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.

翻译：单调布尔函数是一类结构上重要的布尔函数，但其受限形式对可实现非线性度施加了严格限制。本文研究进化计算能否在平衡和非平衡设置下进化出高非线性度的单调布尔函数。我们考虑三种解编码：标准真值表表示、保持汉明权重的平衡真值表编码，以及基于符号树的遗传编程表示。为引导搜索向单调递增函数方向进行，我们引入非单调性惩罚，并将其与针对平衡性和非线性度的适应度函数相结合。实验报告涵盖$n=5$至$n=14$的维度。结果表明，进化搜索能够发现非线性度明显超过多数函数的单调布尔函数，并且在若干案例中接近当前已知的单调函数最佳值。同时，实验揭示了编码之间的显著差异：平衡真值表编码在较大维度上表现较差，而标准真值表和遗传编程编码仍具竞争力，其中遗传编程在测试的最大维度中尤为突出。