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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 42 (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 Unifying Approach to Temporal Constraint Reasoning
 Artificial Intelligence
"... We present a formalism, Disjunctive Linear Relations (DLRs), for reasoning about temporal constraints. DLRs subsume most of the formalisms for temporal constraint reasoning proposed in the literature and is therefore computationally expensive. We also present a restricted type of DLRs, Horn DLRs ..."
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Cited by 37 (10 self)
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We present a formalism, Disjunctive Linear Relations (DLRs), for reasoning about temporal constraints. DLRs subsume most of the formalisms for temporal constraint reasoning proposed in the literature and is therefore computationally expensive. We also present a restricted type of DLRs, Horn DLRs, which have a polynomialtime satisfiability problem. We prove that most approaches to tractable temporal constraint reasoning can be encoded as Horn DLRs, including the ORDHorn algebra by Nebel and Burckert and the simple temporal constraints by Dechter et al. Thus, DLRs is a suitable unifying formalism for reasoning about temporal constraints. 1 Introduction Reasoning about temporal knowledge abounds in artificial intelligence applications and other areas, such as planning [4], natural language understanding [25] and molecular biology [6, 13]. In most applications, knowledge of temporal constraints is expressed in terms of collections of relations between time intervals or time po...
Temporal Representation and Reasoning in Artificial Intelligence: Issues and Approaches
, 2002
"... this paper, we survey a wide range of research in temporal representation and reasoning, without committing ourselves to the point of view of any speci c application ..."
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Cited by 26 (1 self)
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this paper, we survey a wide range of research in temporal representation and reasoning, without committing ourselves to the point of view of any speci c application
Twentyone Large Tractable Subclasses of Allen's Algebra
 ARTIFICIAL INTELLIGENCE
, 1997
"... This paper continues Nebel and Burckert's investigation of Allen's interval algebra by presenting nine more maximal tractable subclasses of the algebra (provided that P != NP), in addition to their previously reported ORDHorn subclass. Furthermore, twelve tractable subclasses are identifi ..."
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Cited by 23 (8 self)
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This paper continues Nebel and Burckert's investigation of Allen's interval algebra by presenting nine more maximal tractable subclasses of the algebra (provided that P != NP), in addition to their previously reported ORDHorn subclass. Furthermore, twelve tractable subclasses are identified, whose maximality is not decided. Four of them can express the notion of sequentiality between intervals, which is not possible in the ORDHorn algebra. All of the algebras are considerably larger than the ORDHorn subclass. The satisfiability algorithm, which is common for all the algebras, is shown to be linear. Furthermore, the path consistency algorithm is shown to decide satisfiability of interval networks using any of the algebras.
Reasoning on Interval and Pointbased Disjunctive Metric Constraints in Temporal Contexts
 Journal of Artificial Intelligence Research
, 2000
"... We introduce a temporal model for reasoning on disjunctive metric constraints on intervals and time points in temporal contexts. This temporal model is composed of a labeled temporal algebra and its reasoning algorithms. The labeled temporal algebra defineslabeled disjunctive metric pointbased co ..."
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Cited by 17 (1 self)
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We introduce a temporal model for reasoning on disjunctive metric constraints on intervals and time points in temporal contexts. This temporal model is composed of a labeled temporal algebra and its reasoning algorithms. The labeled temporal algebra defineslabeled disjunctive metric pointbased constraints, where each disjunct in each input disjunctive constraint is univocally associated to a label. Reasoning algorithms manage labeled constraints, associated label lists, and sets of mutually inconsistent disjuncts. These algorithms guarantee consistency and obtain a minimal network. Additionally, constraints can be organized in a hierarchy of alternative temporal contexts. Therefore, we can reason on contextdependent disjunctive metric constraints on intervals and points. Moreover, the model is able to represent nonbinary constraints, such that logical dependencies on disjuncts in constraints can be handled. The computational cost of reasoning algorithms is exponential in ac...
Reasoning about Action in Polynomial Time
, 1997
"... Although many formalisms for reasoning about action exist, surprisingly few approaches have taken computational complexity into consideration. The contributions of this paper are the following: a temporal logic with a restriction for which deciding satisfiability is tractable, a tractable extension ..."
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Cited by 10 (2 self)
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Although many formalisms for reasoning about action exist, surprisingly few approaches have taken computational complexity into consideration. The contributions of this paper are the following: a temporal logic with a restriction for which deciding satisfiability is tractable, a tractable extension for reasoning about action, and NPcompleteness results for the unrestricted problems. Many interesting reasoning problems can be modelled, involving nondeterminism, concurrency and memory of actions. The reasoning process is proved to be sound and complete. 145 1 Introduction Although many formalisms for reasoning about action exist, surprisingly few approaches have taken computational complexity into consideration. One explanation for this might be that many interesting AI problems are (at least) NPhard, and that tractable subproblems that are easily extracted, tend to lack expressiveness. This has led a large part of the AI community to rely on heuristics and incomplete systems to s...
