From: AAAI Technical Report WS-02-17. Compilation copyright © 2002, AAAI (www.aaai.org). All rights reserved.
Direct Conversions among Multi-Granularity
Temporal Constraints
Claudio Bettini and Simone Ruffini
DSI - Università di Milano, Milan, Italy
bettini@dsi.unimi.it
s.ruffini@infinito.it
Abstract
Temporal constraints are part of the specification of
many complex application frameworks including planning,
scheduling, real-time systems, workflow and cooperative systems. This paper investigates the problem of converting
among quantitative temporal constraints expressed in terms
of different time granularities. An expressive formal model
for time granularities is assumed including common granularities like hours and days as well as user-defined granularities
like business-days and academic semesters.
Keywords: temporal constraints, time granularity, temporal abstraction.
Introduction
Temporal constraints are part of the specification of many
complex application frameworks including activities to be
scheduled, real-time components, temporal data management, and workflow management. Integrated e-commerce
applications are an example of such frameworks where an
advanced and efficient management of temporal aspects may
be a significant advantage over the competition. In these
complex systems the temporal relationships between events
and activities involving different components are often described in terms of different time units. For example, the
time between the receipt of an online order and the forwarding of item orders to the warehouses is described in minutes,
while the time between the shipment of the order and the
delivery is described in business-days.
This paper considers a formalization of the notion of temporal constraint with granularity and investigates the problem of converting constraints in terms of different granularities. This may be desirable for at least two reasons: a)
there are well-known algorithms to process temporal constraint networks where all the constraints are in terms of the
same granularity (DMP91), and b) it may be useful for the
sake of application process monitoring to view all the constraints in terms of a particular granularity.
It will be immediately clear to the reader that the conversion problem is not a trivial one, even if we only consider
common granularities like hours and days. For example, by
saying that one event should occur the next day with respect
to another one, we do not mean that the event should occur
24 hours after the other. Nor it is satisfactory to allow a range
between 1 and 47 hours, as it may be suggested. Indeed 1
and 47 hours are actually the minimum and maximum distance between the occurrence of two events if they occur in
consecutive days. However, a distance of 47 hours may also
be the distance between two events whose occurrence is 2
days apart. This example actually says that an exact conversion does not exist, i.e., the constraint we obtain is not
equivalent to the one we take as input.
Hence, our goal is, given a constraint in terms of a source
granularity , and given a target granularity , to identify
which is the tightest among
the constraint in terms of
those implied by the input constraint or a good approximation of it. Being the tightest means that the set of distances
allowed by the constraint is contained by any other logically
represents the constraint imimpled constraint. If
posing a distance of at least and at most time units of ,
hour is the tightest among
then in the above example
day. Inthe constraints in terms of hour implied by
hour are logdeed, other constraints, as for example,
day, but none of them can exclude
ically implied by
distances between and .
The concept of granularity as an abstraction tool has been
deeply investigated in the AI and DB literature probably
starting from (Hob85). The formalization of time granularity as adopted in this paper has been defined in (BWJ98)
and used in several papers on the subject. Interesting recent work on this topic can also be found in (MPP99;
GL+01; BWJ02). Symbolic formalisms to represent time
granularities have been proposed among others in (LMF86;
CSS94). The complex mathematical relations among granularities and calendars have been studied also in (DR90).
There are few approaches considering temporal constraints
in terms of different granularities (e.g., (Dea89; GL+01)),
but in most cases they are restricted to common granularities. Our goal is to admit powerful constraints and very general user-defined granularities both to accurately model the
new application domains and to select appropriate abstraction levels. A first comprehensive solution to this problem
has been given in (BWJ98) proposing two alternative algorithms for constraint conversion. The main contribution of
this paper with respect to (BWJ98) is the construction of a
set of formulas based on granularity relationships enabling
an efficient implementation of the direct conversion algorithm for a significant set of periodical granularities. As a
secondary contribution, we fix a bug in the original algorithm proposed in (BWJ98) for the case of the constraint
minimum distance equal to zero. A prototype system has
been developed at the University of Milan and used for extensive testing.
