Fuzzing and Verifying RAT Refutations with Deletion Information
- 1. Fuzzing and Verifying RAT Refutations with Deletion Information Walter Forkel, Tobias Philipp, Adri´an Rebola-Pardo, Elias Werner Introduction SAT solvers are common tools in the industry Unfortunately, SAT solvers contain bugs DRAT format is the de facto standard for emitting unsatisfiable proofs F SAT solver SAT UNSAT, P checker Single Point of Failure Are there any unknown bugs in the checker ? Background Tautological clause: clause containing A and ¬A for some variable A Resolvent of C, D with A ∈ C and ¬A ∈ D is (C {A}) ∪ (D {¬A}) Asymmetric Tautology (AT) Asymmetric Literal Addition (ALA): alaF (C) = C ∪ {L | L1, ..., Ln, L ∈ F and Li ∈ C for all 1 ≤ i ≤ n} C is an AT wrt F if there is n ∈ N such that alaF (C) ↑ n is a tautology Resolution Asymmetric Tautology (RAT) C is a RAT upon literal L wrt the formula F if 1. C is an AT wrt the formula F, or 2. L ∈ C, and all resolvents of C with any D ∈ F upon L are AT wrt F DRAT Refutation Finite sequence of addition and deletion instructions to the empty clause Each added clause is a RAT wrt the preceding clauses Our Approach: Fuzzing DRAT Refutations and Check Against a Fully Mechanically Verified Checker 1 2 3 4 5 random formula generation SAT solver checkers classify P Verified Checker in Coq modifier checkers classify P Verified Checker in Coq F unsatisfiable proof P F satisfiable F all accept P inconsistent P inconsistent all reject P Example F = {p, q} {p, q} {p, q} {p, q} P = a {¬r} RAT upon ¬r a {r, p} AT a {r} AT d {p, q} deletion d {¬p, q} deletion d {p, ¬q} deletion d {¬p, ¬q} deletion d {} AT drat-trim and drat-fc classify P as accepted add r to the first clause P = a {¬r, r} RAT upon ¬r a {r, p} AT a {r} AT d {p, q} deletion d {¬p, q} deletion d {p, ¬q} deletion d {¬p, ¬q} deletion d {} AT drat-trim rejects P drat-fc accepts P Our verified checker also accepts P Found a bug in drat-trim Results Formulas: 2900 clauses, 800 variables on average Resulting proofs: 2100 steps on average drat-trim rejects DRAT refutations containing tautological clauses proofcheck rejects DRAT refutations in which unit clauses are deleted DRAT emission procedure in the SAT solver minisat constructs unexpected DRAT refutations of the form (a {})(d C)(a {}) Conclusion DRAT checkers proofcheck and drat-trim contain bugs New fuzzing technique that automatically finds bugs in checkers Our verified DRAT checker is complete Future Work: adapt the checker and fuzzing procedure to other proof formats Code available at github.com/drat-tools Acknowledgement: This work was supported by RiSE, LogiCS and WWTF grant VRG11-005 FLAIRS 30