57 research outputs found

    Quantifier Structure in Search-Based Procedures for QBFs

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    DepQBF 6.0: A Search-Based QBF Solver Beyond Traditional QCDCL

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    We present the latest major release version 6.0 of the quantified Boolean formula (QBF) solver DepQBF, which is based on QCDCL. QCDCL is an extension of the conflict-driven clause learning (CDCL) paradigm implemented in state of the art propositional satisfiability (SAT) solvers. The Q-resolution calculus (QRES) is a QBF proof system which underlies QCDCL. QCDCL solvers can produce QRES proofs of QBFs in prenex conjunctive normal form (PCNF) as a byproduct of the solving process. In contrast to traditional QCDCL based on QRES, DepQBF 6.0 implements a variant of QCDCL which is based on a generalization of QRES. This generalization is due to a set of additional axioms and leaves the original Q-resolution rules unchanged. The generalization of QRES enables QCDCL to potentially produce exponentially shorter proofs than the traditional variant. We present an overview of the features implemented in DepQBF and report on experimental results which demonstrate the effectiveness of generalized QRES in QCDCL.Comment: 12 pages + appendix; to appear in the proceedings of CADE-26, LNCS, Springer, 201

    Evaluating QBF Solvers: Quantifier Alternations Matter

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    We present an experimental study of the effects of quantifier alternations on the evaluation of quantified Boolean formula (QBF) solvers. The number of quantifier alternations in a QBF in prenex conjunctive normal form (PCNF) is directly related to the theoretical hardness of the respective QBF satisfiability problem in the polynomial hierarchy. We show empirically that the performance of solvers based on different solving paradigms substantially varies depending on the numbers of alternations in PCNFs. In related theoretical work, quantifier alternations have become the focus of understanding the strengths and weaknesses of various QBF proof systems implemented in solvers. Our results motivate the development of methods to evaluate orthogonal solving paradigms by taking quantifier alternations into account. This is necessary to showcase the broad range of existing QBF solving paradigms for practical QBF applications. Moreover, we highlight the potential of combining different approaches and QBF proof systems in solvers.Comment: preprint of a paper to be published at CP 2018, LNCS, Springer, including appendi

    Planning as Quantified Boolean Formulae

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    This work explores the idea of classical Planning as Quantified Boolean Formulae. Planning as Satisfiability (SAT) is a popular approach to Planning and has been explored in detail producing many compact and efficient encodings, Planning-specific solver implementations and innovative new constraints. However, Planning as Quantified Boolean Formulae (QBF) has been relegated to conformant Planning approaches, with the exception of one encoding that has not yet been investigated in detail. QBF is a promising setting for Planning given that the problems have the same complexity. This work introduces two approaches for translating bounded propositional reachability problems into QBF. Both exploit the expressivity of the binarytree structure of the QBF problem to produce encodings that are as small as logarithmic in the size of the instance and thus exponentially smaller than the corresponding SAT encoding with the same bound. The first approach builds on the iterative squaring formulation of Rintanen; the intuition behind the idea is to recursively fold the plan around the midpoint, reducing the number of time-steps that need to be described from n to logâ‚‚n. The second approach exploits domain-level lifting to achieve significant improvements in efficiency. Experimentation was performed to compare our formulation of the first approach with the previous formulation, and to compare both approaches with comparative and state-of-the-art SAT approaches. Results presented in this work show that our formulation of the first approach is an improvement over the previous, and that both approaches produce encodings that are indeed much smaller than corresponding SAT encodings, in both terms of encoding size and memory used during solving. Evidence is also provided to show that the first approach is feasible, if not yet competitive with the state-of-the-art, and that the second approach produces superior encodings to the SAT encodings when the domain is suited to domain-level lifting.This work explores the idea of classical Planning as Quantified Boolean Formulae. Planning as Satisfiability (SAT) is a popular approach to Planning and has been explored in detail producing many compact and efficient encodings, Planning-specific solver implementations and innovative new constraints. However, Planning as Quantified Boolean Formulae (QBF) has been relegated to conformant Planning approaches, with the exception of one encoding that has not yet been investigated in detail. QBF is a promising setting for Planning given that the problems have the same complexity. This work introduces two approaches for translating bounded propositional reachability problems into QBF. Both exploit the expressivity of the binarytree structure of the QBF problem to produce encodings that are as small as logarithmic in the size of the instance and thus exponentially smaller than the corresponding SAT encoding with the same bound. The first approach builds on the iterative squaring formulation of Rintanen; the intuition behind the idea is to recursively fold the plan around the midpoint, reducing the number of time-steps that need to be described from n to logâ‚‚n. The second approach exploits domain-level lifting to achieve significant improvements in efficiency. Experimentation was performed to compare our formulation of the first approach with the previous formulation, and to compare both approaches with comparative and state-of-the-art SAT approaches. Results presented in this work show that our formulation of the first approach is an improvement over the previous, and that both approaches produce encodings that are indeed much smaller than corresponding SAT encodings, in both terms of encoding size and memory used during solving. Evidence is also provided to show that the first approach is feasible, if not yet competitive with the state-of-the-art, and that the second approach produces superior encodings to the SAT encodings when the domain is suited to domain-level lifting

    Incremental QBF Solving

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    We consider the problem of incrementally solving a sequence of quantified Boolean formulae (QBF). Incremental solving aims at using information learned from one formula in the process of solving the next formulae in the sequence. Based on a general overview of the problem and related challenges, we present an approach to incremental QBF solving which is application-independent and hence applicable to QBF encodings of arbitrary problems. We implemented this approach in our incremental search-based QBF solver DepQBF and report on implementation details. Experimental results illustrate the potential benefits of incremental solving in QBF-based workflows.Comment: revision (camera-ready, to appear in the proceedings of CP 2014, LNCS, Springer

    Reasons for Hardness in QBF Proof Complexity

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    Quantified Boolean Formulas (QBF) extend the canonical NP-complete satisfiability problem by including Boolean quantifiers. Determining the truth of a QBF is PSPACE-complete; this is expected to be a harder problem than satisfiability, and hence QBF solving has much wider applications in practice. QBF proof complexity forms the theoretical basis for understanding QBF solving, as well as providing insights into more general complexity theory, but is less well understood than propositional proof complexity. We begin this thesis by looking at the reasons underlying QBF hardness, and in particular when the hardness is propositional in nature, rather than arising due to the quantifiers. We introduce relaxing QU-Res, a previous model for identifying such propositional hardness, and construct an example where relaxing QU-Res is unsuccessful in this regard. We then provide a new model for identifying such hardness which we prove captures this concept. Now equipped with a means of identifying ‘genuine’ QBF hardness, we prove a new lower bound technique for tree-like QBF proof systems. Lower bounds using this technique allows us to show a new separation between tree-like and dag-like systems. We give a characterisation of lower bounds for a large class of tree-like proof systems, in which such lower bounds play a prominent role. Further to the tree-like bound, we provide a new lower bound technique for QBF proof systems in general. This technique has some similarities to the above technique for tree-like systems, but requires some refinement to provide bounds for dag-like systems. We give applications of this new technique by proving lower bounds across several systems. The first such lower bounds are for a very simple family of QBFs. We then provide a construction to combine false QBFs to give formulas for which we can show lower bounds in this way, allowing the generation of the first random QBF proof complexity lower bounds
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