Quantified Constraint Satisfaction Problem Research Articles

QCSP Monsters and the Demise of the Chen Conjecture

We give a surprising classification for the computational complexity of the Quantified Constraint Satisfaction Problem over a constraint language Γ, QCSP(Γ), where Γ is a finite language over three elements that contains all constants. In particular, such problems are in P, NP-complete, co-NP-complete, or PSpace-complete. Our classification refutes the hitherto widely believed Chen Conjecture. Additionally, we show that already on a 4-element domain there exists a constraint language Γ such that QCSP(Γ) is DP-complete (from Boolean Hierarchy), and on a 10-element domain there exists a constraint language giving the complexity class Θ P 2 . Meanwhile, we prove the Chen Conjecture for finite conservative languages Γ. If the polymorphism clone of such Γ has the polynomially generated powers property, then QCSP(Γ) is in NP. Otherwise, the polymorphism clone of Γ has the exponentially generated powers property and QCSP(Γ) is PSpace-complete. 1

QCSP on Reflexive Tournaments

We give a complexity dichotomy for the Quantified Constraint Satisfaction Problem \( \mathrm{QCSP}(\mathrm{H}) \) when \( \mathrm{H} \) is a reflexive tournament. It is well known that reflexive tournaments can be split into a sequence of strongly connected components \( \mathrm{H}_1,\ldots ,\mathrm{H}_n \) so that there exists an edge from every vertex of \( \mathrm{H}_i \) to every vertex of \( \mathrm{H}_j \) if and only if \( i\lt j \) . We prove that if \( \mathrm{H} \) has both its initial and final strongly connected component (possibly equal) of size 1, then \( \mathrm{QCSP}(\mathrm{H}) \) is in \( \mathsf {NL} \) and otherwise \( \mathrm{QCSP}(\mathrm{H}) \) is \( \mathsf {NP} \) -hard.

Robust Design Optimization of Permanent Magnet Linear Synchronous Motor Based on Quantified Constraint Satisfaction Problem

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.

Solving quantified constraint satisfaction problems with value selection rules

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.

Quantified Constraint Handling Rules

We shift the QCSP (Quantified Constraint Satisfaction Problems) framework to the QCHR (Quantified Constraint Handling Rules) framework by enabling dynamic binder 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 binder must be formerly known and cannot be built dynamically even if the worst case won't occur. To overcome this issue, we define the new QCHR formalism that allows to build the binder dynamically during the solving. Our QCHR models exhibit state-of-the-art performances on static binder and outperforms previous QCSP approaches when the binder is dynamic.

On the Complexity of the Model Checking Problem

The complexity of the model checking problem for various fragments of first-order logic (FO) has attracted much attention over the last two decades, in particular for the fragment induced by $\exists$ and $\wedge$ and that induced by $\forall, \exists$, and $\wedge$, which are better known as the constraint satisfaction problem and the quantified constraint satisfaction problem, respectively. The former was conjectured to follow a dichotomy between P and NP-complete by Feder and Vardi [SIAM J. Comput., 28 (1998), pp. 57--104]. For the latter, there are several partial trichotomy results between P, NP-complete, and Pspace-complete, and Chen [Meditations on quantified constraint satisfaction, in Logic and Program Semantics, Springer, Heidelberg, 2012, pp. 35--49] ventured a conjecture regarding Pspace-completeness vs. membership in NP in the presence of constants. We give a comprehensive account of the whole field of the complexity of model checking similar syntactic fragments of FO. The above two fragments...

Quantified Constraint Satisfaction Problem on Semicomplete Digraphs

We study the (non-uniform) quantified constraint satisfaction problem QCSP( H ) as H ranges over semicomplete digraphs. We obtain a complexity-theoretic trichotomy: QCSP( H ) is either in P, is NP-complete, or is Pspace-complete. The largest part of our work is the algebraic classification of precisely which semicomplete digraphs enjoy only essentially unary polymorphisms, which is combinatorially interesting in its own right.

