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Backtracking Algorithms for Disjunctions of Temporal Constraints
 Artificial Intelligence
, 1998
"... We extend the framework of simple temporal problems studied originally by Dechter, Meiri and Pearl to consider constraints of the form x1 \Gamma y1 r1 : : : xn \Gamma yn rn , where x1 : : : xn ; y1 : : : yn are variables ranging over the real numbers, r1 : : : rn are real constants, and n 1. W ..."
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Cited by 117 (2 self)
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We extend the framework of simple temporal problems studied originally by Dechter, Meiri and Pearl to consider constraints of the form x1 \Gamma y1 r1 : : : xn \Gamma yn rn , where x1 : : : xn ; y1 : : : yn are variables ranging over the real numbers, r1 : : : rn are real constants, and n 1. We have implemented four progressively more efficient algorithms for the consistency checking problem for this class of temporal constraints. We have partially ordered those algorithms according to the number of visited search nodes and the number of performed consistency checks. Finally, we have carried out a series of experimental results on the location of the hard region. The results show that hard problems occur at a critical value of the ratio of disjunctions to variables. This value is between 6 and 7. Introduction Reasoning with temporal constraints has been a hot research topic for the last fifteen years. The importance of this problem has been demonstrated in many areas of artifici...
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 Constraints: A Survey
, 1998
"... . Temporal Constraint Satisfaction is an information technology useful for representing and answering queries about the times of events and the temporal relations between them. Information is represented as a Constraint Satisfaction Problem (CSP) where variables denote event times and constraints re ..."
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Cited by 26 (1 self)
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. Temporal Constraint Satisfaction is an information technology useful for representing and answering queries about the times of events and the temporal relations between them. Information is represented as a Constraint Satisfaction Problem (CSP) where variables denote event times and constraints represent the possible temporal relations between them. The main tasks are two: (i) deciding consistency, and (ii) answering queries about scenarios that satisfy all constraints. This paper overviews results on several classes of Temporal CSPs: qualitative interval, qualitative point, metric point, and some of their combinations. Research has progressed along three lines: (i) identifying tractable subclasses, (ii) developing exact search algorithms, and (iii) developing polynomialtime approximation algorithms. Most available techniques are based on two principles: (i) enforcing local consistency (e.g. pathconsistency), and (ii) enhancing naive backtracking search. Keywords: Temporal Constra...
Eight Maximal Tractable Subclasses of Allen's Algebra with Metric Time
, 1997
"... This paper combines two important directions of research in temporal resoning: that of finding maximal tractable subclasses of Allen's interval algebra, and that of reasoning with metric temporal information. Eight new maximal tractable subclasses of Allen's interval algebra are presented, ..."
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Cited by 26 (10 self)
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This paper combines two important directions of research in temporal resoning: that of finding maximal tractable subclasses of Allen's interval algebra, and that of reasoning with metric temporal information. Eight new maximal tractable subclasses of Allen's interval algebra are presented, some of them subsuming previously reported tractable algebras. The algebras allow for metric temporal constraints on interval starting or ending points, using the recent framework of Horn DLRs. Two of the algebras can express the notion of sequentiality between intervals, being the first such algebras admitting both qualitative and metric time. 91 1 Introduction Reasoning about temporal knowledge abounds in artificial intelligence applications and other areas, such as planning [ Allen, 1991 ] , natural language understanding [ Song and Cohen, 1988 ] and molecular biology [ Benzer, 1959; Golumbic and Shamir, 1993 ] . However, since even the restricted problem of reasoning with pure qualitative ti...
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 25 (10 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.
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...
The Augmented Interval and Rectangle Networks
"... We augment Allen's Interval Algebra networks and Rectangle Algebra networks by quantitative constraints represented by ST Ps. With the help of polynomial algorithms based on the traditional and weak pathconsistency methods, we prove the tractability of the consistency problem of preconv ..."
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Cited by 9 (0 self)
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We augment Allen's Interval Algebra networks and Rectangle Algebra networks by quantitative constraints represented by ST Ps. With the help of polynomial algorithms based on the traditional and weak pathconsistency methods, we prove the tractability of the consistency problem of preconvex augmented interval networks and stronglypreconvex augmented rectangle networks. Keywords: Temporal and spatial reasoning, Interval Algebra and Rectangle Algebra, constraint networks, complexity. 1 Introduction Temporal and spatial reasoning with constraints is a relevant activity in artificial intelligence. Concerning qualitative temporal reasoning, Allen's Interval Algebra [1] (IA) is one of the most known and used formalisms. Allen takes intervals as primitive temporal entities and considers 13 atomic relations between these intervals (fig. 1). These relations represent all the possible relative positions between two intervals on the rational line. With this formalism we can represent qu...
On solving soft temporal constraints using SAT techniques
 In CP’05, LNCS 3709
, 2005
"... Abstract. In this paper, we present an algorithm for finding utilitarian optimal solutions to Simple and Disjunctive Temporal Problems with Preferences (STPPs and DTPPs) based on Benders ’ decomposition and adopting SAT techniques. In our approach, each temporal constraint is replaced by a Boolean i ..."
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Cited by 8 (3 self)
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Abstract. In this paper, we present an algorithm for finding utilitarian optimal solutions to Simple and Disjunctive Temporal Problems with Preferences (STPPs and DTPPs) based on Benders ’ decomposition and adopting SAT techniques. In our approach, each temporal constraint is replaced by a Boolean indicator variable and the decomposed problem is solved by a tightly integrated STP solver and SAT solver. Several hybridization techniques that take advantage of each solver’s strengths are introduced. Finally, empirical evidence is presented to demonstrate the effectiveness of our method compared to other algorithms. 1