Evolving Boolean functions with specific properties is an interesting optimization problem since, depending on the combination of properties and Boolean function size, the problem can range from very simple to (almost) impossible to solve. Moreover, some problems are more interesting as there may be only a few options for generating the required Boolean functions. This paper investigates one such problem: evolving five-valued spectra Boolean functions, which are the functions whose Walsh-Hadamard coefficients can only take five distinct values. We experimented with three solution encodings, two fitness functions, and 12 Boolean function sizes and showed that the tree encoding is superior to other choices, as we can obtain five-valued Boolean functions with high nonlinearity.

翻译：演化具有特定性质的布尔函数是一个有趣的优化问题，因为根据性质组合与布尔函数规模的不同，该问题可能从非常简单到（几乎）无法求解。此外，某些问题更具研究价值，因为生成所需布尔函数的方法可能非常有限。本文研究了其中一个问题：演化五值谱布尔函数，即其沃尔什-哈达玛系数仅能取五个不同值的函数。我们通过三种解编码方式、两种适应度函数以及12种布尔函数规模进行实验，结果表明树编码优于其他方案，使我们能够获得具有高非线性的五值布尔函数。