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## Reasoning about Temporal Relations: A Maximal Tractable Subclass of Allen's Interval Algebra (1995)

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Venue: | Journal of the ACM |

Citations: | 193 - 8 self |

### Citations

2900 | Maintaining knowledge about temporal intervals
- Allen
- 1983
(Show Context)
Citation Context ...lti-media context [7, 9], diagnosis of technical systems [29], and knowledge representation [20, 37], the representation of qualitative temporal relations and reasoning about them is essential. Allen =-=[2]-=- introduces an algebra of binary relations on intervals (hereafter referred to as Allen's interval algebra) for representing qualitative temporal information and addresses the problem of reasoning abo... |

1137 |
Consistency in networks of relations
- Mackworth
- 1977
(Show Context)
Citation Context ...ation. In particular, he gives an algorithm for computing an approximation to the strongest implied relation for each pair of intervals, which is a simplified version of the pathconsistency algorithm =-=[25]-=-. As already noted by Allen [2], the path-consistency method is in general not sufficient for computing the strongest implied relation for each pair of intervals. Since this problem is NP-hard in the ... |

1026 | Towards a general theory of action and time - Allen - 1984 |

484 |
Networks of constraints: Fundamental properties and applications to picture processing
- Montanari
- 1974
(Show Context)
Citation Context ...hese problems (approximately for all interval relations or exactly for subclasses) is constraint propagation [2, 22, 29, 32, 34, 35] using a slightly simplified form of the path-consistency algorithm =-=[25, 27]-=-. In the following, we briefly characterize this method without going into details, though. In order to do so, we first have to introduce Allen's interval algebra. Allen's interval algebra [2] consist... |

423 | Constraint propagation algorithms for temporal reasoning
- Vilain, Kautz
- 1986
(Show Context)
Citation Context ...f X i;j 's and Y i;j 's in the second group. It should be noted that the above NP-completeness result does not refer to the relation fOE; ��g, which has been used in all NP-completeness proofs so =-=far [16, 17, 35, 36]-=-. Vilain et al [36] have pointed out that this relation was crucial for their NP-completeness result and mention this relation as an instance of a truly disjunctive relation. However, as we have seen ... |

288 |
The complexity of some polynomial network consistency algorithms for constraint satisfaction problems
- Mackworth, Freuder
- 1985
(Show Context)
Citation Context ...Under these assumptions, an 5 Allen [2] gives a composition table for the basic relations. 6 algorithm can be specified that computes the reduced closure of a set of interval formulas in O(n 3 ) time =-=[26, 27]-=-. It should be noted that the path-consistency method provides only an approximationsto ISI. This means that the relations in a path-consistent set contain the strongest implied relations, but the con... |

280 | Temporal Reasoning Based on Semi-Intervals
- Freksa
- 1992
(Show Context)
Citation Context ...problem in general. Subsequent research has concentrated on designing more efficient reasoning algorithms, on identifying tractable special cases, and on isolating sources of computational complexity =-=[10, 13, 14, 15, 16, 17, 22, 23, 28, 29, 31, 32, 33, 34, 35, 36]-=-. However, it is by no means clear whether the tractable cases that have been identified are maximal and whether the sources of computational complexity found are the only ones. We extend these previo... |

256 |
Synthesizing constraint expressions
- Freuder
- 1978
(Show Context)
Citation Context ...only the strongest relations is called the reduced closure of \Theta and is denoted by b \Theta. As can easily be shown, every reduced closure of a set \Theta is path consistent [25] (or 3-consistent =-=[11]-=-), which means that for every three intervals X; Y; Z and for every interpretation = that satisfies (XRY ) 2 b \Theta, there exists an interpretation = 0 that agrees with = on X and Y and in addition ... |

145 | Reasoning about qualitative temporal information
- Beek
- 1992
(Show Context)
Citation Context ...problem in general. Subsequent research has concentrated on designing more efficient reasoning algorithms, on identifying tractable special cases, and on isolating sources of computational complexity =-=[10, 13, 14, 15, 16, 17, 22, 23, 28, 29, 31, 32, 33, 34, 35, 36]-=-. However, it is by no means clear whether the tractable cases that have been identified are maximal and whether the sources of computational complexity found are the only ones. We extend these previo... |

143 |
Expressiveness and tractability in knowledge representation and reasoning
- Levesque, Brachman
- 1987
(Show Context)
Citation Context ...that ISAT(S) is NP-complete. By Corollary 16, it follows that ISAT(S) is NP-complete, which completes the proof. In other words, H presents an optimal tradeoff between expressiveness and tractability =-=[24]-=- in the framework of reasoning about qualitative temporal relations using Allen's interval algebra. 7 Conclusion We have identified a new tractable subclass of Allen's interval algebra, which we call ... |

