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Group Title: Department of Computer and Information Science and Engineering Technical Reports
Title: Resolution of time concepts in temporal databases
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Title: Resolution of time concepts in temporal databases
Series Title: Department of Computer and Information Science and Engineering Technical Report ; 93-004
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Language: English
Creator: Chakravarthy, Sharma
Kim, Seung-Kyum
Publisher: Department of Computer and Information Sciences, University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: January, 1993
Copyright Date: 1993
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Full Text















University of Florida
Computer and Information Sciences


.. -* Department of Computer and Information Sciences
,_ Computer Science Engineering Building
SUniversity of Florida
Gainesville, Florida 32611


Resolution of Time Concepts in
Temporal Databases

Sharma Chakravarthy
Seung-Kyum Kim
email: sharma@snapper.cis.ufl.edu

Tech. Report UF-CIS-TR-93-004
January 1993
(Submitted for publication)










Contents

1 Introduction 2

2 Analysis of previous work 3

3 Temporal validity 5
3.1 Equi-distant discrete time ................. . . . . . ... 5
3.2 A simple data model ................... . . . . . ... 6
3.3 Definition of temporal validity ............... . . . . . .... 8
3.4 Sort of validity . . . . . . . . . . . . . . ... . . . . 10

4 Preservation of multiple past states 12
4.1 Event tim e . . . . . . . . . . . . . . .. . . . . . 13
4.1.1 Time difference between an event and its accompanying facts . . . ... 13
4.1.2 Classification of events .............. . . . . . ...... 13
4.1.3 Multiple pasts generated by the time difference . . . . . . . 15
4.2 Transaction time ................... . . . . . ...... 16
4.2.1 Error corrections ................. . . . . . ...... 16
4.2.2 Delayed updates ................. . . . . . ...... 16
4.2.3 Consolidation of error correction and delayed update . . . . . .... 18
4.3 Orthogonality of event time and transaction time ... . . . . . . 19
4.4 Classification and maximal set of time concepts ...... . . . . .... 21

5 One-dimensional interpretation 23
5.1 History of an object ................... . . . . . ...... 23
5.1.1 Special time points in history .................. ...... .. 24
5.1.2 Lifespan ................... ......... ........ . 25
5.1.3 Snapshot ................... . . . . . ...... 26
5.2 Effect of data manipulation on special time points . . . . . . ..... 26
5.3 Construction of history .................. . . . . . ...... 28

6 Language and Implementation issues 29
6.1 Need for a restriction on history ............. . . . . . ...... 29
6.2 Attribute time-stamped relations ............ . . . . . ...... 29
6.3 Temporal query languages . . . . . . . . . . . . . 31

7 Conclusions 32














Resolution of Time Concepts in Temporal Databases*


Sharma Chakravarthy Seung-Kyum Kim

Database Systems Research and Development Center
Department of Computer and Information Sciences
University of Florida, Gainesville, FL 32611
e-mail: {sharma, skk}@cis.ufl.edu

January 14, 1993




Abstract
In this paper, we resolve the confusion among different time concepts proposed for tempo-
ral databases and demonstrate the adequacy of three well-defined time concepts required for
preserving multiple past states of a database.
We start with a formal definition of temporal validity which we believe is the first and the
most comprehensive quantification of the notion of temporal validity. The notion of temporal
validity is extended to sort of validity with which the confusion among various time concepts
introduced earlier for temporal databases can be dispelled. Then, we discuss the problem of
preserving multiple past states of a temporal database, which leads to the identification of a
maximal set of time concepts and the need for multidimensional representation. It is shown that
the time concept, event time is needed to properly model retroactive and proactive updates, as
it is not possible to model them using only the valid and transaction times as thought earlier.
We also show the adequacy of three time concepts (event time, along with valid and transaction
time) for completely preserving different past states generated by retroactive and proactive
updates, error corrections, and delayed updates. In addition, we define the evolution of an
object (i.e., object's history) along one-dimensional time (valid time), by using the concepts of
temporal validity and sort of validity. We conclude by sketching the implementation of history
for the relational data model.

Index Terms: Temporal databases, valid, event, and transaction times, multiple past states,
retro- and pro-active updates, error correction, delayed update, history of an object.













*This work was supported in part by the NSF Research Initiation Grant (IRI-9011216) and by a Grant from the
Florida High Technology and Industrial Council (UPN# 900911013).










