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The constraint satisfaction problem (CSP) can be formulated as a homomorphism problem between relational structures: given a structure $\mathcal{A}$, for any structure $\mathcal{X}$, whether there exists a homomorphism from $\mathcal{X}$ to $\mathcal{A}$. 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 $W^1_1$ introduced by Skelley (2004). The formalization, together with our previous results (2022), shows that $W_1^1$ 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 $G$. |