A Complete Classification of Tractability in Allen's Algebra Relative to Subsets of Basic Relations
 Proceedings of the 15th International Joint Conference on Artificial Intelligence (IJCAI '97
, 1998
"... We characterise the set of subalgebras of Allen's algebra which have a tractable satisfiability problem, and in addition contain certain basic relations. The conclusion is that no tractable subalgebra that is not known in the literature can contain more than the three basic relations (j), (b) a ..."
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Cited by 7 (0 self)
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We characterise the set of subalgebras of Allen's algebra which have a tractable satisfiability problem, and in addition contain certain basic relations. The conclusion is that no tractable subalgebra that is not known in the literature can contain more than the three basic relations (j), (b) and (b ), where b 2 fd; o; s; fg. This means that concerning algebras for specifying complete knowledge about temporal information, there is no hope of finding yet unknown classes with much expressivity. We also classify completely some cases where we cannot even express complete information (but close to complete), showing that there are exactly two maximal tractable algebras containing the relation (OE ), exactly two containing the relation (OE m m ), and exactly three containing the relation (OE m). The algebras containing (OE ) can express the notion of sequentiality; thus we have a complete characterisation of tractable inference using that notion. 1 Introduction This paper improves o...
Some Observations on Durations, Scheduling and Allen's Algebra
 In Proceedings of the 6th Conference on Constraint Programming (CP'00
, 2000
"... . We continue the study of complexity issues in Allen's algebra extended for handling metric durations. The eighteen known maximal tractable subclasses are studied and we show that three of them can be extended with metric constraints on the durations without sacrificing tractability, while the ..."
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Cited by 5 (3 self)
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. We continue the study of complexity issues in Allen's algebra extended for handling metric durations. The eighteen known maximal tractable subclasses are studied and we show that three of them can be extended with metric constraints on the durations without sacrificing tractability, while the remaining fifteen subclasses become NPcomplete even if we restrict ourselves to the case of fixed lengths on the durations. We also consider the scheduling problem and the main result is that ORDHorn is the only known maximal subclass for which the scheduling problem is tractable. This extends previous tractability results on scheduling in Allen's algebra since we allow far more temporal relations and we allow intervals with uncertain durations. 1 Introduction Representing and reasoning about time has for a long time been acknowledged as one of the core areas of artificial intelligence and a large number of formalisms for temporal constraint reasoning (TCR) have been proposed in the literatur...
Qualitative temporal reasoning about vague events
 In Proceedings of the 20th International Joint Conference on Artificial Intelligence
, 2007
"... The temporal boundaries of many real–world events are inherently vague. In this paper, we discuss the problem of qualitative temporal reasoning about such vague events. We show that several interesting reasoning tasks, such as checking satisfiability, checking entailment, and calculating the best tr ..."
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Cited by 3 (2 self)
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The temporal boundaries of many real–world events are inherently vague. In this paper, we discuss the problem of qualitative temporal reasoning about such vague events. We show that several interesting reasoning tasks, such as checking satisfiability, checking entailment, and calculating the best truth value bound, can be reduced to reasoning tasks in a well–known point algebra with disjunctions. Furthermore, we identify a maximal tractable subset of qualitative relations to support efficient reasoning. 1
Algorithms and Complexity for Temporal and Spatial Formalisms
, 1997
"... The problem of computing with temporal information was early recognised within the area of artificial intelligence, most notably the temporal interval algebra by Allen has become a widely used formalism for representing and computing with qualitative knowledge about relations between temporal interv ..."
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Cited by 3 (2 self)
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The problem of computing with temporal information was early recognised within the area of artificial intelligence, most notably the temporal interval algebra by Allen has become a widely used formalism for representing and computing with qualitative knowledge about relations between temporal intervals. However, the computational properties of the algebra and related formalisms are known to be bad: most problems (like satisfiability) are NPhard. This thesis contributes to finding restrictions (as weak as possible) on Allen's algebra and related temporal formalisms (the pointinterval algebra and extensions of Allen's algebra for metric time) for which the satisfiability problem can be computed in polynomial time. Another research area utilising temporal information is that of reasoning about action, which treats the problem of drawing conclusions based on the knowledge about actions having been performed at certain time points (this amounts to solving the infamous frame problem). One ...