1 Introduction


The temporal database which incorporates time concept into the conventional database has been
investigated by many researchers over the past decade [27]. The most predominant direction of the
previous research has been the extension of the relational data model to a temporal one and/or
the extension of the relational query languages SQL and Quel to temporal versions. Work in this
area includes [4, 2, 25, 12, 19, 20, 21]. Besides the emphasis on the relational model, temporal
extensions to the ER and semantic models have begun to emerge in the literature [17, 9, 21, 29].
The formalization of temporal data model [22, 5, 30, 11] and the physical data organization for
temporal databases [18, 23, 1, 10, 8] have also been investigated.
In our view, earlier attempts at clarifying various time concepts introduced for enriching the
semantics of temporal databases have only added more confusion. Early on, Snodgrass and Ahn
[26] proposed three time concepts, valid time, transaction time, and user-defined time, based on the
classification of databases into rollback databases, historical databases, and temporal databases.
However, the confusion concerning time concepts still persists and leads to the following problems:

1. Classification of time concepts into a dichotomy of reality vs. representation (or real-world
vs. database-world).

2. Employing inappropriate time concepts to capture retroactive and proactive updates.

In this paper, we develop the notions of temporal validity and sort of validity to clarify the first
problem. With these notions we are able to infer that there is no absolute real-world validity but
everything is interpretation-dependent. Also we show that there may be multiple interpretations
for temporal behavior (validity) of a class of data objects. This implies that the notion of valid
time, irrespective of its name, is enough only as far as the validity is concerned. Using the temporal
validity and sort of validity, we define the notion of history, being a representation of evolution of
an object along one-dimensional time.
As another aspect of time concept, we show that valid and transaction times are not sufficient
to model retroactive and proactive updates (the second problem), while they are still adequate for
error corrections and delayed updates. In fact, the event time bearing a revised meaning is required
to capture, in conjunction with the valid time, the precise meaning of retroactive and proactive
updates. We discuss this issue in the context of preservation of multiple past states and argue that
the preservation problem is independent of the temporal validity.
Combining the previous arguments, we conclude that only the valid time is needed for a single
sort of validity (one interpretation) and that multiple valid times can be employed if multiple
interpretations of temporal validity are required. Apart from this, the valid time needs to be
augmented to two- or three-dimensional time by incorporating the event time and/or the transaction










time to preserve multiple past states of databases. In summary, the valid, event, and transaction
times are practically a maximal set of times needed to completely model the temporal database.


The rest of this paper is organized as follows. In section 2, we give a critical review of previous
work dealing with time notions. In section 3, the notions of temporal validity and sort of validity
are presented. In section 4, the problem of preserving multiple past states is discussed. In section 5
and 6, we show one-dimensional interpretation of temporal databases based upon the notion of
history. Finally, in section 7, we make concluding remarks about our work.


2 Analysis of previous work

Of numerous time concepts proposed to date, we select some and highlight the confusion among
them.
Copeland and Maier [7] introduced two types of time, event time and transaction time. Ac-
cording to their description, the event time is the time an event, which gives rise to a change to
an object, happens in real world. And the transaction time is the time the change is recorded
in a database. Although not explicitly mentioned, to our best comprehension, it seems that they
were trying to explain two kinds of temporal validity. One is a real-world validity modeled by the
event time. The other, an approximation of the former, is a database-world validity modeled by
the transaction time. For example, let's assume that a faculty member, Smith was promoted to
associate professor in Aug. 1'' ", and the fact was recorded in a database in Sep. 1'' ". If we adopt
the event time, Smith's associate professor property becomes valid from Aug. 1' ". If we take the
transaction time, as they actually did, it becomes valid from Sep. 1',". We will call, tentatively,
the former validity effective validity and the latter i.. '-i. l',.-[-..-., 1 validity.


Lum and others [18] proposed other kinds of time, 1i,,;iral time and physical time. Basically,
the logical time is the same as the event time in the sense that both are time of real world, i.e.,
the true time with which an object changes. However, the physical time is different from the
transaction time of [7], even though the two times have been treated identically as both are the
recording time of data. The physical time was intended to be a reference time for the logical time
to model retroactive and proactive updates. For example, suppose Smith received a promotion to
associate professor in Aug. 1' "., but the promotion was retroactive from Jan. 1',' ". Then, the fact
is recorded with logical time of Jan. 1', -". and physical time of Aug. 1', ". if indeed recorded then.
We should be able to notice the problem with this approach. What if the fact were recorded in Sep.
or later instead of Aug.? Does that mean Smith received a retroactive promotion in Sep. or later?
In addition, the validity dealt with in [18] is in essence the same with the effective validity of [7],
provided that the event time and the logical time are for modeling of the same real-world validity.










The difference is that in [18] they used an additional time to capture the semantics of retroactive
and proactive updates. We will discuss this problem further in section 4.


The valid time and transaction time proposed by Snodgrass and Ahn are also confusing [26].
They distinguished the two times based on the classification of temporal databases into rollback
databases, historical databases, and temporal databases1. That is, the valid time is the time
employed by the historical database, which is described as a database storing history of relations
as it is best known. The transaction time is described as the time employed by the rollback
database, which is a collection of all the snapshot databases. The historical and rollback databases
are distinguished by error corrections also. That is, errors in historical databases can be rectified,
whereas such corrections are not allowed in rollback databases. Here, we can see that the distinction
between historical databases and rollback databases is not so clear in the sense that a collection of
snapshot databases can be regarded as a history of relations. Further, prohibiting error correction
in rollback databases is not meaningful because erroneous database states should be able to get
corrected anyhow. Accordingly, the definitions of valid and transaction times as attributes to
historical and rollback databases are also unclear.
On the other hand, the role of transaction time in the rollback database is different from that of
their temporal counterpart, which employs both valid and transaction times to support retroactive
and proactive updates as well as preservation of all past database activities. The transaction time
in rollback databases is equivalent to the transaction time of [7] in that rollback databases are in
fact temporal databases using the registration-based validity. In contrast, the transaction time in
their temporal databases is the same with the physical time of [18] in the sense that both times are
used to model retroactive and proactive updates.


