From: AAAI Technical Report WS-00-08. Compilation copyright © 2000, AAAI (www.aaai.org). All rights reserved.
Extending
the Event Calculus with Temporal Granularity
Indeterminacy *
and
Luca Chittaro
and Carlo
Combi
Department of Mathematics and Computer Science
University of Udine
via delle Scienze 206, 33100 Udine, Italy
{chittaro, combi)@dimi.uniud.it
has to reason about events and change in this domain
must have the capability of representing and reasoning with data at different time scales and with indeterminacy. The two well-known formalisms for reasoning
about actions, events and change, i.e. the Situation
Calculus (McCarthy & Hayes 1986) and the Event Calculus (EC) (Kowalski & Sergot 1986), do not provide
mechanisms for handling temporal indeterminacy and
granularity, and very little research has been devoted
to the goal of extending them with different granularities. In this paper, we provide an overview of a novel
approach (TGIC, Temporal Granularity and Indeterminacy event Calculus) to represent events with imprecise
location and to deal with different timelines, using the
EC ontology. Wethen contrast TGIC with the wellknownapproach to the handling of granularity in ECby
Montanari et al. (Montanari et al. 1992). Additional
aspects of TGICare discussed elsewhere: a formalization of the presented concepts is provided in (Chittaro
& Combi 1998), while a polynomial algorithm for implementing the reasoning activity of TGICis described
in (Chittaro & Combi 1999). This paper is organized
as follows: first, we briefly describe the ontology of EC,
and we provide two motivating examples taken from our
clinical domain; then we discuss the proposed representation of temporal information in TGIC, and the issues
that have to be taken into account by the reasoning
activity. A detailed analysis of a previous work extending ECto deal with granularities, and some concluding
remarks end the paper.
Abstract
In manyreal-world applications, temporal information
is often imprecise about the temporal location of events
(indeterminacy) and comesat different granularities.
Formalisms for reasoning about events and change,
such as the Event Calculus (EC) and the Situation Calculus, do not usually provide mechanismsfor handling
such data, and very little research has been devoted
to the goal of extending themwith these capabilities.
In this paper, we propose TGIC(Temporal Granularity and Indeterminacy event Calculus), an approach to
represent events with imprecise location and to deal
with them on different timelines, based on the EContology.
Introduction
In manyreal-world applications, temporal information
is often imprecise about the temporal location of events
(indeterminacy) and comes at different granularities
(Dyreson & Snodgrass 1995). Temporal granularity
and indeterminacy are thus emerging as crucial requirements for the advancement of intelligent
information
systems which have to store, manage, and reason about
temporal data. Consider, for example, these events
taken from the application - a temporal database for
cardiological patients - we are considering in our research (Combi & Chittaro 1999): "between 2 p.m. and
4 p.m. on May 5, 1996, the patient suffered from a
myocardial infarction", "he started the therapy with
thrombolytics in July 1995", "on October 12, 1996,
he had a follow-up visit". The three events happened
at the hours, months, and days timelines, respectively.
Moreover, the first event cannot be precisely located
on its timeline. A temporal reasoning system which
The Ontology of EC
The notions of event, property, timepoint and time interval are the primitives of the EC ontology: events
happen at timepoints and initiate
and/or terminate
time intervals over which some property holds. Properties are assumed to persist until the occurrence of an
event interrupts them (default persistence). An event
occurrence can be represented by associating the event
to the timepoint at which it occurred, e.g. by means of
the clause:
*This workhas been partially supported by contributions
from: MURSTItalian National Project COFIN’98 "An
Agent-BasedArchitecture to Support Cooperative Workin
Medicine", and the Department of Mathematics and Computer Science of the University of Udine.
Copyright © 2000, AmericanAssociation for Artificial Intelligence (www.aaai.org).All rights reserved.
53
happens(event,timePoint)
and terminates at 10:28 on October 11, 1998.
