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Datalog and constraint satisfaction with infinite templates
 In Proceedings of the 23rd International Symposium on Theoretical Aspects of Computer Science (STACS’06), LNCS 3884
, 2006
"... Abstract. On finite structures, there is a wellknown connection between the expressive power of Datalog, finite variable logics, the existential pebble game, and bounded hypertree duality. We study this connection for infinite structures. This has applications for constraint satisfaction with infin ..."
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Cited by 31 (17 self)
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Abstract. On finite structures, there is a wellknown connection between the expressive power of Datalog, finite variable logics, the existential pebble game, and bounded hypertree duality. We study this connection for infinite structures. This has applications for constraint satisfaction with infinite templates, i.e., for all computational problems that are closed under disjoint unions and whose complement is closed under homomorphisms. If the template Γ is ωcategorical, we obtain alternative characterizations of bounded Datalog width. We also show that CSP(Γ) can be solved in polynomial time if Γ is ωcategorical and the input is restricted to instances of bounded treewidth. Finally, we prove algebraic characterisations of those ωcategorical templates whose CSP has Datalog width (1, k), and for those whose CSP has strict Datalog width k.
Linear datalog and bounded path duality of relational structures
 Logical Methods in Computer Science
"... Abstract. In this paper we systematically investigate the connections between logics with a finite number of variables, structures of bounded pathwidth, and linear Datalog Programs. We prove that, in the context of Constraint Satisfaction Problems, all these concepts correspond to different mathemat ..."
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Cited by 15 (2 self)
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Abstract. In this paper we systematically investigate the connections between logics with a finite number of variables, structures of bounded pathwidth, and linear Datalog Programs. We prove that, in the context of Constraint Satisfaction Problems, all these concepts correspond to different mathematical embodiments of a unique robust notion that we call bounded path duality. We also study the computational complexity implications of the notion of bounded path duality. We show that every constraint satisfaction problem CSP(B) with bounded path duality is solvable in NL and that this notion explains in a uniform way all families of CSPs known to be in NL. Finally, we use the results developed in the paper to identify new problems in NL. 1.
Dualities for constraint satisfaction problems
"... In a nutshell, a duality for a constraint satisfaction problem equates the existence of one homomorphism to the nonexistence of other homomorphisms. In this survey paper, we give an overview of logical, combinatorial, and algebraic aspects of the following forms of duality for constraint satisfact ..."
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Cited by 14 (5 self)
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In a nutshell, a duality for a constraint satisfaction problem equates the existence of one homomorphism to the nonexistence of other homomorphisms. In this survey paper, we give an overview of logical, combinatorial, and algebraic aspects of the following forms of duality for constraint satisfaction problems: finite duality, bounded pathwidth duality, and bounded treewidth duality.
Homomorphism Preservation Theorems
, 2008
"... The homomorphism preservation theorem (h.p.t.), a result in classical model theory, states that a firstorder formula is preserved under homomorphisms on all structures (finite and infinite) if and only if it is equivalent to an existentialpositive formula. Answering a longstanding question in fin ..."
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Cited by 10 (0 self)
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The homomorphism preservation theorem (h.p.t.), a result in classical model theory, states that a firstorder formula is preserved under homomorphisms on all structures (finite and infinite) if and only if it is equivalent to an existentialpositive formula. Answering a longstanding question in finite model theory, we prove that the h.p.t. remains valid when restricted to finite structures (unlike many other classical preservation theorems, including the ̷Lo´sTarski theorem and Lyndon’s positivity theorem). Applications of this result extend to constraint satisfaction problems and to database theory via a correspondence between existentialpositive formulas and unions of conjunctive queries. A further result of this article strengthens the classical h.p.t.: we show that a firstorder formula is preserved under homomorphisms on all structures if and only if it is equivalent to an existentialpositive formula of equal quantifierrank.
Homomorphisms and FirstOrder Logic
, 2007
"... We prove that the homomorphism preservation theorem (h.p.t.), a classical result of mathematical logic, holds when restricted to finite structures. That is, a firstorder formula is preserved under homomorphisms on finite structures if, and only if, it is equivalent in the finite to an existentialp ..."
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Cited by 2 (0 self)
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We prove that the homomorphism preservation theorem (h.p.t.), a classical result of mathematical logic, holds when restricted to finite structures. That is, a firstorder formula is preserved under homomorphisms on finite structures if, and only if, it is equivalent in the finite to an existentialpositive formula. This result, which contrasts with the known failure of other classical preservation theorems on finite structures, answers a longstanding question in finite model theory. The relevance of this result, however, extends beyond logic to areas of computer science, including constraint satisfaction problems and database theory; the database connection arises from a correspondence between existentialpositive formulas and unions of conjunctive queries (also known as selectprojectjoinunion queries). A second result of this article strengthens the classical h.p.t. by showing that a firstorder formula is preserved under homomorphisms on all structures if, and only if, it is equivalent to an existentialpositive formula of equal quantifierrank. Unlike traditional proofs of the classical h.p.t., the proof of this stronger “equirank ” theorem is compactnessfree and constructive. While these results are logical in nature, the technical development of the article takes place almost entirely within a combinatorial framework. The concept of treedepth, a graph parameter related to treewidth, plays an important role in our analysis (as a combinatorial counterpart to quantifierrank). We introduce new notions of nhomomorphism and ncore, which approximate the familiar concepts of homomorphism and core “up to treedepth n”. The key technical lemmas take a pair of nhomomorphically equivalent [finite] relational structures and construct corresponding [finite] coretracts which satisfy a certain backandforth property.
