Short Time Periods

Earlier papers (Allen and Hayes 1985,1986) described a compact axiomatic theory which provided a formal basis for temporal reasoning. This theory makes a sharp distinction between time points and periods of time. We show here, by considering possible models of the theory, that it can be slightly extended to give a better fit to intuition. In particular, the extended theory is preserved under changes of temporal granularity. TIMEPERIODS AND MEETING The original period theory (Allen 1984, Allen & Hayes 1986) used periods rather than points as its temporal primitive. (The terminology used earlier was 'interval' rather than 'period'. We have changed to avoid confusion with the mathematical use of 'interval'.) The thirteen possible relationships between periods, including for example overlapping, inclusion, and before are defined in terms of MEETS, which we will write as in infix colon, with p:q:r meaning p:q and q:r. The basic axioms of the theory are then as follows: These five axioms are all the assumptions which the theory makes. For a longer discussion of their implications and justification, see (Allen and Hayes 1986). A key intuition is that propositions are true or false during periods, rather than at points of time. This overcomes the problem of the 'divided instant' (Van Bentham 1982). Consider switching on a light, so that the proposition "light on" is first false, then immediately true. If the timeperioda are thought of as sets of points, then only artificial constructions can avoid the dilemma of there being a point at which the light is neither on nor off (or, worse, both on and off). theory as the 'places' where periods meet. In (Allen & Hayes 1986), a set-theoretic construction is given of points from periods, but we can introduce points directly, and it can be shown that the theory obtained is identical. We will use variable names u,v,w,.. for points, p,q,r,..for periods, and introduce two functions START and END from periods to points. PI forall p,q. p:q iff end(p)-start(q) P2 forall u exists p. u-end(p) It is possible now to show that points, whatever they are, have all the properties one would directly expect, such as being a totally ordered infinite set with no limits.We could also start with points as the primitive idea and define periods as pairs of points from the totally ordered set. In (Allen and Hayes 1986) we show how the natural construction of periods from points by …