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Peter Ladkin

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4 papers

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AAAI Conference 1987 Conference Paper

Models of Axioms for Time Intervals

Peter Ladkin

James Allen and Pat Hayes have considered axioms expressed in first-order logic for relations between time intervals [AllHay85, AllHay87. 1, AllHay87. 2]. One important consequence of the results in this paper is that their theory is decidable [Lad87. 4]. In this paper, we characterise all the models of the theory, and of an important subtheory. A model is isomorphic to an interval structure INT(S) over some unbounded linear order S, and conversely, INT(S), for an arbitrary unbounded linear order S, is a model. The models of the subtheory are similar, but with an arbitrary number of copies of each interval (conversely, all structures of this form are models). We also show that one of the original axioms is redundant, and we exhibit an additional axiom which makes the Allen-Hayes theory complete and countably categorical, with all countable models isomorphic to INT(Q), the theory of intervals with rational endpoints, if this is desired. These results enable us to directly compare the Allen-Hayes theory with the theory of Ladkin and Maddux iLadMad87. 11, and of van Bent hem [vBen83].

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IJCAI Conference 1987 Conference Paper

The Completeness of a Natural System for Reasoning with Time Intervals

Peter Ladkin

James Allen defined a calculus of time intervals by identifying time intervals as pairs of real numbers, and considering binary relations that can hold between such pairs [Alll83]. We call this the Interval Calculus. We consider the system of interval time units defined in [Lad86. 2] (the T U S ), which was intended for the natural representation of real clock time on any scale. We introduce the convex part of the T U S, and show that it may be regarded as a canonical model of the Interval Calculus. We discuss the consequences of this result.

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AAAI Conference 1986 Conference Paper

Primitives and Units for Time Specification

Peter Ladkin

We work in a calculus of intervals, formulated by James Allen for convex intervals, and by ourselves for unions of convex intervals [AZZ2, Lad. Z]. We investigate the primitive relations and operations needed for implementing such calculi in a system which includes some set theory, and which allows the assertional definition of operators in Horn clause fashion. We indicate how standard temporal logic may be rephrased in the interval calculus, and present a formalisation of a system of time units in the interval framework. We are implementing the primitives in the REF IN ETM system’.

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AAAI Conference 1986 Conference Paper

Time Representation: A Taxonomy of Internal Relations

Peter Ladkin

James Allen in [AZZ2]formulated a calculus of convex time intervals, which is being applied to commonsense reasoning by Allen, Pat Hayes, Henry Kautz and others [AZZKuu, AZZHay]. For many purposes in AI, we need more general time intervals. We present a taxonomy of important binary relations between intervals which are unions of convex intervals, and we provide examples of these relations applied to the description of tasks and events. These relations appear to be necessary for such description. Finally, we provide logical definitions of a taxonomy of general binary relations between non-convex intervals.

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