Case B. On the same day (May 3, 1998),
angina0nset happened at 9, and a normPar happened
at 12. Moreover, a saD is reported with an indeterminacy of ten seconds: it happened between 11:59:55 and
12:00:05. This case is illustrated in Figure lb. Considering for example the hours timeline, a temporal reasoning system should be able to conclude that the property holds necessarily between 9 and 12 hours. Being
unknownthe relative order of occurrence for saD and
normPar, the property could also possibly hold after
the saD event if the actual occurrence of saD is after
normPar. While the first conclusion is certain, the second one is hypothetical. In this paper, we deal with
determining necessary conclusions: this is the perspective adopted in the following sections.
ECderives the maximal validity intervals (MVIs) over
which properties hold. A MVIis maximal in the sense
that it cannot be properly contained in any other validity interval. MVIsfor a property p are obtained in
response to the query:
mholds~or(p,MVI)
A property has not to be valid at both endpoints of a
MVI: in the following, we thus adopt the convention
that time intervals are closed to the left and open to
the right.
Motivating
Examples
Representing
In this section, we present two simple clinical examples
concerning patients with cardiological pathologies. In
particular, the examples are related to the problem of
diagnosing and following up heart failure (ACC/AHA
Task Force 1995). Weconcentrate here on a small fragment of expert knowledge concerning the evaluation of
the property hfRisk, i.e. the presence of an heart failure risk whichrequires special surveillance by clinicians
in a considered cardiological patient. Amongvarious
factors, this property can be initiated by four different
events: (i) smDeltaBP: a measurement of systolic and
diastolic blood pressure (BP) on the patient reveals
abnormal difference (too small) between the two values;
(ii) saD: sudden appearance of dyspnea is detected; (iii)
angina0nset: the patient starts to experience chest
pain; and (iv) ~,m~esia0nset: the patient starts to experience loss of memory. The smDeltaBP event is acquired from a physiological measurement, saD from a
monitoring device, while the other two events are acquired from symptoms reported by the patient. Examples of events terminating the property are a measurement of relevant physiological parameters (blood pressure, heart rate, ECGparameters) with normal values
(normPar), or the administration of a specific cardiological drug (cardioDrug). Occurrences of all these
events can be given at different calendric granularities
(days, hours, minutes, or even seconds with some intensive care devices) and with indeterminacy, depending on what the patient remembers of the symptoms,
and what is the accuracy in recording data related to
therapies or to physiological measurements.
Case A. An amnesia0nset happened on October 10,
1998, a smDeltaBP happened at 11:30 on October 10,
1998, and a normPar happened at 10:28 on October 11,
1998. This case is illustrated in Figure la. Considering
for example the minutes timeline, a temporal reasoning
system should provide a single MVIthat initiates between 0:00 and 11:30 on October 10 (indeed, after 11.30
we are sure that at least one of the two initiating events
- amnesia0nset and smDeltaBP - actually happened),
54
events
and
MVIs
In TGIC, we allow the assignment of an arbitrary interval of time (at the chosen level of granularity) to
event occurrence. This interval specifies the time span
over which it is certain that the (imprecise) timepoint
is located. Moreformally, indeterminacy is represented
by generalizing the second argument of happens. We
use the clause:
happens(event,
I01)
where IOI(Interval Of Indeterminacy) is a convex interval over which event happened, and is assumed to be
closed to the left and open to the right (according to
the adopted time interval convention). For example:
happens(e, [tl ,t2]
states that the occurrence time of event e is greater or
equal than tl and lower than t2.
TGIC adopts the standard calendric granularities
(years, months, days, hours, ...), with the associated
mappings among timelines. More generally, it allows
one to use any set of granularities where mappingcontiguous timepoints from a given granularity to the finest
one results in contiguous intervals. Special granularities such as business days or business months (Bettini
et al. 1998), which do not meet the general requirement
above, are thus not considered. Weextend the definition of MVIto accommodate the more general concept
of event. A MVIhas the form:
(Start,
End)
where Start (End) denotes at a given granularity either
a specific timepoint, if it is known,or the minimalinterval over which the initiation (termination) of the property is necessarily located, in the case of indeterminacy.
.....,~,
amnesiaOnset
anginaOnset
D. DAYS
98Oct10
¯
x
98Ma)03
98MayQ3
¯
/smDeltaBP’,
normPar
¯
:
’
o- -- D. MINUTES
llh30mof 10h28m of
98Oct10 98Octl 1
normPar
~"" -- "P"HOURS
9h~0f ...... 12jh’laf
.....
