Generalization and Completeness of Stochastic Local Search Algorithms
Daniel Loscos, Narciso Marti-Oliet, Ismael Rodriguez
TL;DR
The paper presents a fully formal, general framework for stochastic local search (SLS) methods, unifying diverse families such as Genetic Algorithms (GA), Ant Colony Optimization (ACO), and Particle Swarm Optimization (PSO) under a single operational semantics with a large common structure and a small parametric component. It then proves a sweeping result: SLS methods are Turing-complete in general by constructing a GA capable of simulating any Turing machine via a MPCP-based encoding of the machine’s computation history, implying that non-trivial semantic properties are undecidable in general (Rice’s theorem). The authors sketch how these Turing-completeness proofs extend to ACO and PSO, highlighting the deep implications for predictability and verifiability of SLS behavior. They also discuss how this formal framework can enable a formal taxonomy of SLS methods and guide future work on complexity and restricted-memory variants.
Abstract
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.