Verification of a DPLL Transition System in Rocq

2026-07-16

AI Summary

This work formally verifies the correctness, completeness, and termination of DPLL-based SAT solvers. The authors model the DPLL procedure as a rule-driven state transition system and formalize the syntax and semantics of propositional logic in the Rocq proof assistant. They define both a basic DPLL system and an extended variant incorporating the pure literal rule, introducing a strategy abstraction to uniformly characterize solver behavior. Notably, this is the first complete formal verification in Rocq of an abstract DPLL system that includes the pure literal rule. Termination is established using well-founded relations and structural induction. The proposed framework enables the automatic derivation of terminating and sound solvers from strategies satisfying specified conditions, thereby unifying theoretical verification with concrete implementation.

Abstract

We present a formal verification of an abstract transition-system presentation of the Davis-Putnam-Logemann-Loveland (DPLL) procedure in the Rocq proof assistant. Following Nieuwenhuis et al., SAT solving is modeled as a set of rule-based transitions between states rather than as a concrete algorithm. We formalize the syntax and semantics of propositional formulas, define the classical and base DPLL transition systems, and prove their key metatheoretic properties. In particular, we establish correctness and completeness with respect to satisfiability, and we prove termination by showing that the transition relation is well-founded. The formalization extends the original abstract system by also including the pure literal rule. Building on the verified transition system, we introduce an abstract notion of strategy and derive a terminating solver from any strategy satisfying suitable conditions. We then implement a concrete strategy in Rocq and show that it satisfies the strategy specification.

Problem

• DPLL
• formal verification
• transition system
• SAT solving
• Rocq

Innovation

• DPLL
• formal verification
• transition system
• SAT solving
• proof assistant