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

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Venue: | Journal of the ACM |

Citations: | 160 - 9 self |

### BibTeX

@ARTICLE{Nebel95reasoningabout,

author = {Bernhard Nebel and Hans-Jürgen Bürckert},

title = {Reasoning about Temporal Relations: A Maximal Tractable Subclass of Allen's Interval Algebra},

journal = {Journal of the ACM},

year = {1995},

volume = {42},

pages = {43--66}

}

### Years of Citing Articles

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### Abstract

We introduce a new subclass of Allen's interval algebra we call "ORDHorn subclass," which is a strict superset of the "pointisable subclass." We prove that reasoning in the ORD-Horn subclass is a polynomial-time problem and show that the path-consistency method is sufficient for deciding satisfiability. Further, using an extensive machine-generated case analysis, we show that the ORD-Horn subclass is a maximal tractable subclass of the full algebra (assuming<F NaN> P6=NP). In fact, it is the unique greatest tractable subclass amongst the subclasses that contain all basic relations. This work has been supported by the German Ministry for Research and Technology (BMFT) under grant ITW 8901 8 as part of the WIP project and under grant ITW 9201 as part of the TACOS project. 1 1 Introduction Temporal information is often conveyed qualitatively by specifying the relative positions of time intervals such as ". . . point to the figure while explaining the performance of the system . . . "...

### Citations

2285 | 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... |

1000 |
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 ... |

825 | Towards a general Theory of action and time - Allen - 1984 |

423 |
Networks of constraints: fundamental properties and application 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... |

372 | P.: “Constraint propagation algorithms for temporal reasoning: a revised report
- Vilain, Kautz, et al.
- 1989
(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 ... |

264 |
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... |

245 |
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 ... |

235 | 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... |

136 | 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... |

116 |
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 ... |

115 | Planning Using a Temporal World Model
- Allen, Koomen
- 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 ... |

105 |
A Common-Sense 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... |

87 | On binary Constraint Problems - LADKIN, MADDUX - 1994 |

86 |
Linear Time Algorithms for testing the Satisfiability of Propositional Horn formulae
- Downing, 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... |

86 | Complexity and algorithms for reasoning about time: a graph-theoretic 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 ... |

77 |
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... |

62 | 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... |

54 |
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 ... |

43 |
Approximation Algorithms for Temporal Reasoning
- Beek
- 1989
(Show Context)
Citation Context |

40 |
Unit refutation 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 ... |

38 | On Binary Constraint Networks
- Ladkin, Maddux
- 1988
(Show Context)
Citation Context |

30 | Efficient Temporal Reasoning through Timegraphs
- Gerevini, Schubert
(Show Context)
Citation Context |

29 |
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... |

27 |
Telos: Features and formalization
- KOUBARAKIS, MOPOULOS, et al.
(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... |

25 |
Temporarily Distributed Symptoms in Technical Diagnosis
- 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... |

23 |
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... |

13 |
Amine Mounir Alaoui. Managing efficiently temporal relations through indexed spanning trees
- Ghallab
- 1989
(Show Context)
Citation Context |

13 |
Convex Relations Between Time Intervals
- Nökel
- 1990
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Citation Context |

10 |
Towards coordinated temporal multimedia presentations
- 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... |

10 |
Algorithms and complexity for reasoning about time
- Golumbic, Shamir
- 1992
(Show Context)
Citation Context |

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

3 |
The satisfiability of temporal constraint networks
- Vald'ez-P'erez
- 1987
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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... |