Time Representation: A Taxonomy of Internal Relations

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. Introduction James Allen in [Alli?] formulated a calculus of convex time intervals, which is being applied to commonsense reasoning by Allen, Pat Hayes, Henry Kautz and others [AZZKau, AZZHay]. Convex intervals are intuitively those which have no gaps. The term convex comes from topology. Allen’s calculus is a finite relation algebra in the sense of Tarski [Jo Tul, JoTa2, dada]. It has 13 atoms, which Allen enumerates, and hence the algebra has 2ls elements. We refer to the elements of this algebra as convex relations. There are close relations between algorithms used by Allen [Freu], and work in representations of relation algebras [Ma&, corn11 WC present some mathematical results on Allen’s algebra in [LadAdnd. other ways of representing time in AI have been argued for in [ McDer1, McDerZ]. Here, we investigate the binary relations that can hold between intervals which are UniCJnS of convex intervals. We call such relations non-convex relations. These intervals consist intuitively of some (maximal) convex subintervals with convex gaps in between them. We star? by discussing points-based and intervals-based representations of time. We then 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. The combinatorial explosion of possible binary relations between unions-of-convex intervals is dampened by considering only a subset of all possible relations. However, results of the author and Roger Maddux show that there are infinitely many relations definable in the algebra generated by these intervals [LadMadj. The notion of convex interval is definable in the algebra, as are the notions of having exactly (greater than, less than) n maximal convex subintervals, for each n [LadMad]. *This work was partially supported by RADC contract F30@X-84-C0109 and DARPA contract N00014-81-C-0582 Instants, Intervals and the Representation of Periods In [Ladl], we discussed points-based and intervals-based ways of representing time. Project management systems, amongst others, need a way of representing periods of time over which tasks happen, are scheduled, etc. There is a choice to be made between instantbased and interval-based representations of periods: Instants are atomic, indivisible entities which do not overlap, and are usually partially or linearly ordered. The order is usually called later than. Instants have no duration. This notion is used in the semantics of serial or concurrent programming languages with atomic instructions. Instants of time are identified with states of the system, and attached to these instants are propositions which describe the internal, nontemporal structure of the states. Use of instants in this way can be referred to as taking snapshots, and this approach is often taken when the system to be modelled is clocked. All snapshots are then synchronised with the clock, and the problem of determining is reduced to a count of clock interrupts. durations of periods To build periods from instants, we have to specify a range of instants, e.g. period(tl,tz) ES {t : tl < t 5 tz}. There is a question about whether to include endpoints, which we shall refer to again. A more complex, but useful, kind of period can be specified by taking (finite or arbitrary) unions of these basic convex periods. We then have periods which can represent, say, the during which a given process has control of the processor time time-shared environment. Intervals represent time periods directly. Intervals have duration, and are not necessarily indivisible. They are thus an abstraction from the properties of sets of time instants with measure. Thus, there are 12 ways that intervals may be related, excluding equality, e.g. precedes, overlaps, contained in [AZ1211 By contrast, instants can be related by only two, earlier than, later than. To determine what structure we need in this context, we believe it is best to work with the abstraction directly. This position is argued in [Ladl, AZZZ, AZZKau, AZZHay]. We consider the sets-of-points notion as one possible interpretation of intervals. The use of intervals is not restricted to AI. [~am1] defines an ordering on intervals (there referred to as sets of events), in order to prove the correctness of certain concurrency algo360 / SCIENCE From: AAAI-86 Proceedings. Copyright ©1986, AAAI (www.aaai.org). All rights reserved. rithms. Interval representations are also considered in [uBen, Hum, Dow]. There are mathematical constructions that convert point structures to interval structures and vice versa, e.g. the sets-of-points with measure construction [vBen]. See also [LadMad]. In particular, [Hum] resolves some supposed difficulties in the definition of truth of propositions over intervals. Additional reasons we prefer to work with intervals are l intervals provide a natural way of talking about duration, the length of time over which something happens, because they are an abstraction of the properties of periods of time The Choice of Relation Primitives There are too many discrete ways that unions of convex intervals may be related to each other. An exhaustive enumeration is infeasible, because