The results of this paper can be applied in several application areas where multi-granularity temporal constraints
have been shown to be useful: data mining (BW+98), workflow management (BWJ02), and clinical data management
(CC99) among others.
The rest of the paper is organized as follows. In the next
section we introduce the formal notions of granularity and
temporal constraints with granularities. In Section we illustrate a general algorithm for granularity conversion, while in
Section we show how the algorithm can be implemented exploiting the periodicity of granularities. In Section we give
an example of constraint network conversion, and Section
concludes the paper. For lack of space, proofs are omitted,
but can be found in (BR02).
Temporal Constraint Networks with
Granularities
We first define a granularity system, called (General
Granularities on Reals) in (BWJ98), as the set of granularities satisfying the following definition.
set
of
the
Definition. A granularity is a mapping from the
of
reals)
such
that
integers to (i.e., all subsets
for each non-positive
integer , and for all positive integers
, the following two conditions are satisfied:
and with
and imply thateach
real number in
is less than all real numbers in , and
implies .
This is a very general notion of granularity modeling standard ones like hour, day, week and month as well as
more specific ones like academic semester, b-day
(business day), or b-week (business week), and arbitrarily
user-defined granularities. For example, b-week may be
defined by the mapping of each positive integer (index) to a
set of elements of the temporal domain denoting the period
from a Monday through the next Friday every week. Each
set of these sets of elements is called a granule of the granularity. If we decide to model time starting from Monday
2001/1/1, then the index is mapped to the instants denoting
the period from 2001/1/1 through 2001/1/5, forming the first
granule b-week(1), the index is mapped to b-week(2),
denoting 2001/1/8 through 2001/1/12, and so on.
In order to have finite representations of the granularities
suitable to be automatically manipulated, we further restrict
the granularities to those whose granules can be defined as a
periodical pattern with respect to the granules of a fixed bottom granularity. For this purpose we first need to introduce
a granularity relationship.
Definition. A granularity groups periodically into a granularity if
, there exists a (possi1. for each non-empty granule
of
positive
integers
such that
bly
infinite)
set
, and
2. there exist ! #"%$&(' , where ! is less than
the number of granules of , such that for all
$)
+, .- / 43 (5
+43 &*' , if
#
/
2
0
1
and
then
!
!
- / 3 /6071 " .
is the union of
Condition 1 says that any granule
;
for
instance,
assume
some granules of
+8:9; +8=<>
A8 - it is the union
@?;?;?
. The periodicity
of the granules
property
(condition 2)
Densures
3 EF
that
if the !B+C granule after
+8G9Hexists
3 (i.e.,
+8=<(3 ! +8 - ), 3 then it is the union of
"
" ;?@?;?
" . This results in a
periodic “pattern” of the composition of ! granules of in
terms of granules of . The pattern repeats along the time
domain by “shifting” each granule of by " granules of .
The integer " is called the period. Many common granularities are in this kind of relationship, for example, both days
and months group periodically into years. In general, this
relationship guarantees that granularity can be finitely described providing the specification of granules of in terms
of granules of in an arbitrary period and the period value.
For example, b-week can be described by the five business
days in the first week of the time domain, and by the value
for the period. A granularity 1 is called bottom granularity
if 1 groups periodically into each other granularity in the
system.
We can now define a temporal constraint with granularity.
3
I$J&LKNM OQP
PSR with FT and
Definition. Let
a granularity. Then
, called a temporal constraint
with granularity (TCG), is the binary relation on 9 positive
<
integers
96W>X U and<YW>U X ,
9 <@ defined as follows: For positive integers
U #U satisfies
if and only
<YW>X if (1) 96V
W\U XD and V
U
V
U
O[V
U
T .
are both defined, and (2) ZT
WX
function
The V^]
returns the index of the granule of
that includes 1 ] , where 1 is the bottom
In<
9 granularity.
satisfies the
tuitively, to check if a pair of instants U _U
TCG
9 , we < derive the indexes of the granules of
containing U and U , respectively, and then we take the difference. If it is at least and at most , then the pair of instants
9
< For example, the pair
9 <@ is said to satisfy the constraint.
