Analysing the Influence of Reorder Strategies for Cartesian Genetic Programming

TL;DR

This work analyzes how reorder strategies influence Cartesian Genetic Programming by addressing positional bias through genotype reshuffling while preserving phenotype. It revisits Original and Equidistant Reorder and introduces Uniform-Reorder, NegBias-Reorder, and Left-Skewed Reorder, evaluating them on eight benchmarks (four Boolean and four symbolic regression) using Bayesian methods to rank configurations. Results show reordering reliably improves performance over Standard CGP, but no single operator dominates across tasks, and hyperparameters like $p_{reorder}$ add optimization complexity. Across convergence analyses, all reorder variants exhibit broadly similar Fast-to-Slow convergence behavior, suggesting reordering reshapes exploration without altering the fundamental convergence dynamics.

Abstract

Cartesian Genetic Programming (CGP) suffers from a specific limitation: Positional bias, a phenomenon in which mostly genes at the start of the genome contribute to a program output, while genes at the end rarely do. This can lead to an overall worse performance of CGP. One solution to overcome positional bias is to introduce reordering methods, which shuffle the current genotype without changing its corresponding phenotype. There are currently two different reorder operators that extend the classic CGP formula and improve its fitness value. In this work, we discuss possible shortcomings of these two existing operators. Afterwards, we introduce three novel operators which reorder the genotype of a graph defined by CGP. We show empirically on four Boolean and four symbolic regression benchmarks that the number of iterations until a solution is found and/or the fitness value improves by using CGP with a reorder method. However, there is no consistently best performing reorder operator. Furthermore, their behaviour is analysed by investigating their convergence plots and we show that all behave the same in terms of convergence type.

Figures (9)

• Figure 1: Example graph defined by a CGP genotype. The dashed node and connections are inactive due to not contributing to the output.
• Figure 2: Distribution of active nodes over a graph defined by CGP. Example generated over 75 independent runs on the 3-bit multiply Boolean benchmark.
• Figure 3: Distribution of active nodes for Standard compared to Original-Reorder.
• Figure 4: Thought experiment illustrating limitations in EquiDist-Reorder. By reordering the nodes, the optimal solution is harder to obtain. Before reordering nodes, the node $Result$ must mutate its connection from node $B$ to $D$. After reordering, more changes must be made to obtain the optimal solution.
• Figure 5: Probability density function of the beta distribution with $\alpha = 6$ and $\beta=1$.