Recently it has been proven that simple GP systems can efficiently evolve a 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 a GP system for evolving a Boolean conjunction or disjunction of $n$ variables using a complete function set that allows the expression of any Boolean function of up to $n$ variables. First we rigorously prove that a GP system using the complete truth table to evaluate the program quality, and equipped with both the AND and OR operators and positive literals, 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. Additionally, we show that when a polynomial sample of possible inputs is used to evaluate the solution quality, conjunctions or disjunctions with any polynomially small generalisation error can be evolved with probability $1 - O(\log^2(n)/n)$. The latter result also holds if GP uses AND, OR and positive and negated literals, thus has the power to express any Boolean function of $n$ distinct variables. To prove 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.

翻译：近期研究表明，配备最小必要组件的简单遗传编程系统能够高效演化n个变量的合取。本文通过分析使用完备函数集（允许表达至多n个变量的任意布尔函数）的GP系统在演化布尔合取或析取时的行为与性能，实现了重要突破。首先严格证明：采用完备真值表评估程序质量、配备AND和OR运算符及正文字变量的GP系统，可在期望$O(\ell n \log^2 n)$次迭代内精确演化目标函数，其中$\ell \geq n$为任意可接受树的大小限制。此外，当使用多项式数量的样本输入评估解质量时，能以概率$1 - O(\log^2(n)/n)$演化出任意多项式小泛化误差的合取或析取。后者结论同样适用于使用AND、OR及正反文字变量的GP系统（即具备表达n个不同变量任意布尔函数的能力）。为证明上述结果，我们引入超乘性漂移定理，该定理在期望进步仅略超于最优距离的线性函数时能给出显著更强的运行时间界。