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

2025-05-28

AI Summary

Linear Genetic Programming (LGP) lacks a theoretical characterization of the relationship between fitness and genotype, hindering principled algorithm design. Method: We establish a fitness upper-bound analysis framework based on instruction edit distance. Building upon this, we propose the freemut mutation operator, whose theoretical analysis jointly considers instruction count, program size control, and multi-point mutation. Contribution/Results: We first reveal a universal linear relationship between expected fitness and its upper bound, unifying explanations for code bloat and minimal hitting time phenomena. Theoretical analysis shows that moderate increases in instruction count—combined with coordinated control of program size and multi-point mutation—significantly enhance evolutionary efficiency. Empirical validation confirms the accuracy of our theoretical predictions, demonstrating superior convergence speed and solution quality compared to baseline approaches.

Technology Category

• Machine Learning: Evolutionary Learning
• Search and Optimization: Evolutionary Computation
• Natural Language Processing: Code Generation / Program Synthesis from Natural Language

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

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.

Problem

• Modeling fitness-genotype relationship in linear genetic programming
• Explaining bloat effect via instruction editing distance analysis
• Optimizing LGP performance through program size and mutation strategies

Innovation

• Fitness supremum models genotype-fitness relationship
• Instruction editing distance explains bloat effect
• Small program size improves LGP performance