Yet another time notion, the user-defined time [26, 15] seems to be a misnomer. Its definition
is that it is an attribute whose domain consists of date and time values [15] which is supplied by
the user. Unlike previous time concepts, the definition of user-defined time designates it as data
(attribute value), which is not defined in terms of time-stamps. If the time is treated as a data
value, then there are at least two possible semantics: interval time such as 1 hour and chrnl ,,,,iu;,IIl
time (calendar time) such as Jan. 1, 1992. The definition of user-defined time does not indicate
which time it refers to. Indeed, the interval time can be treated as an ordinary data type, and
can be time-stamped with the chronological time in order to represent its validity. In this paper,
the interval time is excluded from the discussion; all time concepts we deal with are chronological
times.
1This "temporal database" has a narrow meaning in their work that it is a database made by consolidating the
rollback database and the historical database. This term should not be confused with our general meaning of temporal
database.










Finally, another confusion arises in the work of Gadia and Yeung [13], in which the valid and
transaction times of [26] were adopted for the demonstration of their n-dimensional, symmetric
time concept. However, the valid and transaction times of [13] were in fact not those of [26], but
the event and transaction times of [7], respectively.


To summarize, we identify several problems with the previous approaches. First, we think there
has been a dichotomous classification of time concepts, i.e., reality vs. representation (or real-world
vs. database-world) [26]. That dichotomy, together with the absence of a proper quantification of
temporal validity, presumably misled us to the point that there is one genuine real-world validity
and other possible database-world validity, if any, such as the registration-based validity. Second, we
can see that the problem of preserving multiple past states has been intermingled with the validity
problem. They need to be separated. Lastly, unlike the previous view [18, 26], combination of valid
and transaction times cannot model retroactive and proactive updates.


3 Temporal validity

In this section, we present the notions of temporal validity and sort of validity. With these notions,
the validity problem discussed in the previous section can be clearly understood.

3.1 Equi-distant discrete time

Throughout this paper we assume an equi-distant discrete time domain T, termed the integer time
domain, having "0" as the origin of time, which is used to represent the temporal validity. The
integer time domain T is isomorphic to the Gregorian calendar time domain. Therefore, without
the loss of generality, we use the integer time and the calendar time interchangeably in this paper.


Now and oo We adopt two indefinite time constants. One is Now, by which we mean a moving
time point that exactly keeps track of the current time. We use the convention that all Now's in
a transaction are substituted with the same current time. This seems to be reasonable unless the
transaction is a long-lived one.
The other indefinite time constant is oo (infinity). The indefinite constant oo denotes a suffi-
ciently remote future time point.


Time interval A time interval is defined to be consecutive time points in T. To be more specific,
a time interval, i is a non-empty subset of the time domain T, i.e., i C T and 1 5 0, such that all
time points in i are consecutive. If i consists of only one time point, it is regarded as being trivially
consecutive.
Two time intervals, ti and tj, are said to be equal if any t in ti is also in ij and vice versa.










Also, ii and tj are said to overlap if there exists any time point t such that t E ii and t E tj. A
set of time intervals is said 1.'- ; ',/ if and only if no interval overlaps one another.
A time interval is alternatively denoted by (ts ~ te), where t, and te are the starting point and
the ending point of the interval, respectively. For example, (5 ~ 9) and (1/1/ ~ 8/31/91) are
time intervals of the integer time system and the Gregorian calendar time system, respectively.


Indefinite time interval The indefinite time interval is defined to be a time interval whose
ending point is the indefinite time constant oo, denoted by (t, ~ oc). The meaning of an indefinite
time interval is that the ending point of an interval is not determined (or not known) as of now.
Thus, the ending point, in fact, can be any time point greater than or equal to t,. In other words,
an indefinite time interval (t, ~ oc) shall be equal to one of definite intervals, {t1}, {ts, t + 1},
{t, t, + 1, t + 2}, and so on.
This interpretation, however, makes problems unnecessarily complicated even though it might
lead to a more precise modeling. We, instead, just follow the mathematical meaning of oo. That
is, an indefinite time interval (ts ~ oc) is simply interpreted as a set of time points, {ts, t, + 1, st +
2, oo}. Note that (t, ~ oc) is more inclusive than another indefinite time interval (ts ~ Now).

