . . 35 3.1.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.3 Design Decisions . . . . . . . . . . . . . . . . . . . . . . . 38 3.1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2 Symbolic Execution . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.1 Restriction Operators . . . . . . . . . . . . . . . . . . . . 42 3.2.2 Data-Flow Summaries by Symbolic Execution . . . . . . 48 3.2.3 Representation . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.3 Complexity and Correctness . . . . . . . . . . . . . . . . . . . . 63 3.3.1 Space Complexity . . . . . . . . . . . . . . . . . . . . . . 63 3.3.2 Correctness . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4 Structural Abstraction . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1.1 Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.1.2 Motivating Example . . . . . . . . . . . . . . . . . . . . . 71 4.1.3 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.1.4 Design Decisions . . . . . . . . . . . . . . . . . . . . . . . 74 4.1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Interprocedural Analysis . . . . . . . . . . . . . . . . . . . . . . 76 v Table of Contents 4.2.1 Side-Effects . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2.2 Abstraction . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2.3 Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2.4 Improvements . . . . . . . . . . . . . . . . . . . . . . . . 83 5 A Decision Procedure for Bit-Vector Arithmetic . . . . . . . . 87 5.1 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.1.1 Expression Simplification . . . . . . . . . . . . . . . . . . 89 5.1.2 Conjunctive Normal Form Encoder . . . . . . . . . . . . 90 5.1.3 Dynamic Conjunctive Normal Form Simplifier . . . . . . 91 5.1.4 Custom-Made SAT Solver . . . . . . . . . . . . . . . . . 93 5.2 Automatic Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.2.1 Manual Tuning . . . . . . . . . . . . . . . . . . . . . . . 94 5.2.2 Parameter Optimization by Local Search . . . . . . . . . 97 5.2.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . 99 5.2.4 Experimental Results . . . . . . . . . . . . . . . . . . . . 101 5.2.5 Best Parameter Configuration Found . . . . . . . . . . . 103 5.3 Improving Performance . . . . . . . . . . . . . . . . . . . . . . . 104 5.3.1 Novel Heuristics . . . . . . . . . . . . . . . . . . . . . . . 104 5.3.2 Prefetching . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6 Prototype and Experiments . . . . . . . . . . . . . . . . . . . . . 107 6.1 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 109 6.2.1 Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.2.2 Saturn’s Results . . . . . . . . . . . . . . . . . . . . . . . 112 6.2.3 Calysto’s Results . . . . . . . . . . . . . . . . . . . . . . 113 6.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 7.1 Extended Static Checking . . . . . . . . . . . . . . . . . . . . . . 119 7.2 Computation of Verification Conditions . . . . . . . . . . . . . . 121 vi Table of Contents 7.3 Interprocedural Analysis . . . . . . . . . . . . . . . . . . . . . . 124 7.4 Decision Procedures for Bit-Vector Arithmetic . . . . . . . . . . 127 7.5 Automatic Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . 130 8 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . 133 8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 8.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 vii List of Tables 3.1 Prerequisite Algorithms and Their Computational Complexities. 38 5.1 Summary of Automatic Tuning Results. . . . . . . . . . . . . . . 103 6.1 Benchmarks Used for Experiments . . . . . . . . . . . . . . . . . 110 6.2 Saturn NULL Pointer Dereference Checking Results. . . . . . . . . 114 6.3 Calysto NULL Pointer Dereference Checking Results. . . . . . . 115 6.4 Calysto Total Runtime Split. . . . . . . . . . . . . . . . . . . . 117 6.5 Saturn Total Runtime Split. . . . . . . . . . . . . . . . . . . . . . 118 viii List of Figures 2.1 Grammar of the F3OL. . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 Control Flow Graph Example and Parental Relations. . . . . . . 17 2.3 Traversal Orders. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Dominance Relations. . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5 Gating Path Expression Examples. . . . . . . . . . . . . . . . . . 26 3.1 The Grammar of a Simple Language. . . . . . . . . . . . . . . . . 34 3.2 Some Sources of Complexity in Symbolic Execution. . . . . . . . 37 3.3 Control Flow Graph and Dominator Tree Subgraphs Used in Ex- ample 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.4 Static Single Assignment of the Function in Example 4. . . . . . 56 3.5 Control Flow Graph and Dominator Tree Used in Example 4. . . 57 3.6 Maximally-Shared Graph. . . . . . . . . . . . . . . . . . . . . . . 61 4.1 Structural Abstraction Example. . . . . . . . . . . . . . . . . . . 73 5.1 Architecture of Spear. . . . . . . . . . . . . . . . . . . . . . . . 88 5.2 MiniSAT 2.0 vs. Manually Tuned Spear. . . . . . . . . . . . . . 96 5.3 Overall Improvements Achieved by Automatic Tuning of Spear. 102 6.1 High-Level Calysto Architecture. . . . . . . . . . . . . . . . . . 108 ix List of Symbols := Assignment operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 ≡ Equivalence of logical expressions . . . . . . . . . . . . . . . . . . . . . . 13 ::= Grammar production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 ∀ a : a Universal quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 ∃ a : a Existential quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 ∃! a : a Exists a unique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 〈 a , a〉 Pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 supp ( a) Support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 [ a ← a] Substitution operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12 [ a : a ← a] Parallel substitution operator. . . . . . . . . . . . . . . . . . . . . . . . . .12 B Boolean set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Zn Set of integers modulo n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14 ⇒ Implication. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14 Semantic entailment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 ite( a , a , a) Logical if-then-else operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 ITE ( a , a , a) If-then-else operator for terms . . . . . . . . . . . . . . . . . . . . . . . . . 15 B Set of basic blocks in a CFG . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Entry Function entry block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Exit Function exit block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 −→ Graph edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 ∗−→ Path of length zero or more . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 +−→ Path of length one or more . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Pred( a) Set of predecessors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Succ( a) Set of successors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 x List of Symbols Ancs( a) Set of ancestors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Ancestor relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Desc( a) Set of descendants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 PropDesc( a) Set of proper descendants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ≺ Proper ancestor relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 PropAncs( a) Set of proper ancestors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Pre[ a] Preorder number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Post [ a] Postorder number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Dominance relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18 Strict dominance relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 m Postdominance relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18 m Strict postdominance relation . . . . . . . . . . . . . . . . . . . . . . . . . 18 idom ( a) Immediate dominator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 ipostdom ( a) Immediate postdominator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 ncpostdom ( a , a) Nearest common postdominator . . . . . . . . . . . . . . . . . . . . . . . 19 DF ( a) Dominance frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 DF ( a)+ Iterated dominance frontier. . . . . . . . . . . . . . . . . . . . . . . . . . . .19 α( a , a) Inverse of Ackermann’s function . . . . . . . . . . . . . . . . . . . . . . . 20 dtl [ a] Dominator tree level. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21 γ ( a , a , a) Gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 GP ( a) Gating path expression of a block . . . . . . . . . . . . . . . . . . . . . 25 γ ( a , a , a) Gating path expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 ∗ Dereference operator and multiplication . . . . . . . . . . . . . . . 35 Λ Placeholder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 γ ( a)| a Gating path expression restriction operator . . . . . . . . . . . . 43 γ| aa Gamma function restriction operator . . . . . . . . . . . . . . . . . . 45 abelem ( a) Absorptive element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 val ( a) Value assigned by a decision procedure . . . . . . . . . . . . . . . . 82 xi Abbreviations AST Abstract Syntax Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 BDD Binary Decision Diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 BMC Bounded Model Checking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 CFG Control Flow Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 CG Call Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 CNF Conjunctive Normal Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14 CPU Central Processing Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 CSE Common Subexpression Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 DF Dominance Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 DT Dominator Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 F3OL Finite Fragment of First-Order Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 FOL First-Order Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 FSM Finite State Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 GCC GNU C compiler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 GSA Gated Single Assignment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23 IDF Iterated Dominance Frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 ILS Iterated Local Search . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 KLOC thousand lines of code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 LLVM Low Level Virtual Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 MLOC million lines of code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 xii Abbreviations PDT Postdominator Tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 QBF Quantified Boolean Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 RAM Random Access Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 RISC Reduced Instruction Set Computer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 RTL Register Transfer Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 SAT Boolean satisfiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 SMT Satisfiability Modulo Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 SSA Static Single Assignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 SWV Software Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 VC Verification Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 xiii List of Algorithms 1 Symbolic Execution Algorithm. . . . . . . . . . . . . . . . . . . . 51 2 Merging Abstract Memory Locations at Dominance Frontiers. . . 52 3 Main Abstraction-Checking-Refinement Loop. . . . . . . . . . . . 80 4 Structural Refinement Algorithm. . . . . . . . . . . . . . . . . . . 82 5 Additions to the Basic Refinement Algorithm. . . . . . . . . . . . 83 xiv Acknowledgments I have been really fortunate to have Alan J. Hu as my Ph.D. advisor. Most of what I learned about how research is done and how the research community works can be attributed to him. His insightful comments and advice immensely improved this thesis. I am extremely grateful to my committee members, Tom Ball and Gail Murphy, for their insightful suggestions for improving the content of this the- sis. Encouraging me to do research on software verification, Tom has played a crucial role in the formulation of my thesis topic. His enthusiasm, wisdom, and knowledge have been a great source of inspiration for me. I also very much appreciate the directness and shrewdness of the career advice he has given me. While I was a research intern at Microsoft Research (MSR), Rustan Leino gave me an unforgettable crash-course in extended static checking, which has impacted much of my later work. Sriram Rajamani provided invaluable sug- gestions and wise feedback on many of my early ideas for my thesis topic, both while I was at MSR and later at the Lipari Summer School. I can freely say that Sriram’s comments saved me months (if not years!). To Madan Musuvathi, I owe my interest in bit-vector arithmetic — his genuine enthusiasm for research is simply contagious. Byron Cook’s interest in my work and his encouragement have been a constant source of inspiration. My internship at MSR created a number of long-lasting personal connections, which seem to multiply over time. Leonardo de Moura and Nikolaj Bjørner became close friends and their constructive criticism was often a guideline in deciding what constitutes a publishable result and what doesn’t. Their incred- ible work on decision procedures really sets the research standard that I strive xv Acknowledgments to achieve. Zvonimir Rakamarić, a close friend and an honest critic, has spent countless hours listening my practice talks. Our discussions and his invaluable feedback have had significant influence on my research. All in all, I have been very fortunate to benefit from his knowledge and intuition. I would also like to thank Jesse Bingham, Shaun Xiushan Feng, Naghmeh Ghafari, and Flavio de Paula for their support, friendship, and valuable feedback on my research. I benefited a lot from a close collaboration with Holger Hoos and Frank Hutter. It has been a joy to work and discuss research with them. Some of the material presented in this thesis is a joint work with the two of them. Under the influence of Holger’s legendary time organization and efficiency, I became a better and more efficient researcher. Many other people influenced my research and/or the presentation of the material in this thesis. Just to mention a few. . . The discussions I had with Aarti Gupta at various conferences kept reassuring me that I was on the right track. Ed Clarke’s excitement and curiosity about my research, as well as the discussions I had with him and his group during my visit to Carnegie Mellon University, were extremely rewarding and motivating. I am also grateful to Bob Brayton, David Dill, Arie Gurfinkel, Sava Krstić, Alan Mishchenko, Ofer Strichman, and Karen Yorav for the career advice, encouragement, and research feedback that they provided me on multiple occasions. My research has been supported by graduate fellowships from Microsoft Research and the University of British Columbia. The Natural Sciences and Engineering Research Council of Canada also supported my research indirectly (through my advisor). I am deeply grateful to all three organizations. Finally, I would like to thank my wife for putting up with me on a long journey through all my academic degrees. All the way, she has been a reliable source of encouragement, support, and motivation. xvi Chapter 1 Introduction 1.1 Motivation Error removal via verification and testing is one of the most time-consuming parts of the software development life cycle — for instance, the Windows XP testing team is larger than the development team [95]. Accordingly, an enormous range of techniques have been developed to support this task. Although existing techniques offer different scalability-precision-automation tradeoffs, program- mers most often rely only upon debuggers, compiler type-checking, and ad-hoc testing. Techniques based on formal underpinnings are rarely used, especially those requiring a non-trivial amount of manual effort. The core of the problem seems to be in the inadequate balance between scalability and precision of automatic software analysis techniques. The basic hypothesis that motivated this thesis was that a highly scalable (to several hun- dred thousand lines of code) interprocedural analysis can be conceived, without sacrificing precision, by exploiting the structure of the program code and cus- tomization of the entire tool chain. Experimental results (Chapter 6) not only confirm the hypothesis, but also suggest that with more research, the approach proposed in this thesis could be pushed further, perhaps even up to applications that have several million lines of code. The two main ideas running through the thesis are exploitation of struc- ture and customization. The motivation to exploit the structure of the problem came from my own observations of how programmers think about program code and how they manage to get large applications, which defy automatic analy- sis, almost correct. In different chapters, structure will represent a different set 1 Chapter 1. Introduction of properties of the problem being solved. In particular: symbolic execution (Chapter 3) preserves the original dataflow dependencies among variables and memory locations in the analyzed program and thus reflects the structure of the original code in the computed symbolic expressions; structural abstraction (Chapter 4) exploits those dependencies for precise, yet scalable, interprocedu- ral analysis; and automatic tuning of decision procedures (Chapter 5) exploits the fine-grained properties of the code being checked. The motivation to cus- tomize the entire tool chain came from my previous experience with decision procedures — it is much harder to design a decision procedure that will perform well on a wide range of problems than a decision procedure that will perform exceptionally well on a specific problem. Design of decision procedures is a complex task that requires intimate knowledge of the problem being solved, un- derstanding of a very wide range of algorithms, and engineering skills. So, why make that task even harder by requiring superb performance on a very broad set of problems? The main barrier that prevented this customization was the simple fact that there are many more classes of problems than there are skilled people that could design highly customized decision procedures. In Chapter 5, I show that a combination of light-weight manual customization and smart auto- matic customization can be very effective, giving up to two orders of magnitude speedup. Taken altogether, exploitation of structure and customization applied at all levels of software analysis give an unprecedented combination of precision and scalability. 1.2 Research Context This section places my research in context of previous work (a much more de- tailed survey of the related work is given in Chapter 7). More precisely, the section discusses the role of abstraction and structure in software analysis (Sec- tion 1.2.1), situates my research in between two of the most important tech- niques for formal software analysis: model checking and static analysis (Section 1.2.2), and concludes by discussing the scalability-precision tradeoff, which is 2 Chapter 1. Introduction the key tradeoff of any verification technique based on formal underpinnings. By verification in this thesis, I mean checking that the program conforms to the specification provided in terms of asserted predicates (assertions) in code. Such assertions can be either written by programmers or inserted automatically. 1.2.1 Abstraction and Structure Despite the increasing popularity of formal methods and significant progress made in the last two decades, the scalability of software formal verification and bug-finding tools is inadequate. The rule of the thumb for production compil- ers is that all algorithms used for code transformations and optimizations must be of almost linear complexity. Higher complexity algorithms are allowed only rarely and only for problems that are known to be localized and of small size in practice. In contrast, formal verification methods are notoriously slow. For instance, model checking [33] is linear with the size of the state space (e.g., [113]), which is exponential (or worse). Lighter-weight methods, like static checking, offer a broader range of precision-performance tradeoffs, although, in general, static analysis is known to be undecidable [85]. This negative result has a far more serious impact on verification than optimization. Sound and very imprecise analyses are often good enough for optimization, but produce an unacceptably high percentage of false positives if used for verification. More precise approaches typically do not even have polynomial computational com- plexity. Steadily increasing computing power certainly helps, but let us not forget that a 1000-fold increase in computing power is not that much if we are dealing with problems of exponential complexity. It is hard to ignore such a huge complexity gap. However, programmers seem to be able to deal somehow with that complexity and write more-or-less usable code. How do they do that? Although no one has yet given a clear answer, it is obvious that abstraction and the structure of the code play an important role. Abstraction has received well-deserved attention, and a large number of pa- pers have been published on the topic. A seminal work on abstraction in the 3 Chapter 1. Introduction context of program analysis was done by Cousot and Cousot [39]. The abstract interpretation framework they invented forms a theoretical basis of essentially all software model checkers [112] and static checkers. Code structure, on the other hand, has received much less attention. In particular, most research on exploiting structure has been focused on function summaries. Reps et al. [109] proposed summarization of functions, as natural abstraction boundaries, and reuse of summaries in different calling contexts. The SLAM software model checker [13] uses Binary Decision Diagrams (BDDs) [24] for the representation of function summaries. The Saturn static checker [135] also uses BDDs for summaries, but in a slightly different way. Summaries in Saturn represent the effect of the function on a single property that is being verified, rather than the predicate abstraction of the effect of the function on a set of variables related to the checked property. Recently, Conway et al. [38] have shown that summaries can be stored and reused for incremental verification of large bodies of code. The focus of this thesis is on improving the scalability of software formal verification by exploiting the structure present in code. The presented work addresses two main bottlenecks: interprocedural path-sensitivity and decision procedures for proving the validity of logical formulas constructed from checked software and given properties (also known as Verification Conditions (VCs)). I show how the code structure can be used to mitigate the computational costs of those two bottlenecks. 1.2.2 Model Checking and Static Checking A model checker is an automatic tool that does more or less brute force ex- ploration of the state space (e.g., [33]). Arguably the most famous software model checkers are SLAM [13] and SPIN [76], which have both been applied to commercial code bases. SLAM is a symbolic model checker designed for the ver- ification of Windows operating system device drivers, which typically have up to 10 thousand lines of code (KLOC). SPIN is a model checker for verification 4 Chapter 1. Introduction of distributed software systems. SLAM verifies a finite abstraction, while SPIN verifies a finite description of the system in the Promela language. The finite approximation of infinite state software systems doesn’t seem to be a serious hurdle for applicability of SLAM and SPIN — SLAM has found its way to driver developers by becoming a part of Microsoft Development Studio, and SPIN seems to be widely accepted too. The static checker Calysto2, which I developed for the experimental evaluation of the innovations presented in this thesis, continues this tradition, and focuses on checking finite approximations of software. Static checkers trade imprecision for scalability. A number of systems have been presented and used commercially. Astrée [22, 41] has been used for veri- fication of some parts of Boeing’s airplane control software. According to the reported results, Astrée scales up to 400 KLOC, but requires some code anno- tations from the programmer and a fairly high level of expertise. The systems verified by Astrée contain no goto statements, no dynamic memory alloca- tion, no recursive calls, no recursive data structures, and no pointer arithmetic. These properties drastically simplify the requirements on the static checker. In contrast, Calysto is designed to be applicable to general-purpose software. MC [57], on the other side of the spectrum, is a relatively imprecise, but very scalable (up to 4.7 million lines of code (MLOC)) static checker. The high rate of false positives is somewhat remedied by a bug probability rating that uses a number of heuristics to try to detect the most likely and the most critical bugs. Calysto, on the other hand, strives to eliminate as many false positives as possible through a very precise analysis. Static checkers have some advantages over model checkers, the most impor- tant one being scalability. Static checkers are able to go over larger code bases and therefore usually find more bugs. Soundness and a proof of correctness 2The name is an intentionally misspelled version of Callisto, who is a nymph in Greek mythology. Callisto was a hunting companion of the goddess of the hunt — Artemis. The focus of the work in this thesis is bug finding, rather than the computation of proofs of correctness. Accordingly, Calysto is a bug hunting companion. 5 Chapter 1. Introduction that the model checkers can give (as opposed to static checkers which are often unsound and cannot give a proof of correctness) seem to be of little value in mainstream software development where the optimization function is to cram as much functionality into an application as cheaply, as quickly, and as correctly as possible. As the emphasis of the presented work is scalable software verification, static checking was a natural path to choose. More precisely, the work focuses on ex- tended static checking [50], which is a combination of static checking and deci- sion procedures targeted at achieving a practical balance between the precision of model checking and scalability of static checking. 1.2.3 Scalability and Precision The focus of this thesis is on a scalable, yet precise, combination of static check- ing with automatic theorem proving for checking properties of general-purpose software. The presented approach is essentially language-independent, but tar- geted towards imperative procedural languages. The proposed analysis is general enough to handle user-provided assertions. The assertions are essentially predicates expressible in the programming lan- guage in which the checked application is written and can be either provided by the programmer or automatically inserted in the code. In order to achieve a high level of precision (and therefore a low false-positive rate), the analysis is field-, path-, and context-sensitive. The computational complexities of the last two properties are fairly high (even intraprocedural path- sensitivity for Boolean programs is P-space complete, while context-sensitivity is exponential3). In addition, path-sensitive analyses require some sort of a decision procedure for checking feasibility of paths. The decision procedures suitable for precise software verification are computationally expensive. The path-sensitivity, context-sensitivity, and decision procedures are the 3 k-CFA (Control Flow Analysis), where k is the maximum call depth in the original call graph that the analysis tracks has exponential complexity for k > 0 [81], [101, pages 187–197]. 6 Chapter 1. Introduction main performance bottlenecks for precise static analyses. The work presented in this thesis takes a holistic approach, considers their interactions, and pro- poses innovative solutions based on exploitation of the structure present in the software: Path-Sensitivity. Chapter 3 shows how local structure in the Control Flow Graphs (CFGs) can be used to generate path-sensitive summaries and VCs in linear time and space. Context-Sensitivity. Chapter 4 proposes structural abstraction — an abstrac- tion for achieving interprocedural path-sensitivity and context-sensitivity. Structural abstraction is based on symbolic relational summaries [40], and exploits the structure of software at the function level. The chapter also discusses the tradeoffs between the precision of summaries and the cost of interprocedural analysis. Decision Procedure. Chapter 5 focuses on a bit-vector arithmetic decision procedure designed especially for checking the validity of VCs. Software VCs have specific low-level structural properties that can be exploited by customizing the decision procedure. Such customization speeds up the main analysis by two orders of magnitude. To summarize, the thesis demonstrates how to exploit the structure present in: 1. the CFG — for efficient generation of compact verification conditions, 2. the Call Graph (CG) — for mitigating the cost of context-sensitivity and achieving interprocedural path-sensitivity, and 3. within the VCs themselves — for efficient path-sensitivity, and scalable validity checking. 7 Chapter 1. Introduction 1.3 Contributions This section outlines the main contributions of the thesis. The contributions are mostly based on exploitation of the structure of checked programs at various levels of granularity. Chapter 3: Structure-Preserving Symbolic Execution. One of the main contributions of this thesis is a novel structure-preserving symbolic execution, which maintains the program dataflow dependencies (structure) in the com- puted intraprocedural summaries. That structure is later exploited during the interprocedural analysis. The proposed symbolic execution algorithm computes fully path-sensitive summaries for each side-effect4 of each function. Besides summaries, the symbolic execution algorithm also computes partial VCs, which are assembled into fully interprocedurally path-sensitive VCs later during the interprocedural analysis. The highlights of this novel intraprocedural symbolic execution algorithm are the following: • The data-flow dependencies in the original program are reflected in the computed function summaries and VCs. Structural abstraction (Chapter 4) exploits this structure for selecting the parts of the formula that need to be refined to refute spurious counterexamples. • The computed symbolic expressions are compact and immutable. Im- mutability, in practice, means faster symbolic execution that is less memory- demanding. • The algorithm exploits the structure of the CFG — more precisely the (post)dominance relation — to avoid expensive recursive memory lookups. Chapter 4: Structural Abstraction and Refinement. Scalable inter- procedural path- and context-sensitive analysis has been a long-standing open 4In this thesis, both the returned value and modifications of the caller’s state will be considered as side-effects. 