Communication Dans Un Congrès Année : 2025

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

Résumé

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 ≥ 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 superlinear in the distance from the optimum.

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Dates et versions

hal-05046799 , version 1 (25-04-2025)

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Citer

Benjamin Doerr, Andrei Lissovoi, Pietro Oliveto. (1+1) Genetic Programming with Functionally Complete Instruction Sets Can Evolve Boolean Conjunctions and Disjunctions with Arbitrarily Small Error. The Thirty-Ninth AAAI Conference on Artificial Intelligence (AAAI-25), 2025, Vancouver (BC), Canada. pp.28740-28740, ⟨10.1609/aaai.v39i27.35131⟩. ⟨hal-05046799⟩
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