17.5.3 Representing Time

In the preceding discussion of events, we did not address the issue of representing the time when the represented events are supposed to have occurred. The representation of such information in a useful form is the domain of temporal logic. This discussion will serve to introduce the most basic concerns of temporal logic along with a brief discussion of the means by which human languages convey temporal information, which among other things includes tense logic, the ways that verb tenses convey temporal information.

The most straightforward theory of time hold that it flows inexorably forward, and that events are associated with either points or intervals in time, as on a timeline. Given these notions, an ordering can be imposed on distinct events by situating them on the timeline. More specifically, we can say that one event precedes another, if the flow of time leads from the first event to the second. Accompanying these notions in most theories is the idea of the current moment in time. Combining this notion with the idea of a temporal ordering relationship yields the familiar notions of past, present and future.

Not surprisingly, there are a large number of schemes for representing this kind of temporal information. The one presented here is a fairly simple one that stays within the FOPC framework of reified events that we have been pursuing. Consider the following examples:

(17.33) I arrived in New York.

(17.34) I am arriving in New York.

(17.35) I will arrive in New York.

These sentences all refer to the same kind of event and differ solely in the tense of the verb. In our current scheme for representing events, all three would share the following kind of representation, which lacks any temporal information:

$ \exists w \text{ ISA}(w, \text{Arriving}) $

The temporal information provided by the tense of the verbs can be exploited by predicating additional information about the event variable w. Specifically, we can add temporal variables representing the interval corresponding to the event, the end point of the event, and temporal predicates relating this end point to the current time as indicated by the tense of the verb. Such an approach yields the following representations for our arriving examples:

$ \exists i,e,w,t ISA(w,Arriving) $

$ \land Arriver(w, Speaker) \land Destination(w, NewYork) $

$$ I n t e r v a l O f(w,i)\land E n d P o i n t(i,e)\land P r e c e d e s(e,N o w) $$

$ \exists i, e, w, t \text{ ISA}(w, Arriving) $

$$ IntervalOf(w,i)\land MemberOf(i,Now) $$

$$ \begin{array}{l}\exists i,e,w,t ISA(w,Arriving)\\ \quad\land Arriver(w,Speaker)\land Destination(w,New York)\\ \quad IntervalOf(w,i)\land EndPoint(i,e)\land Precedes(Now,e)\end{array} $$

This representation introduces a variable to stand for the interval of time associated with the event, and a variable that stands for the end of that interval. The two-place predicate Precedes represents the notion that the first time point argument precedes the second in time; the constant Now refers to the current time. For past events, the end point of the interval must precede the current time. Similarly, for future events the current time must precede the end of the event. For events happening in the present, the current time is contained within the event interval.

Unfortunately, the relation between simple verb tenses and points in time is by no means straightforward. Consider the following examples:

17.36) Ok, we fly from San Francisco to Boston at 10.

(17.37) Flight 1390 will be at the gate an hour now.

In the first example, the present tense of the verb fly is used to refer to a future event, while in the second the future tense is used to refer to a past event.

More complications occur when we consider some of the other verb tenses. Consider the following examples:

(17.38) Flight 1902 arrived late.

(17.39) Flight 1902 had arrived late.

Although both refer to events in the past, representing them in the same way seems wrong. The second example seems to have another unnamed event lurking in the background (e.g., Flight 1902 had already arrived late when something else happened). To account for this phenomena, Reichenbach (1947) introduced the notion of a reference point. In our simple temporal scheme, the current moment in time is equated with the time of the utterance, and is used as a reference point for when the event occurred (before, at, or after). In Reichenbach's approach, the notion of the reference point is separated out from the utterance time and the event time. The following examples illustrate the basics of this approach:

(17.40) When Mary's flight departed, I ate lunch.

(17.41) When Mary's flight departed, I had eaten lunch.

In both of these examples, the eating event has happened in the past, i.e. prior to the utterance. However, the verb tense in the first example indicates that the eating event began when the flight departed, while the second example indicates that the eating was accomplished prior to the flight's departure. Therefore, in Reichenbach's terms the departure event specifies the reference point. These facts can be accommodated by asserting additional constraints relating the eating and departure events. In the first example, the reference point precedes the eating event, and in the second example, the eating precedes the reference point. Figure 17.5 illustrates Reichenbach's approach with the primary English tenses. Exercise 17.9 asks you to represent these examples in FOPC.

This discussion has focused narrowly on the broad notions of past, present, and future and how they are signaled by verb tenses. Of course, languages also have

many other more direct and more specific ways to convey temporal information, including the use of a wide variety of temporal expressions as in the following ATIS examples:

(17.42) I'd like to go at 6:45, in the morning.

(17.43) Somewhere around noon, please.

(17.44) Later in the afternoon, near 6PM.

As we will see in the next chapter, grammars for such temporal expressions are of considerable practical importance in information extraction and question-answering applications.

Finally, we should note that there is a systematic conceptual organization reflected in examples like these. In particular, temporal expressions in English are frequently expressed in spatial terms, as is illustrated by the various uses of at, in, somewhere and near in these examples (Lakoff and Johnson, 1980; Jackendoff, 1983). Metaphorical organizations such as these, where one domain is systematically expressed in terms of another, will be discussed in more detail in Ch. 19.