Bridging Fitness With Search Spaces By Fitness Supremums: A Theoretical Study on LGP

TL;DR

This paper provides a theoretical analysis of linear genetic programming, revealing how fitness relates to program modifications, explaining the bloat effect, and suggesting strategies for improving LGP performance based on instruction editing distance.

Contribution

It introduces a theoretical model linking fitness and genotype in LGP, explaining the bloat effect and minimum hitting time through instruction editing distance.

Findings

Fitness expectation increases with instruction editing distance in LGP.

Bloat occurs due to higher likelihood of producing better offspring by adding instructions.

Small program size and multiple mutations improve LGP performance.

Abstract

Genetic programming has undergone rapid development in recent years. However, theoretical studies of genetic programming are far behind. One of the major obstacles to theoretical studies is the challenge of developing a model to describe the relationship between fitness values and program genotypes. In this paper, we take linear genetic programming (LGP) as an example to study the fitness-to-genotype relationship. We find that the fitness expectation increases with fitness supremum over instruction editing distance, considering 1) the fitness supremum linearly increases with the instruction editing distance in LGP, 2) the fitness infimum is fixed, and 3) the fitness probabilities over different instruction editing distances are similar. We then extend these findings to explain the bloat effect and the minimum hitting time of LGP based on instruction editing distance. The bloat effect happens because it is more likely to produce better offspring by adding instructions than by removing them, given an instruction editing distance from the optimal program. The analysis of the minimum hitting time suggests that for a basic LGP genetic operator (i.e., freemut), maintaining a necessarily small program size and mutating multiple instructions each time can improve LGP performance. The reported empirical results verify our hypothesis.