Resolution Proofs and Skolem Functions in QBF Evaluation and Applications [chapter]

Valeriy Balabanov, Jie-Hong R. Jiang
2011 Lecture Notes in Computer Science  
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
more » ... ula 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. Quantified Boolean formulae (QBF) allow compact encoding of many decision problems, for example, hardware model checking [6], design rectification [17], program synthesis [18], two-player game solving [13], planning [15], and so on. QBF evaluation has been an important subject in both theoretical and practical computer sciences. Its broad applications have driven intensive efforts pursuing effective QBF solvers, despite the intractable PSPACE-complete complexity. Approaches to QBF evaluation may vary in formula representations, solving mechanisms, data structures, preprocessing techniques, etc. As a matter of fact, the advances of DPLL-style satisfiability (SAT) solving make search-based QBF evaluation [5] on prenex conjunctive normal form (PCNF) formulae the most popular approach. As QBF evaluation procedures are much more complicated than their SAT solving counterparts, validating the results of a QBF solver is more critical than that of a SAT solver. The commonly accepted certificate formats to date are mainly resolution proofs and Skolem-function models. More precisely, for a true QBF, a certificate can be in the syntactic form of a cube-resolution proof (e.g., available in solvers QuBE-cert [12] and yQuaffle [20]) or in the semantic form of a model consisting of a set of Skolem functions (e.g., available in sKizzo [1, 2], squolem [9], and Ebddres [9]); for a false QBF, it can be in the syntactic form of a clause-resolution proof (e.g., available in all the above solvers except for sKizzo). Despite some attempts towards a unified QBF proof checker [9], resolution proofs and Skolem-function models remain weakly related. Moreover, the asymmetry between the available certificate formats in the true and false QBF may seem puzzling. From the application viewpoint, Skolem functions are more directly useful than resolution proofs. The Skolem-function model in solving a true QBF may correspond to, for example, a correct replacement in design rectification, a code fragment in program synthesis, a winning strategy in two-player game solving, a feasible plan in robotic planning, etc. Unfortunately, Skolem-function models are currently only derivable with Skolemization-based solvers, such as sKizzo, squolem, and Ebddres. Moreover, the derivation can be expensive as evidenced by empirical experience that Skolemization-based solvers usually take much longer time on solving true instances than false ones. In contrast, searchbased solvers, such as QuBE-cert, can be more efficient and perform more symmetrically in terms of runtime on true and false instances. This paper takes one step closer to a unified approach to QBF validation by showing that, for a true QBF, its Skolem-function model can be derived from its cube-resolution proof of satisfiability and also from its clause-resolution proof of unsatisfiability under formula negation, both in time linear with respect to proof sizes. Consequently, the aforementioned issues are addressed. Firstly, the connection between resolution proofs and Skolem functions is strongly established. Secondly, it practically conceives Skolem-function countermodels for false QBF, and thus yielding a symmetric view between satisfiability and unsatisfiability certifications. Finally, Skolem-function derivation can be decoupled from Skolemization-based solvers and achieved from the more popular search-based solvers, provided that resolution proofs are maintained. A key characteristic of the new derivation is that Skolem functions are generated for variables quantified from outside in, in contrast to the inside-out computation of Skolemization-based solvers. This feature gives the flexibility of computing some Skolem functions of interests, rather than all as in Skolemization-based solvers. Experimental results show that search-based QBF solver QuBE-cert certifies more QBFEVAL instances 1 than Skolemization-based solvers sKizzo and squolem. Almost all of the Skolem-function models (respectively countermodels) are computable, under resource limits, from the cube-resolution proofs of the true cases (respectively clause-resolution proofs of the false cases). On the other
doi:10.1007/978-3-642-22110-1_12 fatcat:c7kf3ypvajgevo36qm6j7ola6a