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 . . . "...

Representing and reasoning about incomplete and indefinite qualitative temporal information is an essential part of many artificial intelligence tasks. An interval-based framework and a point-based framework have been proposed for representing such temporal information. In this paper, we address two fundamental reasoning tasks that arise in applications of these frameworks: Given possibly indefinite and incomplete knowledge of the relationships between some intervals or points, (i) find a scenario that is consistent with the information provided, and (ii) find the feasible relations between all pairs of intervals or points. For the point-based framework and a restricted version of the intervalbased framework, we give computationally efficient procedures for finding a consistent scenario and for finding the feasible relations. Our algorithms are marked improvements over the previously known algorithms. In particular, we develop an O(n 2 ) time algorithm for finding one co...

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...

A class of interval-based temporal languages for uniformly representing and reasoning about actions and plans is presented. Actions are represented by describing what is true while the action itself is occurring, and plans are constructed by temporally relating actions and world states. The temporal languages are members of the family of Description Logics, which are characterized by high expressivity combined with good computational properties. The subsumption problem for a class of temporal Description Logics is investigated and sound and complete decision procedures are given. The basic language TL-F is considered #rst: it is the composition of a temporal logic TL # able to express interval temporal networks # together with the non-temporal logic F # a Feature Description Logic. It is proven that subsumption in this language is an NP-complete problem. Then it is shown how to reason with the more expressive languages TLU-FU and TL-ALCF . The former adds disjunction both at...

While the worst-case computational properties of Allen's calculus for qualitative temporal reasoning have been analyzed quite extensively, the determination of the empirical efficiency of algorithms for solving the consistency problem in this calculus has received only little research attention. In this paper, we will demonstrate that using the ORD-Horn class in Ladkin and Reinefeld's backtracking algorithm leads to performance improvements when deciding consistency of hard instances in Allen's calculus. For this purpose, we prove that Ladkin and Reinefeld's algorithm is complete when using the ORD-Horn class, we identify phase transition regions of the reasoning problem, and compare the improvements of ORD-Horn with other heuristic methods when applied to instances in the phase transition region. Finally, we give evidence that combining search methods orthogonally can dramatically improve the performance of the backtracking algorithm. Contents 1 Introduction 1 2 Allen's...

We study the problems of deciding consistency and performing variable elimination for disjunctions of linear inequalities and disequations with at most one inequality per disjunction. This new class of constraints extends the class of generalized linear constraints originally studied by Lassez and McAloon. We show that deciding consistency of a set of constraints in this class can be done in polynomial time. We also present a variable elimination algorithm which is similar to Fourier's algorithm for linear inequalities. Finally, we use these results to provide new temporal reasoning algorithms for the Ord-Horn subclass of Allen's interval formalism. We also show that there is no low level of local consistency that can guarantee global consistency for the OrdHorn subclass. This property distinguishes the Ord-Horn subclass from the pointizable subclass (for which strong 5-consistency is sufficient to guarantee global consistency), and the continuous endpoint subclass (for whi...

In applications dealing with ordered domains, the available data is frequently indefinite. While the domain is actually linearly ordered, only some of the order relations holding between points in the data are known. Thus, the data provides only a partial order, and query answering involves determining what holds under all the compatible linear orders. In this paper we study the complexity of evaluating queries in logical databases containing such indefinite information. We show that in this context queries are intractable even under the data complexity measure, but identify a number of PTIME sub-problems. Data complexity in the case of monadic predicates is one of these PTIME cases, but for disjunctive queries the proof is non-constructive, using well-quasi-order techniques. We also show that the query problem we study is equivalent to the problem of containment of conjunctive relational database queries containing inequalities. One of our results implies that the latter is \Pi p 2 ...

Many applications|from planning and scheduling to problems in molecular biology| rely heavily on a temporal reasoning component. In this paper, we discuss the design and empirical analysis of algorithms for a temporal reasoning system based on Allen's in uential interval-based framework for representing temporal information. At the core of the system are algorithms for determining whether the temporal information is consistent, and, if so, nding one or more scenarios that are consistent with the temporal information. Two important algorithms for these tasks are a path consistency algorithm and a backtracking algorithm. For the path consistency algorithm, we develop techniques that can result in up to a ten-fold speedup over an already highly optimized implementation. For the backtracking algorithm, we develop variable and value ordering heuristics that are shown empirically to dramatically improve the performance of the algorithm. As well, we show that a previously suggested reformulation of the backtracking search problem can reduce the time and space requirements of the backtracking search. Taken together, the techniques we develop allow a temporal reasoning component tosolve problems that are of practical size. 1.

While the computational properties of qualitative temporal reasoning have been analyzed quite thoroughly, the computational properties of qualitative spatial reasoning are not very well investigated. In fact, almost no completeness results are known for qualitative spatial calculi and no computational complexity analysis has been carried out yet. In this paper, we will focus on the so-called RCC approach and use Bennett's encoding of spatial reasoning in intuitionistic logic in order to show that consistency checking for the topological base relations can be done efficiently. Further, we show that path-consistency is sufficient for deciding global consistency. As a side-effect we prove a particular fragment of propositional intuitionistic logic to be tractable.

In this paper we address the problem of scalability in temporal reasoning. In particular, new algorithms for efficiently managing large sets of relations in the Point Algebra are provided. Our representation of time is based on timegraphs, graphs partitioned into a set of chains on which the search is supported by a rnetagraph 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 presented in this work concern the construction of a timegraph from a given set of relations and are implemented in a temporal reasoning system called TG-II. Experimental results show that our approach is very efficient, especially when the given relations admit representation as a collection of chains connected by relatively few cross-chain links. 1