U #U satisfies 9 >`<@b adc if U and U are within
9 the same
<
if U and U are
day. Similarly, U _U satisfies eO ;f:gdhbi
at most 9 one< hour apart (and their order
< is immaterial). FiY
k
j
d
g
b
l
4
m
f
if
U
is in the next month
nally, U _U satisfies
9
with respect to U .
Definition. A constraint
network
An
(with n granularities) is a
directed graph nw6vL
o 6n p #qsr , where is a finite set of
a set of arcs, p is a mapping from
variables, out
o to the finite sets of temporal constraints
with granularin
is a mapping from
to a possibly bounded
ties, and qsr
periodical set of positive integers.
A set of positive integers is said to be periodical if there
exists a granularity such that thebottom
x yW>X granularity periis defined R . The
odically groups into and M V
set is bounded if an integer z is given such that each value
in the set must be less than z .
Intuitively, a constraint network specifies n a complex temrepresents a
poral relationship where each variable in
specific instant (for example the occurrence time of an event)
X1
[0,0]b-day
[2,2]b-week
[-1,1]b-day
X0
X
X3
[0,0]day
[0,0]b-day
[0,3]day
[1,6]b-day
X2
Z
Figure 1: A constraint network with granularities
Figure 2: A network to illustrate conversion conditions
in terms of the bottom granularity. The domain of each variable is essentially a set of granules of the bottom granularity, represented through the positive integers corresponding
to the granule indexes. The set of TCGs assigned to an edge
is taken as conjunction.
That
is, for each TCG in the set as, the instants assigned to
and
signed to the edge
must satisfy the TCG. Fig. 1 shows an example of a constraint network withN
granularities
,
with no explicit constraint
on domains (q r
P for each variable ).
Conversion of Constraints in Different
Granularities
As we have pointed out in the introduction, in general, given
a TCG, it does not exists an equivalent one in terms of a
different granularity.
9 However, we9 are interested in convertinto a logically implied
ing a< given TCG in < terms of
. Note that a TCG o logically imTCG in terms of
plies a TCG if any pair of time instants satisfying o also
satisfy . Among all the logically implied TCGs we prefer the tightest or a good approximation of it. For example,
and @f:ghbi are logically implied by
both@fGgdhbi
\`ba c
, with the first being preferable since, in this case it
is the tightest, and, indeed, it provides a more precise conversion of the original one. If we only have a total order of granularities with uniform granules, like e.g., minute, hour,
and day, then the conversion algorithm is trivial since fixed
conversion factors can be used. However, if incomparable
granularities like week and month, or granularities with
non-contiguous granules like b-day and b-week are considered, the conversion becomes more9 complex.
Moreover, given an arbitrary TCG , and a granularity < ,
it is not always possible to find a logically implied TCG
in terms of . For example,
\`ba c does not logically
- `=a c no matter what
and are. The reason
imply
is that ^`=a c is satisfied by any two events that happen
during the same day, whether the day is a business day or a
weekend day.
Y
Allowed conversions
In our framework, we allow the conversion of a TCG of a
constraint network into another TCG if the resulting constraint is implied by the set of all the TCGs
in . More
on arc
in a network is
specifically, a TCG
as long as allowed to be converted into is implied by , i.e., for any pair of values
U
and U assatisfies
signed to and respectively, if U _U
also
and U and U belong to a solution of , then U _U
. To guarantee that only allowed conversatisfies sions are performed, we assign to each variable of a constraint network, a periodical set which is the intersection
of the domain of and all the periodical sets defined by the
granularities appearing in TCGs involving . Then, a TCG
on variables and can be converted in terms of a target
is contained in a
granularity if each element in JK
granule of . It is easily seen that this condition guarantees
the existence of an allowed conversion; indeed any pair of
and respectively which
values U and U assigned to
(we only excluded
is part of a solution must be in NK
values which could not satisfy TCGs), and if both values are
contained in granules of , their distance in terms of can
be evaluated.
