Expressing Symmetry Breaking in DRAT Proofs

An effective SAT preprocessing technique is the addition of symmetry-breaking predicates: auxiliary clauses that guide a SAT solver away from needless exploration of isomorphic sub-problems. Symmetry-breaking predicates have been in use for over a decade. However, it was not known how to express the addition of these predicates in proofs of unsatisfiability. Hence, results obtained by symmetry breaking cannot be validated by existing proof checkers. We present a method to express the addition of symmetry-breaking predicates in DRAT, a clausal proof format supported by top-tier solvers. We applied this method to generate SAT problems that have not been previously solved without symmetry-breaking predicates. We validated these proofs with an ACL2-based, mechanically-verified DRAT proof checker and the proof-checking tool of SAT Competition 2014.

[1]  Armin Biere,et al.  PicoSAT Essentials , 2008, J. Satisf. Boolean Model. Comput..

[2]  Allen Van Gelder.  Verifying Propositional Unsatisfiability: Pitfalls to Avoid , 2007, SAT.

[3]  S. Radziszowski.  Small Ramsey Numbers , 2011 .

[4]  Armin Biere,et al.  Inprocessing Rules , 2012, IJCAR.

[5]  Kenneth E. Batcher,et al.  Sorting networks and their applications , 1968, AFIPS Spring Joint Computing Conference.

[6]  Marijn Heule,et al.  Mechanical Verification of SAT Refutations with Extended Resolution , 2013, ITP.

[7]  Igor L. Markov,et al.  Solving difficult SAT instances in the presence of symmetry , 2002, Proceedings 2002 Design Automation Conference (IEEE Cat. No.02CH37324).

[8]  E. Szemerédi,et al.  O(n LOG n) SORTING NETWORK. , 1983 .

[9]  Michal Kouril,et al.  The van der Waerden Number W(2, 6) Is 1132 , 2008, Exp. Math..

[10]  Marijn J. H. Heule,et al.  Dynamic Symmetry Breaking by Simulating Zykov Contraction , 2009, SAT.

[11]  Marijn J. H. Heule,et al.  DRAT-trim: Efficient Checking and Trimming Using Expressive Clausal Proofs , 2014, SAT.

[12]  Igor L. Markov,et al.  Exploiting structure in symmetry detection for CNF , 2004, Proceedings. 41st Design Automation Conference, 2004..

[13]  Michael Frank,et al.  Twenty-Five Comparators Is Optimal When Sorting Nine Inputs (and Twenty-Nine for Ten) , 2014, 2014 IEEE 26th International Conference on Tools with Artificial Intelligence.

[14]  Armin Biere,et al.  Automated Testing and Debugging of SAT and QBF Solvers , 2010, SAT.

[15]  James M. Crawford,et al.  Symmetry-Breaking Predicates for Search Problems , 1996, KR.

[16]  Marijn J. H. Heule,et al.  Verifying Refutations with Extended Resolution , 2013, CADE.

[17]  Allen Van Gelder,et al.  Verifying RUP Proofs of Propositional Unsatisfiability , 2008, ISAIM.

[18]  Nathan Wetzler,et al.  Efficient, Mechanically-Verified Validation of Satisfiability Solvers , 2015 .

[19]  Igor L. Markov,et al.  Efficient symmetry breaking for Boolean satisfiability , 2003, IEEE Transactions on Computers.

[20]  Oliver Kullmann,et al.  On a Generalization of Extended Resolution , 1999, Discret. Appl. Math..

[21]  Marijn J. H. Heule,et al.  Bridging the gap between easy generation and efficient verification of unsatisfiability proofs , 2014, Softw. Test. Verification Reliab..

[22]  Marijn J. H. Heule,et al.  Trimming while checking clausal proofs , 2013, 2013 Formal Methods in Computer-Aided Design.

[23]  Armin Biere,et al.  Blocked Clause Elimination , 2010, TACAS.

[24]  Ian Parberry,et al.  The Pairwise Sorting Network , 1992, Parallel Process. Lett..

[25]  Alexei Lisitsa,et al.  A SAT Attack on the Erdős Discrepancy Conjecture , 2014, SAT.

[26]  Sharad Malik,et al.  Validating SAT solvers using an independent resolution-based checker: practical implementations and other applications , 2003, 2003 Design, Automation and Test in Europe Conference and Exhibition.

[27]  Niklas Sörensson,et al.  An Extensible SAT-solver , 2003, SAT.

[28]  Ian P. Gent,et al.  Symmetry Breaking in Constraint Programming , 2000, ECAI.