3.2 A simple data model

Before presenting a formal definition of temporal validity, we define a simple data model from which
some necessary concepts are drawn. The data model presented here is not intended to be complete.
Its sole objective is to provide a set-theoretic definition of data value.


Object An object is an entity, a relationship among entities, or an attribute of an entity (or a
relationship).


Object set An object set is a set of objects. However, not all arbitrary object sets are meaningful
to us. We are interested in only certain object sets which are pre-classified from the universal object
set, consisting of all conceivable objects in the world, according to some criteria.
For example, we may have three object sets, S1, S2, and S3, each of which is a subset of
the universal object set U. The subsets might be pre-classified and named Person, Faculty, and
Furniture, respectively. Then, an unrelated object, such as a building, will not be included in set
Si. We call such a name a property of the object set. From now on, whenever we say an object
set we mean an object set with property. Also, it is assumed that object sets are pre-classified or
identified by database designers.


Name and value A name is regarded in our work as an identifier for a property. Accordingly,
two different properties have different names. However, an object set may happen to have several










properties, each of which is identified by a distinct name. Further, we do not make any distinction
between name and data value in the simple data model. A value such as 2 may be treated as a
name, and a name such as Smith treated as a value.


Property The definition of property is given below.

A property P of a given object set A is a subset (including the empty set) of A, i.e., P C A.2

An object e, e E A, has a property P if e E P.

We also define the empty property, denoted by e, to be non-existent property. In other words, if an
object has the empty property e, the object is regarded as having no property.
Incidentally, a property should not be confused with an attribute. An attribute in our work, is
simply viewed as a subordinate object which along with other attributes (or subordinate objects)
describes the semantics of an object. Defining various relations among objects, such as coordinate
relations (entity vs. entity) and subordinate relations (entity vs. attributes), is beyond the scope
and objectives of this paper. For details on data/knowledge modeling refer to [6, 3, 24, 14, 28].


Property set A property set, P is a meaningful set of properties that are associated with a given
object set A.
A property set is called a .1'-;,.',/I property set when no element (property) in the set overlaps
the others. That is, if e E Pi, then e Pj, for any Pi,Pj E P such that i 5 j. If a property set
is not disjoint, it is called an -,.'. i/,/,/,',,, property set. When every property in a property set is
guaranteed to have exactly one member, the property set is called an individual property set. The
individual property set is a special case of the disjoint property set. With the individual property
set, each individual object in the object set can be identified.
The empty property e is included in every property set and does not overlap any other property
in the property set.


For example, given an object set, Person = {el,e2,e3,e4,e65,e6, e7}, three properties can be
defined as follows.

Faculty = {e, e2, e3, e4}, Student = {e5, e6}, Unknown = {e7}.

The property Faculty (an object set) is further specialized into two ways as follows.

Instructor = 0, Assistant = {el}, Associate = {e2, e3}, Full = {e4}, Tenured = {e3, e4}.
Jane = {el}, Smith = {e2}, John = {e3}, Miller = {e4}.
2To be more specific, property P is a name of a subset A' C A. However, such a subtle distinction between a
name of the set and the set itself is not so critical as long as the name is an identifier of the set.










The object set Person has a property set, 'Pccupation = {Faculty, Student, Unknown}, while the
object set Faculty has two property sets, PF_Rank = {Instructor, Assistant, Associate, Full, Tenured}
and PFame = {Jane, Smith, John, Miller}.3 An object e2 in object sets, Person and Faculty, has
three properties, Faculty, Associate, and Smith, each of which belongs to property sets, POccupation,
PF ,Rank, and PFP Vame, respectively. PFaName is a disjoint (individual) property set, whereas
PFPFank is an overlapping one. Note that all property sets include the empty property e tac-
itly. The object e7 might have empty properties included in property sets, PF vame and PF_Rank,
implying that the object is no longer a faculty member.
As a matter fact, a property set corresponds to a domain of attribute in terms of the traditional
definition.

3.3 Definition of temporal validity

Informally, the temporal validity of a data value for a given object is the object's possession of
the value along the time dimension. For example, the validity of associate professor property for
Smith's rank is a state of whether or not Smith possesses the associate professor property with
time.


Temporal validity The temporal validity of property P of an object set A is a characteristic
function of P,

VP: (Ax T) {0,1}

given by

VP(e, t) 0 ifeePatt

where e is an object in A, t is a time point in the time domain T, P is in 7, and P is a property
set of A.
If (e, t) is mapped onto 1 by a characteristic function VP, then property P is said to be valid
for object e at time t by VP. It is assumed that for each property Pi in property set P of object
set A, there exists one or more characteristic functions, such as VP,, V2P,, and so on.
The use of a characteristic function provides a flexible mechanism for customizing the definition
of temporal validity. Note that the temporal validity of a property P of an object e is derived
from the interpretation of the characteristic function. Later we will show how this concept of
characteristic functions can be used to define validity for various time concepts. The only necessary
assumption is that such a function does exist.
3It should be noted that the property Faculty (also an object set) is different from the property set PFName
denotationally, as well as semantically. That is, the former is denoted by {e, e2, e3,64} while the latter by
{{e1}, {2}, {3}, {4}}.









