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(1+1) genetic programming with functionally complete instruction sets can evolve Boolean conjunctions and disjunctions with arbitrarily small error
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DOI：10.1016/j.artint.2023.103906.png)
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Recently it has been proven that simple GP systems can efficiently evolve a conjunction of n variables if they are equipped with the minimal required components. In this paper, we make a considerable step forward by analysing the behaviour and performance of a GP system for evolving a Boolean conjunction or disjunction of n variables using a complete function set that allows the expression of any Boolean function of up to n variables. First we rigorously prove that a GP system using the complete truth table to evaluate the program quality, and equipped with both the AND and OR operators and positive literals, evolves the exact target function in O (ln log(2) n) iterations in expectation, where l >= n is a limit on the size of any accepted tree. Additionally, we show that when a polynomial sample of possible inputs is used to evaluate the solution quality, conjunctions or disjunctions with any polynomially small generalisation error can be evolved with probability 1 - O (log(2) (n)/n). The latter result also holds if GP uses AND, OR and positive and negated literals, thus has the power to express any Boolean function of n distinct variables. To prove our results we introduce a super-multiplicative drift theorem that gives significantly stronger runtime bounds when the expected progress is only slightly super-linear in the distance from the optimum. (c) 2023 Published by Elsevier B.V.
Keyword:
Genetic programming
Computational complexity
Theory
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