[Crm-sig] A question about relations between time intervals with fuzzy borders

Christian-Emil Ore c.e.s.ore at iln.uio.no
Mon Aug 24 12:04:22 EEST 2009

Dear all,
It is correct that in his 1983 paper Allen argues that one should 
represent temporal entities  as  intervals  with non-fuzzy boarders.  In 
the paper he discusses a system where the intervals/temporal entities 
are atoms without any internal structure. In the system a set of 
relations between these atoms together with transitivity rules are 
defined. These are used to reason about relations between temporal 
entities and connected propositions.
In the CRM the E2 temporal entities and the properties P114-P140 
correspond to the temporal entities and the relations in Allen’s paper. 
  It is important to note that the CRM is an ontology and contains 
absolutely no rules for deduction or inference.  Thus E2 and P114-P140 
define a static model for Allen’s temporal entities and thus an 
architecture for structures for storing this information, relations and 
abut no deduction system.
As I understand CRM, the class E53 time-span describes the knowledge 
categories we (may) use to describe approximations to some ideal real 
world time-spans (in which temporal entities take place).  Since the 
idea behind time-spans is that they are approximations, time-spans 
should not be connected with fixed start or end points in some system of 
time primitives, that is, a concrete implementation.  Instead we have 
two facts expressed by P81 and P82, which can typically be expressed as 
an inner  ‘interval’ and an outer ‘interval’ on a numeric time-line. On 
a time-line these two intervals will be expressed as four points. The 
start points define an interval   corresponding to an approximation of 
the start point of the time-span based on our given knowledge. 
(correspondingly for the end point). The intervals in the 
time-primitive-structure for the start and end points of a time-span can 
be seen as fuzzy (underspecified) points on the timeline. There is no 
infinite recursion in the model since these intervals are not connected 
to a E53 time-span and E2 temporal entity.
If the length of such an interval is zero, then we have knowledge about 
exact when the time-span started. Thus in the Erlangen version with its 
extra properties it is claimed that we usually can know exactly when a 
time span starts.
Martin Scholz’s diagram:
Allen’s view of temporal entities as atoms and relations corresponds to 
the E2, P114-140 part of the CRM. As I mentioned above the CRM contains 
no deduction rules. In a deduction system based on CRM-compliant 
structures, we have to add the transitivity rules etc from Allen’s 
paper, to do any reasoning about the relations between  instances of E2. 
  The information is simply a graph.  One may very well express a 
discrete time line down to nanoseconds in this structure. So far there 
is no fuzziness at all.
In the CRM the time-span+time primitive is an alternative model to the 
Allen approach. The scope note for E59 is “This class comprises 
instances of E59 Primitive Value for time that should be implemented 
with appropriate validation, precision and interval logic to express 
date ranges relevant to cultural documentation.” In a consistent 
instantiation temporal facts about instances of E2 expressed in 
time-span+time primitive have to be consistent with facts expressed via 
Martin Schlolz writes “Even for X1 and X2 three properties --- P118 
overlaps in with, P119 meets in time with and P120 occurs before --- 
(may) hold, depending on the view on the fuzzy borders.  Allen's 
relations, though, are thought of as mutually exclusive.”  That is 
correct and unproblematic in the CRM since the time primitive structure 
(with fuzzy borders) needs not to be a model or implementation of the 
Allen calculus.   One can however use this in a deduction system. 
Assume that we have the X1 and X2 situation, then we can deduce that one 
and at most one of “P118 overlaps in with, P119 meets in time with and 
P120 occurs before“   holds. If another fact, say P116 is equal in time 
with, then this will cause an impossible situation.  In this case one 
has to check the premises and the deduction chains to see which fact(s) 
caused the inconsistency.

Why use both Allen time capsules and (fuzzy) approximations?
The Allen system expresses information as a multi-dimensional graph and 
the time-span + time-primitive system expresses (in our use) the 
information as pair of intervals on a one-dimensional time line. The 
Allen system is excellent to store relative temporal facts. It is 
possible to define a mapping of the information in an Allen structure 
onto the time line as pairs of inner and outer intervals or equivalent 
as intervals with fuzzy start and end points. In many cases it is 
handier to reason with such under-determined points.  In the opposite 
direction the pairs of intervals (approximations) define sets of 
constrains on the Allen structure.  Jon Holmen and I have worked on this 
in connection with a CIDOC-CRM based temporal deduction system for 


Allen and fuzzyness:
There is a rather mathematical paper on this
Temporal Integrity Constraints with Indeterminacy
Proceedings of the 26th International Conference on Very Large Data Bases
Pages: 441 - 450
Year of Publication: 2000
Authors  Wes Cowley ,Dimitris Plexousakis

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