Efficient Verified (UN)SAT Certificate Checking

Abstract

We present an efficient formally verified checker for satisfiability and unsatisfiability certificates for Boolean formulas in conjunctive normal form. It utilizes a two phase approach: Starting from a DRAT certificate, the unverified generator computes an enriched certificate, which is checked against the original formula by the verified checker. Using the Isabelle/HOL Refinement Framework, we verify the actual implementation of the checker, specifying the semantics of the formula down to the integer sequence that represents it. On a realistic benchmark suite drawn from the 2016 SAT competition, our approach is more than two times faster than the unverified standard tool drat-trim. Additionally, we implemented a multi-threaded version of the generator, which further reduces the runtime.

Theory and Applications of Satisfiability Testing – SAT 2018

“…Given a formula and a clause, it can be decided in polynomial time whether the clause is a resolution asymmetric tautology with respect to the formula and therefore the soundness of DRAT proofs can be checked efficiently. Several formally-verified checkers for DRAT proofs are available [5,6].…”

Everything's Bigger in Texas: "The Largest Math Proof Ever"

“…The size of the combined proof is 200 terabytes in size. However, it has been validated using three formally verified checkers -two of these efforts were by independent groups [5,3,13]. Our ability to produce such proofs and certify them using theorem provers provides high confidence in the correctness of our result.…”

“…However, the size shows that automated tools combined with super computing facilitate solving bigger problems. Moreover, the proof of 200 terabytes can now be validated using highly trusted systems [5,3,13], demonstrating that we can check the correctness of proofs no matter their size.The We answer this question, known as the Boolean Pythagorean triples problem, by encoding it into propositional logic and applying massive parallel SAT solving on the resulting formula. More concretely, we search for the smallest number n such that every coloring of the numbers 1 to n with red and blue results in a monochromatic solution of a 2 + b 2 = c 2 .…”

“…However, the size shows that automated tools combined with super computing facilitate solving bigger problems. Moreover, the proof of 200 terabytes can now be validated using highly trusted systems [5,3,13], demonstrating that we can check the correctness of proofs no matter their size.…”

Beyond DRAT: Challenges in Certifying UNSAT

“…Before this year (2017), the unsatisfiability certificates had to be checked by an independent program, DRAT-trim, which needed to be trusted. Now we have hybrid verifiers (by CruzFilipe et al [2] and Lammich [8], both presented at CADE-26), which are based on an untrusted preprocessor that trims and annotates a DRAT proof, followed by a formally verified (in Coq or ACL2 [2] respectively Isabelle [8]) checker that certifies that the preprocessed proof establishes unsatisfiability of the input problem. Compared to DRAT-trim, which is written in C, this increases the level of trust significantly, without significantly impact on the verification time.…”