8 Chapter 1. Introduction problem. Structural abstraction, one of the main contributions of this thesis, is an effective and novel solution. The proposed interprocedural analysis uses a decision procedure to lazily assemble function summaries into formulas that represent the checked assertion predicate and the conditions under which that assertion is reachable (i.e., global control-flow context). This assembly is done in a lazy manner through structural abstraction and refinement, which both exploit the dataflow dependencies (preserved by the above mentioned symbolic execution) in the original code. The most important aspects of the proposed approach are: • Structural abstraction is the first completely incremental abstraction frame- work for interprocedural analysis — when analyzing a single VC, the de- cision procedure does not need to repeat the same work. • Since the proposed symbolic execution computes summaries for each side- effect, structural refinement can be more precise and, instead of inlining the whole function call, can inline only the needed side-effect. • While abstraction at function interfaces has been done in various forms be- fore (e.g., [126]), structural refinement is, to the best of my knowledge, the first proposed refinement that exploits the dataflow dependencies (struc- ture) in code for refining abstracted VCs. Structural refinement is computationally cheaper than other previously proposed alternatives. More precisely, structural refinement identifies missing subexpressions in time that is linear with the size of the abstract counterexample, while predicate abstraction, for instance, has to solve an NP-complete problem to compute new predicates for refinement. • The core insight of structural abstraction is that decision procedures can be used to effectively perform precise interprocedural analysis. While this insight was exploited before for interprocedural pointer alias analysis [132] (BDDs can also be seen as a decision procedure), it hasn’t been exploited for analysis of program functionality. 9 Chapter 1. Introduction Chapter 5: Customized Bit-Vector Arithmetic Decision Procedure. Proving the logical validity of interprocedurally path-sensitive and bit-precise VCs is computationally very challenging, and requires a high-performance cus- tomized decision procedure. Chapter 5 presents my custom-made decision pro- cedure for VC-solving, named Spear.5 I achieved customization through the design of the decision procedure, simplifications that I built in, and through au- tomatic tuning. To the best of my knowledge, no prior work has been published on automatic tuning of decision procedures. Another important contribution of my work on Spear is a practical proof that a decision procedure based on a Boolean satisfiability (SAT) solver can be highly effective for solving bit-vector arithmetic problems.6 Chapter 6: Prototype and Experimental Evaluation. From a practi- cal perspective, one contribution of this thesis, especially important for readers who wish to implement structure-preserving symbolic execution and structural abstraction, is the description of Calysto’s architecture in Chapter 6. The chapter also provides valuable experimental results that show that the combi- nation of the techniques in Calysto and Spear is highly effective in finding bugs on large real-world applications. 1.4 Organization of the Thesis The background material required for understanding the thesis is presented in Chapter 2. Chapter 3 presents the intraprocedural structure-preserving sym- bolic execution for computation of function summaries and VCs. The presented symbolic execution constructs summaries and VCs in the form of maximally- shared graphs, which preserve the data-flow dependencies in the original code. Chapter 4 presents structural abstraction, which uses a decision procedure to perform interprocedural VC-checking, abstracting away irrelevant side-effects 5Callisto, a hunting companion of the goddess of the hunt, uses spears for hunting. 6Spear won the bit-vector division of the Satisfiability Modulo Theories (SMT) 2007 com- petition. 10 Chapter 1. Introduction of function calls. The work presented in Chapter 4 appeared in [8]. Chapter 5 dives into the details of Spear— my bit-vector decision procedure for soft- ware verification. The material presented in that chapter is rooted in several of my publications on decision procedures [6, 10, 11, 78], but focuses on my in- sights, conclusions, and results that are relevant to software analysis. Chapter 6 is based on [9] and presents a prototype implementation — the Calysto ex- tended static checker — of the analyses presented in the previous chapters. The same chapter also gives experimental results obtained by checking a number of real-world applications. Related work is surveyed in Chapter 7. Finally, the last chapter gives conclusions and discusses future work. 11 Chapter 2 Background This chapter introduces a number of concepts that will be routinely used later. Section 2.1 introduces less standard notations that will be used throughout the thesis. Section 2.2 introduces VCs and the underlying logic. Section 2.3 pro- vides some background on program analysis and compilers. The final section overviews the Low Level Virtual Machine (LLVM) compilation framework used for implementation of the Calysto static checker. The section on LLVM fo- cuses on the details of LLVM that are of special importance for verification. 2.1 Notational Conventions Throughout this thesis, the conditional selection if-then-else operator will be denoted by ITE . More formally: Var = ITE (Ψ,TermT ,TermF ) ≡ (Ψ⇒ Var = TermT ) ∧ (¬Ψ⇒ Var = TermF ) Some other notation used throughout the thesis includes the following. Fre- quently, two objects appear together for logical or practical reasons. Pairs will be denoted by 〈a, b〉. The finite set of variable names in a logical expression ψ is denoted by supp (ψ). For example, supp (x · y + z > 0) = {x, y, z}. The oper- ator φ [a← b] is the substitution operator that substitutes all free instances of a in φ with b. Parallel substitution will be denoted by [range : x← y] — every element x for which the range predicate evaluates to true is substituted with y. The parallel substitution operator performs all substitutions at once. Hence, if 12 Chapter 2. Background y contains x, the substitution will be performed only once. Equality for terms is denoted by =, equality for logical expressions as ≡, the assignment operator as :=, and the grammar production as ::=. Existential quantification will be denoted by ∃; universal, by ∀; and ∃! will stand for “exists a unique.” 2.2 Verification Conditions This section begins by introducing the notion of Verification Conditions (VCs), and proceeds by describing the underlying logic that will be used in this thesis for reasoning about VCs. An example at the end of the section shows a VC constructed from a simple piece of code. VCs are logical formulas, constructed from a system and desired correct- ness properties, such that the validity7 of VCs corresponds to the correctness of the system. Commonly, correctness properties in programs are specified with assertions, which can be either written by programmers, or automatically gen- erated (e.g., [15]). If all the assertions in the program are valid, the program is considered to be correct with respect to the given set of assertions. The VCs in this thesis are quantifier-free First-Order Logic (FOL) formulas over finite structures (Fig. 2.1), namely bit-vectors. The used logic enables us to precisely model both hardware and software, which commonly use bit- vectors to represent data, including boundary behaviors (over- and under-flows), bit-wise operators (AND, XOR,. . . ), and non-linear operators (multiplication, division,. . . ). The finite nature of the logic also makes it decidable (whereas FOL is in general undecidable [30]). More precisely, checking the validity of a formula in the given logic is NP-complete [103]. Any formula in Finite Fragment of First-Order Logic (F3OL) can be repre- sented as a Boolean logic circuit, because Var (resp. Constant) represents finite 7 A logical formula φ is said to be satisfiable if there exists a partial or complete assignment to its variables, such that the formula is true. Otherwise, it is said to be unsatisfiable. A logical formula φ is valid if and only if ¬φ is unsatisfiable, written as φ. 13 Chapter 2. Background Formula ::= ¬Formula | (Formula) | Atom | Formula ⇒ Formula | Formula ∧ Formula | Formula ∨ Formula | Formula ≡ Formula | ite(Formula,Formula,Formula) Atom ::= BoolConstant | BoolVar | Term Rel Term Term ::= Var | Constant | UnaryOp Term | Term BinaryOp Term | ITE (Formula,Term,Term) Rel ::= = | 6= | <u | ≤u | >u | ≥u | <s | ≤s | >s | ≥s UnaryOp ::= ¬ | ZeroExtend | SignExtend | Truncate BinaryOp ::= + | − | ∗ | ÷u | ÷s | Remu | Rems | ∧ | ∨ | Shl | Ashr | Lshr Figure 2.1: Grammar of the F3OL. integers (bit-vectors) from the set Zn8 and BoolVar (resp. BoolConstant) rep- resents Boolean variables (of type B). Such a circuit can be translated into a Conjunctive Normal Form (CNF) formula over Boolean variables by Tseitin’s transform [128] in linear time and by introducing at most a linear number (lin- ear with the size of the circuit) of fresh variables. Such a translation is usually called bit-blasting. The formulas from F3OL can be constructed from the grammar in Fig. 2.1. All the operators are pretty much standard. The operator ¬ (resp. ∧, ∨) is overloaded to represent both logical and bit-wise NOT (resp. AND, OR). The left side of an implication α⇒ β is known as the antecedent (α), while the right side is called the consequent (β). Equality of formulas is denoted by ≡. Unary relations include zero (resp. sign) extension, which extends a bit-vector by zero (resp. its most significant bit). Truncation discards a number of most significant bits so that the operand fits into the type of the result. Binary relations over terms are standard equality (=), disequality (6=), as well as unsigned and signed comparisons. Binary operators include standard addition (+), subtraction (−), multiplication (∗), unsigned and signed division (÷u and ÷s), unsigned and signed remainder (Remu and Rems), bit-wise AND (resp. OR), and shifts (left, 8The set of integers modulo n. 14 Chapter 2. Background arithmetic right, logical right). Two versions of the if-then-else operators will be distinguished: The ite(Ψc,Ψa,Ψb) operator for formulas is syntactic sugar for (Ψc ∧Ψa)∨ (¬Ψc ∧Ψb), whereas the second is the operator for terms: ITE (Ψ,Terma,Termb), is equal to Terma if Ψ ≡ true, and Termb otherwise. Example 1: The following example illustrates a VC computed from some code and a checked property. Let us assume that we want to check the validity of the asserted condition on the last line of: x0 = a + b ; i f (x0 < 0) { x1 = −y0 ; y1 = y0 ; } else { x1 = x0 ; y1 = y0 + x0 ; } assert (0 <= y1 ) ; Using either symbolic execution [82] or the weakest precondition predicate trans- former [52], one can compute the following VC: 0 ≤ ITE (a+ b < 0, y0, y0 + a+ b) The computed verification condition is not valid. One possible falsifying assign- ment is a = 0, b = −1, y0 = −1. Thus, the asserted condition does not hold and either the program or the assertion has to be fixed. 2.3 Program Analysis Many concepts that originated in early work on optimizing compilers are use- ful in software verification. After introducing the basic concepts, this section proceeds with some elementary data structures used for software analysis and 15 Chapter 2. Background transformations. Section 2.3.2 describes the commonly used static single assign- ment forms that have played a major role in the development of compilers and also serve as basic data structures used in this thesis. The subsequent section introduces the problem of aliasing in software analysis. The last section reviews several approaches to interprocedural analysis by summarization. 2.3.1 Basic Definitions Control Flow Graph (CFG) and Related Concepts. A basic block con- sists of a sequence of program statements and a single branch or return state- ment at the end. The flow of control enters the block only at its beginning and leaves it by execution of the branch or return at its end. A basic block B can be split into several basic blocks B0, . . . , Bn connected with unconditional branch statements. Above the basic-block level, functions are the next higher organizational unit. Each function can be represented with a single connected, directed graph G = (N,E, Entry, Exit), called the Control Flow Graph. A function consists of a set of basic blocks B, which are represented by the nodes N in the graph. If there is a branch from some basic block B1 to B2, those two are connected by an edge in the CFG. The set of edges is denoted by E. The start node Entry ∈ N has no incoming edges, while the end node Exit ∈ N has no outgoing edges. An example of a CFG is given in Fig.2.2. It will be assumed that G has a single Exit node. This is not a serious constraint — all return nodes can be easily unified into a single node. We will assume that G contains no blocks unreachable from Entry. Unreachable blocks can never be executed and therefore can be removed without having any effect on the outcome of the execution of the function. A point of definition of a variable will be known simply as definition, and the points at which it is read will be known as uses. If there is an edge connecting two nodes, x −→ y ∈ E, then x and y are called the source and the destination of the edge. A path of length n is a sequence of edges x0 −→ x1 −→ · · · −→ xn, where xi −→ xi+1 ∈ E. If x0 = xn, the path 16 Chapter 2. Background Entry B1 B2B3 B4 B5 B6 Exit Figure 2.2: CFG Example and Parental Relations. Some examples of the parental relations are: Pred(B4) = {B2, B3} (predecessors), Succ(B4) = {B5, B6} (successors), Ancs(B3) = {Entry, B1, B3} (ancestors), PropAncs(B3) = {Entry, B1} (proper ancestors). is a cycle. In the context of the analysis presented in later chapters, CFGs are going to be acyclic. Section 3.1.3 on page 38 explains how the analysis can be extended to cyclic CFGs. Paths of length zero or more are denoted by x ∗−→ y and of length one or more as x +−→ y. Given two basic blocks B1, B2 such that B1 −→ B2, we say that B1 is a predecessor of B2 (B1 ∈ Pred(B2)) and B2 is a successor of B1 (denoted B2 ∈ Succ(B1)). If B1 ∗−→ B2, then B1 is an ancestor of B2 (B1 ∈ Ancs(B2) or B1 B2) and B2 is a descendant of B1 (denoted B2 ∈ Desc(B1)). If the path is of length one or more, we qualify the parental relation with the word “proper”, written as B1 ∈ PropDesc(B2) or B1 ≺ B2 (resp. B2 ∈ PropAncs(B1)). Two basic blocks B1 and B2, such that B1 6 B2 ∧ B2 6 B1, are called siblings. A basic block with more than one successor is called a branch node, 17 Chapter 2. Background and a node with more than one predecessor is called a join node. Basic blocks will be frequently identified by labels (for example, ReturnLabel), and the two notations will be used interchangeably. Two basic types of traversal, preorder and postorder, are frequently used in the analysis of CFGs. Both types of traversal can be defined in terms of a depth-first search that marks each newly discovered node as visited (avoiding revisiting the already marked nodes) and assigns a unique consecutive num- ber to each node. The preorder (resp. postorder) traversal assigns numbers to nodes before (resp. after) recursing on their unmarked descendants. The preorder (resp. postorder) sequence number of a node B will be denoted by Pre[B ] (resp. Post [B ]). By using preorder and postorder sequence numbers, ancestry and descendancy queries in a CFG can be answered in constant time [72]. Let B0, B1, . . . , Bn be a sequence of nodes in the postorder traversal. The reverse-postorder traversal is defined as the inverted sequence Bn, . . . , B1, B0. An important property of reverse-postorder is that all the predecessors of a join node are visited before the node itself. This property will be used later for computation of VCs. A program is considered to be structured if it is composed only of so called structured constructs, like if-then-else, do-while, and statement sequences. The CFG of a structured program has the following property: every subgraph that represents an if-then-else or a loop construct has a single point of entry. Furthermore, all loops in a structured program are strictly nested. Dominance Relations. A node Bi dominates node Bj if and only if all the paths from the Entry node to Bj go through Bi, written as BiBj . If Bi 6= Bj , Bi strictly dominates Bj , denoted by Bi Bj . A node Bj postdominates node Bi if and only if all the paths from Bi to the Exit node go through Bj , written as BjmBi.9 If Bi 6= Bj , Bj strictly postdominates Bi, denoted by Bj mBi. 9The pointed side of the postdominance relation symbol points to the ancestor in B. The reverse holds for the dominance relation symbol. It seemed natural to turn the pointed side of m symbol to the basic block that is being postdominated. Essentially, the postdominance 18 Chapter 2. Background 1 2 83 4 5 7 6 (a) Preorder. 8 6 75 4 2 3 1 (b) Postorder. 1 3 24 5 6 7 8 (c) Rev. Postorder. Figure 2.3: Traversal Orders. A node Bi is an immediate dominator of Bj (idom (Bj)) if Bi Bj and ¬∃Bk ∈ B : Bi Bk Bj . A nearest common dominator of two nodes, ncdom (Bi, Bj), is a node Bk such that Bk Bi ∧ Bk Bj ∧ ¬∃Bh ∈ B : (Bh Bi ∧Bh Bj ∧Bk Bh). An immediate postdominator and a nearest common postdominator are denoted by ipostdom (Bi) and ncpostdom (Bi, Bj) respectively. The Dominance Frontier (DF) of a node B, denoted by DF (B), is a set of nodes {Bx | ∃Bi ∈ Pred(Bx) : B Bi ∧ B 6 Bx}. The DF naturally extends to a set S ⊆ B of basic blocks: DF (S) = ∪Bi∈SDF (Bi). The Iterated Dominance Frontier (IDF), denoted by DF (S)+, is defined as the limit of the increasing sequence of sets of nodes: DF (S)i+1 = DF ( S ∪DF (S)i ) relation is a dual of the dominance relation. If B1mB2 in the direct graph, B1B2 in the reverse graph. 19 Chapter 2. Background Entry B1 B2B3 B4 B5 B6 Exit (a) CFG. Entry B1 B2 B4 B3 B5 B6 Exit (b) DT. Exit B5 B6 B4 B3 B2 Entry B1 (c) PDT. Figure 2.4: Dominance Relations. Some examples of the dominance relations are: Entry B4 (dominance), idom (Exit) = B4 (immediate dominator), B4m B2 (postdominance), ncdom (B5, B2) = Entry (nearest common dominator), DF (B1) = {B4} (dominance frontier). Basic dominance relations are illustrated in Fig. 2.4. Dominator Tree (DT). The dominance relation [104] is a partial order (re- flexive, antisymmetric, and transitive). Furthermore, the set of dominators of a node are linearly ordered by the dominance relation [102]. Thus, the domi- nance relation can be represented by a tree, called the Dominator Tree (resp. Postdominator Tree (PDT) for the postdominance relation). Dominator trees can be computed in O(E · α(E,N)) time [91], where α is the extremely slowly growing inverse of Ackermann’s function. An example of a DT is given in Fig. 2.4(b), and an example of a PDT is given in Fig. 2.4(c). 20 Chapter 2. Background The dominator tree level of a basic blockB, denoted by dtl [B ], is the breadth- first-search level of the corresponding node in the DT. The root node of the dominator tree has level 1, its children 2, and so on. Call Graph (CG). Programs are composed of one or more functions. A call graph is a directed graph Gf = (Nf , Ef ), such that each node f ∈ Nf represents a single function and each edge represents a call. Let fi −→ fj ∈ Ef , then fi is the caller, and fj the callee. Call graphs might be cyclic and can have multiple roots. For languages that support function pointers, a simple pointer analysis can be used for call graph construction. It has been shown that even a very simple field-sensitive version of Steensgaard’s [123] points-to analysis works very well for call graph construction [96]. For data memory locations, on the contrary, cheap analyses tend to be very imprecise, implying that manipulations of data pointers are much more complex than manipulations of function pointers in practice. Abstract Syntax Tree (AST). Although not directly used in the work pre- sented in this thesis, ASTs have been commonly used as a basic code represen- tation in several related works, as discussed in Chapter 7. An AST is a data structure that represents a parsed program. Each node represents an operator or statement and its children represent the operands [4]. Code transformations and analyses are commonly performed on simpler lower-level representations, like CFGs and Static Single Assignment (SSA), which are also more language- independent than ASTs. 2.3.2 Single Assignment Forms Single assignment forms are intermediate representations used in compilers for more efficient optimizations and transformations of the program code. A number of single assignment forms have been proposed. Their common property is that local variables have a single point of definition. This is achieved by introducing a fresh variable name for each definition and propagating the name to the uses 21 Chapter 2. Background dominated by the definition. Merges of values are handled through special functions, which in most cases appear at the beginning of each join block. Static Single Assignment (SSA). The SSA form [110] is a directed flow graph in which each definition defines a distinct name and each use refers to a single definition. In addition, each definition dominates all uses. At join points, the names are merged by the use of φ-functions, which take as arguments a number of definitions. One of the definitions is chosen depending on the predecessor node from which the flow of control enters the join node. However, the φ-functions have no special arguments that signal where the flow of control came from. So, strictly speaking, φ-functions are not really functions, but a non- deterministic choice operator. If the exact definition matters, the analysis that relies on the φ-functions has to track where the flow of control came from, and pick the appropriate definition. All φ functions are considered to be executed instantaneously in parallel at the beginning of the block. The first efficient (quadratic worst case, linear in practice) algorithm for computing SSA was proposed by Cytron et al. [43]. Sreedhar and Gao have more recently proposed a simple and elegant linear algorithm [122]. SSA handles pointers and globals in a slightly unexpected way. All globals are accessed through constant pointers, while all other pointers are handled as local variables. For instance: int ∗a = malloc ( . . . ) ; int ∗b = malloc ( . . . ) ; int ∗ptr ; i f (x ) { ptr = a ; } else { ptr = b ; } is in SSA represented as ptr = φ(a, b). This is nowadays a common approach, 22 Chapter 2. Background which has evolved from the Hashed SSA form proposed by Chow et al. [29]. The approach is commonly found in modern compilers, like the GNU C compiler (GCC) and the LLVM compilation framework [86]. Because of this feature, all pointers (not the abstract memory locations to which they may point!) have unique points of definition. At this point, I introduce the concept of a terminal pointer: Definition 1 (Terminal Pointer). A terminal pointer is a constant pointer to a global, a formal pointer parameter, or a pointer initialized with some memory allocation function (like alloca(), malloc(), realloc(),...). For instance, the symbolic definition of the ptr pointer in the above given example would be ITE (x, a, b), where a and b are terminal pointers. Note that the definition of ptr would be the same if int *ptr; were replaced with the int *ptr = malloc(...); statement. The intraprocedural symbolic execution algorithm for function summariza- tion presented in Chapter 3 defines all pointers in terms of terminal pointers, by performing definition table lookups, and always replacing uses with definitions. The replacement is simplified by the unique point of definition property of SSA. Gated Single Assignment (GSA). SSA is not precise enough for verifi- cation purposes. First, the φ-functions are not really functions. Second, the φ-functions contain no information about the conditions under which a certain definition is chosen. As a result, the GSA form will be used in this thesis. GSA form was first introduced by Ballance et al. [17]. GSA is an appropriate intermediate representation for a number of code transformations [129]. An efficient algorithm (running in O(E · α(E,N)) time) for GSA computation was proposed by Tu and Padua [129]. GSA extends SSA with gating functions that replace the φ-functions. Besides the definitions, the gating function also contains conditions under which the flow of control reaches the block in which the function is placed. Tu and Padua [129] define three types of gating functions: • For each definition, the γ-function represents the condition under which 23 Chapter 2. Background that definition reaches a join node, and it directly replaces the φ-function in SSA. For example, the following piece of code i f (c1 ) { x = 1 ; } else { i f (c2 ) { x = y ; } else { x = z ; } } is translated to x1 = γ (c1, 1, γ (c2, y, z)). • The µ-functions are inserted in the loop headers. These functions choose between the initial and the loop-carried definitions. • The η-function is inserted at loop exits, and it captures the flow of defini- tions from within the loop to subsequent uses [17]. The static analysis proposed in this thesis unrolls loops once and terminates them with an assumption that the loop test has failed, as in ESC/Java [63]. Consequently, µ and η functions become unnecessary. The γ-function is more formally defined as: Definition 2 (γ-Function [129]). A γ-function for a block B is recursively defined by the following grammar: Γ ::= γ (Cond ,Γ,Γ) | Object | ∅ In the grammar, Cond is a Boolean variable. The object can be of an arbitrary type. The second and the third operand of the γ-function must be of the same type. An infeasible path is represented as ∅. 24 Chapter 2. Background As the γ-function is semantically equivalent to the if-then-else construct, it will be interpreted either as a term ITE or a logical formula ite, depending on the context. The terminal symbols in the given grammar of the γ-functions will be simply known as terminals. Terminals in the standard γ-functions are variable definitions. In the previously given example, the terminals are {1, y, z}. All terminals are of the same type. Gating paths are another important concept related to GSA, and they will be heavily used for the construction of VCs: Definition 3 (Gating Path [129]). A gating path of a node B is a path in the CFG from idom (B) to B, such that every node on the path is dominated by idom (B). In the context of the analysis presented in later chapters, CFGs are going to be acyclic, and therefore a gating path is just a path idom (B) +−→ B. Definition 3 is written in a more general way to allow cycles. To represent a set of gating paths, we are going to introduce an expression similar to γ-functions that has only basic-block labels as terminals. These are known as gating path expressions, and will be denoted with an overline (γ), as a mnemonic for the path expression. Definition 4 (Gating Path Expression). Let G be an acyclic CFG. A gating path expression GP (B) of a basic block B ∈ G is an expression that tracks the conditions that select which Bp ∈ Pred(B) would precede B on a path idom (B) ∗−→ Bp −→ B. It is defined recursively as: Γ ::= γ (Cond ,Γ,Γ) | Bp | ∅ where Bp ∈ Pred(). As a special case, GP (Entry) = Entry. Intuitively, one can think of the conditions Cond as switches that determine which path is taken from idom (B) to Bp ∈ Pred(B). Gating path expressions for a given CFG can be easily canonicalized by simplifying expressions of the form γ (c,Bi, Bi) to Bi. As each unconditional 25 Chapter 2. Background Entry B1 T B2 F B3 T B4 F B7 B5 B6 B8 T B9 F Exit T F Figure 2.5: Gating Path Expression Examples. Only blocks Entry, B1, B6, and B7 are terminated with conditional branch statements. Let us denote the corresponding branch conditions as cEntry, cB1 , cB6 , and cB7 . Assume that the left branch is taken if the condition is true, and the right branch other- wise. Some examples of the gating path expressions are: GP (B1) = Entry, GP (B6) = γ (cB1 , B5, B4), GP (B8) = γ ( cEntry, B6, B7 ) , and GP (Exit) = γ ( cEntry, γ (cB6 , B9, B8) , γ (cB7 , B8, B7) ) . 