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.…”

“…Clauses and Models. 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.…”

Efficient Certified Resolution Proof Checking

“…One approach investigated over the years has been to develop formally derived SAT solvers [36,31,32,7]. These works all follow the same underlying idea: formally specify SAT solving techniques within a constructive theorem prover and apply program extraction (an implementation of the Curry-Howard correspondence) to obtain a certified SAT solver.…”

QED at Large: A Survey of Engineering of Formally Verified Software

“…Proof engineering has already had a large impact on program verification in many domains, including those from Section 3.1. Proof engineers have in recent years verified operating system (Klein et al, 2009) and web browser (Jang et al, 2012) kernels, machine learning systems , distributed systems (Woos et al, 2016), quantum circuits (Rand et al, 2017), constraint solvers (Blanchette et al, 2018;3.3…”