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20
Relation algebras in qualitative spatial reasoning
 Fundamenta Informaticae
, 1999
"... The formalization of the “part – of ” relationship goes back to the mereology of S. Le´sniewski, subsequently taken up by Leonard & Goodman (1940), and Clarke (1981). In this paper we investigate relation algebras obtained from different notions of “part–of”, respectively, “connectedness” in var ..."
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Cited by 34 (13 self)
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The formalization of the “part – of ” relationship goes back to the mereology of S. Le´sniewski, subsequently taken up by Leonard & Goodman (1940), and Clarke (1981). In this paper we investigate relation algebras obtained from different notions of “part–of”, respectively, “connectedness” in various domains. We obtain minimal models for the relational part of mereology in a general setting, and when the underlying set is an atomless Boolean algebra. 1
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 32 (18 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.
Reasoning About Temporal Relations: The Tractable Subalgebras Of Allen's Interval Algebra
 Journal of the ACM
, 2001
"... Allen's interval algebra is one of the best established formalisms for temporal reasoning. This paper is the final step in the classification of complexity in Allen's algebra. We show that the current knowledge about tractability in the interval algebra is complete, that is, this algebra c ..."
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Cited by 31 (2 self)
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Allen's interval algebra is one of the best established formalisms for temporal reasoning. This paper is the final step in the classification of complexity in Allen's algebra. We show that the current knowledge about tractability in the interval algebra is complete, that is, this algebra contains exactly eighteen maximal tractable subalgebras, and reasoning in any fragment not entirely contained in one of these subalgebras is NPcomplete. We obtain this result by giving a new uniform description of the known maximal tractable subalgebras and then systematically using an algebraic technique for identifying maximal subalgebras with a given property.
A RelationAlgebraic Approach to the Region Connection Calculus
 Fundamenta Informaticae
, 2001
"... We explore the relationalgebraic aspects of the region connection calculus (RCC) of Randell et al. (1992a). In particular, we present a refinement of the RCC8 table which shows that the axioms provide for more relations than are listed in the present table. We also show that each RCC model leads ..."
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Cited by 19 (0 self)
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We explore the relationalgebraic aspects of the region connection calculus (RCC) of Randell et al. (1992a). In particular, we present a refinement of the RCC8 table which shows that the axioms provide for more relations than are listed in the present table. We also show that each RCC model leads to a Boolean algebra. Finally, we prove that a refined version of the RCC5 table has as models all atomless Boolean algebras B with the natural ordering as the "part  of" relation, and that the table is closed under first order definable relations iff B is homogeneous. 1 Introduction Qualitative reasoning (QR) has its origins in the exploration of properties of physical systems when numerical information is not sufficient  or not present  to explain the situation at hand (Weld and Kleer, 1990). Furthermore, it is a tool to represent the abstractions of researchers who are constructing numerical systems which model the physical world. Thus, it fills a gap in data modeling which often l...
The core of a countably categorical structure
 In Proceedings of the 22nd Annual Symposium on Theoretical Aspects of Computer Science (STACS’05), LNCS 3404
, 2005
"... Abstract. A relational structure is a core, if all its endomorphisms are embeddings. This notion is important for computational complexity classification of constraint satisfaction problems. It is a fundamental fact that every finite structure S has a core, i.e., S has an endomorphism e such that th ..."
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Cited by 17 (13 self)
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Abstract. A relational structure is a core, if all its endomorphisms are embeddings. This notion is important for computational complexity classification of constraint satisfaction problems. It is a fundamental fact that every finite structure S has a core, i.e., S has an endomorphism e such that the structure induced by e(S) is a core; moreover, the core is unique up to isomorphism. We prove that every ωcategorical structure has a core. Moreover, every ωcategorical structure is homomorphically equivalent to a modelcomplete core, which is unique up to isomorphism, and which is finite or ωcategorical. We discuss consequences for constraint satisfaction with ωcategorical templates. 1.
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 14 (7 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.
Determining the consistency of partial tree descriptions
 Artificial Intelligence
"... Abstract. We present an efficient algorithm that decides the consistency of partial descriptions of ordered trees. The constraint language of these descriptions was introduced by Cornell in computational linguistics; the constraints specify for pairs of nodes sets of admissible relative positions in ..."
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Cited by 6 (4 self)
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Abstract. We present an efficient algorithm that decides the consistency of partial descriptions of ordered trees. The constraint language of these descriptions was introduced by Cornell in computational linguistics; the constraints specify for pairs of nodes sets of admissible relative positions in an ordered tree. Cornell asked for an algorithm to find a tree structure satisfying these constraints. This computational problem generalizes the commonsupertree problem studied in phylogenetic analysis, and also generalizes the network consistency problem of the socalled leftlinear point algebra. We present the first polynomial time algorithm for Cornell’s problem, which runs in time O(mn), where m is the number of constraints and n the number of variables in the constraint.
Tractable Approximations for Temporal Constraint Handling
 Artificial Intelligence
, 1999
"... Relation algebras have been used for various kinds of temporal reasoning. Typically the network satisfaction problem turns out to be NPhard. For the Allen interval algebra it is often convenient to use the propagation algorithm. This algorithm is sound and runs in cubic time but it is not compl ..."
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Cited by 5 (1 self)
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Relation algebras have been used for various kinds of temporal reasoning. Typically the network satisfaction problem turns out to be NPhard. For the Allen interval algebra it is often convenient to use the propagation algorithm. This algorithm is sound and runs in cubic time but it is not complete. Here we define a series of tractable algorithms that provide approximations to solving the network satisfaction problem for any finite relation algebra. For algebras where all 3consistent atomic networks are satisfiable, like the Allen interval algebra, we can improve these algorithms so that each algorithm runs in cubic time. These algorithms improve on the Allen propagation algorithm and converge on a complete algorithm. Algebras of relations were studied extensively in the nineteenth century and, together with Frege's logic of quantifiers, form the foundation of modern logic (see [Mad91b, AH91] for some of the history). In the twentieth century algebras of binary relations wer...
Reasoning About Temporal Constraints: Classifying The Complexity In Allen's Algebra By Using An Algebraic Technique
, 2001
"... Allen's interval algebra is one of the best established formalisms for temporal reasoning. We study those fragments of Allen's algebra that contain a basic relation distinct from the equality relation. We prove that such a fragment is either NPcomplete or else contained in some already kn ..."
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Cited by 3 (0 self)
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Allen's interval algebra is one of the best established formalisms for temporal reasoning. We study those fragments of Allen's algebra that contain a basic relation distinct from the equality relation. We prove that such a fragment is either NPcomplete or else contained in some already known tractable subalgebra. We obtain this result by giving a new uniform description of known maximal tractable subalgebras and then systematically using an algebraic technique for description of maximal subalgebras with a given property. This approach avoids the need for extensive computerassisted search.
RCC8 Is Polynomial on Networks of Bounded Treewidth
 PROCEEDINGS OF THE TWENTYSECOND INTERNATIONAL JOINT CONFERENCE ON ARTIFICIAL INTELLIGENCE
"... We construct an homogeneous (and ωcategorical) representation of the relation algebra RCC8, which is one of the fundamental formalisms for spatial reasoning. As a consequence we obtain that the network consistency problem for RCC8 can be solved in polynomial time for networks of bounded treewidth. ..."
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Cited by 1 (0 self)
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We construct an homogeneous (and ωcategorical) representation of the relation algebra RCC8, which is one of the fundamental formalisms for spatial reasoning. As a consequence we obtain that the network consistency problem for RCC8 can be solved in polynomial time for networks of bounded treewidth.