26 Chapter 2. Background branch statement branch Bs can be seen as a special case of a conditional branch statement branch(∗, Bs, Bs), the same simplification rule applies to basic blocks terminated by unconditional branches. Consequentially, given a canonicalized GP (B), the basic blocks Bi ∈ Ancs(B) terminated by unconditional branches will never be in supp (GP (B)), unless Bi ∈ Pred(B). In Chapter 3, we shall perform operations on these gating path expressions (illustrated in Fig. 2.5) to find definitions of variables and abstract memory locations quickly. Memory Access. The work presented in this thesis is based on the single assignment forms. SSA and GSA handle pointers, local variables, and globals in a specific way. To avoid ambiguity, I provide the definitions of some quite standard notions that might have a slightly unexpected meaning in this thesis: • A local variable denotes a location on the stack that holds a value of an arbitrary type, it is visible only within a single function, and it is immutable (because it has a single point of definition). If a function passes its local variable as a parameter to another function, the value of the local variable is copied to the context of the callee. • A pointer is a local variable that holds an address of a memory loca- tion. Pointers inherit all the properties of local variables. In addition, pointers can be dereferenced. The memory location to which a pointer points is being written (resp. read) if the dereferenced pointer is on the left (resp. right) side of the assignment operator. • An abstract memory location is a piece of memory to which one or more pointers can point. Abstract memory locations can be globally visible, and can be modified at multiple points in the program, even in different functions. • A global variable is an abstract memory location. All functions that access a global variable have to access it by dereferencing a pointer. Compilers 27 Chapter 2. Background usually keep pointers to global variables in the symbol table. To simplify the formalization. I assume that all pointers are passed as parameters to functions. Calysto handles global variables by performing symbol table look-ups. 2.3.3 Alias Analysis Aliasing in software occurs when two or more names in the program refer to a single memory location, either on the heap or on the stack. For example, given this code: x = 17 ; ∗y = b ; z = x ; the constant propagation optimization can assign 17 to z if and only if y does not alias x. Pointer analysis answers whether two pointers can refer to the same memory location, while points-to analysis returns a set of abstract memory locations to which a single pointer can point. In general, both analyses are undecidable [85, 106]. This negative result led to a wide range of algorithms with different precision- performance tradeoffs. The plethora of alias analysis algorithms that has been proposed can be classified according to the following properties: Context-Sensitivity. Alias analyses are mostly interprocedural. If the analy- sis differentiates the contexts from which a function is called, it is said to be context-sensitive. Otherwise, it’s context-insensitive. The cost of context- sensitive analysis depends on the used abstract domain. Context-sensitive algorithms that handle bit-vector operations precisely have exponential worst-case run times. Flow-Sensitivity. If an analysis takes into account the flow of control between statements in a function, it is said to be flow-sensitive. Flow-insensitive 28 Chapter 2. Background analyses treat the entire function as a set of statements ignoring the flow. Path-Sensitivity. An analysis that considers the exact path along which a certain statement is executed is known as path-sensitive. For example, given the program: x = 7 ; i f (c ) { ∗y = a ; } . . . i f ( ! c ) { z = x ; /∗ Block B ∗/ } a path-sensitive analysis can infer that z = 7 in block B even if y aliases x. In contrast, a path-insensitive variant must conservatively assume that writing to y can modify the value of x if the two may alias. Path-sensitivity is an expensive property, and path-sensitive analyses typically require some sort of a decision procedure to distinguish feasible from infeasible paths. An analysis that is path-sensitive is also necessarily flow-sensitive. Field-Sensitivity. Field-sensitive analyses distinguish the individual fields wi- thin a structure, while field-insensitive ones fold the entire structure into a single “super-field.” Handling of Recursive Data Structures. Most alias analyses used in prac- tice start with a single abstract location per each allocation site, and refine the information by an iterative algorithm. Such an approach is oblivious to recursive data structures. Early work on the alias analysis of recursive data structures has been done by Chase, Wegman and Zadeck [28]. Later, the analysis of recursive data structures became known as shape analysis [133]. Shape analysis is computationally very expensive [80]. 29 Chapter 2. Background The interprocedural analysis presented in this thesis is context-sensitive, interprocedurally path-sensitive (and therefore also flow-sensitive), and direc- tional, but does not handle recursive structures precisely — rather, it uses a logical memory model [16] and represents the heap by a single memory location per allocation site to avoid the high computational complexity associated with shape analysis. 2.3.4 Summarization for Interprocedural Analysis Summaries are used in software analysis to avoid or mitigate the potentially exponential cost of context-sensitivity. Cousot and Cousot [40] made a classifi- cation of approaches to modular static analysis: Worst-case analysis assumes that absolutely no information is known on the interfaces of modules. This form of analysis is very imprecise, but also scalable and easy to parallelize. Simplification-based approaches simplify the summaries before using them. Simplification can range from existential quantification of local variables to abstractions over various abstract domains [39]. Analysis based on user annotations requires the users to provide interface specifications in the forms of preconditions and postconditions. Users can often also provide assertions and loop invariants. An example of such a system is the Boogie [90] Spec# static checker. Symbolic relational analysis represents summaries as relations, similar to circuit modules. The environment assigns symbolic names to each object that a module can use or modify. The analysis instantiates the names in the environment as relational parameters when the module is used. Such an approach is used in the SLAM software model checker [14], in which finite-state summaries are represented as BDDs [24] and used for image computation. 30 Chapter 2. Background Structural abstraction in Chapter 4 builds upon symbolic relation sum- maries, but also allows various simplifications and abstractions to be performed before the summaries are used. 2.4 Low Level Virtual Machine My experimental static checker Calysto uses the Low Level Virtual Machine (LLVM) compiler framework [86] as a front-end. LLVM consists of a num- ber of front-ends for different languages, an extensive framework of analyses and transformations that operate on the LLVM intermediate representation, an interpreter, and a number of back-ends for several mainstream processors. The LLVM intermediate representation is in SSA form, and the instructions are similar to standard three-address Reduced Instruction Set Computer (RISC) in- structions (e.g., [73]). All abstract memory locations are accessed through load and store instructions, so the standard address-of operator is unnecessary. Several LLVM features are relevant to the work presented in this thesis: 1. The intermediate form is language-independent. 2. LLVM eliminates all the complexity of dealing with numerous official and unofficial standards of C, which is the main verification target for Ca- lysto. 3. LLVM performs a standard batch of semantic and syntactic checks on the program before producing the intermediate code. Calysto assumes that the code is syntactically and semantically checked. 4. Local variables (including pointers) in LLVM SSA have a unique point of definition and are immutable. 5. LLVM handles all address-of, field-access, and array-access constructs with a single instruction — getelementptr. Handling of getelementptr will be abstracted away from the algorithms presented in this thesis. 31 Chapter 2. Background 6. Global variables can be only accessed through constant pointers. 7. Multi-level pointers are flattened by using temporary variables. 8. LLVM handles pointer arithmetic. The features related to pointers are important for the work presented in this thesis, and the rest of this section provides more details. Global Variables. All identifiers used within SSA must have a unique point of definition. If the globals were accessed directly, it would not be possible to enforce the single point of definition requirement. For that reason, globals can be accessed only through constant pointers. Multi-Level Pointers. LLVM expands multi-level (non-recursive) pointers into a number of temporary variables making the analysis simpler and more uniform. All abstract memory locations are read and written by a single pointer dereference. Since Calysto always substitutes definitions for uses, it can handle multi-level pointers. The intraprocedural analysis need not be aware of the multi-level pointers, because they are all expanded into single-level ones. Pointer Arithmetic. LLVM resolves most pointer arithmetic that can be resolved statically without using a decision procedure. Furthermore, it does a number of global optimizations, like constant propagation and strength reduc- tion, that make this resolution more precise. Calysto makes additional efforts to resolve pointer arithmetic through sev- eral optimizations of the constructed summaries. Variable offsets resulting from arbitrary pointer arithmetic are treated in the same way as the variable array indices — they are considered to be zero, in the spirit of the logical memory model [16]. This approximation is unsound (bugs can be missed) and incom- plete (spurious bugs can be reported), but renders the constructed VCs easier for the decision procedures. Section 3.1.3 explains how this approximation can be eliminated at the cost of additional complexity. 32 Chapter 3 Structure-Preserving Symbolic Execution This chapter will focus on the intraprocedural analysis of call-free functions, and the following chapter will extend the intraprocedural into an interproce- dural analysis. Section 3.1 introduces a programming language grammar that reflects the most important features of the LLVM intermediate form, while us- ing γ-functions instead of φ-functions.10 The introduced language makes the presentation more readable. The implementation in Calysto works on the full LLVM intermediate form. The same section also surveys the design decisions, and discusses the main contributions. Section 3.2 introduces a novel structure- preserving symbolic execution algorithm. The structure preservation property is going to be exploited later by structural abstraction in Chapter 4. Section 3.3 discusses the space complexity and correctness of the algorithm. 3.1 Introduction The real implementation of the presented symbolic execution algorithm works on a standard SSA form (as implemented in the LLVM framework). To make the exposition easier to follow, I will use a simpler language. The language picks up the most important features of the LLVM intermediate form and replaces φ-functions with their γ-function counterparts. The language captures all the important features of commonly used programming languages, except for func- 10As previously mentioned, SSA is not precise enough for symbolic execution, so the imple- mentation of the algorithm replaces φ-functions with γ-functions on the fly. 33 Chapter 3. Structure-Preserving Symbolic Execution BasicBlock ::= Terminator | Stmt ; Terminator Terminator ::= branch (BoolVar , LabelT , LabelF ) | branch Label | return IntVar Stmt ::= IntVar := IntExpr | IntVar := ∗IntVar | ∗IntVar := IntVar | BoolVar := BoolExpr | Stmt ; Stmt | assert (BoolVar) | assume (BoolVar) | abort IntExpr ::= IntConst | IntVar BinaryOp IntVar | UnaryOp IntVar | γ (BoolVar ,ΓI ,ΓI) | Function (Params) BoolExpr ::= BoolConst | IntVar Rel IntVar | γ (BoolVar ,ΓB ,ΓB) | BoolVar ∧ BoolVar | BoolVar ∨ BoolVar | ¬BoolVar Params ::= EMPTY | IntVar ,Params ΓB ::= γ (BoolVar ,ΓB ,ΓB) | BoolVar | ∅ ΓI ::= γ (BoolVar ,ΓI ,ΓI) | IntVar | ∅ Figure 3.1: The Grammar of a Simple Language. tion pointers. As explained earlier in Section 2.3.1 on page 21, a simple points-to analysis suffices in practice for resolving the function pointers. The implemen- tation of the algorithm performs such a simple points-to-analysis and explicitly quantifies over all side-effects of functions that can be called at an indirect call site. The maximal number of allowed calls is determined by a user-controllable parameter. The grammar of the language is given in Fig. 3.1. Two types of variables are defined: Boolean type B and integers ZN . Abstract memory locations, function parameters, and function return values can only be of the integer type. As noted in Section 2.3.2, pointers are handled in the same manner as local variables. The 34 Chapter 3. Structure-Preserving Symbolic Execution only point when this distinction matters is when a pointer is dereferenced. A program consists of one or more functions. Each function consists of one or more basic blocks and can have zero or more input parameters. Each ba- sic block contains one or more statements and is terminated with a statement that changes the flow of control: either a branch or a return statement. The branch statement can be conditional or unconditional. The conditional branch jumps to the basic block denoted by label LabelT if BoolVar is true, and to LabelF otherwise. Note that such a branch statement allows the representation of unstructured code. The return statement returns the flow of control and a returned value back to the caller. The dereference operator ∗ is used to repre- sent access to an abstract memory location — pointer read if the dereferenced pointer is on the right side of an assignment, and pointer write if it is on the left. Sequential composition is denoted by the “ ; ” operator. The statements assert(BoolVar) and assume(BoolVar) do not influence the state of the pro- gram if BoolVar is true. If BoolVar is false, assert terminates erroneously and assume terminates with no error [89]: the difference between the two is that erroneous termination says it’s the programmer’s fault that the condition does not hold, while the other type of termination relieves the programmer of any responsibility. The abort statement is syntactic sugar for assume(false). Ex- pressions allow a standard set of logic and arithmetic operators. All relations (Rel), as well as all unary (UnaryOp) and binary operators (BinaryOp) specified in Fig. 2.1 on page 14, are supported. 3.1.1 Sources of Complexity Symbolic execution (also known as symbolic simulation) [82] is a technique that computes symbolic expressions that give the values of all modified locations in a program for all possible execution paths. The technique has numerous applications, e.g., verification of software partial correctness [36, 54, 42], software testing [35, 12], program slicing [93], and equivalence checking between high- and low-level hardware descriptions [83, 34, 61]. 35 Chapter 3. Structure-Preserving Symbolic Execution Symbolic execution is inherently difficult. Even if loops are unrolled (as is often done in software verification), and recursive data structures are imprecisely represented by a single node per allocation site, symbolic execution remains very expensive because: 1. There are a potentially exponential number of paths that need to be con- sidered. 2. Pointer reads and writes can access a potentially large number of abstract memory locations, due to pointer aliasing. 3. Computed symbolic expressions can be of exponential size if represented näıvely. Fig. 3.2 on the next page illustrates some of these difficulties. Accordingly, precise symbolic execution has been considered too expensive to use on large bodies of code. For example, many previously proposed tools considered paths one-by-one (e.g., [42, 138, 26]) and can not handle the join operation. Software model checking tools (e.g., [13, 74]) also typically perform symbolic execution on a single path at a time. This chapter introduces a novel symbolic execution algorithm that addresses the above three problems. I attack the first two problems by exploiting structure in the CFG to keep the analysis local. The third problem could be solved by introducing a linear number of additional variables (as originally proposed by Tseitin [128]) to flatten the computed expressions without blowup. However, flattening computed expressions loses the original structure, which is valuable for later analysis. The approach I use is to represent the computed expressions in the form of maximally-shared graphs, which not only avoids the blowup, but also preserves the original structure of problem. 3.1.2 Assumptions The presented analysis makes several assumptions: 1. gating path expressions are computed for each basic block; 36 Chapter 3. Structure-Preserving Symbolic Execution C1 C2 C3 T C4 F T F C5 T C6 F T F (a) B1 B2 w w w B3 w w w (b) Figure 3.2: These two fragments of CFGs illustrate some sources of complexity in symbolic execution. Nodes denote basic blocks, and edges represent flow of control. Graph (a) illustrates the exponential number of paths possible. Symbolic execution computes exact symbolic expressions for modified locations for all possible paths, so the algorithm must consider all (exponential number of) paths through the code and encode the conditions under which each path is executed. For example, basic block C6 can be reached from C1 or C2 if (c1∧¬c3)∨ (¬c1∧¬c4)∨ (c2∧¬c3)∨ (¬c2∧¬c4), where the branching condition of block Bx is denoted as cx. Graph (b) shows that the number of live definitions of an abstract memory location∗ can be much larger than the cut-width of the CFG. Assume that some abstract memory location, say ∗p, is written on all the paths from B1 to B2, except on one path. The same holds for paths from B2 to B3. Assuming that ∗p is defined in B1, seven possible definitions reach B3. The interactions of pointer reads and writes tend to get even more complex in practice. ∗Local variables always have a unique point of definition, unlike abstract memory locations. 37 Chapter 3. Structure-Preserving Symbolic Execution Data structure Computational complexity Due to gating path expressions O(N · α(E,N)) [129] dominator trees O(N · α(E,N)) [91] iter. dom. frontiers “almost linear” [107] Table 3.1: Prerequisite Algorithms and Their Computational Complexities. 2. dominator and postdominator trees are available; 3. the iterated dominance frontiers are computed; 4. all loops are unrolled at least once and terminated with assume(¬loop test) The computational complexities of the algorithms for computing the required data structures are given in Table 3.1. All loops can be unrolled in linear time by breaking the back edges in the call graph. 3.1.3 Design Decisions To achieve scalability, I made several simplifying assumptions about loops, mem- ory model, and terminal pointers. This section discusses the tradeoffs and pos- sible improvements. Loops. Like ESC/Java [63], Calysto approximates loops by unrolling them once and terminating them with an assumption that the loop test has failed. Although this approach cannot introduce spurious errors, it can result in de- creased code coverage — if the loop test cannot fail after the first iteration, the conjunction of the loop test and its negation is going to be false. Since anything can be implied from false, all assertions reachable from the loop exit block become trivially satisfiable. The approach can be easily extended with a more precise treatment of loops in several ways: 38 Chapter 3. Structure-Preserving Symbolic Execution • By unrolling loops multiple times and omitting the assumption that the loop test has failed. This approach is unsound, but gives higher code coverage. • By using the framework of abstract interpretation [39]. Loops can be replaced with their invariants computed by abstract interpretation before running the presented symbolic execution algorithm, like in [88]. This approach can be expensive and its effectiveness depends on the templates used for invariant guessing. • By considering loop-carried values to be unconstrained. This approach is very imprecise, but sound and simple. The loop-carried values can be simply detected in both GSA and SSA. In GSA, the µ-functions select between the initial and the loop-carried values. In SSA, definitions are joined through the φ-functions. In many imple- mentations of SSA, operands of a φ-function are definition-source pairs, where source is the basic block from which the corresponding definition reaches the φ-function. If a CFG is traversed in the reverse postorder, the definitions that come from sources that haven’t yet been visited are the loop-carried values. • By requiring the users to provide loop invariants. This approach requires a significant amount of manual effort, but is sound, and potentially very precise. Memory Model. The algorithm, as implemented in Calysto, uses a simple memory model, similar to the logical memory model [16], in which ∗(ptr + i) and ∗ptr are assumed to refer to the same object, except that Calysto does distinguish those two locations if the expression simplifier can simplify i to a constant. Such constant offsets are often used for access to structure fields (making the analysis field-sensitive as well), so this added precision is important in practice. Recursive data structures are imprecisely abstracted by 39 Chapter 3. Structure-Preserving Symbolic Execution a single node per allocation site. All other locations in the program (including dynamically allocated structures) are handled precisely. Multi-level pointers and statically known pointer offsets are handled precisely as well. It would be relatively simple to extend the proposed analysis with support for the non-extensional theory of arrays [124] if a more precise memory model is required. The tradeoff between the precision and the additional complexity is unclear at this point and requires further research. Non-Aliased Terminal Pointers. The last design decision worth mention- ing is my assumption that the terminal pointers do not alias each other. This is a reasonable assumption in practice: Pointers returned by the memory alloca- tion functions (like malloc()) can be safely considered unique. The pointers to globals are always constants and therefore unique. The pointers passed to func- tions as parameters frequently do not alias each other, but if they do, Calysto computes a join of the corresponding definitions by explicit existential quantifi- cation over the symbolic definitions of modified abstract memory locations. For instance, let f be a function that calls function g, which takes two pointer parameters. Let p1 and p2 be the pointers, which may alias, that f passes to g as parameters. If g executes ∗p1 = d1 and ∗p2 = d2, where d1 and d2 are arbitrary symbolic expressions, Calysto will represent the side-effects of the call as ∗p1 = ITE (c1, d1, d2) and ∗p2 = ITE (c2, d1, d2), where c1 and c2 are fresh unconstrained Boolean variables. If the may-alias analysis is sound, so is handling of aliased pointer parame- ters. Calysto, however, only checks whether the pointer parameters are aliased in the caller. This could be easily changed by substituting a sound and arbitrar- ily precise may-alias analysis. The tradeoff between the precision and soundness in handling aliased pointer parameters requires further research. 3.1.4 Contributions The main technical contribution of this chapter is an efficient symbolic execution algorithm that preserves the data-flow dependencies (structure) of the original 40 Chapter 3. Structure-Preserving Symbolic Execution program in computed symbolic expressions. Structural abstraction (Ch. 4) uses that structure for fast validity checking of the computed VCs. The proposed symbolic execution algorithm uses gating path expressions (Definition 4 on page 25) to reconstruct only as much high-level structure as needed and does not require a structural analysis (e.g., [98, page 236]). The algorithm exploits the CFG structure for efficiency. The CFG structure exploitation is based on the insight that all the paths by which a basic block B is reachable from the Entry block have to pass through the immediate dominator of B. This property localizes the problem to subgraphs of the dominator tree. My dynamic programming algorithm iterates over the CFG in the reverse post- order, assuring that the immediate dominator of each node is visited before the node itself. After a definition of an abstract memory location is computed for the immediate dominator of some basic block B, it can be redefined only on a small number of paths that reach B. Direct application of this insight would require computation of a definition of every modified location for almost every single block. My algorithm updates definitions in a lazy manner and only for the basic blocks where an update is required. 3.2 Symbolic Execution This section starts by giving a high-level introduction into the construction of VCs. VCs are composed of two components: the control-flow context and the asserted condition. Symbolic representations of both components can be com- puted with the proposed symbolic execution algorithm. After introducing the notion of a VC and describing its components, this section continues with two novel operators (Section 3.2.1) that will be instrumental for performing efficient symbolic execution. The structure-preserving symbolic execution algorithm is introduced in Section 3.2. Section 3.2.3 defines maximally-shared graphs for representation of computed symbolic expressions, while Section 3.2.4 provides a number of examples that illustrate the newly introduced concepts and the dynamics of the proposed algorithm. 41 Chapter 3. Structure-Preserving Symbolic Execution The logical formula representing a VC is composed of two parts: the control- flow context, and the condition being checked. The control-flow context, say ψ, represents the conditions under which a particular statement is reachable. Let us assume that we want to check that an assertion assert(α) can never fail. Intuitively, the VC should be ψ ⇒ α. For example: assume ( ! c1 ) ; i f (c2 ) { assert (c3 ) ; } the control-flow context of the assertion is ¬c1 ∧ c2, and the constructed VC would be: ¬c1 ∧ c2 ⇒ c3. More formally, VCs are defined as follows: Definition 5 (Verification Condition). Let α stand for an asserted condition and ψ for the predicate that specifies the control-flow context, i.e. the conditions under which assert(α) statement is reachable. The VC for the asserted condi- tion is a logical formula φ defined to be equal to ψ ⇒ α. The assertion α holds if and only if φ is valid. Both ψ and α can be computed with the proposed algorithm. The sym- bolic value of α is computed through symbolic execution. To compute ψ, Ca- lysto uses the following trick [87]: For every conditional branch statement branch(c,B1, B2), an assumption assume(c) (resp. assume(¬c)) is prepended to the statements in B1 (resp. B2). The passive statements (like assert(φ) and assume(φ)) that influence the flow of control, but not the state of the program itself, are assumed to write φ to a special Boolean abstract memory location (and abort writes false). Reading that memory location produces the symbolic definition of the control-flow context. 3.2.1 Restriction Operators Even if loops are eliminated, the number of paths in the program can be expo- nential with the size of the program, but is finite. Analyzing paths one by one 42 Chapter 3. Structure-Preserving Symbolic Execution during the symbolic execution is hopelessly inefficient, so I take a different ap- proach. Instead of executing symbolic paths one by one, the proposed algorithm computes symbolic definitions of each variable or abstract memory location in the program on all paths. Once symbolic execution is seen as computation of symbolic definitions, it is obvious that at each use (resp. def ) point in the program we need to con- struct (resp. update) symbolic definitions. Two problems are associated with the construction (resp. update) of symbolic definitions: (1) How to find which definitions matter? (2) How to avoid recursive lookups? This section presents two restriction operators that solve these two problems. One operator is for computing symbolic definitions at the use points, and the other is for updating symbolic definitions after a write to an abstract memory location at the def points. The first operator exploits the dominance relation to reduce the number of definitions that need to be considered. The second oper- ator takes an existing definition, and performs a suitable substitution. Neither of the operators is recursive. The first restriction operator takes a gating path expression γ (representing paths) and a set of location-definition pairs R (representing live definitions) as parameters, and produces a γ-function in which the terminals are from R (new definition at the use point). Definition 6 (Read). Let lower case Greek letters o, . . . , ρ stand for arbitrary objects of some type T . Let R be a set of pairs 〈Bi, o〉, where Bi is a basic block. Each basic block can appear only once in the set R. Let Λ stand for a placeholder for a pair to be substituted later. The restriction of a gating path expression GP (B) (see Definition 4 on page 25) of a basic block B to the set R, denoted as GP (B)|R, is defined as follows: 43 Chapter 3. Structure-Preserving Symbolic Execution γ (Cond ,Γ,Γ)|R = γ (Cond , Γ|R , Γ|R) Bp|R = if ∃〈B, o〉 ∈ R then o elsif ∃〈Bp, pi〉 ∈ R then pi elsif ∃〈Bx, $〉 ∈ R : Bx Bp∧ ∀〈Bi, ρ〉 ∈ R : Bi Bp∧ dtl [Bx ] > dtl [Bi ] then$ else Λ ∅|R = ∅ GP (B)|R attempts to substitute each Bp ∈ Pred(B) with an appropriate definition from B or Bp (if there is no definition in B). If that fails (no definitions from B or Bp in R), it finds the closest dominator Bd of Bp in R, and replaces Bp with the definition from Bd.11 In order to use the restriction operator defined above, the dominators of each node B have to be processed, paired with the appropriate objects, and added to the R before processing B. Fortunately, this is guaranteed by the reverse postorder traversal: Lemma 1. Let Bx, By ∈ B be two basic blocks. If Bx By, then Post [Bx ] > Post [By ]. Proof: Assume the opposite, Post [Bx ] ≤ Post [By ]. The equality part is triv- ially false as the two basic blocks are distinct. For By to be visited first in the reverse post-order traversal, there must exist a path p = Entry ∗−→ By that avoids visiting Bx. From the definition of the dominance relation, it follows 11The initialization step of the symbolic execution algorithm (Algorithm 1 on page 51) assures that there always exists at least one dominating definition in Entry. 44 Chapter 3. Structure-Preserving Symbolic Execution that Bx cannot dominate By, which is a contradiction. The second restriction operator is used for constructing new conditional definitions after a write to an abstract memory location. It takes a γ-function (representing existing symbolic definition), an overwritten location L, and a sub- stitution value y (newly written value) as parameters, and returns a γ-function (new definition) in which all the objects in terminals are modified in the follow- ing way: If a terminal object is equal to L, it is replaced with y. Otherwise, the object is replaced with a placeholder Λ. This operator is used for handling writes to abstract memory locations. Formally, Definition 7 (Write). Let γ be an arbitrary gating function. Let x and y stand for arbitrary objects of the same type T ; and Λ, for a placeholder for an object to be substituted later. The restriction of γ to location L with the substitution element y, denoted as γ|yL is defined as: γ (Cond ,Γ,Γ)|yL = γ (Cond , Γ|yL , Γ|yL) x|yL = y if x ≡ LΛ otherwise ∅|yL = ∅ The example that follows illustrates how the first restriction operator works. More detailed examples in Section 3.2.4 explain how the newly introduced op- erators are used within the symbolic execution algorithm. Example 2: Let Fig. 3.3(a) on page 47 represent a CFG subgraph of some CFG.12 Now, let us compute the definition of ∗p for basic blocks B2 and B7 using the first restriction operator (Definition 6 on page 43). For block B2, it is assumed that 12The focus on a subgraph emphasizes the localization of analysis achieved by using the first restriction operator. 