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

2026-04-19

AI Summary

Monotone Boolean functions are inherently constrained in achieving high nonlinearity, particularly when balancedness is also required. This work presents the first systematic evaluation of three encoding schemes—standard truth tables, Hamming-weight-preserving balanced truth tables, and tree-based genetic programming (GP)—for evolving highly nonlinear monotone Boolean functions, employing a non-monotonicity penalty within a multi-objective fitness framework. Experimental results demonstrate that the proposed approach successfully evolves monotone Boolean functions whose nonlinearity significantly surpasses that of most known constructions and approaches current best-known bounds across multiple dimensions. Notably, in high-dimensional settings (e.g., $n=14$), genetic programming exhibits a clear advantage, exceeding previously established nonlinearity upper limits for monotone functions.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Search and Optimization: Evolutionary Computation
• Machine Learning: Evolutionary Learning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• monotone Boolean functions
• nonlinearity
• balancedness
• evolutionary computation
• Boolean function design

Innovation

• monotone Boolean functions
• evolutionary computation
• nonlinearity
• genetic programming
• solution encodings