Quantified Constraint Satisfaction Research Articles

Hybrid Tractable Classes of Binary Quantified Constraint Satisfaction Problems

In this paper, we investigate the hybrid tractability of binary Quantified Constraint Satisfaction Problems (QCSPs). First, a basic tractable class of binary QCSPs is identified by using the broken-triangle property. In this class, the variable ordering for the broken-triangle property must be same as that in the prefix of the QCSP. Second, we break this restriction to allow that existentially quantified variables can be shifted within or out of their blocks, and thus identify some novel tractable classes by introducing the broken-angle property. Finally, we identify a more generalized tractable class, i.e., the min-of-max extendable class for QCSPs.

Low-level dichotomy for quantified constraint satisfaction problems

Building on a result of Larose and Tesson for constraint satisfaction problems ( CSPs), 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 A L o g t i m e 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 A L o g t i m e . We show that the class of B such that CSP ( B ) is first-order expressible (indeed trivial) is a microcosm for all QCSPs. Specifically, for any B there exists a C — generally not a core — such that CSP ( C ) is trivial, yet QCSP ( B ) and QCSP ( C ) are equivalent under logspace reductions.

Quantified constraint satisfaction and the polynomially generated powers property

The quantified constraint satisfaction probem (QCSP) is the problem of deciding, given a relational structure and a sentence consisting of a quantifier prefix followed by a conjunction of atomic formulas, whether or not the sentence is true in the structure. The general computational intractability of the QCSP has led to the study of restricted versions of this problem. In this article, we study restricted versions of the QCSP that arise from prespecifying the relations that may occur via a set of relations called a constraint language. A basic tool used is a correspondence that associates an algebra to each constraint language; this algebra can be used to derive information on the behavior of the constraint language. We identify a new combinatorial property on algebras, the polynomially generated powers (PGP) property, which we show is tightly connected to QCSP complexity. We also introduce another new property on algebras, switchability, which both implies the PGP property and implies positive complexity results on the QCSP. Our main result is a classification theorem on a class of three-element algebras: each algebra is either switchable and hence has the PGP, or provably lacks the PGP. The description of non-PGP algebras is remarkably simple and robust.

A Constraint Satisfaction Framework for Executing Perceptions and Actions in Diagrammatic Reasoning

Diagrammatic reasoning (DR) is pervasive in human problem solving as a powerful adjunct to symbolic reasoning based on language-like representations. The research reported in this paper is a contribution to building a general purpose DR system as an extension to a SOAR-like problem solving architecture. The work is in a framework in which DR is modeled as a process where subtasks are solved, as appropriate, either by inference from symbolic representations or by interaction with a diagram, i.e., perceiving specified information from a diagram or modifying/creating objects in a diagram in specified ways according to problem solving needs. The perceptions and actions in most DR systems built so far are hand-coded for the specific application, even when the rest of the system is built using the general architecture. The absence of a general framework for executing perceptions/actions poses as a major hindrance to using them opportunistically -- the essence of open-ended search in problem solving. Our goal is to develop a framework for executing a wide variety of specified perceptions and actions across tasks/domains without human intervention. We observe that the domain/task-specific visual perceptions/actions can be transformed into domain/task-independent spatial problems. We specify a spatial problem as a quantified constraint satisfaction problem in the real domain using an open-ended vocabulary of properties, relations and actions involving three kinds of diagrammatic objects -- points, curves, regions. Solving a spatial problem from this specification requires computing the equivalent simplified quantifier-free expression, the complexity of which is inherently doubly exponential. We represent objects as configuration of simple elements to facilitate decomposition of complex problems into simpler and similar subproblems. We show that, if the symbolic solution to a subproblem can be expressed concisely, quantifiers can be eliminated from spatial problems in low-order polynomial time using similar previously solved subproblems. This requires determining the similarity of two problems, the existence of a mapping between them computable in polynomial time, and designing a memory for storing previously solved problems so as to facilitate search. The efficacy of the idea is shown by time complexity analysis. We demonstrate the proposed approach by executing perceptions and actions involved in DR tasks in two army applications.

Quantified Equality Constraints

An equality template is a relational structure with infinite universe whose relations can be defined by Boolean combinations of equalities. We prove a complexity classification for quantified constraint satisfaction problems (QCSPs) over equality templates: These problems are in L (decidable in logarithmic space), NP-hard, or coNP-hard. To establish our classification theorem we combine methods from universal algebra with concepts from model theory.

