Datalog and constraint satisfaction with infinite templates

Bodirsky, Manuel; Dalmau, Víctor

doi:10.1016/j.jcss.2012.05.012

Cited by 23 publications

(15 citation statements)

References 47 publications

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“…We then explain the connection between MMSNP and infinite-domain CSPs: first we syntactically characterise those MMSNP sentences that describe CSPs, by introducing the logic connected MMSNP, and then we show that the dichotomy for MMSNP and the dichotomy for connected MMSNP are equivalent (Section 2.3). In Section 2.4, we revisit the result of Dalmau and Bodirsky [11] that every connected MMSNP sentence is the CSP for an ω-categorical template.…”

Section: Mmsnp and Cspsmentioning

confidence: 99%

“…In this section we first revisit the fact that every connected MMSNP sentence describes a CSP of an ω-categorical structure [11]. The proof uses a theorem due to Cherlin, Shelah, and Shi, stated for graphs in [24]; Theorem 16 below is formulated for general relational structures.…”

Section: Templates For Connected Mmsnp Sentencesmentioning

confidence: 99%

“…[4,3,1,14]. Dalmau and Bodirsky [11] showed that every problem in MMSNP is a finite union of constraint satisfaction problems for ω-categorical structures. These structures can be expanded to finitely bounded homogeneous structures so that they fall into the scope of the mentioned infinite-domain tractability conjecture.…”

Section: Introductionmentioning

confidence: 99%

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A universal-algebraic proof of the complexity dichotomy for Monotone Monadic SNP

Bodirsky

Madelaine

Mottet

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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The logic MMSNP is a restricted fragment of existential second-order logic which allows to express many interesting queries in graph theory and finite model theory. The logic was introduced by Feder and Vardi who showed that every MMSNP sentence is computationally equivalent to a finite-domain constraint satisfaction problem (CSP); the involved probabilistic reductions were derandomized by Kun using explicit constructions of expander structures. We present a new proof of the reduction to finitedomain CSPs which does not rely on the results of Kun. This new proof allows us to obtain a stronger statement and to verify the more general Bodirsky-Pinsker dichotomy conjecture for CSPs in MMSNP. Our approach uses the fact that every MMSNP sentence describes a finite union of CSPs for countably infinite ω-categorical structures; moreover, by a recent result of Hubička and Nešetřil, these structures can be expanded to homogeneous structures with finite relational signature and the Ramsey property. This allows us to use the universal-algebraic approach to study the computational complexity of MMSNP.

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Section: Mmsnp and Cspsmentioning

confidence: 99%

Section: Templates For Connected Mmsnp Sentencesmentioning

confidence: 99%

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A universal-algebraic proof of the complexity dichotomy for Monotone Monadic SNP

Bodirsky

Madelaine

Mottet

2018

Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science

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“…This is mainly because of its significance for constraint satisfaction problems (CSPs). It is expressive enough for formulation of the path consistency algorithm while also not overly complicated so that every ω-categorical CSP expressible in Datalog admits a canonical Datalog program [9].…”

Section: Introductionmentioning

confidence: 99%

Discrete Temporal Constraint Satisfaction Problems

Bodirsky

Martin

Mottet

2018

J. ACM

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Finite-domain constraint satisfaction problems are either solvable by Datalog, or not even expressible in fixed-point logic with counting. The border between the two regimes coincides with an important dichotomy in universal algebra; in particular, the border can be described by a strong height-one Maltsev condition. For infinite-domain CSPs the situation is more complicated even if the template structure of the CSP is model-theoretically tame. We prove that there is no Maltsev condition that characterises Datalog already for the CSPs of first-order reducts of (Q; <); such CSPs are called temporal CSPs and are of fundamental importance in infinite-domain constraint satisfaction. Our main result is a complete classification of temporal CSPs that can be solved in Datalog, and that can be solved in fixed point logic (with or without counting); the classification shows that many of the equivalent conditions in the finite fail to capture expressibility in any of these formalisms already for temporal CSPs.

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“…Bounded strict width k of an ω-categorical template was characterized in [10] by the existence of a quasi-near unanimity polymorphism n of arity k + 1, i.e., n(y, x, . .…”

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confidence: 99%

Sensitive instances of the Constraint Satisfaction Problem

Barto¹,

Kozik²,

Tan³

et al. 2020

Preprint

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We investigate the impact of modifying the constraining relations of a Constraint Satisfaction Problem (CSP) instance, with a fixed template, on the set of solutions of the instance. More precisely we investigate sensitive instances: an instance of the CSP is called sensitive, if removing any tuple from any constraining relation invalidates some solution of the instance. Equivalently, one could require that every tuple from any one of its constraints extends to a solution of the instance.Clearly, any non-trivial template has instances which are not sensitive. Therefore we follow the direction proposed (in the context of strict width) by Feder and Vardi in [13] and require that only the instances produced by a local consistency checking algorithm are sensitive. In the language of the algebraic approach to the CSP we show that a finite idempotent algebra A has a k + 2 variable near unanimity term operation if and only if any instance that results from running the (k, k + 1)-consistency algorithm on an instance over A 2 is sensitive.A version of our result, without idempotency but with the sensitivity condition holding in a variety of algebras, settles a question posed by G. Bergman about systems of projections of algebras that arise from some subalgebra of a finite product of algebras.Our results hold for infinite (albeit in the case of A idempotent) algebras as well and exhibit a surprising similarity to the strict width k condition proposed by Feder and Vardi. Both conditions can be characterized by the existence of a near unanimity operation, but the arities of the operations differ by 1.

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Datalog and constraint satisfaction with infinite templates

Cited by 23 publications

References 47 publications

A universal-algebraic proof of the complexity dichotomy for Monotone Monadic SNP

A universal-algebraic proof of the complexity dichotomy for Monotone Monadic SNP

Discrete Temporal Constraint Satisfaction Problems

Sensitive instances of the Constraint Satisfaction Problem

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