Example
1 Consider the constraint network in Fig. 2. Both
and
are b-day. Hence, the constraint \`ba c
can be converted in terms of b-day since the target granularity b-day
covers a span of time equal to that covered by
and
. Similarly, consider the network in Fig. 1. The
TCG in terms of day cannot be converted in terms of bday or b-week without considering the other constraints
in the network. However, both and evaluate to
the granularity obtained from day by dropping all granules
that are not a business day. Since both b-day and b-week
cover the same span of time as this granularity, the constraint \`ba c can be converted in terms of b-day and
- , respecb-week, obtaining -`ba c and tively.
The conversion algorithm
In Fig. 3 we propose a general
! conversion
y
8 method.
# A It is
based on the #functions
< U and ]" U . Intu 9
denotes the< minimal distance
itively,
U
(in terms of the number of granules
of ) between all pairs
9
and
of instants in a granule of
in# the
th granule after
it,
respectively. For example,
U -$% 6`=a c
,
i.e., the minimum distance in terms of days between two
events that occur in two different business weeks is 3 (one
instant on Friday and the other on Monday). The definition
restricts the considered pairs of instants to those in which the
first instant is included in and the second in . Indeed,
only these
& pairs are candidate solutions for the constraint on
, and this restriction improves the precision
the arc
of the conversion
some inconsistencies.
9 as it< detects
8 #as well
]' U
is the corresponding maxiThe value
9
3
$S&Q'sW>K X M PSR ,
.
V^U
INPUT: a network with a TCG
, with < %$S&JK M4OQP R and
; a target granularity , s.t. U UD$
associated with an arc
K
<
for
or undefined.
OUTPUT: a logically implied TCG
METHOD:
OQP then OQ#P 9
Step 1 if
<
else if then
8 # 9 x x U < else
O 3 ]' U 3
5
Step 2 if
< P
8 P # then
9
else
]" U
Step 3 if either or < is undefined, then return undefined
else return
where
U 9 < < W>,X M V
U
8 9
] U <\< ,W\X !
M V^U
L
q r
'
#
#
9 W> X O ^V U
8]
O[V
U
9
@?;?;?
granularities in the TCGs of , undefined
otherwise,
< W\X where 9 W\
<
X O V^U
R;
U $ _U $ , and V
U
, undefined
otherwise,
< W>X where
O[V
U 9 W>X T R ;
9 W\! X ifx ! 9
<
U $ #U $ , and V^U
9
if ;?@?;? - are the periodical sets corresponding to the
ifx 9
involving node
. (Similarly for
.)
Figure 3: A general method for the conversion of constraints
8
# mum distance.
8 # For example, ]' U -$% 6`=a c
]" U (b-day, 1,day)= 3.
, and
The values of these functions cannot be automatically obtained for general (infinite) granularities, however, they can
be computed efficiently when the involved granularities are
periodic,
cases. The computation of
# Aas in most
8 practical
# A
U and ]' U is more involved when we want
to9 check the condition,
appearing in their specification, i.e.,
<
U $ and U $ . Note that omitting this check leads
to a still sound but less precise conversion. The other condition involving and appears in the input description.
This cannot be ignored since it guarantees an allowed conversion.
3
$ &(' if
P for the inThe method imposes
put constraint. This
is
not
a
limitation
since
any
constraint
eO ;O
on
with
can
be
expressed
as
L
on . When only the lower bound is negative,
it is sufficient to consider its absolute value and treat it exactly as the upper bound except for reversing the sign of the
result. For example, for O
% we derive the upper
bound for day according to the method. Since the absoeO \`ba c as the implied
lute value for O is , we derive
must be treated exactly as for
constraint. The case of
negative values.
