Generalization and Completeness of Stochastic Local Search Algorithms
This provides a foundational theoretical insight into the computational limits of SLS algorithms, which is incremental as it builds on existing concepts to formalize and prove undecidability.
The authors generalized Stochastic Local Search (SLS) heuristics into a formal model and used it to prove the Turing-completeness of SLS algorithms, showing that determining non-trivial input-output properties is undecidable for methods like Genetic Algorithms, Ant Colony Optimization, and Particle Swarm Optimization.
We generalize Stochastic Local Search (SLS) heuristics into a unique formal model. This model has two key components: a common structure designed to be as large as possible and a parametric structure intended to be as small as possible. Each heuristic is obtained by instantiating the parametric part in a different way. Particular instances for Genetic Algorithms (GA), Ant Colony Optimization (ACO), and Particle Swarm Optimization (PSO) are presented. Then, we use our model to prove the Turing-completeness of SLS algorithms in general. The proof uses our framework to construct a GA able to simulate any Turing machine. This Turing-completeness implies that determining any non-trivial property concerning the relationship between the inputs and the computed outputs is undecidable for GA and, by extension, for the general set of SLS methods (although not necessarily for each particular method). Similar proofs are more informally presented for PSO and ACO.