Evolving Boolean Functions with Conjunctions and Disjunctions via Genetic Programming

Recently it has been proved that simple GP systems can efficiently evolve the 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 the GP system for evolving a Boolean function with unknown components, i.e., the function may consist of both conjunctions and disjunctions. We rigorously prove that if the target function is the conjunction of n variables, then the RLS-GP using the complete truth table to evaluate program quality evolves the exact target function in O(ℓ n log2 n) iterations in expectation, where ℓ ≥ n is a limit on the size of any accepted tree. When, as in realistic applications, only a polynomial sample of possible inputs is used to evaluate solution quality, we show how RLS-GP can evolve a conjunction with any polynomially small generalisation error with probability 1 - O(log2(n)/n). To produce 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.