Resolution proofs and Skolem functions in QBF evaluation and applications

  • Authors:

  • Valeriy Balabanov;Jie-Hong R. Jiang

  • Affiliations:

  • Department of Electrical Engineering, Graduate Institute of Electronics Engineering, National Taiwan University, Taipei, Taiwan;Department of Electrical Engineering, Graduate Institute of Electronics Engineering, National Taiwan University, Taipei, Taiwan

  • Venue:
  • CAV'11 Proceedings of the 23rd international conference on Computer aided verification
  • Year:
  • 2011

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Abstract

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 particularly when the certificate is given as a set of Skolem functions. To date the certificate of a true formula can be in the form of either a (cube) resolution proof or a Skolem-function model whereas that of a false formula is in the form of a (clause) resolution proof. The resolution proof and Skolem-function model are somewhat unrelated. This paper strengthens their connection 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 Skolemization-based solvers and computed from standard search-based ones. Fundamentally different from prior methods, our derivation in essence constructs Skolem functions following the variable quantification order. It permits constructing a subset of Skolem functions of interests rather than the whole, and is particularly desirable in many applications. Experimental results show the robust scalability and strong benefits of the new method.