Existentially Restricted Quantified Constraint Satisfaction

The quantified constraint satisfaction problem (QCSP) is a powerful framework for modelling computational problems. The general intractability of the QCSP has motivated the pursuit of restricted cases that avoid its maximal complexity. In this paper, we introduce and study a new model for investigating QCSP complexity in which the types of constraints given by the existentially quantified variables, is restricted. Our primary technical contribution is the development and application of a general technology for proving positive results on parameterizations of the model, of inclusion in the complexity class coNP

Existentially restricted quantified constraint satisfaction

AbstractThe quantified constraint satisfaction problem (QCSP) is a framework for modelling PSPACE computational problems. The general intractability of the QCSP has motivated the pursuit of restricted cases that avoid its maximal complexity. In this paper, we introduce and study a new model for investigating QCSP complexity in which the types of constraints given by the existentially quantified variables, is restricted. Our primary technical contribution is the development and application of a general technology for proving positive results on parameterizations of the model, of inclusion in the complexity class coNP

The Complexity of Quantified Constraint Satisfaction: Collapsibility, Sink Algebras, and the Three-Element Case

The constraint satisfaction probem (CSP) is a well-acknowledged framework in which many combinatorial search problems can be naturally formulated. The CSP may be viewed as the problem of deciding the truth of a logical sentence consisting of a conjunction of constraints, in front of which all variables are existentially quantified. The quantified constraint satisfaction problem (QCSP) is the generalization of the CSP where universal quantification is permitted in addition to existential quantification. The general intractability of these problems has motivated research studying the complexity of these problems under a restricted constraint language, which is a set of relations that can be used to express constraints. This paper introduces collapsibility, a technique for deriving positive complexity results on the QCSP. In particular, this technique allows one to show that, for a particular constraint language, the QCSP reduces to the CSP. We show that collapsibility applies to three known tractable cases of the QCSP that were originally studied using disparate proof techniques in different decades: Quantified 2-SAT (Aspvall, Plass, and Tarjan 1979), Quantified Horn-SAT (Karpinski, Kleine B\"{u}ning, and Schmitt 1987), and Quantified Affine-SAT (Creignou, Khanna, and Sudan 2001). This reconciles and reveals common structure among these cases, which are describable by constraint languages over a two-element domain. In addition to unifying these known tractable cases, we study constraint languages over domains of larger size

Low-level dichotomy for Quantified Constraint Satisfaction Problems

Building on a result of Larose and Tesson for constraint satisfaction problems (CSP s), we uncover a dichotomy for the quantified constraint satisfaction problem QCSP(B), where B is a finite structure that is a core. Specifically, such problems are either in ALogtime or are L-hard. This involves demonstrating that if CSP(B) is first-order expressible, and B is a core, then QCSP(B) is in ALogtime. We show that the class of B such that CSP(B) is first-order expressible (indeed, trivially true) is a microcosm for all QCSPs. Specifically, for any B there exists a C such that CSP(C) is trivially true, yet QCSP(B) and QCSP(C) are equivalent under logspace reductions

Efficient CTL Verification via Horn Constraints Solving

The use of temporal logics has long been recognised as a fundamental approach to the formal specification and verification of reactive systems. In this paper, we take on the problem of automatically verifying a temporal property, given by a CTL formula, for a given (possibly infinite-state) program. We propose a method based on encoding the problem as a set of Horn constraints. The method takes a program, modeled as a transition system, and a property given by a CTL formula as input. It first generates a set of forall-exists quantified Horn constraints and well-foundedness constraints by exploiting the syntactic structure of the CTL formula. Then, the generated set of constraints are solved by applying an off-the-shelf Horn constraints solving engine. The program is said to satisfy the property if and only if the generated set of constraints has a solution. We demonstrate the practical promises of the method by applying it on a set of challenging examples. Although our method is based on a generic Horn constraint solving engine, it is able to outperform state-of-art methods specialised for CTL verification.Comment: In Proceedings HCVS2016, arXiv:1607.0403

CTL+FO Verification as Constraint Solving

Expressing program correctness often requires relating program data throughout (different branches of) an execution. Such properties can be represented using CTL+FO, a logic that allows mixing temporal and first-order quantification. Verifying that a program satisfies a CTL+FO property is a challenging problem that requires both temporal and data reasoning. Temporal quantifiers require discovery of invariants and ranking functions, while first-order quantifiers demand instantiation techniques. In this paper, we present a constraint-based method for proving CTL+FO properties automatically. Our method makes the interplay between the temporal and first-order quantification explicit in a constraint encoding that combines recursion and existential quantification. By integrating this constraint encoding with an off-the-shelf solver we obtain an automatic verifier for CTL+FO

The complexity of quantified constraints using the algebraic formulation

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Generalizing Consistency and other Constraint Properties to Quantified Constraints

Quantified constraints and Quantified Boolean Formulae are typically much more difficult to reason with than classical constraints, because quantifier alternation makes the usual notion of solution inappropriate. As a consequence, basic properties of Constraint Satisfaction Problems (CSP), such as consistency or substitutability, are not completely understood in the quantified case. These properties are important because they are the basis of most of the reasoning methods used to solve classical (existentially quantified) constraints, and one would like to benefit from similar reasoning methods in the resolution of quantified constraints. In this paper, we show that most of the properties that are used by solvers for CSP can be generalized to quantified CSP. This requires a re-thinking of a number of basic concepts; in particular, we propose a notion of outcome that generalizes the classical notion of solution and on which all definitions are based. We propose a systematic study of the relations which hold between these properties, as well as complexity results regarding the decision of these properties. Finally, and since these problems are typically intractable, we generalize the approach used in CSP and propose weaker, easier to check notions based on locality, which allow to detect these properties incompletely but in polynomial time