The complexity of constraint satisfaction games and QCSP

We study the complexity of two-person constraint satisfaction games. An instance of such a game is given by a collection of constraints on overlapping sets of variables, and the two players alternately make moves assigning values from a finite domain to the variables, in a specified order. The first player tries to satisfy all constraints, while the other tries to break at least one constraint; the goal is to decide whether the first player has a winning strategy. We show that such games can be conveniently represented by a logical form of quantified constraint satisfaction, where an instance is given by a first-order sentence in which quantifiers alternate and the quantifier-free part is a conjunction of (positive) atomic formulas; the goal is to decide whether the sentence is true. While the problem of deciding such a game is PSPACE-complete in general, by restricting the set of allowed constraint predicates, one can obtain infinite classes of constraint satisfaction games of lower complexity. We use the quantified constraint satisfaction framework to study how the complexity of deciding such a game depends on the parameter set of allowed predicates. With every predicate, one can associate certain predicate-preserving operations, called polymorphisms. We show that the complexity of our games is determined by the surjective polymorphisms of the constraint predicates. We illustrate how this result can be used by identifying the complexity of a wide variety of constraint satisfaction games.

Non-binary quantified CSP: algorithms and modelling

The Quantified Constraint Satisfaction Problem (QCSP) extends classical CSP in a way which allows reasoning about uncertainty. In this paper I present novel algorithms for solving QCSP. Firstly I present algorithms to perform constraint propagation on reified disjunction constraints of any length. The algorithms make full use of quantifier information to provide a high level of consistency. Secondly I present a scheme to enforce the non-binary pure value rule. This rule is capable of pruning universal variables. Following this, two problems are modelled in non-binary QCSP: the game of Connect 4, and a variant of job-shop scheduling with uncertainty, in the form of machine faults. The job shop scheduling example incorporates probability bounding of scenarios (such that only fault scenarios above a probability threshold are considered) and optimization of the schedule makespan. These contribute to the art of modelling in QCSP, and are a proof of concept for applying QCSP methods to complex, realistic problems. Both models make use of the reified disjunction constraint, and the non-binary pure value rule. The example problems are used to evaluate the QCSP algorithms presented in this paper, identifying strengths and weaknesses, and to compare them to other QCSP approaches.

Existentially restricted quantified constraint satisfaction

The 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.

Worst-case and statistical tolerance analysis based on quantified constraint satisfaction problems and Monte Carlo simulation

This paper deals with the mathematical formulation of tolerance analysis. The mathematical formulation presented in this paper simulates the influences of geometrical deviations on the geometrical behavior of the mechanism, and integrates the quantifier notion (existential quantifier: “there exists”; universal quantifier: “for all”). It takes into account not only the influence of geometrical deviations but also the influence of the types of contacts on the geometrical behavior; these physical phenomena are modeled by convex hulls (compatibility hull, interface hull and functional hull) which are defined in parametric space. With this description by convex hulls, a mathematical expression of the admissible deviations of parts integrates the quantifier notion. This notion translates the concept that a functional requirement must be respected in at least one acceptable configuration of gaps (existential quantifier: “there exists”), or that a functional requirement must be respected in all acceptable configurations of gaps (universal quantifier: “for all”). To compute this mathematical formulation, two approaches based on Quantified Constraint Satisfaction Problem solvers and Monte Carlo simulation are proposed and tested.

The complexity of satisfiability problems: Refining Schaefer's theorem

Schaefer proved in 1978 that the Boolean constraint satisfaction problem for a given constraint language is either in P or is NP-complete, and identified all tractable cases. Schaefer's dichotomy theorem actually shows that there are at most two constraint satisfaction problems, up to polynomial-time isomorphism (and these isomorphism types are distinct if and only if P ≠ NP ). We show that if one considers AC 0 isomorphisms, then there are exactly six isomorphism types (assuming that the complexity classes NP, P, ⊕L, NL, and L are all distinct). A similar classification holds for quantified constraint satisfaction problems.

Value ordering for quantified CSPs

We investigate the use of value ordering in backtracking search for Quantified Constraint Satisfaction problems (QCSPs). We consider two approaches for ordering heuristics. The first approach is solution-focused and is inspired by adversarial search: on existential variables we prefer values that maximise the chances of leading to a solution, while on universal variables we prefer values that minimise those chances. The second approach is verification-focused, where we prefer values that are easier to verify whether or not they lead to a solution. In particular, we give instantiations of this approach using QCSP-Solve's pure-value rule Gent et al. (QCSP-solve: A solver for quantified constraint satisfaction problems. In Proceedings of IJCAI, pp. 138---143, 2005). We show that on dense 3-block problems, using QCSP-Solve, the solution-focused adversarial heuristics are up to 50% faster than lexicographic ordering, while on sparse loose interleaved problems, the verification-focused pure-value heuristics are up to 50% faster. Both types are up to 50% faster on dense interleaved problems, with one pure-value heuristic approaching an order of magnitude improvement.

