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CASL specifications of qualitative calculi
 Spatial Information Theory: Cognitive and Computational Foundations, Proceedings of COSIT’05, LNCS 3693
, 2005
"... Abstract. In AI a large number of calculi for efficient reasoning about spatial and temporal entities have been developed. The most prominent temporal calculi are the point algebra of linear time and Allen’s interval calculus. Examples of spatial calculi include mereotopological calculi, Frank’s car ..."
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Abstract. In AI a large number of calculi for efficient reasoning about spatial and temporal entities have been developed. The most prominent temporal calculi are the point algebra of linear time and Allen’s interval calculus. Examples of spatial calculi include mereotopological calculi, Frank’s cardinal direction calculus, Freksa’s double cross calculus, Egenhofer and Franzosa’s intersection calculi, and Randell, Cui, and Cohn’s region connection calculi. These calculi are designed for modeling specific aspects of space or time, respectively, to the effect that the class of intended models may vary widely with the calculus at hand. But from a formal point of view these calculi are often closely related to each other. For example, the spatial region connection calculus RCC5 may be considered a coarsening of Allen’s (temporal) interval calculus. And vice versa, intervals can be used to represent spatial objects that feature an internal direction. The central question of this paper is how these calculi as well as their mutual dependencies can be axiomatized by algebraic specifications. This question will be investigated within the framework of the Common Algebraic Specification Language (CASL), a specification language developed by the Common Framework Initiative for algebraic specification and development (COFI). We explain scope and expressiveness of CASL by discussing the specifications of some of the calculi mentioned before. 1
Disjunctive Temporal Reasoning in Partially Ordered Models of Time
 In Proceedings of the Seventeenth National Conference on Artificial Intelligence (AAAI2000
, 2000
"... Certain problems in connection with, for example, cooperating agents and distributed systems require reasoning about time which is measured on incomparable or unsynchronized time scales. In such situations, it is sometimes approporiate to use a temporal model that only provides a partial order o ..."
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Cited by 4 (4 self)
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Certain problems in connection with, for example, cooperating agents and distributed systems require reasoning about time which is measured on incomparable or unsynchronized time scales. In such situations, it is sometimes approporiate to use a temporal model that only provides a partial order on time points. We study the computational complexity of partially ordered temporal reasoning in expressive formalisms consisting of point algebras extended with disjunctions. We show that the resulting algebra for partially ordered time contains four maximal tractable subclasses while the equivalent algebra for totalordered time contains two. Keywords: Temporal reasoning, constraint satisfaction, computational complexity. Introduction Many problems in Artificial Intelligence includes temporal reasoning in some form. Research on temporal reasoning has mainly focused on linear models of time, cf. Allen (1983). However, it is clear that more complex time models are needed in a variety...
Refinements and Independence: A Simple Method for Identifying Tractable Disjunctive Constraints
 IN: PROCEEDINGS OF THE 6TH INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING (CP2000
, 2000
"... The constraint satisfaction problem provides a natural framework for expressing many combinatorial problems. Since the general problem is NPhard, an important question is how to restrict the problem to ensure tractability. The concept of independence has proven to be a useful method for constru ..."
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Cited by 3 (2 self)
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The constraint satisfaction problem provides a natural framework for expressing many combinatorial problems. Since the general problem is NPhard, an important question is how to restrict the problem to ensure tractability. The concept of independence has proven to be a useful method for constructing tractable constraint classes from existing classes. Since checking the independence property may be a difficult task, we provide a simple method for checking this property. Our method builds on a somewhat surprising connection between independence and refinements which is a recently established way of reducing one constraint satisfaction problem to another. Refinements have two interesting properties: (1) they preserve consistency; and (2) their correctness can be easily checked by a computerassisted analysis. We show that all previous independence results of the point algebra for totally ordered and partially ordered time can be derived using this method. We also employ t...
Dependency calculus: Reasoning in a general point relation algebra
 KI 2005: ADVANCES IN ARTIFICIAL INTELLIGENCE, PROCEEDINGS OF THE 28TH ANNUAL GERMAN CONFERENCE ON AI
, 2005
"... Reasoning about complex dependencies between events is a crucial task. However, qualitative reasoning has so far concentrated on spatial and temporal issues. In contrast, we present a new dependency calculus (DC) that is created for specific questions of reasoning about causal relations and consequ ..."
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Reasoning about complex dependencies between events is a crucial task. However, qualitative reasoning has so far concentrated on spatial and temporal issues. In contrast, we present a new dependency calculus (DC) that is created for specific questions of reasoning about causal relations and consequences. Applications in the field of spatial representation and reasoning are, for instance, modeling traffic networks, ecological systems, medical diagnostics, and Bayesian Networks. Several extensions of the fundamental linear point algebra have been investigated, for instance on trees or on nonlinear structures. DC is an improved generalization that meets all requirements to describe dependencies on networks. We investigate this structure with respect to satisfiability problems, construction problems, tractable subclassses, and embeddings into other relation algebras. Finally, we analyze the associated interval algebra on network structures.
Branching Allen: Reasoning with intervals in branching time
 Spatial Cognition, Lecture Notes in Computer Science 3343
, 2004
"... Abstract. Allen’s interval calculus is one of the most prominent formalisms in the domain of qualitative spatial and temporal reasoning. Applications of this calculus, however, are restricted to domains that deal with linear flows of time. But how the fundamental ideas of Allen’s calculus can be ext ..."
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Abstract. Allen’s interval calculus is one of the most prominent formalisms in the domain of qualitative spatial and temporal reasoning. Applications of this calculus, however, are restricted to domains that deal with linear flows of time. But how the fundamental ideas of Allen’s calculus can be extended to other, weaker structures than linear orders has gained only little attention in the literature. In this paper we will investigate intervals in branching flows of time, which are of special interest for temporal reasoning, since they allow for representing indeterministic aspects of systems, scenarios, planning tasks, etc. As well, branching time models, i. e., treelike nonlinear structures, do have interesting applications in the field of spatial reasoning, for example, for modeling traffic networks. In a first step we discuss interval relations for branching time, thereby comprising various sources from the literature. Then, in a second step, we present some new complexity results concerning constraint satisfaction problems of interval relations in branching time. 1
Disjunctive Temporal Reasoning in Partially Ordered Models of Time
"... Certain problems in connection with, for example, cooperating agents and distributed systems require reasoning about time which is measured on incomparable or unsynchronized time scales. In such situations, it is sometimes approporiate to use a temporal model that only provides a partial order on ti ..."
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Certain problems in connection with, for example, cooperating agents and distributed systems require reasoning about time which is measured on incomparable or unsynchronized time scales. In such situations, it is sometimes approporiate to use a temporal model that only provides a partial order on time points. We study the computational complexity of partially ordered temporal reasoning in expressive formalisms consisting of point algebras extended with disjunctions. We show that the resulting algebra for partially ordered time contains four maximal tractable subclasses while the equivalent algebra for totalordered time contains two.
Disjunctions, Independence, Refinements
"... An important question in constraint satisfaction is how to restrict the problem to ensure tractability (since the general problem is NPhard). The use of ..."
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An important question in constraint satisfaction is how to restrict the problem to ensure tractability (since the general problem is NPhard). The use of