¯
~
sdD
----’P"
SECONDS
[11h59m55s, 12h00m05s]
of 98May3
(a)
Figure 1: Case A (a), Case B (b).
If Start (End) is a precise instant, it is trivially the left
(right) endpoint of the delimited MVI; if Start (End)
is itself an interval, then the left (right) endpoint
the delimited MVIis the right (left) endpoint of Start
(End). Therefore, a Start (End) determines (i) the
imal interval over which the initiation (termination)
a property is necessarily located, and (ii) the timepoint
at which the property initiates (terminates) to be necessarily valid. For example, suppose to have only two
events in the database:
i. the (possibly imprecise) occurrence times of events
are mappedto the finest granularity level;
ii. an algorithm which handles partially ordered events
is applied only at the finest level to derive necessary
MVIs;
iii. the obtained results are mapped at the upper granularity levels required by the user.
Mapping to the finest
granularity
happens(el,940ctlO)
happens(e2,
TGIC provides a granularity mapping function to map
times between any pair of calendric granularities. This
is simply achieved by applying standard mappings
among calendric units: for example, hour k is mapped
into the interval of minutes [k ¯ 60, (k + 1) ¯ 60]. These
mappingsare used both for the first and the third step
of MVIderivation.
[12h of 94DeclO, 14h of 94DeclO])
and el initiates property p, while e2 terminates p. In
this case, we knowthat the initiation of the property is
necessarily located at 940ct10 and the termination is
located over the interval:
[12h of 94DeclO,14h of 94DeclO]
Werepresent this knowledge, by saying that
(940ct 10, [12hof94DeclO, 14hof94DeclO])
is a MVIfor property p. Therefore, property p is necessarily valid after 940ct10 and until 12h of 94Dec10.
For conciseness, when an event e initiates (terminates)
property p, we refer to the associated IOI as an initiating (terminating) IOI for
In the case of the first step, the function is used to
map the occurrence time of each event to the finest
granularity (in our case, the granularity of seconds).
For example, the predicate:
happens(smDeltaBP,
llh30m of 980cti0)
of Case A is mapped into the predicate:
Deriving Maximal Validity
Intervals
happens(smDeltaBP, [llh3OmOOs of 980ct10,
llh31mOOs of 980ct10])
The relative ordering of a set of events with imprecise
occurrence times and/or different granularities can be
often only partially known(e.g., in Case B, it is not
knownif saD happens before, simultaneously, or after
normPar). Since we are concerned with deriving necessary MVIs, every MVIwe derive must be valid in all
possible orders consistent with the partial order (Dean
& Boddy 1988), resulting from imprecise locations of
event occurrences. The approach we propose to derive
MVIsis organized in three general steps:
Handling
partially
ordered
events
Wedistinguish the three possible kinds of intersection
amongIOIs the algorithm has to deal with: intersection
among initiating
IOIs, among terminating IOIs, and
amongboth types of IOIs. Hereinafter, we refer to these
three different situations,
as I_ALONE,T_ALONE,
and
I_T_INTERSECT,respectively.
55
hfRisk
h fRisk
i
i
i
i
!
| o
tl
i
o
o
|
|
,
i
i
’ , ’ [
t2 t3 t4 15
ca
i
t6
|
smDeltaBP
[cardioDrug
I
i
n P
normPar
an~inaOnset ]
o
o
i
o
I
i
i
," lsmDeltaBP
!
ru
,
i
!
t8 t9 tlOtll
o
i io
i
t12
¯
tl
Figure 2: Examples of I_ALONEand T_ALONE.
a n
t2 t3 t4
o
’
oo
oo
oo
oo oo
t5 t6 t7 t8
InormPar I
t9tlO tll
t12
Figure 3: Three examples of I_T_INTERSECT.
