# Evolving Boolean Functions with Conjunctions and Disjunctions via Genetic Programming

## TL;DR

This paper analyzes the performance of a genetic programming system for evolving Boolean functions composed of conjunctions and disjunctions, providing rigorous runtime bounds and demonstrating effective generalization with limited samples.

## Contribution

It extends previous work by analyzing GP for mixed Boolean functions, introducing a super-multiplicative drift theorem for stronger runtime bounds, and showing effective generalization with polynomial samples.

## Key findings

- RLS-GP evolves conjunctions in expected O(ℓ n log^2 n) iterations.
- GP can achieve low generalization error using polynomial samples.
- Super-multiplicative drift theorem yields stronger runtime bounds.

## Abstract

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 $O(\ell n \log^2 n)$ iterations in expectation, where $\ell \geq n$ 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 $1 - O(\log^2(n)/n)$. 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.