2024/03/11 by Azza Gaysin, Gaysin, Azza · 2 citations
Computer Science · #03F03 (Primary) #68W99 (Secondary) #Advanced Algebra and Logic #F.4.1 #FOS: Mathematics #Logic (math.LO) #Logic, Reasoning, and Knowledge #Logic, programming, and type systems
paper · pdf · doi:10.48550/arxiv.2403.06704
openalex publication_date 2024/03/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The constraint satisfaction problem (CSP) can be formulated as a homomorphism problem between relational structures: given a structure A, for any structure X, whether there exists a homomorphism from X to A. For years, it has been conjectured that all problems of this type are divided into polynomial-time and NP-complete problems, and the conjecture was proved in 2017 separately by Zhuk (2017) and Bulatov (2017). Zhuk's algorithm solves tractable CSPs in polynomial time. The algorithm is partly based on universal algebra theorems: informally, they state that after reducing some domain of an instance to its strong subuniverses, a satisfiable instance maintains a solution. In this paper, we present the formalization of the proofs of these theorems in the bounded arithmetic W11 introduced by Skelley (2004). The formalization, together with our previous results (2022), shows that W11 proves the soundness of Zhuk's algorithm, where by soundness we mean that any rejection of the algorithm is correct. From the known relation of the theory to propositional calculus G, it follows that tautologies, expressing the non-existence of a solution for unsatisfiable instances, have short proofs in G.