45 Chapter 3. Structure-Preserving Symbolic Execution ∗p is read after the ∗p = 2 write statement, while the exact location of the read in B7 is irrelevant. The gating path expressions are: GP (B2) = B1 GP (B7) = γ (c1, B2, B6) The branching conditions of each conditional branch node Bx are denoted as cx. Note that B7 sees the same definition of ∗p from B3 regardless of whether the flow of control passes through B4 or B5. After symbolic execution of B1 and statement ∗p = 2 in B2, the set of location-definition pairs of the abstract memory location ∗p is equal to R = {〈B1, 1〉, 〈B2, 2〉}. From Definition 6 on page 43, it follows that: GP (B2)|R = 2 because 〈B2, 2〉 ∈ R (the first case of B2|R). According to our assumptions on the number of write statements, ∗p cannot change again in B2. The symbolic execution of B3 adds 〈B3, 3〉 to R. The definition of ∗p in B7 can be computed as follows: GP (B7)|R = γ (c1, B2, B6)|R Since 〈B2, 2〉 ∈ R, the restriction quickly determines that the definition that reaches B7 through B2 is equal to 2 (the second case of B2|R). Finding the definition that reaches B7 through B6 triggers the third case of B6|R: As there is no definition in B6, the restriction operator starts traversing the DT bottom- up, until it finds a definition. In this case, the definition is found in B3.13 Thus, the computed symbolic definition of ∗p in B7 is equal to: γ (c1, 2, 3). The handling of a write statement in, for example, block B4 would be slightly more complex. The symbolic execution algorithm that will be presented shortly maintains an important invariant — given a read statement in some block B, the algorithm guarantees that the definition of the corresponding abstract memory 13The initialization step of the main symbolic execution algorithm assures that there always exists a definition in the Entry block of each CFG. 46 Chapter 3. Structure-Preserving Symbolic Execution B1 *p=1; B2 *p=2; T B3 *p=3; F B7 B4 T B5 F B6 (a) CFG B1 B2 B3 B7 B4 B5 B6 (b) DT Figure 3.3: CFG and DT Subgraphs Used in Example 2. Nodes represent basic blocks and edges the flow of control. The edge labels correspond to the branching conditions being true or false. The basic blocks are numbered in the reverse post-order and labeled B1, . . . , B7. Each block B1, . . . , B3 contains one statement that writes some constant to the abstract memory location ∗p. It is assumed that no other statement in the shown CFG subgraph can change ∗p. All other statements are irrelevant for this example and therefore not drawn. The shown CFG and DT graphs are subgraphs of a potentially larger CFG and DT graphs. The dotted edges connect the subgraphs with the corresponding larger graphs. 47 Chapter 3. Structure-Preserving Symbolic Execution location is either available in B or in some dominator (not necessarily proper) of each Bp ∈ Pred(B) — by inserting additional definitions in R at the IDF. Thus, this little example is representative of all possible cases in which the first restriction operator will be used within this thesis. 3.2.2 Data-Flow Summaries by Symbolic Execution This section presents the structure-preserving symbolic execution algorithm. The algorithm requires a dominator tree, iterated dominance frontiers, precom- puted postdominance relation, and the gating path expressions. All these can be done in almost linear time (Table 3.1). The restriction operators presented in Section 3.2.1 are used for computing a unique conditional definition from the definitions of abstract memory locations that reach a use. The first operator can be intuitively understood as a vehicle for resurrecting the single-point-of-definition property of GSA for abstract memory locations. The state of symbolic execution is kept in three tables: 1. The expression table ExpTable keeps the symbolic definitions of local vari- ables. Local variables (including pointers) can be used as unique table indices, because they have a single point of definition. For this reason, each bin of the ExpTable table contains a single entry. 2. Abstract memory locations do not have a single point of definition: they can be (re)defined each time a pointer is written. Thus, multiple defi- nitions can reach a use. To compute precisely which definitions actually reach a use and under which conditions, the algorithm tracks both the locations where a memory location was (re)defined and the correspond- ing definitions. Such location-definition pairs are kept in the expression list ExpList . The expression list contains a list of all active definitions in the form of γ-functions. The fully path-sensitive definition of an abstract memory location ∗p at some statement in a basic block B can be computed by restricting GP (B) with the set of definitions in the ExpList [∗p]. 48 Chapter 3. Structure-Preserving Symbolic Execution 3. The Merge [B] table contains the set of abstract memory locations whose definitions need to be merged once the basic block B is reached. The definitions of abstract memory locations are merged at the dominance frontiers with other definitions reaching the merge point. This merging occurs exactly where the SSA construction would insert φ-functions if abstract memory locations were local variables. Algorithm 2 on page 52 performs the merging. The symbolic execution algorithm frequently uses the following two patterns in the pseudocode: 1. ITE (ExpTable [x] 6= ∅,ExpTable [x] , x) 2. ITE (const (x) , x,ExpTable [x]) The first pattern is used for pointer variables, and its purpose is to replace each use with the corresponding definition. This assures that all pointers are defined in terms of terminal pointers.14 The second pattern uses the const (x) predicate, which is true only if x is a constant, to avoid performing table lookups for constants. Arithmetic and logic operators will be denoted simply as Op. The algorithm computes symbolic expressions x Op y and does not execute the operator Op. Unary operators can be handled in the same way. Algorithm 1 starts with an initialization of abstract memory locations, corre- sponding to terminal pointers, to unconstrained values (unknown initial state). The implementation initializes only the locations actually accessed in the func- tion in a lazy manner. The Merge table is initially empty. Basic blocks are processed in reverse postorder, which guarantees that all dominators of a basic block B are visited before the B itself, and symbolic definitions are always com- puted before the uses. If the processed basic block B is in the IDF of some other block that defines some abstract memory location ∗p, the definitions of ∗p reach- ing B are merged. Statements in the block are processed from the first to the 14Recall that we eliminated recursive data structures from the discussion. 49 Chapter 3. Structure-Preserving Symbolic Execution last instruction sequentially. Lines 7–14 handle local variables and pointers sub- stituting each variable x with its definition, namely ExpTable [x]. The parallel substitution operator is used at Line 16 to substitute multiple variables by their definitions. Since I assume a load-store architecture of the intermediate form, abstract memory locations can never directly appear as operands of φ-function or any other operator. Line 18 constructs the definition of an abstract memory location. The rest of the algorithm deals with pointer writes. First, the defini- tion of the pointer x is found in the ExpTable. For each pointer p that x can alias, the algorithm constructs the currently valid definition Def p of the abstract memory location ∗p. The rationale behind this step is that the new write might not completely overwrite the Def p definition (because x = γ (Cond , . . . , . . .) is conditional), so Def p could become a subexpression of the newly computed definition. In the next step, the stale definitions superceded by the newly con- structed one are deleted from ExpList [∗p]. The new definition of each abstract memory location to which some alias p of x can point to is constructed as fol- lows: First, the definition of x is restricted to p and the definition of the variable on the right hand side of the assignment (v) is used as the substitution value. Second, all the other terminal pointers pi 6= p which x can alias (represented by the Λ placeholder after the restriction) are replaced with Def p — note that this step assumes that the terminal pointers pi and p cannot alias each other. Finally, all basic block in the dominance frontier of B are added to the list of merging points for ∗p. Algorithm 2 is very similar to the innermost loop of Algorithm 1. The definitions of abstract memory locations are simply merged at the dominance frontiers. Precomputation of definitions is not needed as the algorithm has to perform only merge, not write. 3.2.3 Representation The representation of symbolic expressions affects efficiency a lot, and therefore I define the following representation: 50 Chapter 3. Structure-Preserving Symbolic Execution Algorithm 1 Symbolic Execution Algorithm. 1: For all dereferenced terminal pointers p, initialize ∗p : ExpList [∗p] = 〈Entry, ∗p〉 2: For all parameters x of the function ExpTable [x] = x 3: For all Bi ∈ B Merge [Bi] = ∅ 4: for all B ∈ B in reverse postorder on CFG do 5: PerformMerge(B) . Algorithm 2 6: for all assignment statements S in sequential order do 7: if S = (v := x) then 8: ExpTable [v] = ITE(const (x) , x,ExpTable [x]) 9: else if S = (v := Op x) then 10: ExpTable [v] = Op ITE(const (x) , x,ExpTable [x]) 11: else if S = (v := x Op y) then 12: ExpTable [v] = 13: ITE(const (x) , x,ExpTable [x]) Op ITE(const (y) , y,ExpTable [y]) 14: else if S = (v := φ(. . . )) then . Join for local variables 15: map φ to Γ by an operator similar to the one in Def. 6 16: ExpTable [v] = Γ [x ∈ supp (Γ) : x← ExpTable [x]] . Parallel substitution 17: else if S = (v := ∗x) then . Memory read 18: ExpTable [v] = GP (B)|ExpList[∗ITE(ExpTable[x] 6=∅,ExpTable[x],x)] 19: else if S = (∗x := v) then . Memory write 20: d = ITE(const (v) , v,ExpTable [v]) . Get definition of the written value 21: e = ITE(ExpTable [x] 6= ∅,ExpTable [x] , x) . Write through terminal pointers 22: for all pointers p ∈ supp (e) do . Locations x can alias 23: Defp = GP (B)|ExpList[∗p] . Current definition of ∗p 24: end for 25: for all pointers p ∈ supp (e) do . Locations x can alias 26: for all 〈Bi, β〉 ∈ ExpList [∗p] do . Delete stale definitions 27: if BmBi, delete 〈Bi, β〉 from ExpList [∗p] 28: end for 29: ExpList [∗p] = ExpList [∗p] ∪ 〈B, ( e|dp ) [Λ← Defp ]〉 . New definition 30: For each Bi ∈ DF (B)+, Merge [Bi] = Merge [Bi] ∪ ∗p . Merge at IDF 31: end for 32: end if 33: end for 34: end for 51 Chapter 3. Structure-Preserving Symbolic Execution Algorithm 2 Merging Abstract Memory Locations at Dominance Frontiers. 1: procedure PerformMerge(B) 2: if Merge [B] 6= ∅ then 3: for all ∗p ∈ Merge [B] do 4: Def = GP (B)|ExpList[∗p] 5: for all 〈Bi, β〉 ∈ ExpList [∗p] do . Delete stale definitions 6: if BmBi, delete 〈Bi, β〉 from ExpList [∗p] 7: end for 8: ExpList [∗p] = ExpList [∗p] ∪ 〈B,Def 〉 9: end for 10: end if 11: end procedure Definition 8 (Maximally-Shared Graph). Given a graph G = (N,E), let L stand for a labeling function L : N −→ string. Define the arity of a node n, denoted as |n|, as the number of outgoing edges. The outgoing edges are ordered, and the i-th edge of a node n will be denoted as child i(n). Two operator nodes n1 and n2 are defined to be equivalent (n1 , n2) if and only if |n1| = |n2|, L(n1) = L(n2), and: ∀i ∈ [1, |n1|] : child i(n1) , child i(n2) Graph G is maximally-shared if ¬∃n1, n2 ∈ O : n1 6= n2 ∧ n1 , n2. The choice of representation has several consequences: (1) the representation of computed expressions is compact, (2) it is simple to run expression simpli- fication on top of symbolic execution, (3) the computed expressions reflect the structure of the original code. I found these three properties to be important for efficient solving of the VCs. In order to construct such a representation, Algorithm 1 uses a hash table to check for existing definitions before constructing the new ones. This guar- antees that operator nodes are unique. Assuming that all the graph nodes are hashed, variables and constants are represented by a single node. Binary and 52 Chapter 3. Structure-Preserving Symbolic Execution unary operators can construct at most one additional node, while the restriction operators can create more than one additional node. The constructed symbolic expression graph is acyclic. That property triv- ially follows from the single point of definition property that SSA guarantees for both pointers and local variables, and the following facts: the loops carried values are considered to be unconstrained, each basic block is visited only once, and all uses are replaced with definitions. Algorithm 1 also assures that all the pointers are either terminal pointers or entirely defined in terms of terminal pointers. Maximally-shared graphs make the data-flow dependency explicit. Given any two nodes, n and m, from the same maximally-shared graph, we say that n is data-flow dependent on m if there exists a path n ∗−→ m. If neither n ∗−→ m, nor m ∗−→ n, the nodes are data-flow independent, and cannot influence each other. Thus, if we are interested in solving a VC represented by an expression node v in a maximally-shared graph, all nodes that are unreachable from v can be sliced away without impacting v’s validity. Such slicing dramatically improves performance of decision procedures used for validity checking.15 Besides improving performance, slicing also separates out individual side- effects of a function: The proposed symbolic execution algorithm, in general, computes a multi-rooted maximally-shared graph. Every root that corresponds to the returned value, or to a value of an updated abstract memory location visible in the calling context can be seen as a symbolic representation of one side-effect. In effect, symbolic execution slices the function with respect to its side-effects, breaking apart a single function with multiple side-effects into mul- tiple pure functions that have only one side-effect each. Structural abstraction (Chapter 4) operates on such function slices. 15Example 5 on page 60 illustrates this correlation between reachability in maximally-shared graphs and data-flow dependency in programs. 53 Chapter 3. Structure-Preserving Symbolic Execution 3.2.4 Examples Statement 29 of Algorithm 1 might be a bit hard to parse, especially the parallel substitution part. The following example should give a better intuition into how the statement works: Example 3: Let x be a pointer defined with γ-function γ (c1, u, γ (c2, v, γ (c3, z, u))). We shall refer to that γ-function as e. Now, let us consider what happens when the statement ∗x := w is executed. The pointers which x can alias are supp (e) = {u, v, z}. Let us consider the effect of the assignment on the abstract memory location to which u points. Con- struct a new expression by restricting e with {u} and substitute with ExpTable [w]. The result is e|ExpTable[w]u = γ (c1,ExpTable [w] , γ (c2,Λ, γ (c3,Λ,ExpTable [w]))) Effectively, ∗u is overwritten only if c1 ∨ (¬c1 ∧ ¬c2 ∧ ¬c3). In all other cases, the content of the ∗u abstract memory location is left unchanged. The un- changed locations correspond exactly to the Λ placeholder, which is replaced with the currently valid definition of ∗u. Finally, the new definition is added to ExpList [∗u]. The same is repeated for the other two pointers (v and z). The restriction operators introduced in Section 3.2.1 avoid recursive defini- tion lookups, and therefore localize the analysis. The following example illus- trates the localization and the overall dynamics of the algorithm: Example 4: This example illustrates how the symbolic execution algorithm computes defi- nitions of abstract memory locations and local variables of the function repre- sented with the SSA representation in Fig. 3.4 on page 56, while its CFG and DT graphs are shown in Fig. 3.5(a) and 3.5(b) on page 57. The function defines local variables a0, . . . , a5, takes pointer parameters p0 and p1, and accesses two global abstract memory locations through constant pointers g0 and g1. 54 Chapter 3. Structure-Preserving Symbolic Execution F T TF F T F T T F B1 B2 B3 B4 B5 B6 B7 B8 B9 a0 = 0; branch(c1, B3, B2); a1 = ∗p0; ∗p0 = 7; p1 = g1; branch(c2, B3, B6); a1 = φ(a3, a0); branch(c3, B5, B4); branch(c6, B8, B7); p2 = g2; branch(c7, B8, B9); a3 = a2 + 1; branch B5; a4 = φ(a2, a3); p3 = φ(p1, p2); ∗p0 = 0; branch B9; branch B9; a5 = φ(a1, a4); return a5; Figure 3.4: SSA of the Function in Example 4. Each Block is labelled with its name: B1, . . . , B9. The branching conditions of each conditional branch node Bx are denoted as cx. The code that computes the conditions is abstracted away from the example. 55 Chapter 3. Structure-Preserving Symbolic Execution B1 B2 F B3 T T B6 F B5 T B4 F B8 T B7 F B9 F T (a) CFG B1 B2 B3 B9 B6 B4 B5 B7 B8 (b) DT Figure 3.5: CFG and DT of the Function Defined in Example 4. Nodes repre- sent basic blocks and edges the flow of control. The edge labels correspond to the branching conditions being true or false. The basic blocks are numbered in the reverse post-order. The branching conditions of each conditional branch node Bx are denoted as cx. Each cx is a local Boolean variable with a single point of definition. Since those variables have a single point of definition and are immutable, they do not really have much impact on the dynamics of the symbolic execution algorithm and therefore this example abstracts their computation away. 56 Chapter 3. Structure-Preserving Symbolic Execution Gating path expressions for the join blocks are: GP (B3) = γ (c1, B1, γ (c2, B2, ∅)) GP (B5) = γ (c3, B3, B4) GP (B8) = γ (c6, B6, γ (c7, B7, ∅)) GP (B9) = γ (c1, B5, γ (c2, B5, γ (c6, B8, γ (c7, B8, B7)))) The gating path expressions of all other non-join blocks Bi, such that i > 1, are equal to the single element of Pred(Bi). According to Definition 4 on page 25, GP (B1) = B1. The implementation maps φ- to γ-functions as follows: a2 = γ (c1, a0, a1) p2 = γ (c1, p0, p1) p4 = γ (c6, p1, p3) a4 = γ (c3, a2, a3) a5 = γ (c1, a4, γ (c2, a4, a0)) Note that no definition can reach a use along an infeasible path, so ∅ can be simplified away. At Line 1 of Algorithm 1, all modified abstract memory locations are ini- tialized with ExpList [∗m] = {〈B1, ∗m〉}. The unconstrained values represent unknown initial values of the corresponding abstract memory locations. The rest of the example describes the steps of the algorithm executed for each basic block. The algorithm visits basic blocks in the reverse post-order (which corresponds to the numerical order of subscripts of blocks in Fig. 3.5(a)). B1. At the end of the first block, the state of the algorithm is ExpTable [a0] = 0. B2. The second block defines a new local variable a1, writes value 7 to the abstract memory location pointed to by p0, and defines a new pointer p1. After 57 Chapter 3. Structure-Preserving Symbolic Execution this block is symbolically executed, the state of the algorithm is: ExpTable [a1] = ∗p0, ExpList [∗p0] = {〈B2, 7〉, 〈B1, ∗p0〉}, and ExpTable [p1] = g1. Since ∗p0 is written, the algorithm schedules merge operations at DF (B2) + = {B3, B9}, so Merge [B3] = {∗p0}, and Merge [B9] = {∗p0}. B3. At the beginning of B3, the definitions of ∗p0 are merged: GP (B3)|ExpList[∗p] = γ (c1, B1, γ (c2, B2, ∅))|{〈B2,7〉,〈B1,∗p0〉} = γ (c1, ∗p0, 7) The new definition is appended to the expression list: ExpList [∗p0] = {〈B1, ∗p0〉, 〈B2, 7〉, 〈B3, γ (c1, ∗p0, 7)〉}.16 The definition of a2 is added to the expression table: ExpTable [a2] = γ (c1, a0, ∗p0). Notice that a1 was replaced with its defi- nition (∗p0). B4. The symbolic execution of the basic block B4 only updates the table with ExpTable [a3] = γ (c1, a0, ∗p0) + 1. B5. The symbolic execution of B5 is simple — only one definition is added to the expression table: ExpTable [a4] = γ (c3, γ (c1, a0, ∗p0) , γ (c1, a0, ∗p0) + 1). The simplifier in Calysto would simplify that definition to: ExpTable [a4] = γ (c1, a0, ∗p0) + γ (c3, 0, 1). B6. This basic block does not change the state of the algorithm because it contains only the branch statement. B7. Only one new definition is created in this block: ExpTable [p2] = g2. B8. After execution of the first statement in blockB8, we have: ExpTable [p4] = γ (c6, g1, g2). The write statement ∗p3 = 0; is handled by lines 19–31 of Algo- rithm 1. First, the algorithm precomputes the definitions for ∗g1 and ∗g2, which 16Note that B2 and B3 cover all paths B1 −→ {B2, B3} +−→ Bx, but since the proposed algorithm currently does not detect k-postdominance, the definition in B1 will not be erased. 58 Chapter 3. Structure-Preserving Symbolic Execution are unconstrained values (equal to the values at the point of the function call). Second, the restriction is computed as in Example 3: γ (c6, g1, g2)|0g1 = γ (c6, 0,Λ) γ (c6, g1, g2)|0g2 = γ (c6,Λ, 0) After the substitution of the place holder is performed we get: ExpList [∗g1] = {〈B1, ∗g1〉, 〈B8, γ (c6, 0, ∗g1)〉}, and ExpList [∗g2] = {〈B1, ∗g2〉, 〈B8, γ (c6, ∗g2, 0)〉}. Finally, merging of definitions of ∗g1 and ∗g2 is scheduled for B9, so we have Merge [B9] = {∗p0, ∗g1, ∗g2}. B9. The definitions are merged at B9 as follows: GP (B9)|ExpList[∗p0] = (3.1) γ (c1, γ (c1, ∗p0, 7) , γ (c2, γ (c1, ∗p0, 7) , γ (c6, 7, γ (c7, 7, 7)))) which can be simplified to: γ (c1, ∗p0, 7). The definition from B2 was used for predecessors B7 and B8, and the definition from B3 for the predecessor B5. Ob- serve how the restriction operator avoids recomputing the redundant definitions and takes the definition from the nearest dominator that defines the value. As B9 postdominates the definitions from B1, B2, and B3, those three are deleted. The same is repeated for ∗g1 and ∗g2 (the expression for ∗g2 is immediately simplified): GP (B9)|ExpList[∗g1] = (3.2) γ(c1, ∗g1, γ ( c2, ∗g1, γ ( c6, γ(c6, 0, ∗g1), γ ( c7, γ(c6, 0, ∗g1), ∗g1 ))) ) = γ (c1, ∗g1, γ (c2, ∗g1, γ (c6, 0, ∗g1))) GP (B9)|ExpList[∗g2] = (3.3) γ(c1, ∗g2, γ(c2, ∗g2, γ ( c6, ∗g2, γ (c7, 0 ∗ g2) ) )) 59 Chapter 3. Structure-Preserving Symbolic Execution Finally, the definition of a5 is computed and simplified as: ExpTable [a5] = γ (c1, γ (c1, a0, ∗p0) + γ(c3, 0, 1) , γ ( c2, γ (c1, a0, ∗p0) + γ (c3, 0, 1) , a0 ) ) = γ(c1, a0 + γ (c3, 0, 1) , γ ( c2, ∗p0 + γ (c3, 0, 1) , a0 ) ) Although the symbolic expression of the returned value (a5), is relatively com- plex, it actually depends only on one parameter visible in the caller’s context: the abstract memory location to which p0 terminal pointer points to (assuming that the conditions do not depend on any parameters or globals). This completes the example. Equation 3.1 represents the effect of the func- tion on the abstract memory location(s) to which the actual parameter p0 of the function can point. The other two equations (Eq. 3.2 and 3.3) represent the effect of the function on the globals. So, the simulated function has four side-effects: the return value, and modifications of abstract memory locations to which p0, g1, and g2 point. The computed γ definitions can be interpreted as ITE expressions. The following example illustrates both how symbolic execution indirectly performs slicing and how dataflow dependencies (structure) are reflected in the computed symbolic expressions. Example 5: For the following code fragment: x0 = a + b ; i f (x0 < 0) { x1 = −x0 ; y1 = y0 ; } else { x1 = x0 ; y1 = y0 + x0 ; } 60 Chapter 3. Structure-Preserving Symbolic Execution VC <= y1 ITE0 x1 ITE > cond - T x0 + F cond + F y0 T 0 a b Figure 3.6: Maximally-Shared Graph. Nodes are labeled with the correspond- ing operators, variable names, and constants. The Zero node is duplicated to improve the graph layout. The ITE outgoing edges are labeled with cond for the conditional branch, T for the if-branch, and F for the then-branch. assert (0 <= y1 ) ; Calysto would compute the maximally-shared graph shown in Fig. 3.6. Although the maximally-shared graph representation is identical to the standard logical representation, graphs have two interesting additional properties: Explicit Structure. Our VC is dataflow dependent on the variable a, and that is evident from the graph, because there is a path from the node representing the VC to the node representing a. Simple Slicing. In any maximally-shared graph, a node, say x, is dataflow dependent on another node, say y, if and only if there exists a path in the graph from x to y. Otherwise, y is irrelevant for the computation of x. Going back to our example, there exists no path from the node representing the VC to the node representing x1. In effect, x1 can be 61 Chapter 3. Structure-Preserving Symbolic Execution sliced away. More generally, all nodes that are unreachable from the node we are interested in can be sliced away. The graph representation makes such slicing very simple. 62 Chapter 3. Structure-Preserving Symbolic Execution 3.3 Complexity and Correctness The first part of this section focuses on the space complexity of the proposed algorithm. The second part introduces and proves the main correctness invari- ants. 3.3.1 Space Complexity This section discusses the space complexity of the proposed symbolic execution algorithm. The section assumes acyclic CFGs — cyclic CFGs can be reduced to the acyclic ones by replacing loops with appropriate loop invariants. We begin by setting a bound on the size of gating path expressions. Lemma 2. Let B represent a set of basic blocks in an acyclic CFG C. For any B ∈ B, GP (B) can be represented as a maximally-shared graph with at most |B| nodes. Proof: Let C ′ be equal to C with all non-branch statements eliminated. Reverse the edges of C ′ and discard the following nodes and their corresponding outgoing edges: • all nodes unreachable from B, • all nodes reachable from idom (B), and • block B itself, getting C ′′. Reverse the edges of C ′′. The obtained subgraph contains only blocks that are on at least one path idom (B) ∗−→ Bp, where Bp ∈ Pred(B). Graph C ′′ might contain redundant basic blocks, which can be eliminated by repeating the following graph transformation for all blocks Bi 6∈ Pred(B) until no change: • If Bi has two outgoing edges, and they both have the same destination, replace the conditional branch that terminates Bi with an unconditional one and discard one of the outgoing edges. 63 Chapter 3. Structure-Preserving Symbolic Execution • If Bi is terminated by an unconditional branch, disconnect all incoming edges and reconnect them to the successor of Bi, short-circuiting Bi. Dis- card Bi and its outgoing edges. This transformation corresponds to canonicalization of gating path expressions (explained after Definition 4 on page 25). Let Bc denote a set of blocks Bi whose branching conditions ci are used in the canonical GP (B). Graph C ′′′ contains exactly the set of nodes Pred(B) ∪ Bc. The first transformation guarantees that C ′′′ cannot contain any nodes that are not on idom (B) ∗−→ Bp. The second transformation discards only blocks that would be eliminated in the canonical GP (B), and nothing else. Thus, the canonical GP (B) represents exactly the branching conditions on all the paths idom (B) ∗−→ Bp in C ′′′. Now, we only need to translate C ′′′ into the corresponding GP (B). Traverse C ′′′ in postorder, constructing a partial gating path expression E(Bi) for each block Bi ∈ C ′′′ as follows: • If Bi ∈ Pred(B), then E(Bi) = Bi. • If Bi ∈ C ′′′\Pred(B), then Bi must be a branch node (thanks to the pre- viously done transformations). Let Bi be terminated with branch(ci, BT , BF ). Create an expression E(Bi) = γ (ci, BT , BF ). If either of the two basic blocks is not in Pred(B), replace it with ∅. The last step creates at most |C ′′′| expressions. Now replace all Bi 6∈ Pred(B) in each of those expressions with E(Bi). Using structural hashing, create a maximally-shared graph for E(idom (B)). The created graph is semantically equivalent to GP (B), according to Definition 4, and contains at most |B| nodes. The number of terminals of a gating path expression is limited by the maxi- mal number of predecessors any block can have. Thus, the number of terminals is equal to the maximal cut-width of a CFG, which is typically much smaller in practice than the total number of blocks. 64 Chapter 3. Structure-Preserving Symbolic Execution Now we turn our attention to Algorithm 1 and its space complexity. We begin by setting bounds on the size of expressions generated by the two restric- tion operators. The proof of Lemma 2 can also be used to set the bound on the size of any γ-function to |B|, which means that a definition of any variable or pointer can be represented as a maximally-shared graph of size at most |B|. It follows that the first restriction operator (Definition 6 on page 43) can create at most |B| new nodes. Note that the definitions substituted for the terminals of GP (B) were already created elsewhere. Similarly, the second restriction op- erator (Definition 7 on page 45) copies the restricted γ-function and replaces the terminals either with x, or with a placeholder Λ, creating at most |B| new nodes. The following theorem sets pessimistic worst-case bounds on the total size of symbolic expressions that an implementation of Algorithm 1 based on maximally- shared graphs computes for a call-free function. Theorem 1. Given a call-free function f with k operators and |B| basic blocks, an implementation of Algorithm 1 that uses maximally-shared graphs creates at most O(k · |B|2) nodes. Each node represents a variable, a constant, or an operator. Proof: Symbolic execution of any operator can create at most one new node — the result of the operator. Symbolic execution of a GSA γ-function can create at most |B| nodes (proof similar to the proof of Lemma 2). A memory read can create at most |B| nodes, according to the previous analysis. A memory write to some pointer x first reads all the terminal pointers (Line 23 in Algorithm 1), creating at most t · |B| new nodes, where t is the number of the terminal pointers in the definition of x. Line 29 takes the definition of the written pointer x, which is of size at most |B|, and substitutes the definitions computed in the previous step. Hence, a memory write can create at most (t+ 1) · |B| new nodes. The worst case value of t is |B|, because each block can have at most |B|− 1 predecessors. This is a very conservative estimate, because the maximal CFG cut-width is usually much smaller than |B|. 65 Chapter 3. Structure-Preserving Symbolic Execution Thus, we conclude that the symbolic execution of a call-free function f can construct at most O(k · |B|2) nodes in the worst case. This bound is very pessimistic. First, it does not take sharing and expres- sion simplification into account. With common subexpression elimination and simplification, the proposed algorithm in practice usually creates a symbolic rep- resentation of a call-free function that is linear with the number of statements in the function. Second, functions usually have a maximal cut-width that is much smaller than the number of basic blocks. Structured programs without the switch statement tend to have especially small cut-widths [127]. 