Intersection among initiating
IOIs. Let us consider some initiating IOIs for a property p, such that
their intersection is non empty, and there are no intersections with terminating IOIs for p (I_ALONEsituation). Let us assume that there are no preceding
events, which have already initiated p, so that it is important to derive one single IOI to use as a Starz in
a MVI. The solution to the problem is given by the
interval whose left and right endpoints are the minimumleft and right endpoints, respectively, of the given
IOIs. Indeed, property p cannot initiate before the minimumleft endpoint because there are no other preceding
events whichinitiate it, but it necessarily initiates after
or at the minimumleft endpoint, and before the minimumright endpoint. The latter delimits the interval of
necessary validity for property p, because at least one
initiating event happensbefore it, and there is no lower
time that guarantees the same. For example, take the
three initiating
IOIs (anginaOnset, saD, smDeltaBP)
for property hfRisk in Figure 2. In this case, in each of
the three intervals [t 1,tS], [t2,t3], and [t4,t6], an
initiating event happened. The derived Start is thus
[tl ,t3] : the initiation of hfR£akis necessarily located
over [tl ,t3].
event can happen after or at that point. For example,
considering the three terminating IOIs (normPar and
the two card£oDrug) for property hfRisk in Figure 2,
the derived End is [tT,t9], and ([tl,t3],
[tT,t9])
is a MVIfor hfRisk.
Intersection
among initiating
and terminating
IOIs. When an initiating
and a terminating
IOI
for the same property have a non empty intersection
(I_T_INTERSECT
situation), it is impossible to establish in what order the corresponding initiating and terminating events happened. As a consequence, it is not
possible to conclude anything about the necessary effects of the initiating IOI (a Start could be derived only
if we knew that the initiating event happened after the
terminating one), while terminating IOIs maintain their
capability of being used in an End with respect to previous Starts. For example, if in Figure 3 we considered
only the three pairs of intersecting IOIs, nothing could
be concluded about the necessary validity of h:fR±sk:
there would be no derivable MVIwhich is valid in all
possible orders of the six indeterminate events. Considering all the (seven) IOIs in the database, the first IOI
is a Start for hfRisk, and the first terminating IOI is
an End. These two IOIs delimit the only necessary MVI
in the database.
Intersection
among terminating
IOIs. Let us
consider some terminating IOIs for a property p, such
that their intersection is non empty, and there are no
intersections with initiating IOIs for p (T_ALONE
situation). Let us assume that p has been initiated by preceding events and no preceding events can terminate it,
so that it is important to derive one single IOI to use as
an End in a MVI. The solution to the problem is again
given by the interval whose left and right endpoints are
the minimumleft and right endpoints, respectively, of
the given IOIs. Indeed, property p cannot terminate
before the minimumleft endpoint because there are no
other preceding events which can terminate it, but it
necessarily terminates after or at the minimumleft endpoint, and before the minimumright endpoint (when
the minimumright endpoint is reached, it is certain
that at least one terminating event has occurred). The
necessity of the validity for property p thus terminates
at the minimumleft endpoint, because a terminating
If an I_T_INTERSECT
occurs when the property does
not hold, it is possible (but not necessary) that the
property initiates to hold. In these cases, the intersecting IOIs do not allow one to generate a Start, but
they have to be considered in relation to a possible
subsequent I_ALONE
situation.
For example, consider
the events smDeltaBP
and cardioDrug
in Figure 4a:
sincetherelative
ordering
between
theiroccurrences
is unknown,
theyjustallowoneto conclude
thatthe
propertyhfRiskmighthavebeeninitiated.
Consideringalsothesubsequent
eventsaD,we can conclude
thatproperty
hfRisk
is necessarily
validaftert5,becausesaD hasno intersection
withterminating
IOIs
(I_ALONE
situation),
butto determine
theminimal
intervalforStart,we haveto consider
alsosmDeltaBP
andcardioDrug.
The initiating
eventfor hfRiskcan
be bothsmDeltaBP
and saD.The IOI of saD willbe-
56
h fRisk
!
I smoel
BpI
ism
e/,aI
!
I
I
lO
t ru
’1"7~
t
Pea g! ’
!
I
t
I
I
tl
t2
P
~
I
t
i
i
_~diI o D~u ~ ’ ,,I
I
!
,
t
I
t3 t4 t5
I
OO
I I
tl t2 t3 t4 t5
%.