Constraint Satisfaction: A Personal Perspective
"... Attempts at classifying computational problems as polynomial time solvable, NPcomplete, or belonging to a higher level in the polynomial hierarchy, face the difficulty of undecidability. These classes, including NP, admit a logic formulation. By suitably restricting the formulation, one finds the l ..."
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Attempts at classifying computational problems as polynomial time solvable, NPcomplete, or belonging to a higher level in the polynomial hierarchy, face the difficulty of undecidability. These classes, including NP, admit a logic formulation. By suitably restricting the formulation, one finds the logic class MMSNP, or monotone monadic strict NP without inequality, as a largest class that seems to avoid diagonalization arguments. Representative of this logic class is the class CSP of constraint satisfaction problems. Both MMSNP and CSP admit generalizations via alternations of quantifiers corresponding to higher levels in the hierarchy. Examining CSP from a computational point of view, one finds that the polynomial time solvable problems that do not have the bounded width property of Datalog are group theoretic in nature. In general, closure properties of the constraints characterize the complexity of the problems. When one restricts the number of occurrences of each variable, the problems that are encountered relate to deltamatroid intersection. When such a restriction forbidding copying is introduced in the context of inputoutput constraints, one finds nonexpansive mappings as characterizing this restriction. Both deltamatroid intersection and nonexpansive network stability problems yield polynomial
On Datalog vs. LFP
"... Abstract. We show that the homomorphism preservation theorem fails for LFP, both in general and in restriction to finite structures. That is, there is a formula of LFP that is preserved under homomorphisms (in the finite) but is not equivalent (in the finite) to a Datalog program. This resolves a qu ..."
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Abstract. We show that the homomorphism preservation theorem fails for LFP, both in general and in restriction to finite structures. That is, there is a formula of LFP that is preserved under homomorphisms (in the finite) but is not equivalent (in the finite) to a Datalog program. This resolves a question posed by Atserias. The results are established by two different methods: (1) a method of diagonalisation that works only in the presence of infinite structures, but establishes a stronger result showing a hierarchy of homomorphismpreserved problems in LFP; and (2) a method based on a pumping lemma for Datalog due to Afrati, Cosmadakis and Yannakakis which establishes the result in restriction to finite structures. We refine the pumping lemma of Afrati et al. and relate it to the power of Monadic SecondOrder Logic on tree decompositions of structures. 1
On Preservation under Homomorphisms and . . .
, 2006
"... Unions of conjunctive queries, also known as selectprojectjoinunion queries, are the most frequently asked queries in relational database systems. These queries are definable by existential positive firstorder formulas and are preserved under homomorphisms. A classical result of mathematical log ..."
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Unions of conjunctive queries, also known as selectprojectjoinunion queries, are the most frequently asked queries in relational database systems. These queries are definable by existential positive firstorder formulas and are preserved under homomorphisms. A classical result of mathematical logic asserts that the existential positive formulas are the only firstorder formulas (up to logical equivalence) that are preserved under homomorphisms on all structures, finite and infinite. After resisting resolution for a long time, it was eventually shown that, unlike other classical preservation theorems, the homomorphismpreservation theorem holds for the class of all finite structures. In this paper, we show that the homomorphismpreservation theorem holds also for several restricted classes of finite structures of interest in graph theory and database theory. Specifically, we show that this result holds for all classes of finite structures of bounded degree, all classes of finite structures of bounded treewidth, and, more generally, all classes of finite structures whose cores exclude at least one minor.
Descriptive complexity of approximate counting
"... Motivated by Fagin’s characterization of NP, Saluja et al. have introduced a logic based framework for expressing counting problems. In this setting, a counting problem (seen as a mapping C from structures to nonnegative integers) is ’defined ’ by a firstorder sentence ϕ if for every instance A of ..."
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Motivated by Fagin’s characterization of NP, Saluja et al. have introduced a logic based framework for expressing counting problems. In this setting, a counting problem (seen as a mapping C from structures to nonnegative integers) is ’defined ’ by a firstorder sentence ϕ if for every instance A of the problem, the number of possible satisfying assignments of the variables of ϕ in A is equal to C(A). The logic RHΠ1 has been introduced by Dyer et al. in their study of the counting complexity class #BIS. The interest in the class #BIS stems from the fact that, it is quite plausible that the problems in #BIS are not #Phard, nor they admit a fully polynomial randomized approximation scheme. In the present paper we investigate which counting constraint satisfaction problems #CSP(H) are definable in the monotone fragment of RHΠ1. We prove that #CSP(H) is definable in monotone RHΠ1 whenever H is invariant under meet and join operations of a distributive lattice. We prove that the converse also holds if H contains the equality relation. We also prove similar results for counting CSPs expressible by linear Datalog. The results in this case are very similar to those for monotone RHΠ1, with the addition that H has, additionally, ⊤ (the greatest element of the lattice) as a polymorphism.