3.3.2 Correctness When computing a definition of an abstract memory location, Algorithm 1 considers only the definitions reaching B from Pred(B) or immediate dominators of blocks in Pred(B). We start this section with a lemma that explains why this localization is correct. Intuitively, the lemma shows that if a definition in the ExpList precedes a predecessor of B, it must also dominate it. Lemma 3. Let B be a basic block processed at some iteration of the algorithm. Let ∗p be some abstract memory location, and x a term. Then it holds: ∀〈Bw, x〉 ∈ ExpList [∗p] : (∀Bp ∈ Pred(B) : Bw ≺ Bp ⇒ BwBp) Proof: Let us assume the opposite: ∃〈Bw, x〉 ∈ ExpList [∗p] : (∃Bp ∈ Pred(B) : Bw ≺ Bp ∧ Bw 6 Bp) Right after the initialization, the assumption is obviously false because the Entry node has no predecessors. In every subsequent iteration of the algorithm, the definitions are merged at the nodes that belong to the IDF of Bw. With no loss of generality, we can assume that Bw2 is the last merge point that still precedes B. So, the assumption can be rewritten as: ∃Bw2 ∈ DF (Bw)+ : (∃Bp ∈ Pred(B) : Bw Bw2 Bp ∧Bw2 6 Bp) 66 Chapter 3. Structure-Preserving Symbolic Execution If Bw2 = Bp, the assumption is trivially false. As Bw2 is the last merge point, there can not exist another merge point Bw3 such that Bw Bw2 ≺ Bw3 ≺ Bp, so there can be no more join nodes on the path Bw2 +−→ Bp (including Bp). Hence, Bw2 has to dominate Bp. This contradicts the assumption. If Lemma 3 did not hold, the restriction operator would have to recursively traverse the CFG, searching for definitions. Any time a new definition is appended to ExpList , all overwritten (postdom- inated) definitions can be safely deleted. The following lemma proves that such deletions do not violate Lemma 3: Lemma 4. Let 〈B,Def 〉 be a new location-definition pair that is about to be added to ExpList [∗p]. Deleting all 〈Bi, x〉 from ExpList [∗p], such that B m Bi preserves the main invariant of the algorithm (Lemma 3). Proof: Let Bw denote a block postdominated by B. Assume that both blocks write to ∗p. Due to the reverse postorder traversal, all predecessors of B have been already visited. Thus, only its proper descendants or siblings can be visited in the future iterations. Let Bd be a descendant of B. Assume that there exists a path Bw +−→ Bd that avoidsB. Then there exists a pathBw +−→ Bd ∗−→ Exit that avoids visiting B. This contradicts the definition of the postdominance relation. Hence, any path Entry +−→ Bd has to pass through B if it passes through Bw. It follows that if Bw is a dominator of Bd, so is B. This preserves the property. Let Bs be a sibling of B. Obviously, B cannot postdominate Bs or any of its dominators. Consequently, a write to ∗p does not influence reads/writes in Bs. All the entries in the expression table must be well-defined and contain no undefined elements. The only undefined element that is used at several places is the Λ placeholder. Lemma 5 proves that Λ is not in the support of any expressions added to the expression table: 67 Chapter 3. Structure-Preserving Symbolic Execution Lemma 5. Given a non-empty set R, the gating path restrictions in Algorithms 1 and 2 produce a γ-function that contains no Λ placeholder. Proof: The gating path expression for B contains only predecessors of B as terminal symbols. According to Definition 6 on page 43, each predecessor Bp in the gating path expression will be replaced either with the value of an ab- stract memory location ∗p written in B or Bp, or with a value of that location written in the nearest dominator of Bp, or with Λ. However, from Lemma 3, it follows that for any predecessor Bp, either Bp or at least one of its domina- tors is in ExpList [∗p]. Thus, the first restriction operator can never create a Λ subexpression. The second restriction operator can generate Λ subexpressions, but those are replaced by parallel substitution at Line 16. Lemma 3 proves that for any abstract memory location ∗p and any use in some basic block B, the expression list ExpList [∗p] contains only definitions that either dominate or do not even precede the predecessors of B. Since the dominators of any basic block can be linearly ordered and the restriction oper- ator takes the most recent definition reaching each predecessor, it follows that the definition of ∗p is constructed correctly for each predecessor of B. The gat- ing path expression handles all the paths idom (B) ∗−→ Bp −→ B such that Bp ∈ Pred(B). If there exists a definition d in B, all terminals in the gating path expression are replaced with d, and the expression can be trivially simpli- fied to d. Otherwise, the restriction operator takes the definition in Bp, if there is one, or the definition that is in the nearest dominator of Bp that defines the location. Lemma 4 proves that this property is maintained even if postdomi- nated definitions are removed from ExpList . Finally, Lemma 5 shows that each computed symbolic definition is completely defined. 68 Chapter 4 Structural Abstraction The previous chapter dealt with the intraprocedural analysis, while this chapter discusses how the results of the intraprocedural analysis are used for interproce- durally path-sensitive VC validity checking. Section 4.1 provides the intuition behind structural abstraction, motivates the approach with a real-world exam- ple, and outlines the underlying assumptions and design decisions. Further- more, Section 4.1 lists the most important contributions. Section 4.2 starts by enumerating the types of side-effects that each function can have on the caller’s context, and discusses how different types of side-effects interact with structural abstraction. The rest of the section focuses on the abstraction and refinement, and proposes several improvements at the end. 4.1 Introduction In general, proving software VCs requires interprocedural analysis, e.g., of the propagation of data-flow facts. Some properties, like proper nesting of lock- unlock calls, tend to be localized to a single function and are amenable to simpler analysis. Many others, especially pointer-related properties, tend to span through many function calls. To handle the complexity of interprocedural analysis, the software analysis community has developed a number of increasingly expensive abstractions. For instance, path-insensitive analysis does not track the exact path along which a certain statement is executed, while context-insensitive analysis does not differ- entiate the contexts from which a function is called. These abstractions work well in optimizing compilers, but are not precise enough for verification pur- 69 Chapter 4. Structural Abstraction poses. Software verification analysis has to be both path- and context-sensitive (*-sensitive) to keep the number of false errors low. The proposed analysis is both path- and context-sensitive. It is based on an abstraction-checking-refinement framework that exploits the natural function- level abstraction boundaries present in software. 4.1.1 Intuition As discussed in Section 3.2.3, the symbolic execution that computes maximally- shared graphs also indirectly slices the simulated function with respect to its side-effects. Each such slice will be called a summary of a side-effect, and can be seen as a pure function that produces a single result and takes a number of inputs, which are a subset of the parameters passed to the simulated function. Interprocedural analysis of pure functions is relatively simple. When a func- tion call is symbolically executed, formal parameters of the callee have to be substituted with the symbolic definitions of actual parameters, and the abstract memory locations that the callee reads have to be replaced with their symbolic counterparts in the caller’s context. That is the easy part. The harder problem is what to do with the summary itself. According to my experiments, the full expansion of each summary is infeasible. I ran experiments on a number of pro- grams, and even on very small non-recursive programs (3–5 KLOC) full inlining results in an exponential blowup, while on smaller programs it bloats the code size 50–180 times the original size. A better approach, used in this thesis, is to represent summaries with summary operators, which serve only as placehold- ers for the full symbolic representation of the summary, and can be expanded into such a full representation later on demand, if needed. Summary operators are going to be named with function :effect notation, representing the function being summarized and the particular side-effect (more on this in Section 4.2.1). When checking a VC, structural abstraction initially treats all summary operators as unconstrained variables, effectively abstracting away the subex- pressions of the VC. If the decision procedure finds a spurious counterexample, 70 Chapter 4. Structural Abstraction structural refinement finds which summary operator needs to be fully inter- preted, and inlines the symbolic expression represented by the operator. This process is repeated until the VC is either proven valid, or a concrete counterex- ample is found. Experimental results in Chapter 6 show that structural abstraction is very robust and scalable in practice, because it effectively exploits the same natural abstraction boundaries that programmers use when writing a program — func- tions. Programmers organize code into functions and use them as abstractions. They tend to ignore the details of the effects of the function on the caller’s context — the easiest invariant to remember is to remember no invariant at all. 4.1.2 Motivating Example This section gives an example that provides intuition about how structural ab- straction approach solves *-sensitive VCs. The code used in the example is a simplified and slightly modified piece of code from a real application.17 Example 6: Through the example, we shall follow a sequence of steps needed to prove the assertion on line 23. To prove an assertion, we need to prove either that the assertion itself is unreachable, or that it always evaluates to true. 1 int global1 , global2 ; 2 3 // I f ∗data<0, r e t u r n s t r u e and computes 4 // ∗ data=abs (∗ data ) . 5 bool flip ( int ∗data ) { 6 i f (∗ data < 0) { 7 ∗data = −(∗data ) ; 8 return true ; 9 } 10 return fa l se ; 17The example is modified from the bit-vector arithmetic theorem prover Spear. 71 Chapter 4. Structural Abstraction 11 } 12 13 // Assume i n i t i s a pure f u n c t i o n ( no s ide−e f f e c t s ) . 14 int init ( int x ) { 15 // Some e x p e n s i v e computation . . . 16 } 17 18 // I f g l o b a l 1 i s p o s i t i v e and g l o b a l 2 i s nega t i ve , 19 // s c a l e s g l o b a l 1 by abs ( g l o b a l 2 ) . 20 void scale ( ) { 21 global2 = init ( global1 ) ; 22 i f ( flip(&global2 ) ) { 23 assert ( global2 != 0 ) ; // Div by zero check . 24 global1 /= global2 ; 25 } 26 } The VC computed for the assertion at Line 23 is an implication (if Line 23 is reachable, the asserted condition should hold) that can be represented as a maximally-shared graph (Fig. 4.1(a)). Each side-effect is represented by a summary operator. Outgoing edges point to parameters required for computing the result of the particular side-effect. The antecedent contains two nested function calls. The consequent is a simple comparison of zero with the effect of flip on the global variable. The function flip has two effects: it returns a Boolean value and modifies the abstract memory location pointed to by its parameter. In the caller’s context, the side-effects of a function call are represented by a placeholder (summary operator) node. Each summary operator is named with the function :effect notation explained in Section 4.2.1. For example, the return value of a call to flip will be an operator node labeled flip :ret . Each summary operator represents a symbolic expression. Expansion of an operator corresponds to a round of 72 Chapter 4. Structural Abstraction Figure 4.1: Structural Abstraction Example. Summary nodes are structurally refined in the following sequence: flip :ret , flip :global2 , and finally init :ret . The subgraph obtained by the refinement of init :ret is represented by a triangle. For simplicity, these figures do not show pointer references and dereferences. inlining. To be fully context-sensitive, the obvious approach is to completely inline all calls. Such inlining leads to exponential blow-up even on small applications. A better approach is to track the individual effects of a function separately. This fine-grained approach makes it possible to expand only the slice of the called function that is actually in the cone of influence of the verified property. Together with the common subexpression elimination, this approach is more scalable, but does not offer satisfactory performance. The crux of the problem is that interprocedural analysis can’t decide when to stop inlining. After only three refinements, *-sensitive analysis would expand computationally expensive init , rendering the problem much harder for the decision procedure. However, the VC can be proven to be valid after only 73 Chapter 4. Structural Abstraction two refinements Letx = init :ret(global1 ) x < 0⇒ ITE (x < 0,−x, x) 6= 0 which simplifies to true, no matter what init returns. Cases like this appear frequently in practice, especially during *-sensitive verification of data-intensive properties, like checking of assertions or global pointer properties. Structural abstraction initially considers all summary operators to be just unconstrained variables, and gradually refines the maximally-shared graph by expansion of selected summary operators, until the VC becomes valid, or the decision procedure finds a falsifying assignment that does not depend on any summary nodes. 4.1.3 Assumptions Structural abstraction imposes only two requirements: 1. Each summary operator must represent a pure function. The symbolic execution presented in the previous chapter slices simulated functions into a number of pure sub-functions, represented by summary operators. 2. The maximally-shared graph representing a fully-refined VC has to be acyclic and of finite size. As long as there are no cycles in the CG, any VC can be represented as a finite maximally-shared graph (all function calls can be inlined). 4.1.4 Design Decisions The undecidability of software analysis in general and the focus on scalabil- ity motivated me to make a number of simplifying design assumptions. The presented analysis assumes that CFGs are not recursive and that there are no exceptions in the code. Also, the analysis is oblivious to concurrency. The dis- cussion below justifies those simplifying design decisions and addresses possible ways for eliminating them. 74 Chapter 4. Structural Abstraction Recursion. Tail-recursive calls can be transformed into loops [4], and loops can be replaced with expressions that represent their invariants (either manually or through abstract interpretation [39]), which can then be handled by struc- tural abstraction. General recursion (not tail-recursion) cannot be handled by structural abstraction, unless the user provides pre- and post-conditions of all functions involved in a recursive cycle. Exceptions. If the symbolic execution presented in the previous chapter could build the effects of exceptions into the summaries and VCs, structural abstrac- tion could reason about exceptions as well. However, the current analysis does not attempt to be smart about exceptions, and this is left for future work. Concurrency. Recent work on iterative context-bounding by Musuvathi and Qadeer [99] and later work by Bouajjani et al. [23] could be used in combination with structural abstraction: partial order reduction [67] performed directly on the source could generate a sequence of different interleavings of the source (or the intermediate form), which could then in turn be passed to a tool based on structural abstraction, like Calysto. However, it is very unlikely that this would be efficient. A more efficient approach would require a tighter integration of structural abstraction and analysis of different interleavings. 4.1.5 Contributions Structural abstraction and refinement have several advantages over other abs- traction-based approaches to interprocedural analysis, listed below. Incrementality. The abstraction-checking-refinement loop is strictly incre- mental because it always just keeps inlining symbolic expressions in place of summary operators. This is very important in practice because decision pro- cedures use learning to speed up the solving and to avoid revisiting the same search space. So, if that knowledge is discarded between iterations, the decision procedure has to re-learn all the facts it had already learned. This is clearly 75 Chapter 4. Structural Abstraction a waste of resources. Abstractions that learn new facts needed for refinement from the proof of unsatisfiability, like predicate abstraction [13], cannot be com- pletely incremental, because once a formula becomes unsatisfiable, adding new constraints does not change its status. Structural abstraction, on the other hand, uses satisfying18 assignments for refinement, and therefore is fundamen- tally different from other proposed approaches. Cheap Abstraction. Structural abstraction is very cheap, because abstrac- tion is performed by avoiding computation, rather than by computing potentially expensive abstractions. Cheap Refinement. Structural refinement is very efficient as well: given a falsifying assignment, the refinement walks over the maximally-shared graph representing a VC, and if it runs into a summarized node, expands it (refining the VC). Thus, structural refinement performs a number of steps that is at most linear with respect the size of the maximally-shared graph at the current iteration plus the size of the maximally-shared graph of the expanded summary operator. 4.2 Interprocedural Analysis This section explains how the intraprocedural analysis presented in the last chapter can be extended into an interprocedural one (taking the assumptions listed in Sections 3.1.2 and 4.1.3 into account). More specifically, Section 4.2.1 explains handling of different types of side-effects, while Sections 4.2.2 and 4.2.3 introduce structural abstraction and refinement. Finally, Section 4.2.4 suggests possible improvements. 18Falsifying; if we are talking about validity. 76 Chapter 4. Structural Abstraction 4.2.1 Side-Effects Functions can have multiple side-effects. This section begins by discussing sev- eral possible types of side-effects, and explains the function :effect notation for naming summary operators for each type. The second part of this section shows how the intraprocedural symbolic execution presented in the previous chapter can be extended to handle side-effects of function calls. Types of Side-Effects. A called function can influence the caller’s state in multiple ways, for instance by modifying abstract memory locations reachable through pointers passed as parameters, or by influencing the control-flow con- text (through assertions, assumptions, and aborts). The presented analysis distinguishes the following side-effects: Returned value represents the result returned by the called function, and this effect is going to be represented with a keyword ret . So, a summary oper- ator representing the returned value of foo would be denoted as foo :ret . Abstract memory locations that are visible in the caller’s context and mod- ified in the callee’s context cannot be represented with a finite number of summary operators, in general, because software can be an infinite-state system. However, assuming (1) logical memory model, (2) unrolling (or invariants) of loops, and (3) absence of recursion, the number of such modified abstract memory locations becomes finite.19 In the function :effect notation, those effects are going to be represented by expressions from the following grammar: effect ::= ∗eff eff ::= termptr | (eff + Z) | ∗eff 19Even with a more complex memory model based on the theory of non-extensional arrays [124], one could track a possibly infinite number of modified locations, while only a finite number of summary operators would be needed. 77 Chapter 4. Structural Abstraction Where Z is an integer offset, terminal pointers are denoted as termptr , and ∗ is a dereference operator. For instance, if function foo(p) modifies an abstract memory location at p + 8, the corresponding summary operator will be labelled with foo :∗(p + 8 ). Control-flow context of the caller can be impacted by assertions, assump- tions, and aborts executed in the context of the callee. This side-effect is going to be represented with a keyword cfc. Verification conditions are very similar to the control-flow context (CFC) effects, because both can be represented as Boolean summary operators and can’t be dereferenced. Unlike the CFC effects, which need to be assumed in the caller’s context, VCs need to be asserted. This process of pushing the callee’s VCs into the caller’s context is going to be called VC lifting. The simplest (and least effective) technique is to lift VCs all the way to the root of the CG, at which point they are guaranteed to be fully interprocedurally path-sensitive. Section 4.2.4 discusses more advanced techniques for dealing with VCs. Side-Effects and Symbolic Execution. In most cases, the intraprocedural analysis presented in the previous chapter can be easily extended to an inter- procedural analysis. The rest of this section analyzes handling of the function call statement. Handling the effect of a function call on the control-flow context is partic- ularly simple: given a summary operator foo :cfc that represents the effect of calling foo on the caller’s context, symbolic execution only has to conjoin foo :cfc with the control-flow context predicate in the caller’s context. If foo :cfc becomes relevant to any VCs, it can be lazily expanded later. VCs can be handled in the same way if the interprocedural analysis con- structs only one VC for the entire program. However, that is rarely useful in practice — constructing multiple VCs (one VC per assertion and per calling context) usually gives much more fine-grained feedback to the programmer, and 78 Chapter 4. Structural Abstraction facilitates elimination of multiple points of failure in a single run of the static checker. The most important problem of having multiple VCs is that each VC that depends on the calling context has to be lifted and asserted in the calling context. Obviously, this can lead to exponential blowup in the number of lifted VCs. Section 4.2.4 gives the details on how this blowup can be largely avoided in practice. Unlike the previous two side-effects, the return values and the modifications of abstract memory locations can also represent a pointer, which can be derefer- enced in the calling context. To keep the symbolic execution fully precise, such summaries need to be expanded if dereferenced. Calysto expands summaries heuristically — if the number of nodes in a summary is smaller than user-defined bound, the summary is expanded, otherwise the dereferenced pointer is consid- ered to be a terminal pointer. In the worst case, such an expansion can cause an exponential blowup, but such cases seem to be rare in practice — it is rare that a long sequence of function calls all modify the same memory location in a complex manner. It is very important to note that the expansion of dereferenced summary operators is a design tradeoff, not a feature of structural abstraction — one could imagine modeling pointer reads and writes with the theory of ar- rays, which would make the VCs harder for the decision procedure, but would avoid the potential exponential blowup during symbolic execution. 4.2.2 Abstraction The proposed approach follows the general paradigm of automatic, counter- example-guided, abstraction refinement [32], but unlike typical CEGAR ap- proaches, structural abstraction and refinement operations are entirely struc- tural, and the refinement works incrementally on abstract counterexamples (rather than concretizing the abstract counterexample, proving it spurious, and then analyzing the proof). Locations modified by a function call (either indi- rectly through a pointer, or directly via returned values) are initially considered 79 Chapter 4. Structural Abstraction to be unconstrained variables. Those unconstrained variables are incremen- tally refined until the formula represented by the graph becomes valid, or the falsifying assignment does not depend on any unconstrained variables. Incre- mental refinement performs structural refinement on maximally-shared graphs through expansion of summary operators. The refinement step replaces an un- constrained variable with a subgraph that represents the summary expression and the edges that were pointing to the unconstrained variable are relinked to point to the newly constructed expression. Algorithm 3 is a high-level rendition of the structural abstraction-refinement cycle. Algorithm 3 Main Abstraction-Checking-Refinement Loop. The checked VC is represented by a root F in the maximally-shared graph. The algorithm en- codes F (calling encode()) on the fly into formula f and passes it to the decision procedure (solve()). Summary nodes are encoded as unconstrained variables. If the decision procedure proves f valid, we are done. Otherwise, refinement takes F and the table of current assignments to variables in supp (F ), and returns true if the graph was refined, and false otherwise. If the graph was not refined, then all the summary nodes related to the falsifying assignment have been expanded, and the main loop terminates. Otherwise, the abstraction-checking-refinement cycle continues. Since maximally-shared graphs are acyclic, the algorithm nec- essarily terminates. 1: Let F be a node in the maximally-shared graph representing some VC. 2: f = encode(F ) 3: while ¬solve(f) do . solve returns false if a falsifying assignment is found 4: if ¬refine(F, current solution) then 5: Report a bug and exit. 6: end if 7: end while 8: Report VALID and exit. The algorithm interacts gracefully with incremental decision procedures — each expansion of a summary node replaces only a single node with the ex- 80 Chapter 4. Structural Abstraction pression represented by the summary node, monotonically increasing the set of constraints. The abstraction is done in the encode and refine functions in the cheapest way possible — by avoiding computation, or more precisely, by avoiding expan- sion of summary operators. Other abstractions, like predicate abstraction [13], typically perform relatively expensive recomputation of the abstractions in each abstraction-checking-refinement cycle. This lazy approach to interpretation of function summaries resembles the intuition behind lazy proof explication [62], a technique used to bridge between different theories in a theorem prover. The shared intuition is to abstract away expensive reasoning — expanding a function summary or solving a sub-theory query — as unconstrained variables, and then constrain them lazily, only as needed to refute solutions to the abstracted problem. The specifics of what to abstract and how to refine, of course, are different, since I am solving a different problem. 4.2.3 Refinement The first few iterations of the main loop of Algorithm 3 will likely return false counterexamples, since the initial abstraction is usually very crude. So, the refinement algorithm has to identify very quickly a set of summary nodes that are relevant to the falsifying assignment. The algorithm attempts to minimize the number of expanded summaries to avoid expensive computation. Given a falsifying assignment, my refinement scheme searches the graph and selects a single summary node to expand, thereby refining the model. In particular, the algorithm starts traversing the formula from the VC root. During the traversal, the algorithm detects don’t-care values — values that are irrelevant to the current solution and can therefore be ignored. I use the concept of absorptive element to formalize don’t-care values: Definition 9 (Absorptive Element). If there exists an element a for some op- erator ?, such that ∀x : a ? x = a, then a is an absorptive element of ?, denoted 81 Chapter 4. Structural Abstraction as abelem (?) = a. For instance, abelem (∧) = false and abelem (∗) = 0. If the decision procedure returns a falsifying assignment, each node F in the graph representing the checked VC has some assigned value, which will be denoted as val (F ). If F is an operator ?, the algorithm checks val (x) for each operand x of F . If val (x) is an absorptive element of ?, it is a sufficient explanation of the value of F in the falsifying assignment (the other operand is a don’t-care). Hence, it suffices to refine only x. The refinement procedure is given in Algorithm 4. As is usual for graph traversal, visited nodes are marked during traversal to avoid re-visiting nodes; marking is not shown in the pseudocode. Algorithm 4 Structural Refinement Algorithm. F is a node in the maximally- shared graph, and x and y are its operands. The return value indicates whether a summary has been expanded. 1: function refine(graph node F , values assigned to nodes) 2: if F is a summary node then 3: expand the summary for F ; return true 4: else if F is a leaf node then 5: return false 6: else if F ≡ x ? y then 7: if val (x) = abelem (?) then 8: return refine(x) 9: else if val (y) = abelem (?) then 10: return refine(y) 11: else 12: return refine(x) or refine(y) 13: (The or is lazy: if either call succeeds, the other is skipped.) 14: (The order is arbitrary. Either x or y can be refined first.) 15: end if 16: end if 17: end function Operators like implication and if-then-else can be rewritten in terms of simple 82 Chapter 4. Structural Abstraction operators (AND, OR) that have absorptive elements. The implementation of the refinement algorithm in Calysto avoids such rewrites for efficiency, and directly applies customized rules to both implication and if-then-else, according to the following algorithm: Algorithm 5 Additions to the Basic Refinement Algorithm. 