UBC Theses and Dissertations

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UBC Theses and Dissertations

Continual pattern replication Munro, James Ian

Abstract

This thesis continues the studies of A. Waksman (1969) in the repeated generation of finite strings in a one dimensional array of finite automata. Waksman handles this problem by the use of a "modulo arithmetic" algorithm. This is shown to be very restrictive with regard to the number of characters permitted in the output string. In fact, it is shown that unless the length of the string which is to be repeated is of the form p(formula omitted) where p is prime, only one output character is permitted. This of course makes the process quite meaningless. For this reason, a new algorithm is developed. This is referred to as the wheel algorithm, since there is an obvious analogy between it and a wheel, with the output string on its circumference rolling along the array and leaving the imprint of the characters in the string behind it in the same way that a wheel leaves tire tracks. The number of states required for such an algorithm is large and so the binary wheel algorithm is introduced. By using this algorithm, in which an output state is represented as a string of bits in several cells, the number of states required, in addition to the n output states, can be reduced to about 4 log₂ n. Both the wheel and binary wheel algorithms are then extended to the two dimensional and finally the d-dimensional cases.

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