Valid period A valid period, T of property P for object e, characterized by a characteristic
function VP, is defined to be a set of time points t, t E T, such that

{t | VP(e,t)= 1}.

In other words, a valid period is a time interval or a set of disjoint time intervals over which
a given property is valid. A valid period can be pictorially represented as shown in Figure 1.
Figure l(a) shows a valid period of a property, associate professor for an object, Smith, while
Figure l(b) shows a recurring valid period of a property, $100 (which is in a property set, Stock-
price) for an object, say IBM stock.
In addition, the domain of valid periods (including recurring ones) can be denoted by 2T, the
power set of the time domain T.
With the valid period, the temporal validity can be extended as follows.

V'P: (A x 2) {0, 1}

given by

V'P( 1 if r= {t VP(t)=
P(e,) 0 otherwise

where e E A, a valid period r E 2 and 2T is the domain of valid periods.
We call functions such as V'P extended characteristic functions. Note that existence and defi-
nition of an extended characteristic function, V'P depends on VP.

1/4/i, 6/5/89

Associate

(a)



5/3 5/6 5/9 5/13 5/15

$100.00 $100.00 $100.00

(b)


Figure 1: Valid periods

Due to the possibility of multiple characteristic functions for a property, an object may have
several different valid periods for a property. To exhibit this aspect, we define two characteristic
functions, VIP, and V2P1, for a property P1, where P1 is associate professor.










VIP(e, t) :


1. Map to 1 if e receives a promotion letter to associate professor and t is equal to
the promotion date written on the letter.

2. Map to 0 if e receives a promotion letter to other rank than associate professor or
a letter of dismissal, and t is equal to the date written on either letter.

3. Map to 1 if e is an associate professor at t 1 and t is different from the date of
(2) if e gets either letter.

V2P (e, t) :

1. Map to 1 if e receives a promotion letter to associate professor and t is equal to
the date the letter was -'. ,, d1

2. Map to 0 if e receives a promotion letter to other rank than associate professor or
a letter of dismissal, and t is equal to the date either letter was signed.

3. Map to 1 if e is an associate professor at t 1 and t is different from the date of
(2) if e gets either letter.

For the two different characteristic functions shown above, we have different valid periods of Smith's
associate professor property as shown in Figure 2.


1/4/<" 6/5/89
ViP1 : I -I
Associate

12/29/-8 6/7/89
V2Pl : I I
Associate


Figure 2: Two different valid periods of associate-professor property


3.4 Sort of validity

While the temporal validity deals with each individual property, the sort of validity concerns the
totality of properties in a property set. A sort of validity is, in effect, an interpretation of temporal
behavior of data objects.


Definition: Given a disjoint property set, P = {Pi, P2, Pn} and an object set A, a sort of
validity, Sp for P is defined to be a set of characteristic functions,










Sp {V=Pi, VP2, VkPn}


such that if ViPm(, t) = 1, VIPm E Sp, then for all VP, E Sp, V,Pw # VIPm, VPw(e,t) = 0.
According to the above definition, for a set of characteristic functions to be a legitimate sort of
validity, the following conditions need to be satisfied.

1. At most one characteristic function maps (e, t) onto 1.

2. The property set P is disjoint.

The first condition is necessary to prevent an object from having more than one property
simultaneously, which belong to a property set. For example, in general it does not make sense for
a faculty member to have two different ranks, say, associate and full, at the same time. In fact,
this condition is the minimum requirement that the characteristic functions defined should satisfy.
The second condition is imposed because if the property set is indeed overlapping, the first
condition cannot be satisfied semantically. For example, if the property set is 'PF-anJk defined
previously, an object with property Tenured most likely possesses another property, Associate or
Full, too. Thus, if the characteristic functions are defined so as to fulfill the first condition over
such an overlapping property set, the resultant validity will be semantically wrong. The second
condition is actually a necessary condition for the first one.
In addition, it should be noted that none of the characteristic functions in a sort of validity may
map (e,t)'s to 1. In that case, it is assumed that the object e does not exist at time t.4


Single-sort/multisort validity The definition of sort of validity and the possibility of multiple
characteristic functions for a property lead to multiple sorts of validity. For example, assume that
given a property set P = {P1, P2, four characteristic functions, V1IP, V21, V1P2, and V2P2 are
defined. Then, there may be four sorts of validity out of the characteristic functions, provided that
each combination satisfies the conditions of sort of validity. Two of them are given below.

81p = {V1P1,V1P2}
82- = {V2P1,V2P2}

When an object set has multiple sorts of validity, as shown above, we say the object set has
multisort validity. If only one sort of validity is defined, the object set is said to have a '",.1. -sort
validity. For example, if we define VIP, and V2P2 in a similar manner to ViP1 and V2P1, respectively,
where P2 is full professor, then we may have two different sequences of valid periods for two sorts
of validity, Sp and S2p, as shown in Figure 3.
4This assumption might be an over-simplification in that the existence of an object does not depend on the fact
whether the object has a certain property. The simplification, however, keeps the discussion manageable at the
expense of modeling accuracy.










