Existing spatio-temporal datasets do not usually consist of explicit assertions of events occurrences or process activity. Rather, they often contain elements describing changes of objects’ properties over time, from which events and processes can be inferred. For example, movements of objects can be identified from data on the position of such objects at different time instants. Similarly, the expansion of a built-up region can be inferred from data describing its spatial extensions at different times.
Deciding on whether data best reproduce the intended denotation of events or pro- cesses is a crucial issue for the development of a theoretical framework which is supposed to be implemented to operate on real datasets. On the one hand, a piece of data represent- ing a single change affecting a feature (e.g., an increase in area between two distinct time
instants t1 and t2) is not sufficient to infer the occurrence of an event, since information
on the feature’s area after t2would be required to determine the precise temporal bound-
ary which is expected for an event. On the other hand, such a piece of data would not reproduce the density and homogeneity characteristics associated with a process. That is, whereas it can be inferred that an expansion process was active for some time between t1
and t2, it is not possible to ensure that the process was active during the whole interval.
The approach taken for building this logical framework consists of abstracting away from the lack of information for representing the activeness of process. This is to say that, if it is known that a geographic feature changes from instant t1to t2and nothing is known
about the period between them, it is assumed that the process which characterises this change is active at all time instants from t1to t2. The activeness of a process is represented
by the predicates Active-At(p,t) and Active-On(p, i). Whilst the former determines that
a process of type p is active at a time instant t, the latter specifies that a process is active on a time interval i, meaning it is going on at every time instant within that interval.
Since the former is defined in a low level manner, that is, closer to the way data is rep- resented, a discussion on its definition shall be given in Section 4.10.2, after introducing some other required. Whereas the latter is defined as follows.
D 4.5 Active-On(p, i) ≡de f ∀t[ In(t, i) → Active-At(p,t) ]
Beyond the predicates to represent the activeness of a process, it is also convenient to define a predicate which verifies whether a process is inactive at a certain time instant or on a given time interval. The latter is particularly useful for defining the predicate which determine event occurrences (presented later in this section) and the predicate which spec- ifies whether a process proceeds (presented in Section 4.9). It should be noticed that there is a difference between a process being inactive during an interval and it not being active, that is, Inactive-On(p, i) 6≡ ¬ Active-On(p, i). For instance, if an interval includes some
parts where a process is active and others where it is not active, then¬ Active-On(p, i)
will hold, but Inactive-On(p, i) should not hold. Hence, the predicates denoting a process
inactivity are defined as follows.
D 4.6 Inactive-At(p,t) ≡de f ¬Active-At(p,t)
D 4.7 Inactive-On(p, i) ≡de f ∀t[ In(t, i) → ¬Active-At(p,t) ]
nation of the process (i.e., when the goal in initiating it is realised) denotes the occurrence of an event. Therefore, in this logical framework, an event token is modelled as a chunk of a process bounded by temporal discontinuities, meaning that the process is inactive on both time intervals which meets and is met by the interval on which the event occurs.
Furthermore, since it is accepted that events can also be regarded as constituent of processes, this might lead to a continuous cycle. Thus it is important to distinguish event tokens which cannot contain other events of the same type. These are called primitive event tokens. Expressly, if a primitive event e occurs on an interval i, there is no sub- interval of i on which an event of the same type of e occurs.
It can be seen that allowing the existence of nested event tokens of the same type implies that events can also affected by gaps, since an event token is necessarily bounded by temporal discontinuities, as discussed above. Therefore another property of primitive event tokens is that they are not affected by temporal gaps, meaning that the process of which it is made must be active throughout the whole interval on which the event is said to occur. The predicate to represent a primitive event occurrence is defined in D4.8, whilst the representation of non-primitive ones is discussed in Section 4.9.
D 4.8 Occurs-On-Prim(e, i) ≡de f ∃vpbi′i′′[
e= event(v, f ) ∧ p = process(b, f ) ∧
Is-Chunk-Of(v, b) ∧ (∀t[In(t, i) → Active-At(p,t)]) ∧ Meets(i′, i) ∧ Met-by(i′′, i) ∧
Inactive-On(p, i′) ∧ Inactive-On(p, i′′)]
Figure 4.1 exhibits a stretch of the timeline to illustrate possible primitive event tokens for an event type e (bold lines). In this figure, 3 occurrences of the same event type e are shown (on the intervals i1, i2 and i3). On the other hand, no event of type e occur (as a
primitive token) on intervals i4, i5or i6, as such occurrences would not be in accordance
Chapter 4 86 Logical Framework
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Occurs-On-Prim(e,i{
{
{
1) Occurs-On-Prim(e,i{
{
2) Occurs-On-Prim(e,i{
3)¬Occurs-On-Prim(e,i4) ¬Occurs-On-Prim(e,i5) ¬Occurs-On-Prim(e,i6)
Figure 4.1: Stretch of the timeline exhibiting primitive event tokens of type e (bold lines). 3 distinct occurrences of the same event type (e) are shown (on intervals i1, i2 and i3,
respectively. Primitive event tokens are modelled as bounded by temporal discontinuities, containing no temporal gaps, and no event of the same type can occur in a sub-interval of the interval on which it occurs. Thus no primitive event token of type e is identified on intervals i4, i5or i6.
Notice that the possibility of two event tokens of distinct types occurring in parallel was not ruled out. That is, given two intervals i1and i2on which tokens of two different
event types occur, propositions Overlap(i1, i2) or During(i1, i2) might hold. For example,
since different event types can affect the same geographic feature, such events could rep- resent an object which expands and rotate over the same period of time. Similarly, the definitions presented here do not rule out that an event token can be followed by another token of a different type, that is, proposition Meets(i1, i2) might hold where i1 and i2are
time intervals on which two events of different types occur (even if they affect the same geographic feature).