Marko Maliković

Proving Correctness of a KRK Chess Endgame Strategy by SAT-based Constraint Solving

Chess endgame strategies describe, in a concise and intuitive way, the rules that a player should follow to ensure a win (or a draw). Endgame strategies are useful for both computer and human players. Their correctness can be proved in several ways. In this article we present one of them: a computer assisted proof based on reduction to propositional logic, more precisely to SAT. We focus on a strategy for the KRK endgame. The reduction to SAT is performed by using a constraint solving system URSA. The relevant lemmas produced SAT instances with hundreds or even thousands of variables and clauses, but URSA still successfully handled them. We would like to emphasise that this is the first computer-assisted high-level proof of the correctness of a strategy for some chess endgame. The presented methodology can be applied to other endgames and other games as well. Therefore, the point of this article is not only presenting a proof of correctness of an endgame strategy, but also presenting a new methodology for computer-assisted reasoning about chess endgames.

Proving Correctness of a KRK Chess Endgame Strategy by Using Isabelle/HOL and Z3

We describe an executable specification and a total correctness proof of a King and Rook vs King (KRK) chess endgame strategy within the proof assistant Isabelle/HOL. This work builds upon a previous computer-assisted correctness analysis performed using the constraint solver URSA. The distinctive feature of the present machine verifiable formalization is that all central properties have been automatically proved by the SMT solver Z3 integrated into Isabelle/HOL, after being suitably expressed in linear integer arithmetic. This demonstrates that the synergy between the state-of-the-art automated and interactive theorem proving is mature enough so that very complex conjectures from various AI domains can be proved almost in a “push-button” manner, yet in a rich logical framework offered by the modern ITP systems.