1/4/s', 6/5/89 6/6/89 5/31/92
S1p : I I I
Associate Full


12/29/Sf 6/7/89 6/8/89 5/-s/92
2 : I I
Associate Full


Figure 3: Two sorts of validity



On the other hand, the remaining combinations {V2P1, V1p2} and {ViP1, V2p2} are not likely to
be a sort of validity since as shown in Figure 4 during some time period an object may have more
than one property at the same time.


12/29/S- 6/7/89
{V2P1, VP2}: I I
Associate
6/6/89 5/31/92
I I
Full


Figure 4: A disallowed combination of characteristic functions

Lastly, we want to emphasize that the sort of validity (as well as the temporal validity) is
developed for an object set. In other words, a sort of validity defined for one object set, in general,
does not apply to other object sets. For example, the sort of validity Sip is not likely to be
meaningful for an object set Furniture.


4 Preservation of multiple past states

As the temporal database is intended to capture all past database states including the current one,
preserving past states is essential. However, as we shall see, the preservation of past states is not
guaranteed by just not deleting previously entered data. There are two independent causes that
have a bearing on the preservation. The first is the retroactive update, and the second is the error
correction and delayed update. The reason why the preservation can be incomplete is that these
operations generate multiple pasts. If a temporal database system cannot record and retrieve all
the past states, we will get to lose some past states, that is, the preservation of the past will be
incomplete.










[25] is regarded as the first comprehensive treatment of the preservation of past states, although
the aspect of retroactive update was mentioned earlier in [18]. A common problem in their work is
that they made use of the transaction time (the physical time) to deal with retroactive and proactive
updates. In this section, we show that retroactive and proactive updates cannot be captured by
the transaction time, and define another time concept, event time to model these operations.

4.1 Event time

An event can be viewed as an abrupt (I.,I,. of database statess. In our work the event has a
slightly tailored meaning that it generates certain relevant facts, or more specifically, it is a cause
of changes to properties of database objects. Thus, when an event happens, it is naturally assumed
that the event changes certain properties of database objects. For example, a promotion event for
a faculty member generates a new fact, i.e., a new rank, and a relocation event generates a new
address. In other words, the two events change properties of a faculty member, to new ones. Now
we are interested in the time when such an event happens. We call the time an event happens event
time.
Incidentally, it should be pointed out that a fact (property) is not necessarily generated by
only one event. That is, two or more different events can generate the same fact. For example, an
address, Gainesville, may be generated by an event of district adjustment such as annexation, as
well as an ordinary relocation event. As a matter of fact, the heterogeneity of events may interfere
with the interpretation of events recorded in a database. However, we do not pursue that issue any
further in our current work. We simply assume that all events changing a property of an object
are of the same kind.


4.1.1 Time difference between an event and its accompanying facts

Although it is preferred that facts generated by an event become valid as soon as the event happens,
there may be situations in which the time a fact becomes valid differs from the time the event
happens (i.e., event time). Based on this time difference, we can classify events into three categories,
on-time events, retroactive events, and proactive events. When a database is updated with a fact
generated by a retroactive event, we call the update a retroactive update. Similarly, if a fact is
generated by a proactive event, we call the update a proactive update.


4.1.2 Classification of events

On-time events This is the most general case. For an event classified into this category, its
relevant facts become valid immediately after the event happens. Alternatively, it may be thought
5Such a change need not be an instant one. An event may happen through a fairly long period of time. Or a
seemingly instant event may be viewed as a long term one if the time granularity is sufficiently magnified.



















(a)





(b)


8/89
I. 9
Associate *.. Full

'11/89
1 -
Promotion Event

8/89
1 --


Valid Time



Event Time


Associate Full

11/89
------- ( 10/89)
Associate
(c)
8/89
I (11/89 co)
Associate Full
Event Time

Figure 5: Retroactive event


of as facts become valid simultaneously with the event. For example, if a promotion event which
promotes a faculty member Smith to an associate professor happened on 11/89, unless otherwise
stated, his rank of associate professor would become valid on 11/89.6


Retroactive events For some events, the time a fact becomes valid may be prior to the time
at which the fact is generated, i.e., the time an event generating the fact occurs. We call such
events retroactive events. For example, Smith's promotion might be a retroactive one so that the
promotion became valid from 8/89 while the promotion itself was proclaimed in 11/89. The time
difference is depicted in Figure 5(a). It should be noted that even though the time the fact becomes
valid precedes the time of the event, the fact is never known until the event occurs.


