The temporal boundaries of many real–world events are inherently vague. In this paper, we discuss the problem of qualitative temporal reasoning about such vague events. We show that several interesting reasoning tasks, such as checking satisfiability, checking entailment, and calculating the best truth value bound, can be reduced to reasoning tasks in a well–known point algebra with disjunctions. Furthermore, we identify a maximal tractable subset of qualitative relations to support efficient reasoning. 1

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Abstract: The paper presents time-related part of PSI 1 theoretical framework. In comparison to other theories of time based on interval logics our approach presents the advancement by introducing fuzziness of time intervals as transition periods at beginnings and endings. It is argued that, though quite simple (discrete, linear, and anisotropic), our theoretical model is expressive enough to be used as a logical formalism for reasoning about stochastic, unpredictable, weakly defined action and process flows. A metric and a rich set of axiomatic relationships among time intervals are introduced for that. Further on, a means for modeling and reasoning about singular, repeated, regular events and actions having phases and vague durations is elaborated. Presented theory of time is used for modeling and reasoning about events, environmental influences, happenings, and actions while planning and scheduling in our simulations of dynamic engineering design processes. 1

Abstract — Fuzzy qualitative temporal relations have been proposed to reason about events whose temporal boundaries are ill– defined. Although the corresponding reasoning tasks are in the same complexity class as their crisp counterparts, in practice, the scalability of fuzzy temporal reasoners may be insufficient for applications which require a high expressivity and deal with a large number of events. On the other hand, transitivity rules can be used to make sound, but incomplete inferences in polynomial time, utilizing a variant of Allen’s path–consistency algorithm. The aim of this paper is to investigate how this polynomial time algorithm can be improved without altering its time complexity. To this end, we establish a characterization of 2–consistency of fuzzy temporal relations and provide transitivity rules which are significantly stronger than those resulting from straightforwardly generalizing transitivity rules for crisp temporal relations. We furthermore provide experimental evidence for the effectiveness of our improved algorithm.

Abstract — Fuzzy temporal interval relations have been defined to support temporal knowledge representation and reasoning in the presence of vagueness. The most important impediment to use these fuzzy relations in real–world applications is the lack of a characterization that is both easy to implement and computationally efficient. In this paper, we provide such a characterization for the important class of piecewise linear fuzzy time intervals, which covers all types of fuzzy time intervals that we are likely to encounter in applications. Furthermore, we discuss a more elegant characterization for the special case of trapezoidally shaped fuzzy intervals. I.

Traditional approaches to temporal reasoning assume that time periods and time spans of events can be accurately represented as intervals. Real–world time periods and events, on the other hand, are often characterized by vague temporal boundaries, requiring appropriate generalizations of existing formalisms. This paper presents a framework for reasoning about qualitative and metric temporal relations between vague time periods. In particular, we show how several interesting problems, like consistency and entailment checking, can be reduced to reasoning tasks in existing temporal reasoning frameworks. We furthermore demonstrate that all reasoning tasks of interest are NP–complete, which reveals that adding vagueness to temporal reasoning does not increase its computational complexity. To support efficient reasoning, a large tractable subfragment is identified, among others, generalizing the well–known ORD Horn subfragment of the Interval Algebra (extended with metric constraints).