3
When
OQP and/or
P the conversion
method simply keeps these constants also for the TCG in
the target granularity. It is easily seen that
.9 this always
is bounded,
leads to an implied TCG. However, when
more
precisely
into fithe P constants may be converted
3
P j gdlbm f -
, where
nite values. For example,
month-b2000 denotes
the months3 before year 2000, is
3
P \c =a i , while P in this last TCG
converted into
may be safely substituted with the number of years from the
beginning of the time line and year 2000. This refinement is
left as an optimization which may be useful in implementing
the method.
Theorem 1 The general conversion method is correct: Any
constraint obtained as output is implied by the original network.
Implementation
In this section, we show how the conversion algorithm can
be implemented. Essentially, this can be done by exploiting
the periodicity of granularities and variable domains. First
of all, we need to ensure that the conversion is allowed, as
required by the condition specified on the INPUT in Fig. 3.
Secondly, we
need
# y to compute
8 #theA values of the distance
functions
U and ]" U for arbitrary parameters.
Excluding illegal conversions
From Section , we know that the conversion of a constraint
<
is alon the distance from to with target granularity
<
. If we consider
lowed
if each element in K
is in
<
< as the
periodical set induced
it is easily seen
that
< by ,and
<
.
this condition holds if Intersection of periodical sets can be easily implemented by
considering their common period, and similarly can be implemented equivalence. (See (BR02) for details.)
Computation of distance functions
# A
8
# A
The computation of
U and ]' U following
the formulas in their definition may be computationally very
inefficient. Our choice is to devise an efficient computa 9 and
tion
which return values equivalent to
8 method
] ! < respectively, assuming that the conditions U $
, U
$ , appearing in the definitions of the distance
Origin
day
Destination
hour
week
hour
bday
hour
bweek
hour
day
bday
day
bweek
week
day
day
week
year
day
year
month
month
year
bweek
bday
bday
bweek
bweek
week
bday
day
bweek
day
Computation of mindist(), maxdist()
23
mindist(day, k, hour) = O 3 maxdist(day, k, hour) = O 23
mindist(week, k, hour) =
O 3 maxdist(week, k, hour) = 23 O
3
mindist(bday, k, hour) = O
3 23 _ DIV
3 maxdist(bday, k, hour) = DIV
23
O
mindist(bweek, k, hour) =
3 maxdist(week, k, hour) = 3 O 3_
mindist(day, k, bday) = if MOD
_ 3 then O
else
3 O HO
DIV _
maxdist(day, k, bday) = if 3 MOD
_ 3 then O
DIV
else
3 HO
mindist(day, k, bweek) = 3 DIV
maxdist(day, k, bweek) = DIV
3
mindist(week, k, day) = 3 O
maxdist(week, k, day) = mindist(day, k, week) = DIV
3 maxdist(day, k, week) =
DIV23 35_
mindist(year, k, day) =
%
O DIV
O 3 23
maxdist(year, k, day) = 23 DIV
mindist(year, k, month) = \ O 3 O
maxdist(year, k, month) = \
\
mindist(month, k, year) = DIV
3
maxdist(month, k, year) =
DIV
3 \
mindist(bweek, k, bday) = 3 O
maxdist(bweek, k, bday) = mindist(bday, k, bweek) = DIV
3 maxdist(bday, k, bweek) = DIV mindist(bweek, k, week) = maxdist(bweek, k, week) =3
_
mindist(bday, k, day) = 3 3# DIV
3 #
maxdist(bday, k, day) =
DIV 23
mindist(bweek, k, day) = 3 O
maxdist(bweek, k, day) = functions, are always satisfied. Note that, if this is not the
case, we derive a constraint which is looser than the optimal
constraint, but still logically implied by the input constraint
network. Based on extensive experiments we have seen
that the resulting approximation is very good, and hence the
trade-off between efficiency and precision favors our solution, except for special critical applications.
The method we have implemented is based on a table lookup for each pair of source and target granularities.
