The Cut Tool for QCSP

Quantified Constraint Satisfaction Problems (QCSP) are often claimed to be adapted to model and solve problems such as twoplayer games, planning under uncertainty, and more generally problems in which the goal is to control a dynamic system subject to uncontrolled events. This paper shows that for a quite large class of such problems, using standard QCSP or QCSP+ is not the best approach. The main reasons are that in QCSP/QCSP+, (1) the underlying notion of system state is not explicitly taken into account, (2) problems are modeled over a bounded number of steps, and (3) algorithms search for winning strategies defined as memoryfull policy trees instead of winning strategies defined as memoryless mappings from states to decisions. This paper proposes a new constraint-based framework which does not suffer from these drawbacks. Experiments show orders of magnitude improvements when compared with QCSP/QCSP+ solvers.

The quantified constraint satisfaction problem (QCSP) is a natural and useful generalization of the constraint satisfaction problem (CSP) in which both universal and existential quantification of variables is permitted. Because the CSP and QCSP are in general intractable, much effort has been directed towards identifying restricted cases of these problems that are tractable in polynomial time. In this paper, we investigate restricted cases of the QCSP having 2-semilattice polymorphisms. We prove a complete classification of 2-semilattice polymorphisms, demonstrating that each gives rise to a case of the QCSP that is either tractable in polynomial time, or coNP-hard.

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.

In this article, the QCSP (Quantified Constraint Satisfaction Problems) framework is shifted to the QCHR (Quantified Constraint Handling Rules) framework by enabling dynamic generation of quantifiers and access to user-defined constraints. QCSP offers a natural framework to express PSPACE problems as finite two-players games. But to define a QCSP model, the alternation of quantifiers must be formerly known and cannot be built dynamically even if the worst case will not occur. To overcome this issue, the new QCHR formalism, that allows to generate quantifiers dynamically during the solving, is defined. QCHR models exhibit state-of-the-art performances on a priori fixed alternation of quantifiers and outperforms previous QCSP approaches when the generation of quantifiers is dynamic.

The Quantified Constraint Satisfaction Problem(QCSP) is a generalization of the classical constraint satisfaction problem in which some variables can be universally quantified. This additional expressiveness can help model problems in which a subset of the variables take value assignments that are outside the control of the decision maker. Typical examples of such domains are game-playing, conformant planning and reasoning under uncertainty. In these domains decision makers need explanations when a QCSP does not admit a winning strategy. We present an approach to defining preferences amongst the requirements of a QCSP, and an approach to finding most preferred explanations of inconsistency based on preferences over relaxations of quantifiers and constraints. This paper unifies work from the fields of constraint satisfaction, explanation generation, and reasoning under preferences and uncertainty.

Existentially restricted quantified constraint satisfaction

We consider the quantified constraint satisfaction problem (QCSP) which is to decide, given a structure and a first-order sentence (not assumed here to be in prenex form) built from conjunction and quantification, whether or not the sentence is true on the structure. We present a proof system for certifying the falsity of QCSP instances and develop its basic theory; for instance, we provide an algorithmic interpretation of its behavior. Our proof system places the established Q-resolution proof system in a broader context, and also allows us to derive QCSP tractability results.

This study focuses on the robust design optimization of permanent magnet linear synchronous motors (PMLSM) considering the effect of uncertainties (noise factors) such as manufacturing errors, installation errors and working wears. A robust design optimization approach based on quantified constraint satisfaction problem (QCSP) is proposed. It establishes a robust PMLSM thrust ripple model considering noise factors by QCSP, and obtains the optimal design parameters that have lower thrust ripple and smaller thrust ripple fluctuation within their error range by branch-and-prune algorithm. First, an analytical model of thrust ripple for PMLSM is established. Second, the robust QCSP model of thrust ripple is established by quantifier concept based on analytical model. It defines the design parameters and noise variables as existential and universal quantifiers, respectively, defines the variables of QCSP model as quantifier parameters that are composed of existential and universal quantifiers, then QCSP model are established by quantifier parameters. Third, the algorithm of branch-and-prune based on interval arithmetic is proposed to optimize the QCSP model. This approach is proven to be robust and superior compared with traditional deterministic approach and another robust approach. Finally, Finite element analysis and motor prototype experiments confirm the feasibility and validity of the proposed method.