Efficient handling of universally quantified inequalities

This paper introduces a new framework for solving quantified constraint satisfaction problems (QCSP) defined by universally quantified inequalities on continuous domains. This class of QCSPs has numerous applications in engineering and technology. We introduce a generic branch and prune algorithm to tackle these continuous CSPs with parametric constraints, where the pruning and the solution identification processes are dedicated to universally quantified inequalities. Special rules are proposed to handle the parameter domains of the constraints. The originality of our framework lies in the fact that it solves the QCSP as a non-quantified CSP where the quantifiers are handled locally, at the level of each constraint. Experiments show that our algorithm outperforms the state of the art methods based on constraint techniques.

Introduction to the special issue on quantified CSPs and QBF

Constraint satisfaction problems (CSPs) and satisfiability problem (SAT) are very successful frameworks that are used to model and solve a wide variety of combinatorial problems. However, there are classes of problems containing uncertainty that arise in areas such as contingent planning, adversarial game playing, control design, and model checking that cannot be expressed within these frameworks. Typically, such problems involve decisions or events that are beyond the control of the problem solving agent and thus cannot be handled using standard (existentially quantified) variables. Quantified CSPs and quantified Boolean formulae (QBF), which are the extensions of CSPs and SAT that allow for universally quantified variables, make it possible to model and reason with such problems, as well as other problems that contain “bounded uncertainty”. As a result, these frameworks have been attracting significant interest in recent years. The main advances have been achieved in the area of QBF where numerous solvers have been implemented and real problems of considerable size have been tackled. There is also a significant body of work on quantified numerical constraints over continuous domains, while research works on quantified constraint satisfaction problems (QCSPs) with discrete finite domains have started to emerge. With this special issue of Constraints we aim to group together and reflect the state of the art in these rapidly developing areas of research. The papers in this special issue concern the three areas of reasoning with quantified constraints highlighted above.

Relatively quantified constraint satisfaction

The constraint satisfaction problem (CSP) is a convenient framework for modelling search problems; the CSP involves deciding, given a set of constraints on variables, whether or not there is an assignment to the variables satisfying all of the constraints. This paper is concerned with the more general framework of quantified constraint satisfaction, in which variables can be quantified both universally and existentially. We study the relatively quantified constraint satisfaction problem (RQCSP), in which the values for each individual variable can be arbitrarily restricted. We give a complete complexity classification of the cases of the RQCSP where the types of constraints that may appear are specified by a constraint language.

PREPROCESSING QUANTIFIED CONSTRAINT SATISFACTION PROBLEMS WITH VALUE REORDERING AND DIRECTIONAL ARC AND PATH CONSISTENCY

The Quantified Constraint Satisfaction Problem (QCSP) is an extension of the CSP that can be used to model combinatorial problems containing contingency or uncertainty. It allows for universally quantified variables that can model uncertain actions and events, such as the unknown weather for a future party, or an opponent's next move in a game. Although interest in QCSPs is increasing in recent years, the development of techniques for handling QCSPs is still at an early stage. For example, although it is well known that local consistencies are of primary importance in CSPs, only arc consistency has been extended to quantified problems. In this paper we contribute towards the development of solution methods for QCSPs in two ways. First, by extending directional arc and path consistency, two popular local consistencies in constraint satisfaction, to the quantified case and proposing an algorithm that achieves these consistencies. Second, by showing how value ordering heuristics can be utilized to speed up computation in QCSPs. We study the impact of preprocessing QCSPs with value reordering and directional quantified arc and path consistency by running experiments on randomly generated problems. Results show that our preprocessing methods can significantly speed up the QCSP solving process, especially on hard instances from the phase transition region.

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 in 1979), Quantified Horn-SAT (Karpinski, Kleine Büning, and Schmitt in 1987), and Quantified Affine-SAT (Creignou, Khanna, and Sudan in 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.

Solving quantified constraint satisfaction problems

We make a number of contributions to the study of the Quantified Constraint Satisfaction Problem (QCSP). The QCSP is an extension of the constraint satisfaction problem that can be used to model combinatorial problems containing contingency or uncertainty. It allows for universally quantified variables that can model uncertain actions and events, such as the unknown weather for a future party, or an opponent's next move in a game. In this paper we report significant contributions to two very different methods for solving QCSPs. The first approach is to implement special purpose algorithms for QCSPs; and the second is to encode QCSPs as Quantified Boolean Formulas and then use specialized QBF solvers. The discovery of particularly effective encodings influenced the design of more effective algorithms: by analyzing the properties of these encodings, we identify the features in QBF solvers responsible for their efficiency. This enables us to devise analogues of these features in QCSPs, and implement them in special purpose algorithms, yielding an effective special purpose solver, QCSP-Solve. Experiments show that this solver and a highly optimized QBF encoding are several orders of magnitude more efficient than the initially developed algorithms. A final, but significant, contribution is the identification of flaws in simple methods of generating random QCSP instances, and a means of generating instances which are not known to be flawed.