1: if F ≡ (x⇒ y) and val (x) ≡ false then return refine(x) 2: else if F ≡ (x⇒ y) and val (y) ≡ true then return refine(y) 3: else if F ≡ (x⇒ y) return refine(x) or refine(y) 4: 5: if F ≡ ITE (c, x, y) and val (c) ≡ true return refine(c) or refine(x) 6: else if F ≡ ITE (c, x, y) return refine(c) or refine(y) Returning to the example in Fig. 4.1 on page 73, in 4.1(a), the checker treats the placeholder nodes as unconstrained variables and finds a falsifying assignment where flip :ret is true and the != is false. Algorithm 5 will derive the refinement in 4.1(b), where a possible falsifying solution gives the init :ret node a negative value. Next, the algorithm might choose to expand the flip :global2 node, yielding the refinement in 4.1(c), which is valid. Structural abstraction is able to avoid the expensive expansion of the init :ret node. Unlike other approaches, this approach to refinement does not require a the- orem prover. The downside is that such a refinement might be less precise and result in more refinement cycles. However, each refinement cycle only adds ad- ditional constraints to the decision procedure incrementally, making the solving phase more efficient as well. 4.2.4 Improvements Experiments revealed that the two most significant bottlenecks of näıvely im- plemented structural abstraction are: (1) lifting of VCs to the calling context exhausts available memory on large programs, and (2) the total run time of a static checker based on näıve structural abstraction is impractically long (a cou- 83 Chapter 4. Structural Abstraction ple of days for a ∼100 KLOC program). This section discusses implemented and further possible improvements. The improvements that have been already im- plemented significantly improve the overall performance and decrease memory consumption. The first part of this section focuses on the lifting of VCs, the sec- ond one on better refinement heuristics that localize the analysis, dramatically improving the overall performance, and the third part discusses the specifics of the implementation of common subexpression elimination in Calysto. Lifting. A simple, but effective, approach that avoids complete lifting of VCs to the root of the CFG is to avoid lifting VCs that are completely defined in terms of constants and local variables. If such a VC is valid, it is going to be valid in any context (and vice versa). Calysto uses this approach and manages to handle several hundred KLOC without exceeding the memory resources. The only downside of this approach is that the checker can report bugs in unreach- able code. For that reason, Calysto relies on the LLVM dead-code elimination pass and also implements an additional pass that eliminates functions that can never be called. This additional pass can be configured through user-controlled switches. For instance, a paranoid user who is checking a library might want to check all functions, even those with internal linkage that are not visible from the outside. In my own experience, programs either contain trivially unreachable functions, or almost all functions are reachable. Thus, these dead-code elimina- tion passes are sufficient to eliminate almost all cases when the checker would report a warning in unreachable code. An even better approach would be to check each VC before lifting — if a VC is already valid, there is no need to lift it any further. Likewise, if a falsifying assignment does not depend on any parameters or globals, the checker could report a bug (the same argument as above, about warnings in unreachable code, applies). This technique, dubbed localized structural abstraction, would require significantly less memory resources (because most VCs can be proven with very local reasoning, at least in my experience), and would also be applicable to checking of libraries, where unconstrained interfaces result in a large number of 84 Chapter 4. Structural Abstraction false positives. However, achieving incrementality with this technique would be much harder, because one would need to carry around the state of the theorem prover. With a specially designed theorem prover, this could be an interesting tradeoff, definitely requiring further research. Heuristics. The order in which branches are refined in Algorithm 4 actu- ally does matter. An especially effective heuristic that I found is to refine the consequents of implications before the antecedents. With that optimization, Calysto runs approximately 2–3 orders of magnitude faster. The rest of the section explains why this heuristic is so effective. Each time a VC φ is lifted to the calling context, a new VC is constructed that includes the control-flow context in the callee ψ, so we get ψ ⇒ φ. Further lifting prepends more antecedents, resulting in a formula that looks like: ψ0 ⇒ · · · ⇒ ψn ⇒ φ. The number of antecedents is equal to the number of calls in the call chain, and usually, there is some expression sharing among different antecedents. In practice, if programmers perform any checking (for instance, that a pointer is not NULL), usually those checks will be very local to the point where the checked property matters (for instance, where the pointer is dereferenced). So, very often properties can be proven with a very local analysis. If that is the case, an abstraction that avoids analyzing too many contexts actually works faster because it can prove the property to hold with very local reasoning. Re- fining consequents before antecedents achieves exactly that effect and localizes the analysis. It is very possible that other heuristics would improve the performance of structural abstraction even further, but that is left for future research. Merging Common Summary Operators. Elimination of common subex- pressions can also be applied to summary operators, at least those that summa- rize deterministic functions. If a function is non-deterministic, Calysto flags all the side-effects that are non-deterministic with a special flag. Flagged sum- 85 Chapter 4. Structural Abstraction mary operators can never be merged, because they could evaluate to a different result even if all the inputs are identical. In other words, summary operators are merged if and only if they represent the same fully deterministic side-effect and they have (structurally) equivalent operands. Common sources of non- determinism are calls to external functions (like random()), for which the code is not available for analysis. 86 Chapter 5 A Decision Procedure for Bit-Vector Arithmetic The previous two chapters discussed the intraprocedural analysis for computing function summaries and the structural abstraction that assembles and proves interprocedural VCs. Structural abstraction often avoids the full expansion of a VC, and makes the interprocedural path-sensitive analysis feasible. How- ever, even if the analysis is localized (as discussed in Section 4.2.4), there could be hundreds of thousands of VCs that a decision procedure has to (dis)prove. The large number of VCs and their complexity (due to interprocedural path- sensitivity) put a heavy load on the decision procedure. This chapter discusses my bit-vector arithmetic decision procedure, Spear, which I designed especially for software analysis. While structural abstraction in Calysto achieves the precision of interprocedural path-sensitivity, Spear provides the raw speed required to make the Calysto static checker practical. More specifically, this chapter focuses on the design of an application-specific decision procedure that is capable of exploiting the properties and structure of Calysto’s VCs. The presented insights and contributions generalize to other problem do- mains, as demonstrated by experimental results in Section 5.2.3. Even though my main motivation for designing Spear was to solve VCs produced by Ca- lysto as quickly as possible, thanks to a highly modular and configurable de- sign, Spear has been (automatically) optimized for a number of different classes of problems. The end result of such automatic optimization is superior perfor- 87 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic Figure 5.1: Architecture of Spear. In stand-alone mode, the solver can take as input a formula either in the logic described in Fig. 2.1 on page 14 or in the SMT QF BV standard format [108] through a translator provided with Spear. Simplified formulas are translated into CNF and passed to the custom-made SAT solver. The solver takes the formula and a suitable parameter configuration from a simple database. During solving, the solver occasionally calls the CNF simplifier. mance on a number of different classes of problems. 5.1 Architecture This section describes Spear’s architecture, illustrated in Fig. 5.1. The main components — expression simplifier, CNF encoder, dynamic CNF simplifier, and the custom-made SAT solver — are discussed in this section, while the role of a simple database with search parameter configurations will be clarified in Section 5.2. 88 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic 5.1.1 Expression Simplification Spear simplifies expressions in two steps: First, common subexpression elim- ination merges (structurally) equivalent expressions. Second, a term-rewriting engine simplifies the expressions starting at the leaves of the maximally-shared graph representing the formula and moving toward the root. The engine per- forms operations like constant-propagation (e.g., a+ 0 = a), constant-collection (e.g., a + 1 + 2 = a + 3), simple deduction (e.g., a < b ∧ a > b = false), redundancy elimination (e.g., ITE (φ, a, a) = a), partial canonicalization (e.g., ite(φ, φ ∧ a, b) = ite(φ, a, b)), strength-reduction (e.g., 3 ∗ a = a << 1 + a), and so on. This section explains how these two steps work in Spear. Common Subexpression Elimination (CSE) is a simple, but very impor- tant optimization for decision procedures based on SAT solvers. Spear elimi- nates common subexpressions by simple structural hashing. The following ex- ample illustrates the importance of CSE: Example 7: Let us assume that we have a bit-vector arithmetic formula a ∗ b 6= a ∗ b, where ∗ represents multiplication, while a and b are bit-vectors of length N . If such a formula is bit-blasted and passed to a SAT solver, even the best SAT solvers available today20 cannot solve the formula in a reasonable amount of time for N > 11. However, with trivial CSE, we get something like: x = a ∗ b ∧ x 6= x, which becomes trivial even for very large N . That is a trivial example that shows that CSE can exponentially speed-up de- cision procedures based on SAT solvers. The term rewriting engine in Spear is based on approximately 160 hard- coded simplification rules21, which can be broadly classified into several cat- 20As of May 2008. 21Different versions of Spear have different number of simplification rules. In general, I remove a rule when I discover that it is already covered by a combination of simpler rules, and add a rule when I notice an opportunity for simplification. 89 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic egories: constant-propagation, constant-collection, simple deduction, redun- dancy elimination, partial canonicalization, and strength reduction. In general, the engine is not recursive: it looks only at the operator and its operands, and very rarely recurses deeper into the subexpressions.22 This was a deliberate de- cision to speedup the simplifier — such an approach is a good-enough solution for the common cases appearing in software analysis, leaving the hard cases to the SAT solver later. While this customization gives superior performance on Calysto’s VCs, it is relatively easy to come up with small hard problems that defy Spear’s simplifier. 5.1.2 Conjunctive Normal Form Encoder Programs contain non-linear operators, and to be bit-precise, one must have a decision procedure that supports them. A number of different methods have been developed for linear bit-vector arithmetic, but few of them are applicable to non-linear operators. The usual approach is bit-blasting: Variables are encoded as bit-vectors of suitable size, and operators are replaced by digital circuits corresponding to that operator. In effect, VCs become large digital circuits, which can be converted to CNF using Tseitin’s transform [128] and given to a SAT solver. Numerous circuits have been proposed for each standard operation. Choos- ing the right circuit for CNF encoding is a little-researched but important prob- lem — properly selected circuit can easily make the decision procedure an order of magnitude faster. The heuristic I found most effective is to use gate-optimal circuits, i.e., circuits that have the minimal number of gates. Such circuits tend to generate the fewest variables during the encoding, thereby avoiding flooding the SAT solver with redundant variables, which tend to confuse the solver’s decision heuristics. For example, the encoder creates an instance of Guild’s array divider [70] circuit for each division operator that takes two variable inputs. Guild’s array 22 Some more complex rules require the engine to peek several subexpressions deep. 90 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic divider is very compact, as well as regular, making it an ideal choice for the CNF encoding. Operations with constants are encoded using a wide variety strength-reduction techniques [131], e.g., signed division of a 32-bit n by 13 is done as follows: n = n+ ((n Ashr 31) ∧ 12) q = (n Ashr 1) + (n Ashr 4) q = q + (q Ashr 4) + (q Ashr 5) q = q + (q Ashr 12) + (q Ashr 24) q = q Ashr 3 REMAINDER = n− 13q QUOTIENT = q + ((REMAINDER + 3) Ashr 4) For the meaning of operators, see Fig. 2.1 on page 14. Although the encoder tries to be gate-optimal, it can still create redundant variables, especially on the interfaces between multiple operators. The following section describes an effective way to eliminate such redundancies — dynamic CNF simplification. 5.1.3 Dynamic Conjunctive Normal Form Simplifier Developing Spear and HyperSAT [6], I experimented with a large number of static and dynamic CNF simplification techniques. The only two techniques that I found to consistently improve performance on a wide set of industrial instances are: variable elimination [55] and elimination of satisfied clauses and falsified literals [56]. This section focuses on the variable elimination technique, which is the more important of the two for Calysto’s VCs. Other techniques that I have tried did not consistently improve the solver’s performance. Some of those techniques are elimination of subsumed clauses [55], the pure-literal rule23 [45], symmetry-breaking [60], and equivalence pre- processing [130]. I also developed a number of novel techniques that I haven’t 23Called the affirmative-negative rule in the original paper. 91 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic found particularly effective so far: identification of clusters in formulas for lo- calizing the search process, the B-cubing [6] learning technique, and dynamic don’t-care analysis that was supposed to prevent the solver case-splitting on irrelevant variables24. However, I haven’t researched the full potential of those three techniques. Variable elimination is a simple and effective technique based on resolution. For example, given a variable v that appears in only two clauses25, (D1∨¬v) and (D2 ∨ v), where D1 and D2 are disjunct of literals, variable v can be eliminated and those two clauses can be replaced with a resolvent (D1 ∨D2). The technique was originally proposed by Eén and Biere [55] and imple- mented in SatELite. SatELite heuristically selects variables for elimination, and eliminates them by resolving each clause in which a selected variable ap- pears as a positive literal with each clause in which the variable appears as a negative literal. In the worst case, such resolution can quadratically increase the number of created clauses, so in practice, the number of clauses that can be generated is bounded heuristically. SatELite’s variable elimination heuristics are not tuned to Spear’s encoder and the structural properties of Calysto’s VCs. Extensive automatic tuning (Section 5.2) optimized the heuristics used for variable elimination as follows: Order of Elimination. Variable elimination in Spear sorts variables for elim- ination by the number of occurrences in all clauses in the clause database (both original and learned clauses), and starts with the rarest variables. Type of Elimination. Only original clauses are resolved, while the learned clauses that contain the eliminated variable (either as a positive, or a negative literal) are simply discarded. Frequency of Elimination. The dynamic simplifier runs variable elimination 24Done in collaboration with Leonardo de Moura and Nikolaj S. Bjørner. 25A literal is a Boolean variable or its negation. A cube (resp. clause) is a conjunction (resp. disjunction) of literals in which each variable appears at most once. 92 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic only after restarts, but not necessarily after each restart. In total, variable elimination can be run at most sixteen times. Preconditions for Elimination. A variable v can be only eliminated if all the following conditions hold: Clause Cutoff. Variable v appears in at most four clauses. Literal Cutoff. The total sum of the number of literals of all clauses in which v appears is less than one hundred. Max Clause Number Increment. The total number of new clauses af- ter the resolution is allowed to be only up to four times as large as the number of original clauses. 5.1.4 Custom-Made SAT Solver The core of Spear is a DPLL-style [44] SAT solver. Spear incorporates a number of novel optimizations, as described later in Section 5.3. However, one of the most important features of Spear and its core is configurability: every single search parameter, which is typically hard-coded in most off-the- shelf decision procedures, can be set on the command line. The full flexibility and power of this configurability becomes obvious in the following section on automatic tuning. 5.2 Automatic Tuning The work presented in this section is a joint work with Frank Hutter, Holger H. Hoos, and Alan J. Hu [78]. This section surveys the problems related to manual tuning, which is the traditional way of optimizing decision procedures, discusses the local search technique used for automatic optimization of Spear, gives experimental results, and finally discusses the best automatically found parameter configuration for solving Calysto’s VCs. 93 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic In practice, tuning a decision procedure for a specific class of problems is a challenging task. A high-performance decision procedure typically uses nu- merous heuristics that interact in complex ways. Some examples from the SAT-solving world include decision variable and phase selection, clause dele- tion, next watched literal selection, and initial variable ordering heuristics (e.g., [120, 97, 20]). The behavior and performance of these heuristics is typically con- trolled by parameters, and the complex effects and interactions between these parameters render their tuning extremely challenging. In this section, I explain how ParamILS, a recent parameter optimization tool developed by Hutter et al. [79], was used for optimization and develop- ment of Spear. ParamILS optimized Spear for a number of different classes of problems, computing a single parameter configuration for each class. These search parameter configurations are stored in the simple database (see Fig. 5.1 on page 88). Users can select different parameter configurations for different problem classes through command-line switches. Calysto automatically pre- selects the configuration optimized for its VCs. The presented technology of- fers significant performance improvements (up to two orders of magnitude) and avoids tedious and time-consuming manual tuning. 5.2.1 Manual Tuning After the first version of Spear was written and its correctness thoroughly tested26, I spent one week on manual performance optimization, which involved: 26Every release of the SAT solver in Spear was tested with 30 million random CNF in- stances. That number seems to be sufficient to uncover even the most elusive correctness bugs. Frequently, a few thousand random instances suffices to reveal a bug in a SAT solver. If Spear’s SAT solver found a test instance to be unsatisfiable, the instance was also checked with another SAT solver. The encoding of more complex operators (multiplication, division, remainder) was tested with approximately two million different tests for each bit-width. In effect, all complex operations that take operands with twenty or fewer bits were tested exhaus- tively. As complex operators are implemented through the simpler ones (addition, bit-wise operators,. . . ), the simpler operators were implicitly tested as well. The expression simplifier has not been thoroughly tested, but I carefully reviewed that code multiple times. A more 94 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic (i) optimization of the implementation, resulting in a speedup by a constant factor, with no effects on the search parameters, and (ii) manual optimization of roughly twenty search parameters, most of which were hard-coded and scattered around the code at the time. The manual parameter optimization was a slow and tedious process done in the following manner: I collected several medium-sized benchmark instances which Spear could solve in at most 1000 seconds and attempted to come up with a parameter configuration that would result in a minimum total runtime on this set. The benchmark set was very limited and included several medium-sized Bounded Model Checking (BMC) and some small Software Verification (SWV) instances generated by the Calysto static checker.27 Such a small set of test instances facilitates fast development cycles and experimentation, but has many disadvantages. Quickly, it became clear that implementation optimization gave more con- sistent speedups than parameter optimization. Even on such a small set of benchmarks, the variations due to different parameter settings were huge (2–3 orders of magnitude). Given the costly and tedious nature of the process, no fur- ther manual parameter optimization was performed after finding a configuration that seemed to work well on the chosen test set. Fig. 5.2 on the following page compares the performance of this manually tuned version of Spear against MiniSAT 2.0 [56], the winner of the industrial category of the 2005 SAT Competition and of the 2006 SAT Race. The two sets of instances used for this experiment, BMC and SWV, will be introduced in detail in Section 5.2.3 As can be seen from the runtime correlation plots shown in Fig. 5.2, both solvers perform quite similarly for BMC and easy SWV. For difficult SWV instances, however, MiniSAT clearly performs better. This seems to be the effect of focusing the manual tuning on a small number of easy systematic approach — testing through randomly generated bit-vector benchmarks — is left for future work. 27Small instances were selected because Calysto tends to occasionally generate very hard instances that would not be solved within a reasonable amount of time without automatic tuning. 95 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic (a) Bounded Model Checking (BMC) (b) Software Verification (SWV) Figure 5.2: MiniSAT 2.0 vs. Manually Tuned Spear. (a) The two solvers perform comparably on BMC instances, with average runtimes of 298 seconds (MiniSAT) vs. 341 seconds (Spear) for the instances solved by both algorithms. (b) Performance on easy and medium software verification instances is compa- rable, but MiniSAT scales better for harder instances. The average runtimes for instances solved by both algorithms are 30 seconds (MiniSAT) and 787 seconds (Spear). instances. For most decision procedures, the process of finding default (or hard-coded) parameter settings resembles the manual tuning described above. During the typical development process of a heuristic solver, certain heuristic choices and parameter settings are tested incrementally, typically using a modest collection of benchmark instances that are of particular interest to the developer. Many choices and parameter settings thus made are “locked in” during early stages of the process, and frequently, only a few parameters are exposed to the users of the finished solver. Furthermore, most users of these tools do not change these settings, and when they do, they typically apply the same manual approach. Not surprisingly, this manual configuration and tuning approach usually fails to 96 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic realize the full performance potential of a heuristic solver. 5.2.2 Parameter Optimization by Local Search This section gives a quick overview of ParamILS [79] — the tool used for auto- matic tuning of Spear. More specifically, the section focuses on the underlying ParamILS algorithm, which is motivated by the following manual parameter tuning technique often used by algorithm developers: • Start with some parameter configuration • Iteratively, modify one algorithm parameter at a time, keeping the mod- ification if performance on a given benchmark set improves and undoing it otherwise. • Terminate when no single parameter modification yields an improvement, or when the best configuration found so far is considered “good enough”. Notice that this is essentially a simple hill-climbing local search process, and as such it will typically terminate in a locally, but not globally, optimum parameter configuration, in which changing any single parameter value will not achieve any performance improvement. However, since parameters of heuristic algorithms are usually not independent, changing two or more parameter values at the same time may still improve performance. The problem of local optima is ubiquitous in local search, and many ap- proaches have been developed to effectively deal with them; one of these ap- proaches is Iterated Local Search (ILS) [105, 77], which provides the basis for ParamILS. ILS essentially alternates a subsidiary local search procedure (such as simple hill-climbing) with a perturbation phase, which lets the search escape from a local minimum. Additionally, an acceptance criterion is used to decide whether to continue the search from the most recently discovered local minimum or from some earlier local minimum. More precisely, starting from some initial parameter configuration, ParamILS first performs simple hill-climbing search 97 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic until a local minimum c is reached, and then it cycles through the following phases: 1. apply perturbation (in the form of multiple random parameter changes); 2. perform simple hill-climbing search until a new local minimum c′ is reached; 3. accept the better of the two configurations c and c′ as the starting point of the next cycle. ParamILS thus performs a biased random walk over locally optimum pa- rameter configurations. To determine the better of two configurations, it can use arbitrary scalar performance metrics, including expected runtime, expected solution quality (for optimization algorithms), or any other statistic on the per- formance of the algorithm to be tuned when applied to instances from a given benchmark set. This benchmark set is called the training set, in contrast to the test sets that are used later for evaluating the final parameter configurations ob- tained from ParamILS. As is customary in the empirical evaluation of machine learning algorithms, training and test sets are strictly disjoint. Clearly, the choice of the training set has important consequences for the performance of ParamILS. Ideally, the training set is homogeneous, i.e., the impact of parameter settings on the performance of the algorithm to be tuned is similar for all instances in the set. Frequently in practice, instances produced from a specific domain (here, software) and by a specific tool (here, Calysto) are fairly homogeneous. A more advanced version of ParamILS that was used for Spear tuning adaptively chooses the number of training instances to use for each parameter setting: while poor settings can be discarded after a few algorithm runs, promis- ing ones are evaluated on more instances. This mechanism avoids over-fitting to the instances in the training set. (For details, see [79].) 98 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic 5.2.3 Experimental Setup The automatic tuning technique described in the previous section was success- fully used to optimize Spear for a number of different problem domains, namely: Calysto instances, BMC [31], graph coloring [66], and quasigroup completion [68]. This section presents the experimental results on the first two mentioned problem domains. After introducing the training and test instances, this section specifies the experimental environment, and finally quickly lists the heuristic search parameters of the optimized version of Spear. Benchmark Sets. The BMC set consists of 754 IBM bounded model checking instances created by Zarpas [137], and the SWV set is comprised of 604 verifi- cation conditions generated by Calysto, as described in the previous chapters. Both instance sets, BMC and SWV, were split 50:50 into disjoint training and test sets. Only the training sets were used for tuning, and all the reported results are for the test sets. Experimental Environment. All experiments were carried out on a cluster of 55 dual 3.2GHz Intel Xeon PCs with 2MB cache and 2GB Random Access Memory (RAM), running OpenSuSE Linux 10.1. Reported times are Central Processing Unit (CPU) times per single CPU. Runs are terminated after 10 CPU hours or when they run out of memory and start swapping; both of these conditions are counted as time-outs. Search Parameters. The availability of automatic parameter tuning encour- aged me to parameterize many aspects of Spear. The first automatically tuned version exposed only a few important parameters, such as restart frequencies and variable priority increments. The results of automated tuning of those first versions of Spear prompted me to expose more and more search param- eters, up to the point where not only every single hard-coded parameter was exposed, but also a number of new parameter-dependent features were incorpo- rated. This process not only significantly improved Spear’s performance, but 99 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic also has driven the development of Spear itself. The resulting version of Spear used for the experiments reported in the following has 26 parameters: • 7 types of heuristics (with the number of different heuristics available shown in parentheses): – Variable decision heuristics (20) – Heuristics for sorting learned clauses (20) – Heuristics for sorting original clauses (20) – Resolution ordering heuristics (20) – Phase selection heuristics (7) – Clause deletion heuristics (3) – Resolution heuristics (3) • 12 double-precision floating point parameters, including variable and clause decay, restart increment, variable and clause activity increment, percent- age of random variable and phase decisions, heating/cooling factors for the percentage of random choices, etc. • 4 integer parameters which mostly control restarts and variable/clause elimination • 3 Boolean parameters which enable/disable simple optimizations such as the pure literal rule For each of Spear’s floating point and integer parameters, I chose lower and upper bounds on reasonable values and considered a number of values spread uniformly across the respective interval. This number ranges from three to eight, depending on my intuition about importance of the parameter. The total number of possible combinations after this discretization is 3.78 × 1018. By exploiting some dependencies between parameters, I reduced the number of distinctive configurations to 8.34× 1017. 