The complexity of counting quantifiers on equality languages

An equality language is a relational structure with infinite domain whose relations are first-order definable in equality. We classify the extensions of the quantified constraint satisfaction problem over equality languages in which the native existential and universal quantifiers are augmented by some subset of counting quantifiers. In doing this, we find ourselves in various worlds in which dichotomies or trichotomies subsist.

Quantified Constraints and Containment Problems

The quantified constraint satisfaction problem $\mathrm{QCSP}(\mathcal{A})$ is the problem to decide whether a positive Horn sentence, involving nothing more than the two quantifiers and conjunction, is true on some fixed structure $\mathcal{A}$. We study two containment problems related to the QCSP. Firstly, we give a combinatorial condition on finite structures $\mathcal{A}$ and $\mathcal{B}$ that is necessary and sufficient to render $\mathrm{QCSP}(\mathcal{A}) \subseteq \mathrm{QCSP}(\mathcal{B})$. We prove that $\mathrm{QCSP}(\mathcal{A}) \subseteq \mathrm{QCSP}(\mathcal{B})$, that is all sentences of positive Horn logic true on $\mathcal{A}$ are true on $\mathcal{B}$, iff there is a surjective homomorphism from $\mathcal{A}^{|A|^{|B|}}$ to $\mathcal{B}$. This can be seen as improving an old result of Keisler that shows the former equivalent to there being a surjective homomorphism from $\mathcal{A}^\omega$ to $\mathcal{B}$. We note that this condition is already necessary to guarantee containment of the $\Pi_2$ restriction of the QCSP, that is $\Pi_2$-$\mathrm{CSP}(\mathcal{A}) \subseteq \Pi_2$-$\mathrm{CSP}(\mathcal{B})$. The exponent's bound of ${|A|^{|B|}}$ places the decision procedure for the model containment problem in non-deterministic double-exponential time complexity. We further show the exponent's bound $|A|^{|B|}$ to be close to tight by giving a sequence of structures $\mathcal{A}$ together with a fixed $\mathcal{B}$, $|B|=2$, such that there is a surjective homomorphism from $\mathcal{A}^r$ to $\mathcal{B}$ only when $r \geq |A|$. Secondly, we prove that the entailment problem for positive Horn fragment of first-order logic is decidable. That is, given two sentences $\varphi$ and $\psi$ of positive Horn, we give an algorithm that determines whether $\varphi \rightarrow \psi$ is true in all structures (models). Our result is in some sense tight, since we show that the entailment problem for positive first-order logic (i.e. positive Horn plus disjunction) is undecidable. In the final part of the paper we ponder a notion of Q-core that is some canonical representative among the class of templates that engender the same QCSP. Although the Q-core is not as well-behaved as its better known cousin the core, we demonstrate that it is still a useful notion in the realm of QCSP complexity classifications.

Constraint Satisfaction with Counting Quantifiers

We initiate the study of constraint satisfaction problems (CSPs) in the presence of counting quantifiers $\exists^{\geq j}$ which assert the existence of at least $j$ elements such that the ensuing property holds. These are natural variants of CSPs in the mould of quantified CSPs (QCSPs). Namely, $\exists^{\geq 1}:=\exists$ and $\exists^{\geq n}:=\forall$ (for the domain of size $n$). We observe that a single counting quantifier $\exists^{\geq j}$ strictly between $\exists$ and $\forall$ already affords the maximal possible complexity of QCSPs (which have both $\exists$ and $\forall$), namely, being Pspace-complete for a suitably chosen template. Therefore, to better understand the complexity of this problem, we focus on restricted cases for which we derive the following results. First, for all subsets of counting quantifiers on clique and cycle templates, we give a full trichotomy---all such problems are in P, NP-complete, or Pspace-complete. Second, we consider the problem with exactly two quantifiers: $\exists^{\geq 1}:=\exists$ and $\exists^{\geq j}$ ($j \neq 1$). Such a CSP is already NP-hard on nonbipartite graph templates. We explore the situation of this generalized CSP on graph templates, giving various conditions for both tractability and hardness. For quantifiers $\exists^{\geq 1}$ and $\exists^{\geq 2}$, we give a dichotomy for all graphs, namely, the problem is NP-hard if the graph contains a triangle or has girth at least 5, and is in P otherwise. We strengthen this result in the following two ways. For bipartite graphs, the problem is in P for forests and graphs of girth 4, and is Pspace-hard otherwise. For complete multipartite graphs, the problem is in L, NP-complete, or Pspace-complete. Finally, using counting quantifiers we solve the complexity of a concrete QCSP whose complexity was previously open.

