CGP runtime analysis: O(n D^5) for conjunctions, exponential for XOR
First rigorous runtime analysis reveals speedups from accepting equally fit solutions.
Duc-Cuong Dang, Roman Kalkreuth, and Andre Opris have published the first runtime analysis of Cartesian Genetic Programming (CGP) in evolving Boolean functions, accepted at PPSN 2026. CGP uses graph-based program representations, making it popular for practical genetic programming. The authors derive asymptotic bounds for building conjunctions of n inputs using at most D >= n-1 binary gates. With strict survival selection, the expected number of fitness evaluations is O(n D^5). When non-strict selection is enabled—accepting equally good solutions, even those with non-contributing gates—the bound improves to O(n D^4). This formalizes a speedup previously observed only empirically.
The analysis also reveals a negative result: CGP requires exponential time to evolve an exclusive disjunction (XOR), highlighting inherent difficulty for certain Boolean functions. Experimental results complement the theory, showing that using incomplete training sets further reduces the average number of fitness evaluations while retaining strong generalization. The work provides a foundational theoretical framework for understanding CGP's search dynamics, bridging a gap between empirical practice and rigorous proof. For practitioners, these insights offer guidance on selection strategies and training set design to optimize evolutionary program synthesis.
- CGP evolves conjunctions in O(n D^5) expected evaluations with strict selection; O(n D^4) with non-strict selection.
- Evolving XOR requires exponential time, indicating strong problem-specific difficulty.
- Incomplete training sets reduce evaluations by up to 40% while maintaining generalization ability.
Why It Matters
First theoretical foundation for CGP performance guides selection strategies and training set design in evolutionary programming.