Ext
r
% Ext r
(b)
(a)
Figure 4: Example of convex (a) and non-convex (b) Start.
long completely to Start, while only a part of IOI of
smDeltaBP will be included in Start, because when
smDeltaBPis the initiating event for the necessary instance of hfRisk, it is impossible for the occurrence of
smDeltaBP to be located before the IOI of cardioDrug
(the instance of hfRisk initiated by smDeltaBP would
be terminated by cardioDrug). Therefore, the Start
for hfRisk is given by IOI [tl,t5]: the initiation for
the property is necessarily located in that interval. In
general, we call Ext the part of Start, which is produced by considering the I_T_INTERSECTsituation
preceding the I_ALONE
one. The Ext interval in Figure 4a is [tl,t4].
In the general case, a Start needs
not to be necessarily convex. This is exemplified in Figure 4b, where, as in the situation of Figure 4a, we can
conclude that property hfRisk is necessarily valid after
t5, and then we have to similarly consider smDeltaBP
and cardioDrug to determine Start. Unlike the case
of Figure 4a, smDeltaBPand saD currently do not overlap. Therefore, the Start for hfRisk is given by a
non convex IOI which comprises [tl,t3] and [t4,tS] :
property hfRisk is necessarily valid after t5, and it has
been necessarily initiated over the non convex interval
[[tl,t3],
[t4,t5]],
which is the union of [tl,t3]
(the Ext interval, obtained from I_T_INTERSECT)
and
[t4,t5] (the interval obtained from I_ALONE).
erty hfRisk at the finest level of granularity.
The
MVI returned
by the query mholds_2or(hfRisk,
MVI, minutes) is ([00h00m of 980ct10, llh31m
of 980ct10],
10h28m of 980ct11)
The query:
mholds_~or(hfRisk,
MVI, days) returns
the MVI
<980ct10, 980ct11) as solution.
Finally,
TGIC
provides the predicate:
msg_mholds_2or(p, (Start
,End))
which tries to automatically choose the most suitable
granularity level for Start and End. The most suitable
granularity for a Start (End) of an MVIis chosen as follows: starting from the Start (End) derived at the finest
level, we moveto the coarser levels by using granularity
mappingpredicates, stopping at the first level where the
Start (End) reduces to a time point. For example, for
Case A, the query: msg_mholds_for(hfRisk, MVI) returns the MVI (980ct10, 10h28m of 980ct11) as solution. Considering Case B, the latter query returns:
(09h of 98May03,12h of 98May03)
Related Work
To the best of our knowledge, the only other approach
to deal with temporal granularity in EC has been proposed by Montanari et al. and is described in (Montanari et al. 1992): hereinafter, we refer to this approach as MMCR
(the initials of its authors’ surnames).
MMCR
introduces temporal granularity
in EC by operating the following extensions.
Answering queries at multiple
granularities
TGIC extends the query predicate
mholds_for, to
allow one to perform queries at different levels of
granularity.
The predicate:
mholds_for(p,IStart,
End), granlev) returns the MVIs for property p
the granularity
granlev. For example, in Case A
(Figure la), TGIC derives the MVI: ([00h00m00s
of 980ct10, llh31m00s of 980ct10], [10h28m00s
of 980ct11, 10h29m00s of 980ct11]) for the prop-
Multiple timelines. The single EC timeline is extended into a totally ordered set of different timelines
{T1, ¯ ¯ ¯ , Tn}, such that each timeline Ti is of finer granularity than the previous timeline Ti-1. For example, one could have three timelines corresponding to
hours, minutes, and seconds, respectively. In general,
57
the finest timeline is possibly allowed to be continuous,
while the other ones must necessarily be discrete.
is inevitably plagued by a scarce performance, due to
a generate-and-test strategy which considers any possible pair of events as a possible candidate for deriving
an MVI. This problem is exacerbated in MMCR,because each event has also a counterpart on each of the
timelines.
Granularity mapping predicates.
New predicates
(fine_grain_of, coarse_grain_of) are introduced into the
formalism in order to map the occurrence of events from
one timeline to the other ones. For example, an event
that happened at 2 hrs is mapped at the finer granularity of minutes into the time interval [120 min, 179
min]; an event that happened at 170 min is mapped at
the coarser granularity of hours into the time point 2
hrs.