Beyond Q-Resolution and Prenex Form: A Proof System for 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.

Tractability of quantified temporal constraints to the max

A temporal constraint language is a set of relations that are first-order definable over (ℚ;<). We show that several temporal constraint languages whose constraint satisfaction problem is maximally tractable are also maximally tractable for the more expressive quantified constraint satisfaction problem. These constraint languages are defined in terms of preservation under certain binary polymorphisms. We also present syntactic characterizations of the relations in these languages.

Ultra-weak solutions and consistency enforcement in minimax weighted constraint satisfaction

The task at hand is that of a soft constraint problem with adversarial conditions. By amalgamating the weighted and quantified constraint satisfaction frameworks, a Minimax Weighted Constraint Satisfaction Problem (formerly Quantified Weighted Constraint Satisfaction Problem) consists of a set of finite domain variables, a set of soft constraints, and a min or max quantifier associated with each of these variables. We formally define the framework, suggest three solution concepts, and propose a complete solver based on alpha-beta pruning techniques. We discuss in depth our novel definitions and implementations of node, arc and full directional arc consistency notions to help reduce search space on top of the basic tree search with alpha-beta pruning for solving ultra-weak solutions. In particular, these consistencies approximate the lower and upper bounds of the cost of a problem by exploiting the semantics of the quantifiers and reusing techniques from both Weighted and Quantified Constraint Satisfaction Problems. Lower bound computation employs standard estimation of costs in the sub-problems used in alpha-beta search. In estimating upper bounds, we propose two approaches based on the Duality Principle: duality of quantifiers and duality of constraints. The first duality amounts to changing quantifiers from min to max, while the second duality re-uses the lower bound approximation functions on dual constraints to generate upper bounds. Experiments on three benchmarks comparing basic alpha-beta pruning and the six consistencies from the two dualities are performed to confirm the feasibility and efficiency of our proposal.

Sensitivity Analysis in Quantified Interval Constraint Satisfaction Problems

Interval is an alternative to probability distribution in quantifying uncertainty for sensitivity analysis (SA) when there is a lack of data to fit a distribution with good confidence. It only requires the information of lower and upper bounds. Analytical relations among design parameters, design variables, and target performances under uncertainty can be modeled as interval-valued constraints. By incorporating logic quantifiers, quantified constraint satisfaction problems (QCSPs) can integrate semantics and engineering intent in mathematical relations for engineering design. In this paper, a global sensitivity analysis (GSA) method is developed for feasible design space searching problems that are formulated as QCSPs, where the effects of value variations and quantifier changes for design parameters on target performances are analyzed based on several proposed metrics, including the indeterminacy of target performances, information gain of parameter variations, and infeasibility of constraints. Three examples are used to demonstrate the proposed approach.

Searching Feasible Design Space by Solving Quantified Constraint Satisfaction Problems

In complex systems design, multidisciplinary constraints are imposed by stakeholders. Engineers need to search feasible design space for a given problem before searching for the optimum design solution. Searching feasible design space can be modeled as a constraint satisfaction problem (CSP). By introducing logical quantifiers, CSP is extended to quantified constraint satisfaction problem (QCSP) so that more semantics and design intent can be captured. This paper presents a new approach to formulate searching design problems as QCSPs in a continuous design space based on generalized interval, and to numerically solve them for feasible solution sets, where the lower and upper bounds of design variables are specified. The approach includes two major components. One is a semantic analysis which evaluates the logic relationship of variables in generalized interval constraints based on Kaucher arithmetic, and the other is a branch-and-prune algorithm that takes advantage of the logic interpretation. The new approach is generic and can be applied to the case when variables occur multiple times, which is not available in other QCSP solving methods. A hybrid stratified Monte Carlo method that combines interval arithmetic with Monte Carlo sampling is also developed to verify the correctness of the QCSP solution sets obtained by the branch-and-prune algorithm.