In such as Real-time Scheduling Problems with uncertain interrupt operation, Quantified Constraint Satisfaction Problems were introduced to model and solve Constraint Satisfaction Problems involving uncertainty or uncontrollability on the value taken by some variables. In the former work, we propose to implement the schedulability test by using the algorithm QCSP-SchTest. In this paper, we analysis the search procedure in QCSP-SchTest and give a new algorithm QCSP-SchTest2 which combines the standard backtracking search with a repair-based procedure. A search state is best described as a strategy where we try to set the existential variables so that for all values of the universals there is a solution.

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

Solving a quantified constraint satisfaction problem (QCSP) is usually a hard task due to its computational complexity. Exact algorithms play an important role in solving this problem, among which backtrack algorithms are effective. In a backtrack algorithm, an important step is assigning a variable by a chosen value when exploiting a branch, and thus a good value selection rule may speed up greatly. In this paper, we propose two value selection rules for existentially and universally quantified variables, respectively, to avoid unnecessary searching. The rule for universally quantified variables is prior to trying failure values in previous branches, and the rule for existentially quantified variables selects the promising values first. Two rules are integrated into the state-of-the-art QCSP solver, i.e., QCSP-Solve, which is an exact solver based on backtracking. We perform a number of experiments to evaluate improvements brought by our rules. From computational results, we can conclude that the new value selection rules speed up the solver by 5 times on average and 30 times at most. We also show both rules perform well particularly on instances with existentially and universally quantified variables occurring alternatively.

In a multi-objective combinatorial optimization (MOCO) problem, multiple objectives must be optimized simultaneously. In past years, several constraint-based algorithms have been proposed for finding Pareto-optimal solutions to MOCO problems that rely on repeated calls to a constraint solver. Understanding the properties of these algorithms and analyzing their performance is an important problem. Previous work has focused on empirical evaluations on benchmark instances. Such evaluations, while important, have their limitations. Our paper adopts a different, purely theoretical approach, which is based on characterizing the search space into subspaces and analyzing the worst-case performance of a MOCO algorithm in terms of the expected number of calls to the underlying constraint solver. We apply the approach to two important constraint-based MOCO algorithms. Our analysis reveals a deep connection between the search mechanism of a constraint solver and the exploration of the search space of a MOCO problem.

Variable and value ordering heuristics for the job shop scheduling constraint satisfaction problem

The standard constraint satisfaction problem over an arbitrary finite domain can be expressed as follows: given a first-order sentence consisting of a conjunction of predicates, where all of the variables are existentially quantified, determine whether the sentence is true. This problem can be parameterized by the set of allowed constraint predicates. With each predicate, one can associate certain predicate-preserving operations, called polymorphisms, and the complexity of the parameterized problem is known to be determined by the polymorphisms of the allowed predicates. In this paper we consider a more general framework for constraint satisfaction problems which allows arbitrary quantifiers over constrained variables, rather than just existential quantifiers. We show that the complexity of such extended problems is determined by the surjective polymorphisms of the constraint predicates. We give examples to illustrate how this result can be used to identify tractable and intractable cases for the quantified constraint satisfaction problem over arbitrary finite domains.

A wide variety of problems, ranging from planning and scheduling tasks in industry to making autonomous decisions in smart environments, can be modelled as constraint satisfaction problems (CSPs). While this general mathematical framework facilitates the reuse of algorithms for many different types of problems, the fact that CSPs are NP-complete gives them an exponential worst-case computational complexity. This complexity imposes severe restrictions on the size of the problem, as virtually all applications require problems to be solved in a timely fashion. However, despite being NP-complete problems, CSPs can often be solved efficiently in practice, making this worst-case complexity largely uninformative. The structure of a problem and of the search space have been used to identify classes of tractable problems and design more efficient solving techniques with better theoretical guarantees. Unfortunately, the worst-case bounds of these algorithms are hardly more informative when it comes to their practical efficiency, and exploiting this structure is challenging because efficient algorithms do not exist for many related tasks. This dissertation studies the structure of problems and their search space and how these influence the search times of algorithms. It provides additional insights into the computational complexity and practical efficiency of search algorithms, which can, for instance, be used to select the best algorithm for a given problem and aid in the design of more efficient solving techniques. This work also contributes towards improving the solving efficiency more directly by proposing improved ways to exploit the structure of a problem and the search space algorithmically.