Theorem 1 The number of relations between unions of convex intervals is at least ezponential in the number of maxconsubints. In fact, a much sharper result is true (see [LadMadj), but we intend only to establish infeasibility here. Proof: l interval notation is extensible: complexifying time structure doesn’t lead to changes in syntax; whereas point notation isn’t extensible, in that the number of points needed to specify a time structure varies with the complexity of the structure, e.g. we need 2 x n points to specify the union of n convex periods l there are unresolved difficulties with the endpoints of time periods, whether specified in point structures or interval structures. These need to be resolved before any implementation of time is attempted, but the difficulties are treated in a more ad hoc manner by points-based models of time. Notions such as temporal conjunction of propositions may be expressed in interval formulations [Hum], but not easily (so far) in standard points-based temporal logic. The interval approach allows all possible relations between endpoints, whereas such points-based approaches as [McDerl, McDeri?] have to chose a convention which is then hard-wired into the semantics. In terms of [AU?], the We prove the theorem by enumerating the relations between two intervals with n maxconsubints and using an inductive argument . Consider two intervals which are the unions of 2 convex subintervals each. Suppose the first subinterval of each is entirely disjoint from the second subinterval of the other. Then each first maxconsubint precedes the second maxconsubint of the other. The intervals are related in 132 ways, including equality, since the first maxconsubints can be related in 13 ways, including equality, and so can the second maxconsubints. When we consider that the first maxconsubint of each may be related by other than precedence to the second maxconsubint of the other, e.g. they may overlap, or meet, we see there are more than 132 relations overall. Now consider two intervals with n + 1 maxconsubints, such that the first n maxconsubints of each interval all precede the final maxconsubint of the other. By the inductive hypothesis, there are more than 13” ways the subintervals consisting of the first n maxconsubints may be related. The final two maxpoints-based approach has to choose between dreeedence (don’t include the endpoint), or overlapping (include the endpoint), and usually rules out meeting for any intervals. See [Hum] for another example involving temporal conjunction. consubints may be related in 13 ways, and therefore the total number of possible relations is more than 13n+1, Again, when we consider that the final two maxconsubints may be related by overlaps or meets to the penultimate maxconsubint of the other, we notice there are many more relations than just those we enumerated in the proof. Relation Primitives for Unions of Convex Intervals Hence we have established the base and the inductive steps, and we draw the conclusion of the induction. End of Proof. Intervals which are unions of convex intervals occur naturally. For example, any recurring time period can be represented: we can regard the period MONDAYS as being composed of each individual Monday, LABOR-DAYS is likewise the union of convex intervals, consisting of each individual Labor day, the period of the regular weekly meeting with the boss is also a union of convex intervals, each of them the period of single meeting. These kinds of intervals seem to be among the most useful of the non-convex cases, and since we have reason to hope that knowledge gleaned from considering the convex case will transfer in part, we consider unions of convex intervals in detail. We develop further the definition of time units, which include examples such as the above, in [Lada]. An interval which is a union of convex intervals looks like this 1 To avoid the combinatorial explosion implied by the theorem, our basic relations don’t depend on the number of maxconsubints. It is intuitively plausible that we don’t need relations that depend on the number of maxconsubints for expressing properties of time periods associated with actions, tasks, events or propositions. However, the relation algebra generated by the relations we consider is still infinite, and still enables us to define the class of intervals with exactly n subintervals, for each n [LadMad]. The approach we take will generalise the convex relations, by introducing functors that generate non-convex relations from convex relations, by enumerating new subclassifications of relations that weren’t there in the convex case, and by enumerating the various relations that arise from considering just the first and last maxconsubints. Additionally there is one relation, bars, which is not obtained by generalising the convex case in some way. This interval i has three “parts”, i.e. maximal convex subintervals, which we call mazconaubints. We obtain the following relations between unions of convex intervals: KNOWLEDGE REPRESENTATION / 36 1 l those generated by the functors mostly, always, partially, sometimes, and disjunction from convex relations (always may be defined in terms of mostly)