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 a GP system for evolving a Boolean function with unknown components, i.e. the target function may consist of both conjunctions and disjunctions. We rigorously prove that if the target function is the conjunction of n variables, then a GP system using the complete truth table to evaluate program quality evolves the exact target function in O(ℓn log² 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 solution quality, conjunctions with any polynomially small generalisation error can be evolved 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.