TY - GEN

T1 - Long Proofs of (Seemingly) Simple Formulas

AU - Mikša, Mladen

AU - Nordström, Jakob

PY - 2014

Y1 - 2014

N2 - In 2010, Spence and Van Gelder presented a family of CNF formulas based on combinatorial block designs. They showed empirically that this construction yielded small instances that were orders of magnitude harder for state-of-the-art SAT solvers than other benchmarks of comparable size, but left open the problem of proving theoretical lower bounds. We establish that these formulas are exponentially hard for resolution and even for polynomial calculus, which extends resolution with algebraic reasoning. We also present updated experimental data showing that these formulas are indeed still hard for current CDCL solvers, provided that these solvers do not also reason in terms of cardinality constraints (in which case the formulas can become very easy). Somewhat intriguingly, however, the very hardest instances in practice seem to arise from so-called fixed bandwidth matrices, which are provably easy for resolution and are also simple in practice if the solver is given a hint about the right branching order to use. This would seem to suggest that CDCL with current heuristics does not always search efficiently for short resolution proofs, despite the theoretical results of [Pipatsrisawat and Darwiche 2011] and [Atserias, Fichte, and Thurley 2011].

AB - In 2010, Spence and Van Gelder presented a family of CNF formulas based on combinatorial block designs. They showed empirically that this construction yielded small instances that were orders of magnitude harder for state-of-the-art SAT solvers than other benchmarks of comparable size, but left open the problem of proving theoretical lower bounds. We establish that these formulas are exponentially hard for resolution and even for polynomial calculus, which extends resolution with algebraic reasoning. We also present updated experimental data showing that these formulas are indeed still hard for current CDCL solvers, provided that these solvers do not also reason in terms of cardinality constraints (in which case the formulas can become very easy). Somewhat intriguingly, however, the very hardest instances in practice seem to arise from so-called fixed bandwidth matrices, which are provably easy for resolution and are also simple in practice if the solver is given a hint about the right branching order to use. This would seem to suggest that CDCL with current heuristics does not always search efficiently for short resolution proofs, despite the theoretical results of [Pipatsrisawat and Darwiche 2011] and [Atserias, Fichte, and Thurley 2011].

UR - http://www.scopus.com/inward/record.url?scp=84958552506&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-09284-3_10

DO - 10.1007/978-3-319-09284-3_10

M3 - Article in proceedings

AN - SCOPUS:84958552506

SN - 9783319092836

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 121

EP - 137

BT - Theory and Applications of Satisfiability Testing, SAT 2014 - 17th International Conference, Held as Part of theVienna Summer of Logic, VSL 2014, Proceedings

PB - Springer Verlag,

T2 - 17th International Conference on Theory and Applications of Satisfiability Testing, SAT 2014, Held as Part of the Vienna Summer of Logic, VSL 2014

Y2 - 14 July 2014 through 17 July 2014

ER -