GFO Part I Basic Principles
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5.1 Time

The GFO approach of time is inspired by Brentano's ideas (11) on continuum, space and time. Following this approach, chronoids are not defined as sets of points, but as entities sui generis.11Every chronoid has exactly two extremal and infinitely many inner time boundaries which are equivalently called time-points. Time boundaries depend on chronoids (i.e., they have no independent existence) and can coincide.

Starting with chronoids, we introduce the notion of time region as the mereological sum of chronoids, i.e., time regions consist of non-connected intervals of time. Time entities, i.e., time-regions and time-points, share certain formal relations, in particular the part-of relation between chronoids and time regions, the relation of being an extremal time-boundary of a chronoid, and the relation of coincidence between two time-boundaries.

Dealing with the coincidence of time boundaries is especially useful if two processes are to be modeled as ``meeting'' (in the sense of Allen's relation ``meets''). In our opinion, there are at least three conditions that a correct model must fulfill:

there are two processes following one another immediately, i.e., without any gaps,
there is a point in time where the first process ends, and
there is a point in time where the second process begins.

If, as is common practice, intervals of real numbers are used for modeling time intervals (with real numbers as time points), there are four possibilities for modeling the meeting-point:

  1. The first interval is right-closed and the second is left-closed. This allows for two options with regard to the overlap of both intervals:

    The intervals do not overlap. This conflicts with conditon (a), because a new interval can be placed between the final point of the first and the starting-point of the second interval.
    The intervals overlap at the meeting-point. This raises, however, the possibility of contradictions between properties of the first, and properties of the second process (cf. the examples below).
  2. The first interval is right-open and the second one is left-closed. However, this conflicts with condition (b).
  3. The first interval is right-closed and the second one left-open. This conflicts with condition (c).
  4. The first interval is right-open and the second left-open. This variant fails on both conditions (b) and (c).
In contrast, the approach via the glass continuum allows for two chronoids to follow immediately, one after another, and to have proper starting- and ending-``points'' by allowing their boundaries to coincide. The coincidence relation entails that there is no time difference between the coinciding time boundaries, while maintaining their status as two different entities. This way, conditions (a), (b) and (c) are fulfilled. Let us consider additional examples below.

``She drew a line with her fountain pen until there was no more ink left.'' What do the conditions (a) - (c) mean in this example?

There is no gap where there is no ink in the pen or ink in the pen.
There is a final point where the pen is not empty.
There is an initial point where the ink pen is empty.

``Student $X$ changed his course of study from physics to computer sciences by filling out the appropriate form.'' What do (a) - (c) mean in this example?

There is no gap where $X$ studies nothing.
There is a final point where $X$ is a student of physics.
There is a first point where $X$ is a student of computer sciences.

Robert Hoehndorf 2006-10-18


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