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25
Constraint Satisfaction with Countable Homogeneous Templates
 IN PROCEEDINGS OF CSL’03
, 2003
"... For a fixed countable homogeneous structure we study the computational problem whether a given finite structure of the same relational signature homomorphically maps to . This problem is known as the constraint satisfaction problem CSP( ) for and was intensively studied for finite . We show that ..."
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Cited by 42 (19 self)
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For a fixed countable homogeneous structure we study the computational problem whether a given finite structure of the same relational signature homomorphically maps to . This problem is known as the constraint satisfaction problem CSP( ) for and was intensively studied for finite . We show that  as in the case of finite  the computational complexity of CSP( ) for countable homogeneous is determinded by the clone of polymorphisms of . To this end we prove the following theorem which is of independent interest: The primitive positive definable relations over an !categorical structure are precisely the relations that are invariant under the polymorphisms of .
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 39 (21 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.
The complexity of temporal constraint satisfaction problems
 J. ACM
"... A temporal constraint language is a set of relations that has a firstorder definition in (Q; <), the dense linear order of the rational numbers. We present a complete complexity classification of the constraint satisfaction problem (CSP) for temporal constraint languages: if the constraint langu ..."
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Cited by 33 (22 self)
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A temporal constraint language is a set of relations that has a firstorder definition in (Q; <), the dense linear order of the rational numbers. We present a complete complexity classification of the constraint satisfaction problem (CSP) for temporal constraint languages: if the constraint language is contained in one out of nine temporal constraint languages, then the CSP can be solved in polynomial time; otherwise, the CSP is NPcomplete. Our proof combines modeltheoretic concepts with techniques from universal algebra, and also applies the socalled product Ramsey theorem, which we believe will useful in similar contexts of
Constraint Satisfaction Problems with Countable Homogeneous Templates
"... Allowing templates with infinite domains greatly expands the range of problems that can be formulated as a nonuniform constraint satisfaction problem. It turns out that many CSPs over infinite templates can be formulated with templates that are ωcategorical. We survey examples of such problems in ..."
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Cited by 24 (10 self)
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Allowing templates with infinite domains greatly expands the range of problems that can be formulated as a nonuniform constraint satisfaction problem. It turns out that many CSPs over infinite templates can be formulated with templates that are ωcategorical. We survey examples of such problems in temporal and spatial reasoning, infinitedimensional algebra, acyclic colorings in graph theory, artificial intelligence, phylogenetic reconstruction in computational biology, and tree descriptions in computational linguistics. We then give an introduction to the universalalgebraic approach to infinitedomain constraint satisfaction, and discuss how cores, polymorphism clones, and pseudovarieties can be used to study the computational complexity of CSPs with ωcategorical templates. The theoretical results will be illustrated by examples from the mentioned application areas. We close with a series of open problems and promising directions of future research.
Reducts of Ramsey structures
"... One way of studying a relational structure is to investigate functions which are related to that structure and which leave certain aspects of the structure invariant. Examples are the automorphism group, the selfembedding monoid, the endomorphism monoid, or the polymorphism clone of a structure. Su ..."
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Cited by 23 (12 self)
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One way of studying a relational structure is to investigate functions which are related to that structure and which leave certain aspects of the structure invariant. Examples are the automorphism group, the selfembedding monoid, the endomorphism monoid, or the polymorphism clone of a structure. Such functions can be particularly well understood when the relational structure is countably in nite and has a rstorder de nition in another relational structure which has a nite language, is totally ordered and homogeneous, and has the Ramsey property. This is because in this situation, Ramsey theory provides the combinatorial tool for analyzing these functions { in a certain sense, it allows to represent such functions by functions on nite sets. This is a survey of results in model theory and theoretical computer science obtained recently by the authors in this context. In model theory, we approach the problem of classifying the reducts of countably in nite ordered homogeneous Ramsey structures in a nite language, and certain decidability
The complexity of equality constraint languages
 CORNELL UNIVERSITY
, 2006
"... We apply the algebraic approach to infinitevalued constraint satisfaction to classify the computational complexity of all constraint satisfaction problems with templates that have a highly transitive automorphism group. A relational structure has such an automorphism group if and only if all the c ..."