100 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic 5.2.4 Experimental Results In practice, one typically cares about excellent performance on only a specific class of instances, such as BMC or SWV. This section gives experimental results of tuning Spear for these two specific sets of problems. Since users usually care most about an algorithm’s total runtime, ParamILS used average (arithmetic mean) runtime as the tuning optimization objective. The training cutoff time was set to 300 seconds, which according to Spear’s internal book-keeping mechanisms turned out to be sufficient for exercising all techniques implemented in the solver. In order to speed up the optimization, 95 hard BMC instances, which could not be solved by Spear with its default parameter configuration within one hour, were removed from the training set, leaving 287 instances for training. The tuner ran on the cluster for three days in the case of SWV and for two days for BMC on the training set. Fig. 5.3 on the next page shows the total effect of automatic tuning by comparing the performance of manually and automatically tuned Spear on the test set. For both the BMC and SWV sets, the scaling behavior of the tuned version is much better, and on average, large speedups are achieved — by a factor of 4.5 for BMC and 500 for SWV. Spear with the default settings even times out on four SWV instances after 10 000 seconds, while the tuned version solves every single instance in less than 20 seconds. Table 5.1 on page 103 summarizes the performance of MiniSAT 2.0 and Spear with both the default and automatically tuned parameter settings. No- tice that the versions of Spear specifically tuned for BMC and SWV clearly outperform MiniSAT: for BMC, Spear solves two additional instances and is faster by a factor of three on average; for SWV, the speedup factor is over 100. The presented automatic tuning approach is both very effective and signifi- cantly simplifies the design of decision procedures: Heuristics are hard to design and evaluate, and because of that most modern decision procedures are brittle and (manually) over-tuned to specific classes of problems. Automatic tuning frees the decision procedure designers to invent novel heuristics, eliminates the 101 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic (a) Bounded Model Checking (BMC) (b) Software Verification (SWV) Figure 5.3: Overall Improvements Achieved by Automatic Tuning of Spear. The scatter plots (a) and (b) compare the manually engineered default parameter configuration vs. the automatically optimized version for the BMC and SWV sets. Results are on test sets disjoint from the instances used for parameter optimization. (a) The default timed out on 90 instances after 10 000 seconds, while the tuned configuration solved four additional instances. For the instances that the default solved, mean runtimes are 341 seconds (default) and 75 seconds (tuned), a speedup factor of 4.5. (b) The default timed out on four instances after 10 000 seconds, the tuned configuration solved all instances in less then 20 seconds. For the instances that the default solved, mean runtimes are 787 seconds (default) and 1.35 seconds (tuned), a speedup factor of over 500. 102 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic Solver BMC SWV #(solved) runtime for solved #(solved) runtime for solved MiniSAT 2.0 289/377 360.9 302/302 161.3 Spear default 287/377 340.8 298/302 787.1 Spear tuned 291/377 113.7 302/302 1.5 Table 5.1: Summary of Automatic Tuning Results. For each solver and instance set, #(solved) denotes the number of instances solved within a CPU time of 10 hours, and the runtimes are the arithmetic mean runtimes for the instances solved by that solver. If an algorithm solves more instances, the shown average runtimes include more, and typically harder, instances. Note that the averages in this table differ from the runtimes given in the captions of Figures 5.2 and 5.3, because averages are taken with respect to different instance sets: for each solver, this table takes averages over all instances solved by that solver, whereas the figure captions state averages over the instances solved by both solvers compared in the respective figure. need for tedious manual experimentation, and manages to find effective param- eter configurations automatically, taking the properties and structure of the given class of instances into account. 5.2.5 Best Parameter Configuration Found The analysis of the best parameter configuration found for Calysto’s VCs gave some insights into their properties. According to the experimental results, the SWV instances prefer an activity-based heuristic that resolves ties by picking the variable with a larger product of occurrences. This heuristic might seem too aggressive, but helps the solver to focus on the most frequently used common subexpressions. It seems that a relatively small number of expressions play a crucial role in (dis)proving each verification condition, and this heuristic quickly narrows the search down to such expressions. The SWV instances also favored very aggressive restarts (first after only 50 conflicts), which in combination with 103 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic our experimental results shows that most such instances can be solved quickly if the right order of variables is found. A simple phase selection heuristic (always assign false first) seems to work well for SWV, and also produces more natural bug traces (small values of variables in the satisfying assignments). The VCs used for tuning correspond to NULL-pointer dereferencing checks, and this phase selection heuristic attempts to propagate NULL values first (all false), which explains its effectiveness. Calysto’s VCs prefer no randomness at all, which is probably the result of joint development of Calysto and Spear as a highly optimized tool chain for software verification. The BMC instances seem to prefer a significantly different parameter config- uration. Since the focus of this thesis is on the software analysis, the discussion of the best parameter configuration found for the BMC set is omitted and can be found in [78]. 5.3 Improving Performance The automatic tuning would not be very effective without having a choice of a wide range of heuristics and search parameters to tweak. The quality of choice is equally important as the quantity — if all heuristics are very similar, or if the search parameters do not have much impact on the dynamics of the search process, the size of combinatorial space is going to be pointlessly inflated. Such inflation of the combinatorial space dramatically reduces the effectiveness of automatic tuning. This section begins by describing several novel heuristics that I implemented in Spear, and ends with a quick description of another optimization technique used in Spear— memory latency hiding. 5.3.1 Novel Heuristics Spear features many novel heuristics and this section describes those that au- tomatic tuning identified as a good match for certain classes of problems. 104 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic The standard activity-based (e.g., [56]) heuristic seems to be the heuristic of choice for picking variables for case-splitting. Spear uses slightly more complex versions of the activity-based heuristic: ties are resolved by picking the variable either with the maximal product, or with the maximal sum of positive and neg- ative literal occurrences. Although Spear has been tuned for numerous classes of problems, the tuner consistently picks these two variants of the activity-based heuristic, although I implemented more than twenty different decision heuristics. One of the most robust phase selection heuristics is the minimal-work heuris- tic [6]. After picking a variable for case-splitting, Spear picks the phase of the variable so as to minimize the amount of work that needs to be done, by picking the phase that satisfies a longer list of watched clauses. Although this heuristic does not perform well on Calysto’s VCs, the tuner frequently identified it as the best heuristic on a number of other classes of problems. Interestingly, the best order for variable elimination seems to be highly de- pendent on the properties and structure of the problem class. The tuner picked a different ordering heuristic for each class of problems. This might imply that the heuristics for ordering the variables for elimination are still immature, and that more research might be needed. The heuristics for deletion of learned clauses seem to behave in a similar way: The tuner identified six out of seventeen clause sorting heuristics as good matches for various problem classes. The biggest surprise was the best config- uration the tuner found for graph coloring problems — the clauses that most frequently participate in conflicts are deleted first. That finding contradicts all common explanations of how and why clause deletion should work. 5.3.2 Prefetching According to the profiling I have done, Spear spends 60–80% of its time just waiting for the clauses to be fetched into the CPU cache. To hide this cache latency, I redesigned several algorithms in Spear, so as to be more predictable. 105 Chapter 5. A Decision Procedure for Bit-Vector Arithmetic Afterward, I inserted memory prefetch instructions28 at key places. For example, the conflict analysis algorithm in MiniSAT [56] uses a first-in- first-out algorithm, while a first-in-last-out algorithm produces the same result and has a more predictable memory access pattern. Inserting memory prefetches can also easily hurt the performance if there are too many outstanding prefetches or if the fetched memory locations are actually rarely used. For these reasons, it is important to properly guard the prefetch instructions and execute them only when there is a high probability that the fetched memory location will actually be needed. The amount of guarding has to be balanced with the cost of branch mispredictions, which can easily wipe out the advantage of prefetching. With these techniques, I managed to speedup Spear ∼30% on an AMD Athlon 64 X2 Dual Core 4600+ processor, which I used for development. The speedups on other processors were less remarkable, most likely due to different cache architectures and different costs of prefetches and mispredicted branches. 28Using the GCC builtin prefetch built-in function. 106 Chapter 6 Prototype and Experiments This chapter begins by explaining how the structure-preserving symbolic ex- ecution (Chapter 3), structural abstraction (Chapter 4), and a custom-made bit-vector arithmetic decision procedure (Chapter 5) are architected together in the Calysto static checker. Each component has been carefully designed with the others in mind, resulting in a harmonious, balanced, and well-performing system. The second part of the chapter presents the experimental results obtained on a wide variety of real-world applications. Despite the high cost of interprocedu- ral path-sensitivity and bit-precision, experimental results show that Calysto achieves scalability comparable to a less-precise static checker, Saturn [135], which was especially designed to be highly scalable. The experimental results show that Calysto performs well on a wide variety of real-world applications, but also show that the symbolic execution and VC-checking components are nicely balanced (roughly 50% of time spent in each). Intuitively, that balance suggests that the architecture is well-conceived. 6.1 Architecture The high-level architecture of Calysto is shown in Fig. 6.1. Calysto is de- signed as a compiler pass, in the spirit of Hoare’s “verifying compiler” grand challenge [75]: it accepts the compiler’s intermediate representation, in SSA form [43], performs various verification checks, issues bug reports and warnings, and then passes semantically unmodified SSA on to the compiler back-end. De- signing a static checker in this manner has the obvious advantage of language 107 Chapter 6. Prototype and Experiments Figure 6.1: High-Level Calysto Architecture. independence, but also helps to check for errors in the compiler front-end (and any other compiler passes that precede the Calysto pass). Supporting a differ- ent programming language requires only a different front-end and, if required, a name demangler to improve the legibility of bug reports. Internally, the Calysto system consists of three stages, supported by the decision procedure, Spear. The first stage is a lightweight function pointer alias analysis. It constructs (a sound approximation of) the call graph, including indirect calls through function pointers. The function pointer alias analysis is sound, flow-sensitive, and context-insensitive. This pass requires negligible resources, yet is precise enough in practice. The next stage is the symbolic execution presented in Chapter 3. Calysto symbolically executes functions in the analyzed program, computing symbolic definitions for each modified variable and memory location. These symbolic def- initions are used to create VCs. The symbolic execution machinery allows gen- erating VCs for any assertion at any point in the program. Calysto currently supports user-supplied assertions (written as Boolean expressions in whatever programming language the compiler front-end is parsing), but in the spirit of static checking also automatically generates VCs to check that each pointer dereference cannot be NULL. The last stage has a dual purpose: (1) it performs structural abstraction 108 Chapter 6. Prototype and Experiments and refinement, as described in Chapter 4, and (2) it also filters away all VCs corresponding to the same property within the same function. For instance, if a pointer that can be NULL is dereferenced at many different places within the same function, and that function can be called from many different contexts, Calysto will emit only one bug report per context. Such a filtering approach avoids overloading the programmer with reports that correspond to the same issue. The third stage also emits a detailed graphical trace for each falsified VC; if the falsified VC depends on any global variables, the trace is given all the way from the root of the call graph (the main function). Although traces are detailed, reading them still requires an intimate understanding of the underlying property checking engine — Calysto’s simplifier occasionally simplifies such large parts of formulas that it is very hard to build a complete mental model of the trace. Improving the usability of Calysto is left for future work. The actual validity-checking of VCs is done by Spear, explained in detail in the previous chapter. In short, Spear is a sound and complete decision procedure that supports Boolean logic, bit-vector operations, and bit-accurate arithmetic. Unlike other static checkers, which use general-purpose SAT solvers or theorem provers, Spear is custom-designed for the software VCs generated by Calysto, optimizing performance. 6.2 Experimental Results This section presents the experimental results of the combination of techniques presented in earlier chapters and prototyped in Calysto and Spear. More precisely, after introducing the benchmarks used for experiments, I give the experimental results of both Saturn and Calysto, and discuss my experiences with both static checkers. 109 Chapter 6. Prototype and Experiments Benchmark LOC (total) LOC (code) Modules Bftpd 1.8 4532 3306 1 Bftpd 1.9.2 4602 3368 1 HyperSAT 1.7 9123 6022 1 Spin 4.3.0 28394 20481 1 OpenSSH 4.6p1 81908 45304 11 INN 2.4.3 122727 71102 46 NTP 4.2.4p2-RC5 185865 74230 10 NTP 4.2.5p66 192019 74277 9 BIND 9.4.1p1 393318 184204 26 OpenLDAP 2.4.4a 374266 223595 27 TOTAL 1406754 685408 133 Table 6.1: Benchmarks Used for Experiments. The second and third columns show the number of lines of code before and after preprocessing (“LOC(code)” does not include comments, empty lines, and pragma-disabled code). The fourth column gives the number of compilation units produced by LLVM’s front-end. 6.2.1 Benchmarks To evaluate Calysto, I checked a number of publicly available, real-world ap- plications: the OpenSSH remote access server and client, the InterNetNews (INN) system, the Network Time Protocol (NTP) system, the Berkeley Internet Name Domain (BIND) DNS system, and the OpenLDAP Lightweight Directory Access Protocol system. Those benchmarks are the largest open-source benchmarks that I could successfully compile with both LLVM’s front-end and with the Sat- urn [135] static checker, which I used for comparison. I also used some smaller applications where I was able to get particularly prompt and precise feedback from the developers: the Bftpd FTP server, the HyperSAT Boolean satisfi- ability solver, and the Spin explicit-state model-checker. Table 6.2.1 lists the benchmarks. 110 Chapter 6. Prototype and Experiments I checked the automatically generated assertions that dereferenced point- ers cannot be NULL. This is an excellent property to use to evaluate a static checker because: (i) pointers are often passed through a long sequence of calls, which necessitates interprocedural analysis, (ii) pointer manipulation in pro- grams depends on both data- and control-flow of the program, exercising all the components of a static checker, and (iii) pointers are dereferenced very fre- quently in code — the number of produced VCs is probably larger than what would be generated by any other property (proper locking, for instance), and the sheer number of VCs pushes static checkers to their limits. Initially, I started sending raw reports to developers, but quickly found that developers were very unwilling to separate out real bugs from false positives (in- advertently validating my research goal of minimizing false positives!). I began filtering the reports myself, omitting all reports that I could prove infeasible. All remaining reports were sent to the developers. At that point, I ran into an unexpected problem: the developers would either take a very defensive stance, claiming that a particular bug was either irrelevant or very improbable, or would take a very cautious stance, fixing everything in the code just to be safe, without much thought about whether a bug was feasible or not. To be as rigorous as possible in my reporting of experiments, I have defined a bug strictly as follows: • Only a dereference of a pointer which is either uninitialized or NULL is considered a bug. • There must exist a feasible path from the point where the pointer was initially defined to the point where it was dereferenced. For Calysto’s evaluation, I also required that if any globals are included in the trace, the trace must be given all the way from the main function (root of the call graph). For Saturn’s evaluation, I waived this constraint, because Saturn does not produce precise enough interprocedural traces. • Every feasible NULL pointer dereference was considered a bug, no matter how improbable or irrelevant it might be. 111 Chapter 6. Prototype and Experiments • Many applications contained pointer checks. Usually, such checks exit if the pointer is NULL and print/log an appropriate message. In this case, the NULL pointer is never dereferenced, so failed pointer checks were not counted as bugs. In other words, only dereferences that would cause a segmentation fault were considered bugs. • If a pointer ptr was guaranteed not to be NULL, then I also assumed that pointers with offset ptr + i can never be NULL either. The likelihood of an integral overflow (ptr > 0∧ ptr+ i = 0) is extremely remote, and software developers do not take such reports seriously. Checking that the offset is within allowed bounds is a different property, not to be confused with NULL-pointer checking. For all reports that I could not prove to be false, I asked for a feasibility confir- mation from the developers. Reports that neither I nor developers could prove to be feasible or infeasible are classified as unknown. 6.2.2 Saturn’s Results Saturn [135], an inspiration for Calysto, was designed for scalability from the start. Despite Calysto’s more precise, more expensive analysis, can it match Saturn’s proven scalability? This section presents the results of running Saturn v1.1.29 on the selected 29 I am comparing against the most recent, most up-to-date version of Saturn available. An earlier version of Saturn reported low false error rates while checking for NULL pointer dereferences, although for a much less stringent notion of what constitutes a true bug [53]: in that paper, inconsistencies in whether a pointer is checked for NULL on different paths were included as real bugs; I am using the stricter definition described earlier. Unfortunately, the developers of Saturn have told me that they believe that the current version of Saturn is no longer as effective at identifying NULL pointer dereferences as the earlier version, that the previous version no longer exists, that they could not re-create it, and that they are not planning to update their NULL analysis to where they have confidence in it once again [2]. Thus, the only apples-to-apples comparison I can make, with the same definition of bugs, the same benchmarks, and the same machines, is with the current version of Saturn, which might not represent Saturn in the best possible light. Fortunately, the central point of the 112 Chapter 6. Prototype and Experiments benchmarks. I used only Saturn’s NULL pointer analysis, which finds possible NULL pointer dereferences, and the best known parameter settings. Saturn’s tutorial recommends using a 60 second timeout per function, so all the exper- iments presented in tabular form were obtained using the 60 second timeout. Experimental results for Saturn are presented in Table 6.2. 6.2.3 Calysto’s Results For Calysto, I used the default options, with a 10 second timeout per VC and limiting the number of VCs per function to 500, effectively setting the timeout per function to 5000 seconds. With a 5000 second timeout, Saturn produced the same results on Bftpd and Ntp and ran out of 16 GB of memory on all other benchmarks. Experimental results for Calysto are presented in Table 6.3 on page 115. 6.2.4 Discussion Saturn’s traces were significantly harder to interpret because the tool does not produce the complete trace and because I do not have the same level of familiar- ity with Saturn as I do with Calysto. Interestingly, there is very little overlap between the bugs reported by the two tools. Calysto tends to report either violations of C library properties, which Saturn frequently misses (presumably because of incomplete descriptions of C library functions), or very long traces, sometimes spanning through 10–15 functions, which Saturn misses due to lack of interprocedural path-sensitivity. On the other hand, most of bugs that Ca- lysto missed were due to assertion violations (once an assertion is violated, all the code after it becomes unreachable) and loop handling — if a loop can never be executed only once, then the conjunction of the control-flow context and an comparison is whether Calysto achieves comparable scalability to Saturn, despite performing analyses that are, by design, more precise and therefore presumably more expensive. Perhaps an earlier version of Saturn would have had precision closer to that of Calysto, but that question is moot. 113 Chapter 6. Prototype and Experiments Benchmark LOC Saturn v1.1 Reports Bugs Unkn. FP Rate Time [s] Bftpd 1.8 3306 3 3 0 0% 129.51 Bftpd 1.9.2 3368 3 3 0 0% 105.17 HyperSAT 1.7 6022 4 0 0 100% 647.21 Spin 4.3.0 20481 15 6 2 54% 2129.04 OpenSSH 4.6p1 45304 14 0 1 100% 3707.81 INN 2.4.3 71102 288 ∗12 34 96% 9879.69 NTP 4.2.4p2-RC5 74230 8 0 0 100% 326.44 NTP 4.2.5p66 74277 10 0 0 100% 319.33 BIND 9.4.1p1 184204 951 0 0 100% 14984.72 OpenLDAP 2.4.4a 223595 163 ∗14 50 88% 8098.48 TOTAL 685408 1459 38 87 97% 40226.40 Table 6.2: NULL Pointer Dereference Checking Results. “LOC” indicates the number of lines of true (after preprocessing) code. “Reports” is the total number of warnings produced on the benchmark. “Bugs” is the number of true bugs found. Starred (∗) bug numbers represent my best-effort estimation when I could not get confirmations from developers. Bug numbers without the star have been confirmed by the developers of the corresponding benchmark. “Unknown” shows the number of reports that could not be proved either feasible or infeasible. “FP Rate” gives the false positive rate, calculated as 1 −#Bugs/(#Reports − #Unknowns). “Time” is the total runtime in seconds. Experiments were on a dual Opteron 2.8 GHz with 16 GB RAM. 114 Chapter 6. Prototype and Experiments Benchmark LOC Calysto v1.5 Reports Bugs Unkn. FP Rate Time [s] Bftpd 1.8 3306 12 11 0 9% 3.14 Bftpd 1.9.2 3368 5 4 0 20% 2.86 HyperSAT 1.7 6022 0 0 0 0% 14.57 Spin 4.3.0 20481 0 0 0 0% 6858.10 OpenSSH 4.6p1 45304 4 1 0 75% 8995.64 INN 2.4.3 71102 10 ∗6 1 34% 1312.33 NTP 4.2.4p2-RC5 74230 30 26 0 14% 558.16 NTP 4.2.5p66 74277 13 4 3 56% 493.39 BIND 9.4.1p1 184204 5 ∗2 3 0% ]2436.88 OpenLDAP 2.4.4a 223595 20 15 2 27% 200.02 TOTAL 685408 99 69 9 23% ]20875.09 Table 6.3: Calysto NULL Pointer Dereference Checking Results. Starred (∗) bug numbers represent my best-effort estimation when I could not get confir- mations from developers. Bug numbers without the star have been confirmed by the developers of the corresponding benchmark. ] indicates that on BIND, Calysto’s runtime does not include instances on which Calysto failed to complete — it ran out of memory on 8 compilation units (taking an additional 6263.57 sec), and timed out in one day on one compilation unit. Experiments were on a dual Opteron 2.8 GHz with 16 GB RAM. assumption that the loop test has failed is false, implying that all the code after the loop is unreachable (and therefore unchecked). After I reported Bftpd, Spin, and NTP bugs, the developers immediately fixed all of them in the next release. Thanks to prompt responses from the Bftpd and NTP developers, I managed to check the new versions that fixed all the bugs found in the previous version. In the new versions, Calysto found new bugs, which have also been fixed in the meantime. The most frequent causes of Saturn’s false positives were: lack of interpro- cedural path-sensitivity, incomplete specifications of C library functions (for 115 Chapter 6. Prototype and Experiments instance, passing a NULL pointer to the free function is allowed), and specific code patterns that seemed like inconsistencies to Saturn. Saturn’s results on BIND are especially interesting. Bind’s code is among the highest quality code of all open-source applications I have seen so far — almost every single pointer is checked before dereferencing and complex data structures are checked for con- sistency before usage. This ubiquitous checking apparently confused Saturn’s inconsistency analysis because every single report was provably false. The ma- jor sources of Calysto’s false positives were: missing specifications of external functions, broken cycles in the call graph, and C type unsafety. The runtimes of both static checkers are comparable. Saturn is faster on some; Calysto, on others. Bind was particularly problematic for Calysto — it ran out of memory while analyzing 8 and timed out (1 day) on 1 compilation unit. The timeout was caused by a performance bug (failing to re-use certain cached results) in Calysto’s interprocedural analysis. I also analyzed how much time the two checkers spend in theorem-prover calls. The results are in Tables 6.4 and 6.5. Calysto spends almost 50% of its time in theorem prover calls, even with a small timeout (10 s) and a fast bit-vector arithmetic prover. The amount of time spent in the theorem prover calls is unsurprising, given how difficult the computed VCs are. Despite using a slower theorem prover [78], Saturn spends a much smaller fraction of its time in theorem prover calls. As mentioned earlier, Saturn performs expensive simpli- fication using BDDs during static analysis, whereas I use a fast and incomplete expression simplifier and maximally-shared graphs. Also, because of Saturn’s approximate interprocedural analysis, Saturn’s VCs are likely much simpler. The different tool design shows up in the different time proportions, but both tools end up being usably fast. 116 Chapter 6. Prototype and Experiments Calysto v1.5 Benchmark Total time [s] Spear [s] Percentage Bftpd 1.8 3.14 1.25 39.8% Bftpd 1.9.2 2.86 0.88 30.7% HyperSAT 1.7 14.57 0.10 0.6% Spin 4.3.0 6858.10 473.50 6.9% OpenSSH 4.6p1 8995.64 8167.36 90.7% INN 2.4.3 1312.33 14.77 1.1% NTP 4.2.4p2-RC5 558.16 56.38 10.1% NTP 4.2.5p66 493.39 58.03 11.7% BIND 9.4.1p1 ]2436.88 980.48 40.2% OpenLDAP 2.4.4a 200.02 181.90 90.9% TOTAL ]20875.09 9934.65 47.5% Table 6.4: Calysto Total Runtime Split. The Spear column shows the time spent in the theorem prover, with the next column showing the percentage of the total runtime. 117 Chapter 6. Prototype and Experiments Saturn v1.1 Benchmark Total time [s] MiniSAT [s] Percentage Bftpd 1.8 129.51 17 13.1% Bftpd 1.9.2 105.17 8 7.6% HyperSAT 1.7 647.21 232 35.8% Spin 4.3.0 2129.04 457 21.4% OpenSSH 4.6p1 3707.81 988 26.6% INN 2.4.3 9879.69 2104 21.2% NTP 4.2.4p2-RC5 326.44 82 25.1% NTP 4.2.5p66 319.33 80 25.0% BIND 9.4.1p1 14984.72 1141 7.6% OpenLDAP 2.4.4a 8098.48 949 11.7% TOTAL 40226.40 6058 15.0% Table 6.5: Saturn Total Runtime Split. Saturn’s theorem prover is the SAT solver MiniSAT. 118 Chapter 7 Related Work This chapter provides a survey of closely related literature, and puts the work presented in this thesis in the context of previous research. 7.1 Extended Static Checking The work in this thesis has been largely motivated by the work on extended static checking. That line of work is represented by several extended static checkers: ESC/Modula-3 [49], ESC/Java [63], and Boogie [90]. The focus of that body of work is on the intraprocedural analysis. The interprocedural anal- ysis is achieved through interface invariants (pre- and post-conditions) that must be provided by the user. This approach is particularly interesting for object ori- ented languages (Java,C#,. . . ), in which quantification over object interfaces can be used to prove complex properties about the architecture and intended usage of the interfaces. The work presented in this thesis focuses on the explicit interprocedural analysis that requires no interface invariants. The only require- ment is that the properties to be checked are specified. These properties can be specified once per language/library and reused for other applications written in the same language and/or based on the same library. Another approach to software static analysis is to design a light-weight anal- ysis and then use various heuristics to rank the probability that a bug found by the analysis is really a bug. Such an approach was used in MC [71, 58], which is a relatively imprecise, but very scalable C/C++ static checker. In contrast, I designed Calysto to be very precise. By exploiting the code structure at several levels of granularity, I showed that even a very precise checker can scale 119 Chapter 7. Related Work to several hundred thousand lines of code. My hope is that further research will make Calysto-like approaches competitive to MC in terms of scalability. Saturn [135, 134] is the project most similar in spirit to Calysto because it relies on a SAT solver, requires no user-provided invariants, and performs interprocedural analysis through function summarization. However, there are many important differences: 1. Calysto exploits the code structure at multiple levels of granularity: Symbolic execution preserves the dataflow dependencies in the original code and also exploits the structure of the CFG for faster construction of symbolic expressions. Structural abstraction relies upon the high-level code structure (functions) to perform abstraction and refinement. Finally, Spear, with ParamILS’s help, exploits the fine-grained structural prop- erties of the VCs to achieve superior performance. Saturn, on the other hand, makes no effort to maintain or exploit the structure. 2. The summaries in Saturn are represented as tuples of Finite State Machine (FSM) transition relations, input predicates, output predicates, and a set of modified objects. In that respect, the summaries are almost equivalent in functionality to the summaries computed by SLAM [13]. The FSM relation is specialized for a specific property being verified. In the case of kernel locks [135], the analysis allocates only a single bit per lock. The summaries computed by the interprocedural analysis in Chapter 4 are significantly more expressive (cf. Fig. 2.1 on page 14) and represent all the effects of the function. 