A Verified SAT Solver Framework with Learn, Forget, Restart, and Incrementality

Abstract

We developed a formal framework for conflict-driven clause learning (CDCL) using the Isabelle/HOL proof assistant. Through a chain of refinements, an abstract CDCL calculus is connected first to a more concrete calculus, then to a SAT solver expressed in a functional programming language, and finally to a SAT solver in an imperative language, with total correctness guarantees. The framework offers a convenient way to prove metatheorems and experiment with variants, including the Davis-Putnam-Logemann-Loveland (DPLL) calculus. The imperative program relies on the two-watched-literal data structure and other optimizations found in modern solvers. We used Isabelle's Refinement Framework to automate the most tedious refinement steps. The most noteworthy aspects of our work are the inclusion of rules for forget, restart, and incremental solving and the application of stepwise refinement.

Formalizing Bachmair and Ganzinger's Ordered Resolution Prover

“…Early work was carried out by Shankar [22] and Persson [18]. Some of our own efforts are also related: completeness of unordered resolution using semantic trees by Schlichtkrull [20]; completeness of a Gentzen system by Blanchette, Popescu, and Traytel [9]; and completeness of CDCL by Blanchette, Fleury, Lammich, and Weidenbach [6]. We refer to our earlier papers for further discussions of related work.…”

“…We use the same library of clauses (Clausal _ Logic.thy) as for the verified SAT solver by Blanchette et al [6], which is also part of IsaFoL. Atoms are represented by a type variable a, which can be instantiated by arbitrary concrete types-e.g., numbers or first-order terms.…”

A Verified SAT Solver Framework with Learn, Forget, Restart, and Incrementality

“…The main novelties of our framework are the inclusion of rules for forget, restart, and incremental solving and the application of stepwise refinement to transfer results. The original version of this paper was presented at the eighth edition of the International Joint Conference on Automated Reasoning (IJCAR 2016) in Coimbra, Portugal [Blanchette et al, 2016].…”

A verified prover based on ordered resolution

“…It is also the first example of the application of refinement in a first-order context. This methodology has been used to verify SAT solvers [6,29], which decide the satisfiability of propositional formulas, but first-order logic is semidecidable-sound and complete provers are guaranteed to terminate only for unsatisfiable (i.e., provable) clause sets. This complicates the transfer of completeness results across refinement layers.…”