Clause/Term Resolution and Learning in the Evaluation of Quantified Boolean Formulas

Abstract

Resolution is the rule of inference at the basis of most procedures for automated reasoning. In these procedures, the input formula is first translated into an equisatisfiable formula in conjunctive normal form (CNF) and then represented as a set of clauses. Deduction starts by inferring new clauses by resolution, and goes on until the empty clause is generated or satisfiability of the set of clauses is proven, e.g., because no new clauses can be generated.\ud In this paper, we restrict our attention to the problem of evaluating Quantified Boolean Formulas (QBFs). In this setting, the above outlined deduction process is known to be sound and complete if given a formula in CNF and if a form of resolution, called “Q-resolution”, is used. We introduce Q-resolution on terms, to be used for formulas in disjunctive normal form. We show that the computation performed by most of the available procedures for QBFs –based on the Davis-Logemann-Loveland procedure (DLL) for propositional satisfiability– corresponds to a tree in which Q-resolution on terms and clauses alternate. This poses the theoretical bases for the introduction of learning, corresponding to recording Q-resolution formulas associated with the nodes of the tree. We discuss the problems related to the introduction of learning in DLL based procedures, and present solutions extending state-of-the-art proposals coming from the literature on propositional satisfiability. Finally, we show that our DLL based solver extended with learning, performs significantly better on benchmarks used in the 2003 QBF solvers comparative evaluation

Conformant Planning as a Case Study of Incremental QBF Solving

“…Search-based QBF solving [8] is a generalization of the DPLL algorithm [9] for SAT. Modern search-based QBF solvers implement a QBF-specific variant of conflictdriven clause learning (CDCL) for SAT, called QCDCL [13,23,27,36]. In QCDCL the variables are successively assigned until an (un)satisfiable subcase is encountered.…”

Long-Distance Resolution: Proof Generation and Strategy Extraction in Search-Based QBF Solving

“…Search-based QBF solving is an extension of the DPLL algorithm [2,3]. Given a QBF ψ = P.φ, the idea of QCDCL [4,8,15,16] is to dynamically generate and add derived clauses to the matrix φ. If ψ is false, then the empty clause 2 will finally be generated.…”

“…conjunctions of literals, from solutions by function analyze solution. We refer to related literature on cube learning [4,5,8,10,16].…”

“…The left part of Table 1 shows the number of instances solved in the benchmark set from the QBF evaluation 2012 (QBFEVAL'12-pre), 3 which was preprocessed by Bloqqer. 4 Table 2. Number of resolution steps (in units of 1,000) in refutations of selected formulas in the family ϕt from Section 3.…”

On Conflicts and Strategies in QBF

“…Existing QBF solvers reveal an interesting dichotomy. Some solvers traverse the search space by assigning values to the individual variables similarly as a backtracking algorithm would [30,11,21]. Other solvers expand quantifiers into logical connectives according to their semantics [20,6,14].…”