134 |
A commonsense theory of time
- Allen, Hayes
- 1985
(Show Context)
Citation Context ...can be obtained from the authors or by anonymous ftp from duck.dfki.uni-sb.de as /pub/papers/RR-93-11.programs.tar.Z. 2 Other underlying models of the time line are also possible, e.g., the rationals =-=[5, 21]-=-. For our purposes these distinctions are not significant, however. 3 Basic Interval Sym- Pictorial Endpoint Relation bol Example Relations X before Y OE xxx X \Gamma ! Y \Gamma , X \Gamma ! Y + , Y a... |

127 | Planning Using a Temporal World Model
- Allen, F, et al.
- 1983
(Show Context)
Citation Context ...he relative positions of time intervals such as ". . . point to the figure while explaining the performance of the system . . . " Further, for natural language understanding [3, 30], general=-= planning [4, 6]-=-, presentation planning in a multi-media context [7, 9], diagnosis of technical systems [29], and knowledge representation [20, 37], the representation of qualitative temporal relations and reasoning ... |

98 | Complexity and Algorithms for Reasoning about Time: A Graph-Thoretic Approach
- Golumbic, Shamir
- 1993
(Show Context)
Citation Context ...f X i;j 's and Y i;j 's in the second group. It should be noted that the above NP-completeness result does not refer to the relation fOE; ��g, which has been used in all NP-completeness proofs so =-=far [16, 17, 35, 36]-=-. Vilain et al [36] have pointed out that this relation was crucial for their NP-completeness result and mention this relation as an instance of a truly disjunctive relation. However, as we have seen ... |

98 | On binary constraint problems - Ladkin, Maddux - 1994 |

93 |
Linear time algorithms for testing the satisfiability of propositional Horn formulae
- Dowling, Gallier
- 1984
(Show Context)
Citation Context ...D ��(\Theta) in order to decide I-satisfiability of \Theta, the claim follows. Based on this result and the fact that the best known satisfiability algorithm for propositional Horn theories is lin=-=ear [8]-=-, it is possible to give an upper bound for deciding ISAT(H). Given a set of interval formula \Theta with n distinct intervals, we assume as usual that j\Thetaj 2 O(n 2 ). Theorem 6 ISAT(H) can be dec... |

87 | Exact and approximate reasoning about temporal relations
- Beek, Cohen
- 1990
(Show Context)
Citation Context ...problem in general. Subsequent research has concentrated on designing more efficient reasoning algorithms, on identifying tractable special cases, and on isolating sources of computational complexity =-=[10, 13, 14, 15, 16, 17, 22, 23, 28, 29, 31, 32, 33, 34, 35, 36]-=-. However, it is by no means clear whether the tractable cases that have been identified are maximal and whether the sources of computational complexity found are the only ones. We extend these previo... |

63 | Terminological reasoning with constraint networks and an application to plan recognition
- Weida, Litman
- 1992
(Show Context)
Citation Context ... Further, for natural language understanding [3, 30], general planning [4, 6], presentation planning in a multi-media context [7, 9], diagnosis of technical systems [29], and knowledge representation =-=[20, 37]-=-, the representation of qualitative temporal relations and reasoning about them is essential. Allen [2] introduces an algebra of binary relations on intervals (hereafter referred to as Allen's interva... |

59 |
Temporal reasoning and planning
- Allen
- 1991
(Show Context)
Citation Context ...he relative positions of time intervals such as ". . . point to the figure while explaining the performance of the system . . . " Further, for natural language understanding [3, 30], general=-= planning [4, 6]-=-, presentation planning in a multi-media context [7, 9], diagnosis of technical systems [29], and knowledge representation [20, 37], the representation of qualitative temporal relations and reasoning ... |

49 |
Unit refutations and Horn sets
- Henschen, Wos
- 1974
(Show Context)
Citation Context ...itively, the path-consistency method performs positive unit resolution, i.e., unit resolution involving only positive unit clauses, a resolution strategy that is refutation complete for Horn theories =-=[18]. If-=- a clause C is derivable by positive unit resolution from\Omega\Gamma we write\Omega ` U + C. In the following, we assume that the clauses C 2 ��(OE) are minimal, i.e., there exists no clause C 0 ... |

46 | Approximation Algorithms for Temporal Reasoning
- Beek
- 1989
(Show Context)
Citation Context |

