


Overview
General Formal Ontology (GFO)

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.^{11}Every chronoid has exactly two extremal and
infinitely many inner time boundaries which are
equivalently called timepoints.
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 nonconnected intervals of time.
Time entities, i.e., timeregions and timepoints, share certain
formal relations, in particular the
partof relation between chronoids and time regions, the relation of being
an extremal timeboundary of
a chronoid, and the relation of coincidence between two timeboundaries.
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:
 (a)
 there are two processes following one another immediately,
i.e., without any gaps,
 (b)
 there is a point in time where the first process ends, and
 (c)
 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 meetingpoint:
 The first interval is rightclosed and the second is
leftclosed. This allows for two options with regard to the overlap of
both intervals:
 (i)
 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 startingpoint of the second
interval.
 (ii)
 The intervals overlap at the meetingpoint. This raises,
however, the possibility of contradictions between properties of the first,
and properties of the second process (cf. the examples
below).
 The first interval is rightopen and the second one is
leftclosed. However, this conflicts with condition (b).
 The first interval is rightclosed and the second one
leftopen. This conflicts with condition (c).
 The first interval is rightopen and the second leftopen. 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?
 (a)
 There is no gap where there is no ink in the pen or ink
in the pen.
 (b)
 There is a final point where the pen is not empty.
 (c)
 There is an initial point where the ink pen is empty.
``Student changed his course of study from
physics to computer sciences by filling out the appropriate form.''
What do (a)  (c) mean in this example?
 (a)
 There is no gap where studies nothing.
 (b)
 There is a final point where is a student
of physics.
 (c)
 There is a first point where is a student
of computer sciences.
Robert Hoehndorf
20061018