Proactive events In contrast to retroactive events, a fact may become valid at a later time than
the time at which the event generating the fact occurs. We call such events proactive events. As
an example, when it is decided that a research grant is to be awarded to an institute (i.e., an event
has now happened), the grant usually will not be effective until a later time. The time difference

6Note that the interpretation concerning when the event happened is determined by a characteristic function
discussed in section 3.










7/92
----------IM-----
Not Effective" Effective

10/9"1

Grant-Award Event

Figure 6: Proactive event


for the proactive event is shown in Figure 6. Other examples for the proactive event may be drawn
from a temporal database for weather forecasting. Whenever a forecasting is made (or a fact is
generated), the fact will not be valid until some time later.


4.1.3 Multiple pasts generated by the time difference

The time difference inherent to a retroactive event is one reason for the generation of multiple
pasts. Figure 5(a) depicts a retroactive promotion of Smith. Until 8/89, Smith's rank is no doubt
Associate. After the promotion on 11/89, his rank is Full. But, what about from 8/89 to 10/89?
Before the promotion, the rank during that period was Associate, whereas after the promotion,
the rank was revised to Full. As a result, we have two different pasts during that time period.
Figure 5(b) and (c) show possible representations of the retroactive update.
Figure 5(b) represents the resultant database state after the update, and is in fact the most
reasonable representation for one-dimensional temporal databases. A shortcoming of this represen-
tation is that the intermediate database state, Associate, during 8/89 through 10/89 disappears
and cannot be retrieved. Moreover, the fact that the promotion was retroactive cannot be recalled.
Figure 5(c) shows an alternative in which two histories are kept, one each for the updated
and the previous state. In this approach, the updated state can be retrieved as usual, and the
previous state also can be retrieved if necessary. In addition, the fact that the recorded event was
a retroactive one can be represented in some way and recalled.
A problem with the second approach shown in Figure 5(c) is that one time (or one-dimensional
time) is not sufficient to uniquely qualify a data value at a given time point. For example, at 10/89
Smith's rank may be interpreted as either Associate or Full. However, taking a closer look at the
multiple histories, we can see each history has its own '- .'o',I effective period in terms of event
time. For example, until 11/89 in event time, only the history of the upper figure in Figure 5(c)
will be effective. Such an event time period is shown on the right side of each history. As such, the
event time is now able to be a qualifier to single a history out of multiple histories. Once a history
is selected, then we can examine it to get a data value at a specific time point. Consequently, if
two time values, one each in the valid time and the event time, are given, we are able to retrieve










a unique data value from a temporal database without losing past states generated by the time
difference of retroactive events.

4.2 Transaction time

In section 2, we discussed the meaning of transaction time and the different roles it played in
temporal databases. The transaction time, in our work, is simply the time at which data values
(facts, properties) are recorded in a database and its role is just to supplement the valid time to
preserve multiple pasts generated by error corrections and delayed updates.


4.2.1 Error corrections

The error correction operation may be the most conspicuous one among those operations generating
multiple pasts. Assume that Smith was promoted to associate professor on 1/' ., but the rank was
wrongly recorded as full professor. The error was found later and corrected on 9/$"'- Figure 7(a)
illustrates that situation. The error correction induces two different pasts. During 1/t". through
8/- ". we have an incorrect past in which Smith's rank is viewed as full professor, whereas from 1/. -.
up to now we have a correct past7 where the rank is viewed as associate professor. Figure 7(b) and
(c) show two possible representations for the correction.
Figure 7(b) represents the corrected, most up-to-date state of database, but nothing else. What
we lost in the representation is the incorrect past, i.e., rank of full professor during 1/, -. through
8/ ", which cannot be retrieved after the correction. Moreover, there is no way to recall the fact
that an error was corrected, unless it is stored somewhere else. For many applications, this loss of
past information may not matter at all. However, applications requiring strict audit need to keep
even erroneously created states.
Figure 7(c) shows an alternative, similar to Figure 5(c), where both pasts are maintained. With
this approach we can retrieve the incorrect past information if needed, in addition to the corrected
one. Also, we can discern whether and when an error correction was made. Again, one-dimensional
time is not sufficient to uniquely qualify a data value at a given time point. For example, on 5/ ".
Smith's rank may be construed as either Full or Associate.


4.2.2 Delayed updates

As a fact is generated in real world and recorded in a database, there exists inevitable time delay
between the generation time and the recording time. This delay may vary from a few milliseconds
to several months or more. Due to the delay we, by no means, can avoid temporal inconsistency or
inaccuracy between real world and a modeled database-world, as long as information is retrieved
7Note that the currently correct past is not necessarily indeed correct. It may be corrected again later.