39 | On binary constraint networks
- Ladkin, Maddux
- 1989
(Show Context)
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32 | Efficient temporal reasoning through timegraphs
- Gerevini, Schubert
- 1993
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31 |
Models of axioms for time intervals
- Ladkin
- 1987
(Show Context)
Citation Context ...can be obtained from the authors or by anonymous ftp from duck.dfki.uni-sb.de as /pub/papers/RR-93-11.programs.tar.Z. 2 Other underlying models of the time line are also possible, e.g., the rationals =-=[5, 21]-=-. For our purposes these distinctions are not significant, however. 3 Basic Interval Sym- Pictorial Endpoint Relation bol Example Relations X before Y OE xxx X \Gamma ! Y \Gamma , X \Gamma ! Y + , Y a... |

30 |
Telos: Features and Formalization
- Koubarakis, Mylopoulos, et al.
- 1989
(Show Context)
Citation Context ... Further, for natural language understanding [3, 30], general planning [4, 6], presentation planning in a multi-media context [7, 9], diagnosis of technical systems [29], and knowledge representation =-=[20, 37]-=-, the representation of qualitative temporal relations and reasoning about them is essential. Allen [2] introduces an algebra of binary relations on intervals (hereafter referred to as Allen's interva... |

26 |
Temporally Distributed Symptoms in Technical Diagnosis, volume 517
- Nokel
- 1991
(Show Context)
Citation Context ...e performance of the system . . . " Further, for natural language understanding [3, 30], general planning [4, 6], presentation planning in a multi-media context [7, 9], diagnosis of technical sys=-=tems [29]-=-, and knowledge representation [20, 37], the representation of qualitative temporal relations and reasoning about them is essential. Allen [2] introduces an algebra of binary relations on intervals (h... |

24 |
The Interpretation of Temporal Relations in Narrative
- Song, Cohen
- 1988
(Show Context)
Citation Context ...litatively by specifying the relative positions of time intervals such as ". . . point to the figure while explaining the performance of the system . . . " Further, for natural language unde=-=rstanding [3, 30]-=-, general planning [4, 6], presentation planning in a multi-media context [7, 9], diagnosis of technical systems [29], and knowledge representation [20, 37], the representation of qualitative temporal... |

14 |
and A Mounir Alaoui Managing Efficiently Temporal Relations Through Indexed Spanning Trees
- Ghallab
- 1989
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13 |
Convex relations between time intervals
- Nökel
- 1989
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11 |
Algorithms and complexity for reasoning about time, in
- Golumbic, Shamir
- 1992
(Show Context)
Citation Context |

10 |
Towards coordinated temporal multimedia presentation
- Feiner, Litman, et al.
- 1993
(Show Context)
Citation Context ...point to the figure while explaining the performance of the system . . . " Further, for natural language understanding [3, 30], general planning [4, 6], presentation planning in a multi-media con=-=text [7, 9]-=-, diagnosis of technical systems [29], and knowledge representation [20, 37], the representation of qualitative temporal relations and reasoning about them is essential. Allen [2] introduces an algebr... |

7 | Linear time algorithms for testing the satis of propositional Horn formulae - Dowling, Gallier - 1984 |

4 |
Complexity of temporal reasoning with disjunctions of inequalities
- Gerevini, Schubert
- 1993
(Show Context)
Citation Context |

3 |
The satisfiability of temporal constraint networks
- Vald'ez-P'erez
- 1987
(Show Context)
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2 | Amine Mounir Alaoui. Managing eciently temporal relations through indexed spanning trees - Ghallab - 1989 |

1 |
PPP: Personalized plan-based presenter
- Andr'e, Graf, et al.
- 1993
(Show Context)
Citation Context ...point to the figure while explaining the performance of the system . . . " Further, for natural language understanding [3, 30], general planning [4, 6], presentation planning in a multi-media con=-=text [7, 9]-=-, diagnosis of technical systems [29], and knowledge representation [20, 37], the representation of qualitative temporal relations and reasoning about them is essential. Allen [2] introduces an algebr... |

1 |
Logic programming and graph rewriting
- Gallier, Raatz
- 1985
(Show Context)
Citation Context ...s 3 and 4, ORD\Omega [\Omega [ f(c 6= d); (asb); (a 6= b)g must be unsatisfiable. Since a set of Horn clauses is unsatisfiable iff it contains an unsatisfiable subset with exactly one negative clause =-=[12]-=-, it follows that ORD\Omega [ \Omega [f(asb)g, ORD\Omega [\Omega [f(asb); (a 6= b)g, or ORD\Omega [\Omega [f(c 6= d); (asb)g is already unsatisfiable. If one of the former two cases holds, then the cl... |