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Cited by 16 (12 self)
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We apply the algebraic approach to infinitevalued constraint satisfaction to classify the computational complexity of all constraint satisfaction problems with templates that have a highly transitive automorphism group. A relational structure has such an automorphism group if and only if all the constraint types are Boolean combinations of the equality relation, and we call the corresponding constraint languages equality constraint languages. We show that an equality constraint language is tractable if it admits a constant unary or an injective binary polymorphism, and is NPcomplete otherwise.
On the scope of the universalalgebraic approach to constraint satisfaction
, 2009
"... The universalalgebraic approach has proved a powerful tool in the study of the computational complexity of constraint satisfaction problems (CSPs). This approach has previously been applied to the study of CSPs with finite or (infinite) ωcategorical templates. Our first result is an exact charact ..."
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Cited by 12 (9 self)
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The universalalgebraic approach has proved a powerful tool in the study of the computational complexity of constraint satisfaction problems (CSPs). This approach has previously been applied to the study of CSPs with finite or (infinite) ωcategorical templates. Our first result is an exact characterization of those CSPs that can be formulated with (a finite or) an ωcategorical template. The universalalgebraic approach relies on the fact that in finite or ωcategorical structures A, a relation is primitive positive definable if and only if it is preserved by the polymorphisms of A. In this paper, we present results that can be used to study the computational complexity of CSPs with arbitrary infinite templates. Specifically, we prove that every CSP can be formulated with a template A such that a relation is primitive positive definable in A if and only if it is firstorder definable on A and preserved by the infinitary polymorphisms of A. We present applications of our general results to the description and analysis of the computational complexity of CSPs. In particular, we present a polymorphismbased description of those CSPs that are firstorder definable (and therefore can be solved in polynomialtime), and give general hardness criteria based on the absence of polymorphisms that depend on more than one argument.
Qualitative temporal and spatial reasoning revisited
 In 16th EACSL Annual Conference on Computer Science and Logic (CSL’07
, 2007
"... Abstract. Establishing local consistency is one of the main algorithmic techniques in temporal and spatial reasoning. In this area, one of the central questions for the various proposed temporal and spatial constraint languages is whether local consistency implies global consistency. Showing that a ..."
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Cited by 8 (4 self)
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Abstract. Establishing local consistency is one of the main algorithmic techniques in temporal and spatial reasoning. In this area, one of the central questions for the various proposed temporal and spatial constraint languages is whether local consistency implies global consistency. Showing that a constraint language Γ has this “localtoglobal ” property implies polynomialtime tractability of the constraint language, and has further pleasant algorithmic consequences. In the present paper, we study the “localtoglobal ” property by making use of a recently established connection of this property with universal algebra. Specifically, the connection shows that this property is equivalent to the presence of a socalled quasi nearunanimity polymorphism of the constraint language. We obtain new algorithmic results and give very concise proofs of previously known theorems. Our results concern wellknown and heavily studied formalisms such as the point algebra and its extensions, Allen’s interval algebra, and the spatial reasoning language RCC5. 1
Oligomorphic clones
 Algebra Universalis
, 2007
"... Abstract. A permutation group on a countably infinite domain is called oligomorphic if it has finitely many orbits of finitary tuples. We define a clone on a countable domain to be oligomorphic if its set of permutations forms an oligomorphic permutation group. There is a close relationship to ωcat ..."
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Cited by 6 (6 self)
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Abstract. A permutation group on a countably infinite domain is called oligomorphic if it has finitely many orbits of finitary tuples. We define a clone on a countable domain to be oligomorphic if its set of permutations forms an oligomorphic permutation group. There is a close relationship to ωcategorical structures, i.e., countably infinite structures with a firstorder theory that has only one countable model, up to isomorphism. Every locally closed oligomorphic permutation group is the automorphism group of an ωcategorical structure, and conversely, the canonical structure of an oligomorphic permutation group is an ωcategorical structure that contains all firstorder definable relations. There is a similar Galois connection between locally closed oligomorphic clones and ωcategorical structures containing all primitive positive definable relations. In this article we generalise some fundamental theorems of universal algebra from clones over a finite domain to oligomorphic clones. First, we define minimal oligomorphic clones, and present equivalent characterisations of minimality, and then generalise Rosenberg’s five types classification to minimal oligomorphic clones. We also present a generalisation of the theorem of Baker and Pixley to oligomorphic clones. This is a updated postprint version of an article with the same title that appeared in