Improving Evaluation of Recombination-based Cartesian Genetic Programming

Cartesian Genetic Programming (CGP) is among the practical and popular forms of Genetic Programming as it uses a graph-based representation of programs. This paper presents a first runtime analysis of CGP in evolving Boolean functions using complete training sets. We prove an asymptotic bound $O(n D^5)$ for the expected number of fitness evaluations of CGP to construct a conjunction of $n$ inputs using at most $D \geq n-1$ binary gates, a minimal function set, and even with a strict survival selection. When the non-strict selection is used, the bound is improved to $O(n D^4)$. Our analysis reveals interesting characteristics of CGP induced search, which have been only observed empirically. In particular, enabling the acceptance of equally good solutions, including those with connected gates non-contributing to fitness, can lead to a speedup, and consequently a better asymptotic time bound. In contrast to conjunctions, we also prove a negative result which shows that CGP requires exponential time to evolve an exclusive disjunction. Experiments evolving conjunctions complement our theoretical findings. The use of incomplete training sets is found to further reduce the average number of fitness evaluations while maintaining a good level of generalisation.

We analyze the effect of optimizing the initial population of genetic programming (GP) for symbolic regression (SR) on the accuracy and complexity of solutions. We compare three well-established random initialization methods as well as initialization with small optimized solutions from exhaustive symbolic regression (ESR) using a GP/SR implementation which is based on the multi-objective evolutionary algorithm NSGA-II. We compare the final Pareto fronts found with each initialization method on twelve synthetic problems of varying complexity and one real-world dataset. We find no significant differences in accuracy or model complexity among the initialization methods. The initial advantage of initialization with ESR disappears after only a few generations. Our results show that, given similar diversity in the initial population, the effect of the initialization method in GP-based symbolic regression on the final Pareto front is negligible.

Mathematical formulas serve as a language through which humans communicate with nature. Discovering mathematical laws from scientific data to describe natural phenomena has been a long-standing pursuit of humanity for centuries. In the field of artificial intelligence, this challenge is known as the symbolic regression problem. Among existing symbolic regression approaches, Genetic Programming (GP) based on evolutionary algorithms remains one of the most classical and widely adopted methods. GP simulates the evolutionary process across generations through genetic mutation and crossover. However, mutations and crossovers in GP are entirely random. While this randomness effectively mimics natural evolution, it inevitably produces both beneficial and detrimental variations. If there existed a metaphorical God capable of foreseeing which genetic mutations or crossovers would yield superior outcomes and performing targeted gene editing accordingly, the efficiency of evolution could be substantially improved. Motivated by this idea, we propose in this paper a symbolic regression approach based on gene editing, termed GESR. In GESR, we trained two "hands of God" (two BERT models). Among them, the first leverages the BERT's masked language modeling capability to guide the mutation of genes (expression symbols). The other BERT model guides the crossover of individual genes by predicting the crossover point. Experimental results demonstrate that GESR significantly improves computational efficiency compared with traditional GP algorithms and achieves strong overall performance across multiple symbolic regression tasks.