Unified QBF certification and its applications

Quantified Boolean formulae (QBF) allow compact encoding of many decision problems. Their importance motivated the development of fast QBF solvers. Certifying the results of a QBF solver not only ensures correctness, but also enables certain synthesis and verification tasks. To date the certificate of a true formula can be in the form of either a syntactic cube-resolution proof or a semantic Skolem-function model whereas that of a false formula is only in the form of a syntactic clause-resolution proof. The semantic certificate for a false QBF is missing, and the syntactic and semantic certificates are somewhat unrelated. This paper identifies the missing Herbrand-function countermodel for false QBF, and strengthens the connection between syntactic and semantic certificates by showing that, given a true QBF, its Skolem-function model is derivable from its cube-resolution proof of satisfiability as well as from its clause-resolution proof of unsatisfiability under formula negation. Consequently Skolem-function derivation can be decoupled from special Skolemization-based solvers and computed from standard search-based ones. Experimental results show strong benefits of the new method.

[1]  Luca Pulina,et al.  Evaluating and certifying QBFs: A comparison of state-of-the-art tools , 2009, AI Commun..

[2]  Salil P. Vadhan,et al.  Computational Complexity , 2005, Encyclopedia of Cryptography and Security.

[3]  Jie-Hong Roland Jiang,et al.  Resolution Proofs and Skolem Functions in QBF Evaluation and Applications , 2011, CAV.

[4]  Marco Schaerf,et al.  An Algorithm to Evaluate Quantified Boolean Formulae and Its Experimental Evaluation , 2002, Journal of Automated Reasoning.

[5]  Roderick Bloem,et al.  Fault Localization and Correction with QBF , 2007, SAT.

[6]  G. S. Tseitin.  On the Complexity of Derivation in Propositional Calculus , 1983 .

[7]  Marco Benedetti,et al.  Evaluating QBFs via Symbolic Skolemization , 2005, LPAR.

[8]  Nachum Dershowitz,et al.  Bounded Model Checking with QBF , 2005, SAT.

[9]  Jussi Rintanen,et al.  Constructing Conditional Plans by a Theorem-Prover , 1999, J. Artif. Intell. Res..

[10]  Hans Kleine Büning,et al.  Resolution for Quantified Boolean Formulas , 1995, Inf. Comput..

[11]  Sharad Malik,et al.  Conflict driven learning in a quantified Boolean Satisfiability solver , 2002, ICCAD 2002.

[12]  Armando Tacchella,et al.  Clause/Term Resolution and Learning in the Evaluation of Quantified Boolean Formulas , 2006, J. Artif. Intell. Res..

[13]  Marco Benedetti.  Extracting Certificates from Quantified Boolean Formulas , 2005, IJCAI.

[14]  Daniel Kroening,et al.  A First Step Towards a Unified Proof Checker for QBF , 2007, SAT.

[15]  Sanjit A. Seshia,et al.  Combinatorial sketching for finite programs , 2006, ASPLOS XII.

[16]  Jie-Hong Roland Jiang,et al.  Interpolating functions from large Boolean relations , 2009, 2009 IEEE/ACM International Conference on Computer-Aided Design - Digest of Technical Papers.

[17]  Amir Pnueli,et al.  Automatic Hardware Synthesis from Specifications: A Case Study , 2007, 2007 Design, Automation & Test in Europe Conference & Exhibition.

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

[19]  S. Malik,et al.  Validating the result of a quantified Boolean formula (QBF) solver: theory and practice , 2005, Proceedings of the ASP-DAC 2005. Asia and South Pacific Design Automation Conference, 2005..