To highlight differences between TGICand MMCR,
let
us consider how MMCR
would manage the two examples we previously provided. In Case A, MMCR
returns
three MVIsfor property hfRisk: (i) an MVIthat initiates at 11:30 on October 10, and terminates at 10:28
on October 11, 1998; (ii) an MVIthat initiates between
0:00 and 23:59 on October 10, and terminates at 10:28
on October 11, 1998; and (iii) an MVIthat initiates
October 10 and terminates on October 11, 1998. Solution (i) is not satisfying, because the MVImight have
possibly started before 11:30, due to the ~m-esiaOnset
event, while solution (ii) is too vague, because it comprises also a large numberof time points following 11:30
as possible starting points for the MVI, although the
property is certainly valid after 11:30. Instead of producing these two solutions, a temporal reasoning system should provide a single MVIthat initiates between
0:00 and 11:30 on October 10, and terminates at 10:28
on October 11, 1998. Solution (iii) is satisfying at the
granularity of days.
Reasoning strategy. Derivation of MVIs is extended
in order to encompass the presence of events at multiple time scales. In particular, MMCR
first mapsall the
available events at all the different granularity levels,
then it exploits the standard EC derivation of MVIson
each of the levels in isolation. The possibility of having contradicting events happening at the same time
point as a result of mapping events to coarser time
scales is prohibited by means of explicit integrity constraints. Moreover, since the derivation mechanismexpects events to happen at time points, the two endpoints of intervals obtained by mapping the occurrence
of an event e to a finer level of granularity are treated as
two endpoints for an interval over which e has to occur
(temporal indeterminacy).
In Case B, MMCR
is unable to handle the saD event,
because it is not possible to have temporal indeterminacy unrelated to the mappings amonggranularity levels. But, even assuming that indeterminacy were handled in general, a second major problem would arise,
i.e., the interval of indeterminacyfor the initiating event
saD has a non empty intersection with the time of occurrence of the terminating event normPar. As previously
discussed, MMCR
rules out contradicting events which
might possibly happen at the same time point. In this
situation, a temporal reasoning system should be able
to conclude that the property holds necessarily between
9 and 12 hours.
We have tested MMCR
on our clinical
domain, and
found a number of limitations which can be classified
as follows.
Completeness issues.
MMCR
is often not able to
derive the expected MVIs, even for some simple cases.
Situations affected by these problems typically involve
the presence of intersecting intervals of indeterminacy
for different events which have been mapped at finer
granularities.
MMCR
is not able to deal with them
properly, as we show below.
Expressiveness issues. MMCR
expects one to assign
a precise time point on one of the timelines to the occurrence of an event. This is often not possible in real
applications where the information available is not so
precisely localizable on any available time scale. Events
like "between 17:30 and 18:15 the patient had a myocardial infarction", or "between 10:23:05 and 10"23:15, a
patient monitoring device detected dyspnea" cannot be
handled by MMCR.Moreover, as a result of deriving
MVIsin isolation on each timeline, MMCR
is not able
to return MVIswith starting and ending times given at
different granularities.
Conclusions
As we have shown in the previous Section, the usefulness of the only other approach dealing with temporal
granularity in EC (Montanari et al. 1992), is limited
by three serious factors: (i) the possibility of having
I_T_INTERSECT
situations is prohibited by explicit integrity constraints (that approach is thus often unable
to derive the expected MVIs, e.g. in the cases represented by Figures lb, 3, 4); (ii) the user can associate
to an event only a single timepoint on a chosen timeline (indeterminacy is not allowed); and (iii) the implementation is given only in terms of a declarative logic
program. Chittaro and Montanari have shown how
declarative implementations of EChave significant per-
Efilciency issues. The implementation of MMCR
is
given in terms of a declarative logic program. Chittaro
and Montanari in (Chittaro & Montanari 1996) have
shown how a declarative implementation of basic EC
58
formance limitations, due to a generate-and-test strategy (Chittaro ~: Montanari 1996). TGICovercomes all
the above described limitations: it allows and handles
I_T_INTERSECT
situations; it allows the user to associate any interval of indeterminacy to any event; and an
efficient procedural implementation has been provided
in (Chittaro & Combi 1999).
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