QCSP Based Schedulability Test of Real-Time System

The problem of schedulability test of real-time task in scheduling system has been addressed. In such as real-time scheduling problems with uncertain interrupt operation, Quantified Constraint Satisfaction Problems (QCSP) were introduced to model and solve CSP involving uncertainty or uncontrollability on the value taken by some variables. In this paper, we propose to a novel approach to schedulability test based on the QCSP model. Based on the QCSP model of scheduler and interrupter in the real-time scheduling system, we transform the schedulability test into the satisfability problem. Finally, we design the algorithm QCSP-SchTest for judging the satisfability of a schedule S(α). This leads to a new schedulability test method without considering the specific scheduling algorithm or strategy.

Product Quality Characteristics Robust Optimization Design Based on Quantified Constraint Satisfaction Problem

Product quality characteristics robust optimization design is a constraint solving problem with uncertainties.A robust optimization method based on quantified constraint satisfaction problem(QCSP) is proposed,where,the design parameter variation is expressed as the universal variable.The upper and lower bounds of the objective functions and constraint functions are calculated with the effect of the universal variable,and the robustness indicators are set according to designer’s preference.Considering the model features,a combination algorithm is applied to solve the QCSP model.Interval analysis algorithm is employed to narrow the search space of the design variables,thus valid domain box of design variables is obtained which satisfy the general optimization design constraints.The shuffled frog-leaping algorithm is used to search the Pareto optimal solutions which satisfy the robustness indices;in addition,the information entropy theory-based approach is applied to select the optimal solution.The efficiency and effectiveness of the proposed method are illustrated by the product quality characteristics robust optimization design of turbine compressor diffuser.

Statistical Tolerance Analysis Based on Constraint Satisfaction Problems and Monte Carlo Simulation

This paper deals with the mathematical formulation of tolerance analysis. The mathematical formulation presented simulates the influences of geometrical deviations on the geometrical behavior of the mechanism, and integrates the quantifier notion. We propose a mathematical formulation of tolerance analysis which simulates the influences of geometrical deviations on the geometrical behavior of the mechanism, and integrates the quantifier notion. To compute this mathematical formulation, two approaches based on Quantified Constraint Satisfaction Problem solvers and Monte Carlo simulation are proposed and tested.

Integrating Standard Dependency Schemes in QCSP Solvers

Quantified constraint satisfaction problems (QCSPs) are an extension to constraint satisfaction problems (CSPs) with both universal quantifiers and existential quantifiers. In this paper we apply variable ordering heuristics and integrate standard dependency schemes in QCSP solvers. The technique can help to decide the next variable to be assigned in QCSP solving. We also introduce a new factor into the variable ordering heuristics: a variable’s dep is the number of variables depending on it. This factor represents the probability of getting more candidates for the next variable to be assigned. Experimental results show that variable ordering heuristics with standard dependency schemes and the new factor dep can improve the performance of QCSP solvers.

The Complexity of Positive First-Order Logic without Equality

We study the complexity of evaluating positive equality-free sentences of first-order (FO) logic over a fixed, finite structure B . This may be seen as a natural generalisation of the nonuniform quantified constraint satisfaction problem QCSP( B ). We introduce surjective hyper-endomorphisms and use them in proving a Galois connection that characterizes definability in positive equality-free FO. Through an algebraic method, we derive a complete complexity classification for our problems as B ranges over structures of size at most three. Specifically, each problem either is in L, is NP-complete, is co-NP-complete, or is Pspace-complete.