3. Besides summarization, Saturn makes no other attempts to avoid or mit- igate the potentially exponential cost of context-sensitivity. For simple properties, like lock-checking, the overhead of context-sensitivity could be low enough that the brute-force approach works well, but it is very unlikely that it will scale to more general properties. 4. Saturn’s intraprocedural analysis differs from the one presented in Chapter 120 Chapter 7. Related Work 3 in that it relies on BDDs to simplify the guards, rather than on structure sharing. I decided to omit the BDD-based simplification step because SAT solvers are extremely efficient at detection of conflicts and propagation of resulting implications. Calysto performs extensive simplification of the VCs during the construction of the summary graph, but all those simplifications are very cheap to perform. Furthermore, Saturn makes no attempt to reuse the structure present in the CFG for more efficient generation of VCs. Astrée [22, 41] is a precise and fairly scalable static checker intended for safety-critical software. Such software is typically implemented as clean, heav- ily restricted code (see discussion in Section 1.2.2 on page 5), and that signifi- cantly simplifies the requirements on the static checker. Calysto is designed to be a general-purpose checker, and as such has to deal with type-unsafety, unstructured code, and dynamically allocated memory. The focus of the Astrée project is also different from the work presented in this thesis. Astrée is based on the framework of abstract interpretation [39], while Calysto uses almost no abstraction, and is completely based on structure exploitation. The choice of SSA/GSA form as the starting point for computation of VCs sets Calysto apart from the other mentioned tools. For example, Boogie re- names variables on the AST, creates fresh definitions in predecessors of the basic block that contains the use, and applies the weakest precondition predi- cate transformer to compute the VCs. Being based on SSA/GSA forms, Ca- lysto enjoys the known benefits of single assignment forms (e.g., language independence, single point of definition of local variables, easy manipulation and transformation of the intermediate form, and so on.). 7.2 Computation of Verification Conditions The related work on the computation of VCs can be classified into two subfields: symbolic execution and improving the quality of the VCs. 121 Chapter 7. Related Work Probably the closest approach to the intraprocedural analysis presented in Chapter 3, is that of Kölbl and Pixley [83]. They considered symbolic execution in the context of equivalence checking of higher-level hardware descriptions in C++. They assume that hardware designers are aware that the code is going to be compared to Register Transfer Level (RTL) code, so most of the code is very low-level, and pointers are used sparingly30. The authors use a heuristic scheduling algorithm for symbolic execution and merging of different paths. My approach offers an alternative with an optimal merging strategy — all alterna- tive definitions that reach a use are merged by the proposed restriction operator for gating path expressions. In addition, write merges are scheduled at a mini- mal set of join points, which is efficiently computed by the means of the iterated dominance frontier. Kölbl and Pixley keep the definitions of abstract memory locations in a list together with corresponding guards, minimized with a variant of BDDs. This is exactly the same approach as the one used in Saturn. Their internal representation is similar to the maximally-shared graphs in Calysto. Their results, although obtained on much smaller benchmarks than used in this thesis, are similar to mine. Namely, they report that, in practice, the size of computed intraprocedural summaries is linear with the number of instructions. The authors do not address interprocedural analysis in the paper. Leino [87] presented an approach to the generation of efficient VCs based on the weakest precondition predicate transformer. The weakest precondition can handle only structured code. That work was later extended to unstructured programs [18] through explicit encoding of the CFG. Their introduction of fresh variables to encode the CFG transitions seems similar to the introduction of fresh variables with the purpose of structuring the graph [59]. The analysis presented in this thesis handles unstructured code directly without introducing redundant variables. 30While I was working on my masters thesis, I wrote a number of such models in SystemC. In my experience, those models abstract the bit-level details, and do not significantly differ from the RTL descriptions. Most often, those models are finite-state, meaning that there is no dynamic memory allocation. 122 Chapter 7. Related Work Another line of relevant work is focused on the generation of high-quality VCs. Flanagan and Saxe proposed a passifying and renaming transformation [64] for the generation of VCs. Their transformation produces quadratic-sized VCs in the worst case, and the authors claim the size is linear in practice. Later, Leino [87] shed more light on the passifying component of the trans- formation. The renaming component is similar to Tseitin’s transform [128], which introduces fresh variables to avoid exponential blowup of a circuit-to- CNF translation. The SSA form is based on the same principle. Introduction of fresh variables, both in Tseitin’s transform and the VC generation, effec- tively flattens the logical formula. This flattening hides the structure of the problem. The approach proposed in this thesis also uses Tseitin’s transform for encoding maximally-shared graphs to CNF, however structural abstraction and refinement are done before the flattening, while the VCs are still in the form of maximally-shared graphs. As described in Section 5.1.1, Spear’s expression simplifier also works on maximally-shared graphs. In other words, the approach proposed in this thesis delays the flattening so as to exploit as much structure available in the original problem (and reflected in maximally-shared graphs) as possible. Another difference is that Leino’s approach encodes the variables required for trace reconstruction directly into the VC [89], while my approach keeps the code locations and variable names in summary graph nodes. If a satisfying assignment is found, it can be easily mapped back to the source code. This avoids the introduction of additional variables. Lahiri, in his thesis [84], explored small model properties and the application of positive equality for more efficient solving of the VCs coming from processor verification. Both approaches can be combined with the approach presented in this thesis. 123 Chapter 7. Related Work 7.3 Interprocedural Analysis Reps et al. [109] presented a polynomial-time algorithm for interprocedural, finite, distributive, subset problems (IFDS). Some of the analyses that fit in that class are: constant propagation, pointer analysis, reaching definitions, possibly- uninitialized variables, etc. Unfortunately, the summaries computed in Chapter 3 are neither distributive, nor subset problems. For instance, aliasing of pointers passed to a function makes the summary non-distributive. As the focus of this thesis is on getting as precise and as scalable analysis as possible, abstracting the summaries to the level required by the IFDS framework seemed inappropriate. SLAM [13] is based on predicate abstraction, and the summaries are tran- sition relations with respect to the set of chosen predicates. In SLAM, sum- maries are represented as BDDs. The predicate abstraction suffices for proving control-flow intensive properties, and it seems to be especially suitable for device drivers. Calysto doesn’t abstract much (currently only loops, recursive data structures, and exceptional control-flow). So, encoding the transition relation as BDDs would result in serious blowup (especially on multipliers). Structural abstraction abstracts functions in the cheapest way possible — by avoiding computation, or more precisely, by avoiding expansion of summary operators. Other abstractions, like predicate abstraction [13], typically per- form relatively expensive recomputation of the abstractions in each abstraction- checking-refinement cycle. This lazy approach to interpretation of function summaries resembles the intuition behind lazy proof explication [62], a technique used to bridge between different theories in a theorem prover. The shared intuition is to abstract away expensive reasoning — expanding a function summary or solving a sub-theory query — as unconstrained variables, and then constrain them lazily, only as needed to refute solutions to the abstracted problem. The specifics of what to abstract and how to refine, of course, are different, since I am solving a different problem. Taghdiri’s specification inference [126] computes a coarse abstraction of each 124 Chapter 7. Related Work function, abstracting the side-effects of function calls in a similar way as struc- tural abstraction. If a counterexample is found, her refinement analyzes each function that participates in the counterexample, attempting to prove that there exists an intraprocedural path consistent with the trace. If there is no such path, the proof is mined for additional predicates which are then used to refine the abstraction of the function. Compared to structural abstraction, specification inference is is significantly more expensive because it cannot be made com- pletely incremental — once the trace is found infeasible, all the reasoning has to be discarded, and started anew in the next iteration. Structural abstraction and refinement were carefully designed to be completely incremental and avoid discarding learned facts after each abstraction-checking-refinement iteration. Shortly after publication of my work on structural abstraction [8], Anand et al. [5] published work that is similar to structural abstraction, but in the context of software testing. The main idea presented in both works is to use summariza- tion to avoid the worst-case cost of interprocedural analysis and let the decision procedure inline or expand only what it needs. However, the approaches to the problem are different: Anand et al. approach the problem from the software testing perspective, and I approach the problem from the extended static check- ing perspective. These two approaches, when taken together, strongly suggest that letting the decision procedure handle interprocedural analysis might be the right way to go, and also together provide more insight into this rather impor- tant problem than either approach alone. The rest of this section provides a detailed comparison. One of the key assumptions Anand et al. make is that the code is deter- ministic — unlike structural abstraction, which initially abstracts side-effects with unconstrained variables, their approach relies upon uninterpreted func- tions. Although such an approach gives more opportunities for elimination of common subexpressions that involve uninterpreted functions31, it is not neces- 31Calysto can also merge two summary operators corresponding to the same side-effect of the same function, if and only if the summarized function is deterministic and the operands of 125 Chapter 7. Related Work sarily sound: two different calls to the same non-deterministic function with the same parameters can produce two different results.32 Three important notions (local path constraint, definition predicate, and calling-context predicate) that Anand et al. introduce also have their coun- terparts in the approach proposed in this thesis. The local path constraint is similar to the intraprocedural control-flow context that the symbolic exe- cution presented in Chapter 3 computes, the only difference being that my structure-preserving symbolic execution computes the control-flow context for all the paths in a function, while the local path constraint presents a conjunc- tion of predicates that hold on a single path in the symbolic execution tree. The definition predicate asserts equality between the summary operator and the corresponding abstract memory location in the calling context; instead of constructing such an equality, I rather expand the summary operator in a lazy manner, inlining the corresponding symbolic expression. Finally, their calling- context predicate is similar to the control-flow context, with the difference that Calysto computes it in the summarized form for all the paths. Anand et al. incompletely characterize my earlier work on structural ab- straction [8]. Their paper contrasts the demand-driven compositional symbolic execution to structural abstraction, claiming that inlining in structural abstrac- tion leads to combinatorial explosion. The truth is that both their approach, as well as structural abstraction, have worst-case exponential cost — only the dynamics of the approaches are different: • In structural abstraction, the decision procedure (in my work Spear) con- trols which summary operator needs to be expanded through the process of structural refinement (Section 4.2.3 on page 81). The decision proce- dure can and does learn about the newly expanded summary operators, both summary operators are structurally equivalent, as described in Section 4.2.4 on page 83. 32According to my experience, non-deterministic functions are actually quite frequent in real code. The most frequent sources of non-determinism are: calls to external functions, dependencies on the actual pointer values, dependencies on time/network/files, and calls to random function. 126 Chapter 7. Related Work attempting to avoid unnecessary expansions. Even if a summary operator (corresponding to a single side-effect) is expanded, the nested summary operators are not. Those nested operators might or might not be expanded in the subsequent iterations of the abstraction-checking-refinement loop. • In demand-driven compositional symbolic execution, the exponential be- havior is hidden in quantifiers — the modern SMT decision procedures, like Z3 [47], handle quantifiers by an incomplete heuristic quantifier instan- tiation (e.g., [46]). Depending on the underlying theory and implemented heuristics, such instantiation can incur exponential blowup, or even worse, fail to terminate33. 7.4 Decision Procedures for Bit-Vector Arithmetic The Nelson-Oppen [100] framework is probably the most frequently used ap- proach for proving the validity of VCs. The framework is a method for com- bining decision procedures. The theories have to meet certain requirements (see [94] for an overview), in order to be combined with other theories. Fortu- nately, most theories used in practice, like Presburger arithmetic, the theory of uninterpreted functions, and the theory of arrays, meet the requirements. De Moura and Bjørner [47] combined the Nelson-Oppen framework with the theory of bit-vectors in the Z3 automated theorem prover. Their implementation uses one SAT solver for case-splitting and another one as the bit-vector arithmetic theory solver. The theory solver does bit-blasting, in a similar way to Spear. The difference is that their theory solver only propagates the bit-vector the- ory facts, and does not infer conflicts, which are entirely handled by the core case-splitting SAT solver. The Nelson-Oppen framework requires such a design. Spear, on the other hand, handles both fact propagation and case-splitting 33Z3 uses a heuristical fail-safe bound on the number of instantiations, so it always termi- nates. However, other SMT provers might handle instantiation in a less safe way. 127 Chapter 7. Related Work with a single, highly-tuned, SAT solver. In general, the Nelson-Oppen frame- work framework has certain advantages over the Spear-like approaches: (1) It is easier to combine the theory of bit-vectors with other theories, like the theory of uninterpreted functions. (2) It is easier to incorporate complex term-rewriting techniques, which are very important for certain classes of problems, like equiv- alence of implementations of cryptographic algorithms. In spite of the greater flexibility of the Nelson-Oppen framework, it seems that Spear-like approach is faster on instances with complex predicate structure34. Barrett, Dill, and Levitt [19] proposed a decision procedure for subset of bit-vector arithmetic operations based on Shostak’s method35 for combining theories [117]. Their decision procedure uses term rewriting to avoid bit-blasting as much as possible. Spear is based on similar principles, but relies upon a SAT solver, rather than a framework for combining theories. They implemented their approach in the SVC automated theorem prover. SVC has not been actively developed for many years now, so a direct comparison to the modern bit-vector arithmetic decision procedures, like Z3 and Spear, would be meaningless. Bryant et al. [25] recently suggested an abstraction for bit-vector arithmetic based on an alternation of under- and over-approximations. At this point, it is unclear how effective that abstraction is, because their decision procedure is not publicly available and did not participate at the SMT competition. Z3 solves the hardest examples presented in [25] in 10–14 seconds, while Bryant et al. report that their technique requires 1000+ seconds. Of course, this is not an apples-to-apples comparison, because the experiments were not done on the same machine. On the same examples, Spear requires 100–1000+ seconds, depending on the parameter configuration used. The alternation between under- and over-approximations, if proven effective, could be easily implemented in Spear. 34Spear won the bit-vector division of the SMT 2007 competition, ahead of Z3. 35 Shostak’s method for combining theories has been a source of controversy for a long time. A number of authors pointed out the errors in Shostak’s method (e.g., [111, 116, 37]). According to Detlefs et al. [51, page 386], the consensus in the community is that Shostak’s method is a refinement of the Nelson-Oppen method. 128 Chapter 7. Related Work Ganesh and Dill [65] proposed an abstraction-refinement technique for rea- soning about the non-extensional theory of arrays [124] in combination with a bit-vector decision procedure based on bit-blasting. They implemented their approach in the STP decision procedure, which relies upon an expression sim- plifier that is very similar to the one in SVC. Their technique for handling arrays could be incorporated in Spear, which would help Calysto to handle arrays more precisely. STP seems to be particularly optimized for instances that are a long sequence of conjuncts, because it performs Gaussian elimination as a simplification step. Testing tools, which analyze a single path at time, tend to generate such a long sequence of conjuncts, but extended static checking tools, like Calysto, generate instances with very complex propositional structure. Gaussian elimination is ineffective for such complex instances. If needed, Gaus- sian elimination could be easily added as a preprocessing step in Spear. Unlike Spear, STP does not support all standard bit-vector arithmetic operators (for instance, variable shifts). Babić and Musuvathi [11] proposed two techniques for handling modular arithmetic constraints: The first one is for linear constraints and is based on translation to integer linear programming. My later experiments showed that the leading integer linear programming solvers, like CPLEX36, perform extremely poorly on such problems. More precisely, CPLEX failed to return after 2 weeks of continuous running on a problem that Spear solves in several seconds. It is still an open question how well an SMT solver would perform on such a translation. The second proposed technique is for non-linear constraints and is based on Newton’s p-adic lifting. Newton’s p-adic lifting is somewhat complex because it requires computation of Jacobians, a square matrix of par- tial derivatives. A simpler way to emulate p-adic lifting is simply to force a SAT solver to case-split on the least significant bits first and proceed towards the most significant bits. According to my experiments, this heuristic works very well on hand-crafted examples with many non-linear operations and very simple propositional structure. Unfortunately, such a heuristic p-adic lifting em- 36http://www.ilog.com/ 129 Chapter 7. Related Work ulation does not perform well on the real-world instances, like those generated by Calysto, that have very complex propositional structure. Getting p-adic lifting (either the Jacobian-based version or the heuristic emulation) to work well on the real-world instances is still an open problem of an immense practical importance. 7.5 Automatic Tuning There are almost no publications on automated parameter optimization for de- cision procedures for formal verification. Seshia [114] explored using support vector machine classification to choose between two encodings of difference logic into Boolean SAT. The learned classifier was able to choose the better encoding in most instances he tested, resulting in a hybrid encoding that mostly domi- nated the two pure encodings. The only other work I am aware of is that of Andrei Voronkov — he used a less systematic approach for tuning the strategies of his Vampire first-order theorem prover. However, he has not published that work. There is, however, a fair amount of previous work on optimizing SAT solvers for particular applications. For example, Shtrichman [118] considered the in- fluence of variable and phase decision heuristics (especially static ordering), restriction of the set of variables for case splitting, and symmetric replication of conflict clauses on solving BMC problems. He evaluated seven strategies on the Grasp SAT solver, and found that static ordering does perform fairly well, although no parameter combination was a clear winner. Later, Shacham and Zarpas [115] showed that Shtrichman’s conclusions do not apply to zChaff’s less greedy VSIDS heuristic on their set of benchmarks, claiming that Shtrich- man’s conclusions were either benchmark- or engine-dependent. Shacham and Zarpas evaluated four different decision strategies on IBM BMC instances, and found that static ordering performs worse than VSIDS-based strategies. Lu at al. [92] exploited signal correlations to design a number of ATPG-specific tech- niques for SAT solving. Their technique showed roughly an order of magnitude 130 Chapter 7. Related Work improvement on a small set of ATPG benchmarks. The application of machine learning techniques for improving the scalabil- ity of decision procedures has been more thoroughly researched. For instance, Grumberg et al. [69] used machine learning to learn good BDD variable or- derings from a number of sample orderings. The underlying idea — to use AI-techniques to improve performance — is the same, but the underlying tech- nology used and the goals are very different. The underlying technology that Grumberg et al. use is machine learning, while the automatic tuner used for Spear optimization is a local search engine that attempts to quickly discard bad parameter configurations. The purpose of automatic tuning of Spear was to recognize the common (structural) properties of a class of instances and find suitable search parameters and heuristics that can effectively take advantage of those properties. In contrast, Grumberg et al. dynamically try to come up with a good BDDs-ordering for each individual problem. Greumberg et al.’s work was later extended by [27], but the underlying tuning technology and goals re- mained the same: According to my understanding, Carbin’s work [27] chooses sample BDD-orderings in each iteration by reusing the information learned from the previous iterations. In the context of FOL theorem proving, machine learning is frequently used to detect a small set of axioms that are likely to be sufficient for proving that a given conjecture is a theorem (e.g., [125, 48]). SMT decision procedures and SAT solvers actually use similar heuristic techniques for selecting the variables for case-splitting (e.g., [97]) and for quantifier instantiation (e.g., [46]). The automated parameter optimization tool used in my study has been re- cently introduced by Hutter et al. [79]; however, that work was more focused on theoretical properties of the algorithm and did not consider an application to a state-of-the-art solver for real-world problems. That work and the study pre- sented here complement each other and also address two different communities. Very broadly, automated parameter optimization can be seen as as a stochastic optimization problem that can be solved using a range of generic and specific methods [121, 3, 21]. However, these are either limited to algorithms with con- 131 Chapter 7. Related Work tinuous parameters or algorithms with a small number of discrete parameters. 132 Chapter 8 Conclusions and Future Work 8.1 Conclusions Possibly the most important contribution of this thesis is that it draws the attention of the software analysis research community to, in my opinion, a somewhat neglected research direction — exploitation of structure that is so pervasive in all human-made systems. Humans need structure to solve complex problems. Recognizing that structure and exploiting it is essential if precise software analysis tools are to scale to multi-million line programs. The main hypothesis of this thesis — that the structure of programs can be exploited for achieving a highly scalable and precise analysis — has been proven. First, I showed how structure-preserving symbolic execution can compute sum- maries and VCs that reflect the dataflow dependencies in the original code. The presented symbolic execution also exploits the graph properties of CFGs for fast identification of live definitions. Second, I proposed structural abstraction that exploits the natural abstraction boundaries in programs (functions) for abstrac- tion, and structural refinement that uses the dataflow dependencies (preserved by my symbolic execution) for fast refinement. Third, I showed that at a finer level of granularity, automatic tuners can effectively exploit the properties of a specific class of instances, and adjust the search parameters accordingly, even when the combinatorial search space spanned by the parameters is enormous (> 1018). Finally, I provided a description of the architecture of my experimental 133 Chapter 8. Conclusions and Future Work extended static checker, Calysto, that implements the techniques presented in this thesis. Calysto’s results on a number of real-world benchmarks show that the proposed combination of techniques is both very scalable and sufficiently precise. Furthermore, the results encourage further research. 8.2 Future Work While this thesis has proposed novel and effective solutions for interprocedural software analysis and bit-vector decision procedures, there are still many chal- lenges ahead. In this section, I discuss areas that, in my opinion, need more research. AlthoughCalysto is very precise in terms of interprocedural path-sensitivity and handling of machine operations (bit-precise), further improvements are needed in several areas: • Currently, Calysto knows nothing about concurrency. Recent work on iterative context-bounding by Musuvathi and Qadeer [99] and later work by Bouajjani et al. [23] could be used in combination with structural abstraction: partial order reduction [67] performed directly on the source could generate a sequence of different interleavings of the source (or the intermediate form), which could then in turn be passed to a tool based on structural abstraction, like Calysto. However, it is very unlikely that this would be efficient. A more efficient approach would require a tighter integration of structural abstraction and analysis of different interleavings. • Calysto, like ESC/Java, handles loops by unrolling them and assuming that the loop test has failed (see Section 3.1.3). The major drawback of that approach is decreased code coverage. One promising, but not suf- ficiently researched direction is a combination of abstract interpretation techniques and extended static checking, as proposed by Leino and Lo- gozzo [88]. It is unclear how well such a combination would perform with structural abstraction on non-Java code. Another possibility is to delegate 134 Chapter 8. Conclusions and Future Work handling of loops to the decision procedure, by incorporating widening and narrowing operators directly into the decision procedure. • Arrays could be easily handled more precisely by modifying the proposed structure-preserving symbolic execution to use array read and write op- erators instead of computing the symbolic definitions of abstract memory locations. However, it is very possible that the combination of bit-vector arithmetic and the theory of arrays would make VCs much harder and neg- atively impact the scalability of the entire system. The tradeoffs would need to be evaluated experimentally. • Structural abstraction could be significantly improved as well. First, by better and more precise refinement heuristics, and second, by checking each VC before it is lifted to the calling context (see 4.2.4). Decision procedures, especially for bit-vector arithmetic, are a dynamic re- search area. Spear shows that SAT-solver-based decision procedures for bit- vector arithmetic can be a good choice for problems that have complex propo- sitional structure. However, further improvements are needed. In my opinion, the most important research directions are the following: • One of the downsides of Spear’s architecture is that it is hard to add support for new theories. The most important theory that is currently missing is the theory of non-extensional arrays [124]. Other theories that would be immensely useful for other domains are mixed integer linear programming (e.g., [119]), and the theory of floating-point. Nelson-Oppen framework [100] offers more flexibility than Spear-architecture. However, more research is needed on incorporating the theory of bit-vectors into that framework. • Adding support for quantifiers to Spear would also be very useful, espe- cially for computing bit-precise loop invariants. One interesting direction would be to use a Quantified Boolean Formula (QBF) solver instead of a SAT solver for Spear’s core. 135 Chapter 8. Conclusions and Future Work • Calysto frequently generates a very long sequence of VCs. Often, adja- cent VCs in that sequence share some common subexpressions. Solving these VCs individually wastes computation because the decision proce- dure has to re-learn facts about those shared subexpressions over and over again. Solving them all together exceeds the resources. Since VCs are constructed on-the-fly, through structural abstraction and refinement, it is almost impossible to guess which expressions are going to be shared ahead of time. 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Thesis/Dissertation
2008-11
10.14288/1.0051356
eng
Computer Science
Vancouver : University of British Columbia Library
University of British Columbia
Attribution-NonCommercial-NoDerivatives 4.0 International
http://creativecommons.org/licenses/by-nc-nd/4.0/
Graduate
Software verification
Decision procedures
Software structure
Software analysis
Structural abstraction
Exploiting structure for scalable software verification
Text
http://hdl.handle.net/2429/1502