Reasoning about temporal information is an important task in many areas of Artificial Intelligence. In this paper we address the problem of scalability in temporal reasoning by providing a collection of new algorithms for efficiently managing large sets of qualitative temporal relations. We focus on the class of relations forming the Point Algebra (PA-relations) and on a major extension to include binary disjunctions of PA-relations (PA-disjunctions). Such disjunctions add a great deal of expressive power, including the ability to stipulate disjointness of temporal intervals, which is important in planning applications. Our representation of time is based on timegraphs, graphs partitioned into a set of chains on which the search is supported by a metagraph data structure. The approach is an extension of the time representation proposed by Schubert, Taugher and Miller in the context of story comprehension. The algorithms herein enable construction of a timegraph from a given set of PA-r...

. Information about the relative size of spatial regions is often easily accessible and, when combined with other types of spatial information, it can be practically very useful. In this paper we combine a simple framework for reasoning about qualitative size relations with the Region Connection Calculus RCC-8, a widely studied approach for qualitative spatial reasoning with topological relations. Reasoning about RCC-8 relations is NP-hard, but a large maximal tractable subclass of RCC-8 called b H8 was identified. Interestingly, any constraint in RCC-8 \Gamma b H8 can be consistently reduced to a constraint in b H8 , when an appropriate size constraint between the spatial regions is supplied. We propose an O(n 3 ) time path-consistency algorithm based on a novel technique for combining RCC-8 constraints and relative size constraints, where n is the number of spatial regions. We prove its correctness and completeness for deciding consistency when the input contains topological ...

. 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 polynomial-time approximation algorithms. Most available techniques are based on two principles: (i) enforcing local consistency (e.g. path-consistency), and (ii) enhancing naive backtracking search. Keywords: Temporal Constra...

Information about the size of spatial regions is often easily accessible and, when combined with other types of spatial information, it can be practically very useful. In this paper we introduce four classes of qualitative and metric size constraints, and we study their integration with the Region Connection Calculus RCC-8, a widely studied approach for qualitative spatial reasoning with topological relations. Reasoning about RCC-8 relations is NP-hard, but three large maximal tractable subclasses of RCC-8, called b H8 , C8 and Q8 respectively, have been identied. We propose an O(n 3 ) time path-consistency algorithm based on a novel technique for combining RCC-8 relations and qualitative size relations forming a Point Algebra, where n is the number of spatial regions. This algorithm is correct and complete for deciding consistency when the topological relations are either in b H8 , C8 or Q8 , and has the same complexity as the best known method for deciding consistency...

Contents 1 Introduction 6 1.1 Temporal Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Constraint Programming . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 Constraint problems and constraint satisfaction . . . . . . 7 1.2.2 Algorithms to solve constraints . . . . . . . . . . . . . . . 9 1.3 Temporal reasoning and Constraint Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.1 Temporal Reasoning with metric information . . . . . . . 14 1.3.2 Qualitative approach based on Allen's interval algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.3 Mixed approaches . . . . . . . . . . . . . . . . . . . . . . 15 2 Temporal Reasoning and Constraint Programming 16 2.1 Temporal Constraints with metric information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.1 A first order language . . . . . . . . . . . . . . . . . . . . 16 2.1.2 The original Temporal Constraint Problem . .

We address the problems of determining consistency and of finding a solution for sets of 3-point relations expressing exclusion of a point from an interval, and for sets of 4-point relations expressing interval disjointness. Availability of these relations is an important requirement for dealing with the sorts of temporal constraints encountered in many AI applications such as plan reasoning. We prove that consistency testing is NP-complete and finding a solution is NP-hard. Keywords: temporal reasoning, complexity of reasoning, planning, reasoning with disjunctions The work of the first author was carried out in part during a visit at the Computer Science Department of the University of Rochester (NY) supported by the Italian National Research Council (CNR), and in part at IRST in the context of the MAIA project and CNR projects "Sistemi Informatici e Calcolo Parallelo" and "Pianificazione Automatica". The second author was supported by Rome Lab Contract F30602-91-C-0010. 1 Introd...

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In a previous paper, we have proposed a fuzzy extension of Allen’s Interval Algebra, called IA fuz,which is able to model soft temporal constraints and uncertainty in a unified way. In this paper, we address the problem of finding tractable subalgebras of IA fuz, as it has been done in the classical case. In particular, we show that the fuzzy extensions of classical SAc and SA subalgebras are tractable sub-algebras of IA fuz.