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Scaling of program functionality

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Abstract

The distribution of fitness values (landscapes) of programs tends to a limit as the programs get bigger. We use Markov chain convergence theorems to give general upper bounds on the length of programs needed for convergence. How big programs need to be to approach the limit depends on the type of the computer they run on. We give bounds (exponential in N, N log N and smaller) for five computer models: any, average or amorphous or random, cyclic, bit flip and four functions (AND, NAND, OR and NOR). Programs can be treated as lookup tables which map between their inputs and their outputs. Using this we prove similar convergence results for the distribution of functions implemented by linear computer programs. We show most functions are constants and the remainder are mostly parsimonious. The effect of ad-hoc rules on genetic programming (GP) are described and new heuristics are proposed. We give bounds on how long programs need to be before the distribution of their functionality is close to its limiting distribution, both in general and for average computers. The computational importance of destroying information is discussed with respect to reversible and quantum computers. Mutation randomizes a genetic algorithm population in \(\frac{1}{4}(l+1)(\hbox{log}\,(l)+4)\) generations. Results for average computers and a model like genetic programming are confirmed experimentally.

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Acknowledgements

I would like to thank Jeffrey Rosenthal, Tom Westerdale, James A. Foster, Riccardo Poli, Ingo Wegener, Nic McPhee, Michael Vose, Jon Rowe, Wolfgang Banzhaf, Tina Yu and the anonymous referees.

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  1. Mathematical and Biological Sciences and Computing and Electronic Systems, University of Essex, Essex, CO4 3SQ, UK

    W. B. Langdon

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Appendix A: Summary

Appendix A: Summary

The distribution of outputs produced by all computers converges to a limiting distribution as their (linear) programs get longer. We provide a general quantitative upper bound (2.30a I a, where I is the number of instructions and a is the length of programs needed to store every possible value in the computer’s memory, Sect. 5). Tighter bounds are given for four types of computer. There are radical differences in their convergence rates. The length of programs needed for convergence depends heavily on the type of computer, the size of its (data) memory N and its instruction set.

The cyclic computer (Sect. 6) converges most slowly, ≤0.35 × 22N, for large N. In contrast the bit flip computer (Sect. 7) takes only \(\frac{1}{4}(N+1)(\hbox{log}\,(m)+4)\) random instructions (m bits in output register). In both computers, the distributions of outputs and of functions converge at the same rate to a uniform limiting distribution.

In Sect. 8 we introduced a random or amorphous, model of computers. This represents the average behaviour over all computers (cf. NFL [1]). It takes less than (15.3 + 2.30m)/log I random instructions to get close to the uniform output limit. The limiting distribution contains only functions that are constants. Again convergence is exponential with 90% of programs of length 1.6 × n2N yielding constants (n is the size of the input register in bits).

Section 9 shows the output of programs comprised of four common Boolean operators converges to a uniform distribution within \(\frac{1}{2}N (\hbox{log}\,(m) + 4)\) random instructions. The importance of the pragmatic heuristic of write protecting the input register, is highlighted, since without it there are no “interesting” functions in the limit of large programs.

In Sect. 10 we showed quantum and reversible computers do not have synchronising sequences and consequently the behaviour of their long programs is radically different from that of conventional computers.

Section 11 shows the number of generations (\(\frac{1}{4}(l+1)(\hbox{log}(l)+4)\)) needed for mutation alone to randomise a bit string genetic algorithm (chromosome of l bits).

Practical GP fitness functions will converge faster than the distribution of all functions, since they typically test only a small part of the whole function. Real GP systems allow rapid movement about the computer’s state space and so appear to be close to the bit flipping (Sect. 7) and four Boolean instruction (Sect. 9) models. We speculate rapid O(|test set|N log m) convergence in fitness distributions may be observed.

The number of minimal solutions to XOR (even-2 parity) grows quadratically in memory size (cf. Sect. 5.3) but this corresponds to a rapid fall in proportion as memory is increased.

In the Boolean linear systems considered, complex functions are very rare even in short programs and appear to reach a peak in their frequency near l = N/m. This suggests a size limit of O(N/m) might be beneficial to linear GP. The peak is followed by exponential decline, with the same exponent (\(\approx \sqrt{2/N^{3}}\)) as the other non-trivial functions. Since \(\sqrt{2/N^{3}} < I\), the number of solutions grows exponentially with program length l. It is also appears that the frequency of complex functions decreases dramatically as the number of operations needed to created them from the program’s inputs increases. I.e. most functions are parsimonious.

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Langdon, W.B. Scaling of program functionality. Genet Program Evolvable Mach 10, 5–36 (2009). https://doi.org/10.1007/s10710-008-9065-y

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