Monotone but Exciting: On Evolving Monotone Boolean Functions with High Nonlinearity

TL;DR

This study explores the use of evolutionary algorithms to generate monotone Boolean functions with high nonlinearity, demonstrating success across various encodings and dimensions up to 14.

Contribution

It introduces a non-monotonicity penalty and compares three solution encodings, showing evolutionary methods can surpass traditional nonlinearity benchmarks for monotone functions.

Findings

Evolutionary search can find monotone Boolean functions with high nonlinearity.

Genetic programming encoding performs well in larger dimensions.

Balanced truth table encoding is less effective at higher dimensions.

Abstract

Monotone Boolean functions are a structurally important class of Boolean functions, but their restricted form imposes strong limitations on achievable nonlinearity. In this paper, we investigate whether evolutionary computation can evolve monotone Boolean functions with high nonlinearity, both in the balanced and imbalanced settings. We consider three solution encodings: the standard truth table representation, a balanced truth table encoding that preserves Hamming weight, and a symbolic tree-based genetic programming representation. To guide the search toward monotone increasing functions, we introduce a non-monotonicity penalty and combine it with fitness functions targeting balancedness and nonlinearity. Experimental results are reported for dimensions from $n=5$ to $n=14$. The results show that evolutionary search can discover monotone Boolean functions with nonlinearities clearly exceeding those of majority functions, and in several cases approaching the best currently known values for monotone functions. At the same time, the experiments reveal substantial differences between encodings: the balanced truth table encoding performs poorly for larger dimensions, while the standard truth table and genetic programming encodings remain competitive, with genetic programming becoming especially relevant in the largest tested dimensions.