Simulating Strong Practical Proof Systems with Extended Resolution

Abstract

Proof systems for propositional logic provide the basis for decision procedures that determine the satisfiability status of logical formulas. While the well-known proof system of extended resolution—introduced by Tseitin in the sixties—allows for the compact representation of proofs, modern SAT solvers (i.e., tools for deciding propositional logic) are based on different proof systems that capture practical solving techniques in an elegant way. The most popular of these proof systems is likely DRAT, which is considered the de-facto standard in SAT solving. Moreover, just recently, the proof system DPR has been proposed as a generalization of DRAT that allows for short proofs without the need of new variables. Since every extended-resolution proof can be regarded as a DRAT proof and since every DRAT proof is also a DPR proof, it was clear that both DRAT and DPR generalize extended resolution. In this paper, we show that—from the viewpoint of proof complexity—these two systems are no stronger than extended resolution. We do so by showing that (1) extended resolution polynomially simulates DRAT and (2) DRAT polynomially simulates DPR. We implemented our simulations as proof-transformation tools and evaluated them to observe their behavior in practice. Finally, as a side note, we show how Kullmann’s proof system based on blocked clauses (another generalization of extended resolution) is related to the other systems.

Advanced tools and methods for treewidth-based problem solving

“…This question is still not completely solved, since Sat solvers consist of additional heuristics and a sophisticated interplay between different components. However, there are certainly connections to the underlying proof system that is actually used in CDCL-based solving [Elffers et al, 2018;Gocht et al, 2019;Kiesl et al, 2020], but also structural properties seem to play a role, e.g., [Samer and Szeider, 2009;Ansótegui et al, 2019]. Recently, it was argued and shown empirically that additionally the underlying hardware and optimization towards this hardware is certainly a contributing factor as well [Fichte et al, 2020e].…”

Clause Redundancy and Preprocessing in Maximum Satisfiability

“…been shown to give rise to very powerful proof systems, going far beyond resolution [22,23,30]. In contrast to viewing clause redundancy through the lens of logical entailment, the redundancy criteria developed in this line of work are based on a semantic implication relationship between formulas, making them desirably efficient to decide and at the same time are guaranteed to merely preserve satisfiability rather than logical equivalence.…”