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12
Schaefer’s theorem for graphs
, 2011
"... Schaefer’s theorem is a complexity classification result for socalled Boolean constraint satisfaction problems: it states that every Boolean constraint satisfaction problem is either contained in one out of six classes and can be solved in polynomial time, or is NPcomplete. We present an analog of ..."
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Schaefer’s theorem is a complexity classification result for socalled Boolean constraint satisfaction problems: it states that every Boolean constraint satisfaction problem is either contained in one out of six classes and can be solved in polynomial time, or is NPcomplete. We present an analog of this dichotomy result for the propositional logic of graphs instead of Boolean logic. In this generalization of Schaefer’s result, the input consists of a set W of variables and a conjunction Φ of statements (“constraints”) about these variables in the language of graphs, where each statement is taken from a fixed finite set Ψ of allowed quantifierfree firstorder formulas; the question is whether Φ is satisfiable in a graph. We prove that either Ψ is contained in one out of 17 classes of graph formulas and the corresponding problem can be solved in polynomial time, or the problem is NPcomplete. This is achieved by a universalalgebraic approach, which in turn allows us to use structural Ramsey theory. To apply the universalalgebraic approach, we formulate the computational problems under consideration as constraint satisfaction problems (CSPs) whose templates are firstorder definable in the countably infinite random graph. Our method to classify the computational complexity of those CSPs produces many statements of independent mathematical interest.
Quantified Constraints and Containment Problems
, 2008
"... We study two containment problems related to the quantified constraint satisfaction problem (QCSP). Firstly, we give a combinatorial condition on finite structures A and B that is necessary and sufficient to render QCSP(A) ⊆ QCSP(B). The required condition is the existence of a positive integer r s ..."
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We study two containment problems related to the quantified constraint satisfaction problem (QCSP). Firstly, we give a combinatorial condition on finite structures A and B that is necessary and sufficient to render QCSP(A) ⊆ QCSP(B). The required condition is the existence of a positive integer r such that there is a surjective homomorphism from the power structure Ar to B. We note that this condition is already necessary to guarantee containment of the Π2 restriction of QCSP, that is Π2CSP(A) ⊆ Π2CSP(B). Since we are able to give an effective bound on such an r, we provide a decision procedure for the model containment problem with nondeterministic doubleexponential time complexity. Secondly, we prove that the entailment problem for quantified conjunctivepositive firstorder logic is decidable. That is, given two sentences ϕ and ψ of firstorder logic with no instances of negation or disjunction, we give an algorithm that determines whether ϕ → ψ is true in all structures (models). Our result is in some sense tight, since we show that the entailment problem for positive firstorder logic (i.e. quantified conjunctivepositive logic plus disjunction) is undecidable.
Quantified Constraint Satisfaction and the Polynomially Generated Powers Property (Extended Abstract)
"... Abstract. 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 intractability of t ..."
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Abstract. 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 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 threeelement algebras: each algebra is either switchable and hence has the PGP, or provably lacks the PGP. The description of nonPGP algebras is remarkably simple and robust. 1
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 gener ..."
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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.
Computational complexity of constraint satisfaction
 CiE 2007. LNCS
, 2007
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The Complexity of Abduction for Equality Constraint Languages
"... Abduction is a form of nonmonotonic reasoning that looks for an explanation for an observed manifestation according to some knowledge base. One form of the abduction problem studied in the literature is the propositional abduction problem parameterized by a structure Γ over the twoelement domain. I ..."
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Abduction is a form of nonmonotonic reasoning that looks for an explanation for an observed manifestation according to some knowledge base. One form of the abduction problem studied in the literature is the propositional abduction problem parameterized by a structure Γ over the twoelement domain. In that case, the knowledge base is a set of constraints over Γ, the manifestation and explanation are propositional formulas. In this paper, we follow a similar route. Yet, we consider abduction over infinite domain. We study the equality abduction problem parameterized by a relational firstorder structure Γ over the natural numbers such that every relation in Γ is definable by a Boolean combination of equalities, a manifestation is a literal of the form (x = y) or (x = y), and an explanation is a set of such literals. Our main contribution is a complete complexity characterization of the equality abduction problem. We prove that depending on Γ, it is ΣP 2complete, or NPcomplete, or in P.
On the complexity of the model checking problem
"... The model checking problem for various fragments of firstorder logic has attracted much attention over the last two decades: in particular, for the fragment induced by ∃ and ∧ and that induced by ∀, ∃ and ∧, which are better known as the constraint satisfaction problem and the quantified constrai ..."
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The model checking problem for various fragments of firstorder logic has attracted much attention over the last two decades: in particular, for the fragment induced by ∃ and ∧ and that induced by ∀, ∃ and ∧, which are better known as the constraint satisfaction problem and the quantified constraint satisfaction problem, respectively. These two fragments are in fact the only ones for which there is currently no known complexity classification. All other syntactic fragments can be easily classified, either directly or using Schaefer’s dichotomy theorems for SAT and QSAT, with the exception of the positive equality free fragment induced by ∃,∀, ∧ and ∨. This outstanding fragment can also be classified and enjoys a tetrachotomy: according to the model, the corresponding model checking problem is either tractable, NPcomplete, coNPcomplete or Pspacecomplete. Moreover, the complexity drop is always witnessed by a generic solving algorithm which uses quantifier relativisation (for example, in the coNPcomplete case, the model has a constant e to which all ∃ quantifiers may be relativised). Furthermore, its complexity is characterised by algebraic means: the presence or absence of specific surjective hyperoperations among those that preserve the model characterise the complexity. Our classification methodology relies on this suitably tailored algebraic approach