Proof complexity of Mal'tsev CSP

Abstract: Constraint Satisfaction Problems (CSPs) form a broad class of combinatorial problems, which can be formulated as homomorphism problems between relational structures. The CSP dichotomy theorem classifies all such problems over finite domains into two categories: NP-complete and polynomial-time, see Zhuk (2017), Bulatov (2017). Polynomial-time CSPs can be further subdivided into smaller subclasses. Mal'tsev CSPs are defined by the property that every relation in the problem is invariant under a Mal'tsev operation, a ternary operation $\mu$ satisfying $\mu(x, y, y) = \mu(y, y, x) = x$ for all $x, y$. Bulatov and Dalmau proved that Mal'tsev CSPs are solvable in polynomial time, presenting an algorithm for such CSPs (2006). The negation of an unsatisfiable CSP instance can be expressed as a propositional tautology. We formalize the algorithm for Mal'tsev CSPs within bounded arithmetic $V^1$, which captures polynomial-time reasoning and corresponds to the extended Frege proof system. We show that $V^1$ proves the soundness of Mal'tsev algorithm, implying that tautologies expressing the non-existence of a solution for unsatisfiable instances of Mal'tsev CSPs admit short extended Frege proofs. In addition, with small adjustments, we achieved an analogous result for Dalmau's algorithm that solves generalized majority-minority CSPs -- a common generalization of near-unanimity operations and Mal'tsev operations.