# G
and
Each
a formula for
U
8 table
# : entry contains
]' U having as the only variable. Intuitively, each
formula is based on the relationship between the structures
of the two granularities. In some cases the formulas are trivial, but this is not true
For
in order to
in! general.
example,
8
r 7U , it is sufficient
derive the formula U
to consider that each year is made of 12 months; indeed that
formula is DIV \ .
# 8 ,
Less intuitive is how to evaluate U :r
since we have to take into account noncontiguous granules.
# y U
3
O
DIV DIV
Table 1: Computation of
and
8
]"
! y U
In Table 1 we report some of the formulas involving a quite
rich set of granularities.
There are pairs of source and target granularities for
which is particularly complex to derive a direct formula. This happens in particular when the granules of
the source granularity have different sizes in terms of the
target
!granularity,
as 8 for example, in the computation of
U r 7U . Our choice in this case is to precompute the value of the distance functions for some values of (less than the number of granules of the source
granularity in the common period of the# two
granularities).
8 U
r 7U . For example,
in
order
to
compute
8 # 8 ]" U r 7U , we precompute them for
and
based on their definition, and obtain all other val;?@?;?
ues by the formulas in Table 2.
A library of formulas for common granularities can definitely speed up the conversion process. For cases in which
formulas do not exist yet, and are not trivial to derive, we
may consider an alternative conversion algorithm based on
8 #3
r 7U 2MOD
35_ >3 8 DIV > 8 DIV
63 U r 7U MOD \
maxdist(month, k, day) = ]
DIV >
235_ 3 DIV 8 8 8 U r 7 U and ] U r 7 U Table 2: Computation of Origin
month
Destination
day
Computation of mindist(), maxdist()
# mindist(month, k, day) = U "
!
the relationship of each of the source and target granularities
with the bottom granularity. A preliminary version of such
an algorithm was illustrated in (BWJ98), and a refinement
is proposed in the extended version of this paper (BR02);
However, that method introduces an additional source of approximation.
Example 2 Suppose we want to derive from the network in
Fig. 1 an implied network in terms
& 9 @of business
< days.
@ This
and
must
means the constraints on arcs
be converted. Considering -$#% , the lower bound
is 2, hence we must compute
U -3 % #6 -`ba c .
Looking at Table 1, this value 8 is # O
. For
'
]
U
- % 6
b
`
a
c
the lower
bound,
we
have
3 . Accordingly to the algorithm, we obtain the
new TCG -`ba c . In Example 1, we have seen that,
< @
considering the whole network, the TCG on arc
can be converted in terms of b-day and b-week, despite
its granularity is day. Its conversion accordingly to the algorithm and Table 1 is -`ba c , and the resulting implied
network in terms of b-day is shown in Fig. 4.
X1
[6,14]b−day
[−1,1]b−day
X0
X3
[0,3]b−day
[1,6]b−day
X2
Figure 4: Implied network in terms of b-day
Similarly, we can obtain an implied network in terms of
b-week shown in Fig. 5. In this case,
1 the9 constraints
1 < ,
to be& converted
are those on arcs
< ;
and
. Accordingly to the algorithm, we ob -% , 6
- , and - % , respectain eO
tively.
"
!
X1
[−1,1]b−week
[2,2]b−week
X0
X3
[0,1]b−week
X2
Given a constraint network, we can derive a logically implied network in terms of a target granularity by applying
the conversion algorithm to each constraint for which a conversion into is allowed. The following example shows the
derivation of an implied network following the steps of the
algorithm and the implementation techniques illustrated in
the previous section.
#
[0,2]b−week
Application of direct conversions
Figure 5: Implied network in terms of b-week
Conclusion
In this paper we proposed an algorithm and implementation techniques for the conversion of temporal constraints
in terms of different granularities. By applying the algorithm, it is possible to derive a new constraint network in
terms of a single target granularity which is implied by the
original set of constraints. The conversion algorithm can
also be integrated with arc and path consistency techniques
for constraint propagation (DMP91; BWJ02), leading to the
derivation of implicit constraints, and to the refinement of
existing ones.
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