(1+1) Genetic Programming With Functionally Complete Instruction Sets Can Evolve Boolean Conjunctions and Disjunctions with Arbitrarily Small Error

• URL: http://arxiv.org/abs/2303.07455v1
• Date: Mon, 13 Mar 2023 20:24:21 GMT
• Title: (1+1) Genetic Programming With Functionally Complete Instruction Sets Can Evolve Boolean Conjunctions and Disjunctions with Arbitrarily Small Error
• Authors: Benjamin Doerr, Andrei Lissovoi, Pietro S. Oliveto
• Abstract summary: GP systems can evolve a conjunction of $n$ variables if they are equipped with the minimal required components. We show that a GP system can evolve the exact target function in $O(ell n log2 n)$ in expectation, where $ell geq is a limit on the size of any accepted tree.
• Score: 16.075339182916128
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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