1/85


/


(a)







(b)


I D
Assistant Full

1/85

Assistant "-Associate

.9/85
1 -
Corrected

1/85
i >


ValLid time


Valid time


Transaction time


Assistant Associate

1/85 9/85
I I------
(c)
1/85
I 1
Assistant Associate

Figure 7: Error correction


from the stored data. Suppose that Smith was promoted to full professor in 8/89 and the fact was
not recorded until 11/89. Then, until 11/89 we would never know the fact of his promotion, no
matter how ingenious methods were employed. The best we can do is to either minimize such a
delay or keep all the past incorrect database states so as to be brought back whenever an inspection
or audit is necessary.
Figure 8 shows such a delayed update. An event (or a fact) happened in 8/89, which promoted
Smith to full professor, is recorded in 11/89. As a result, during 8/89 through 10/89 the database
keeps an out-dated data value, Associate. When one records the delayed update as if the event
has been occurred just now, the resultant valid period will be Figure 8(b). The drawback of this
approach is obvious. The discrepancy between the intended valid period and the valid period of
stored data will be perpetuated.
Figure 8(c) is an alternative in which the fact is recorded as valid from 8/89 even if the recording
is carried out in 11/89. This is similar to the approach taken in error correction shown in Figure 7(b).
An advantage of this approach is that once recorded, the resultant database correctly reflects the
intended validity. However, as in the case of error correction, we will lose an incorrect past state of
the database; i.e., the incorrect state during 8/89 through 10/89 will disappear after the update.
Figure 8(d) presents another alternative, in which we have two histories. We keep the incorrect
past state as well as the correct one. By doing so, we are able to alleviate, to a large extent, the
problem of delayed update. Until the delayed data is recorded, we cannot help getting incorrect










8/89
V. m- Intended valid period
(a) Associate '-. Full
'.11/89
'~ --------- -
Updated


11/89
(b) I Delayed update 1


(c)


Associate Full

8/89
I -I


Delayed update 2


Associate Full

11/89
S-------->.- Incorrect valid period
Associate
(d)
(d) 8/89
I Correct valid period
Associate Full

Figure 8: Delayed update


information from the database. However, once it is recorded, we will be able to retrieve the correct
one. In addition, we can always access and roll back to the incorrect past state. But, analogous
to the previous cases, one-dimensional time is not sufficient in this approach to uniquely qualify a
data value at a given time point. For example, in 9/89 Smith's rank may be interpreted as either
Associate or Full.


4.2.3 Consolidation of error correction and delayed update

Even though the error correction and the delayed update may seem to have slightly different
temporal behavior, they have one thing in common. That is, the time of error correction and
the time of (delayed) update are the transaction time. Needless to say, the time when an error is
corrected or a data value is updated is the time when either operation is performed on the database.
This sharing leads to consolidation of error correction and delayed update over the transaction time,
as depicted in Figure 9(a).
Figure 9(a) shows a series of update, correction, and delayed update operations. In 1/$', the
rank was updated to a wrong value, Full. The wrong value was corrected to Associate in 9/' .
Again, the rank was changed to Full in 8/89, but recording of the value was delayed until 11/89.
Figure 9(b), (c), and (d) show multiple histories generated by the operations. Figure 9(b) and (c)










1/85 8/89
* | _


| vaiiua Time
Assistant :'. Full Full
SAssociate
(a)

.9/85 1.1/89
\i "| Transaction Time
Updated Corrected Updated (Delayed)



1/85 9/85
(b) ---- ------------- 8/85)
Assistant Full

1/85 11/89
(c) I I ---- (9/85 10/89)
Assistant Associate

1/85 8/89
(d) I I (11/89 co)
Assistant Associate Full
Transaction Time

Figure 9: Consolidation of error correction and delayed update


reflect the effect of the correction made in 9/', Before the correction the value (rank) is viewed as
Full. After the correction the value is rectified to Associate as shown in Figure 9(c). The effect of
the delayed update is reflected in Figure 9(c) and (d). Before the update at 11/89, the data value
is wrongly seen (from 8/89) as Associate. After the update a new history shown in Figure 9(d) is
obtained.
Similarly to the earlier discussion on event time, in Figure 9 we can see that each history has
its own ,'-.i,',I effective period in terms of transaction time. For example, during 9/',- through
10/89 in transaction time, only the history of Figure 9(c) will be effective. Such a transaction time
period is shown on the right side of each history. Thus, if two time points, one in the valid time
and the other in the transaction time are given, a single data value can be retrieved without losing
past database states generated by error corrections and delayed updates.

4.3 Orthogonality of event time and transaction time

It is interesting to see in previous work [26, 25] that the retroactive update and the delayed update
have been treated in the same way; in other words, the two were not differentiated at all. In
their work, the time a fact is generated and the time it is recorded are indistinguishable and
represented by the transaction time. In this section, we have established the difference between


,1










7/92
------------ me
Not Effective ..*'*' Effective

10/91'

Grant-Award Event



12/91 7/92 10/92


Valid Time



Event Time



Transaction Time


Figure 10: Misinterpretation of a proactive event




the two operations and shown why the notion of event time is required to model retroactive and
proactive updates.
To comprehend the inadequacy of transaction time, let's compare the examples shown in Fig-
ure 5 and Figure 8. In the first case, Smith got a promotion on 11/89, which was retroactively
effective from 8/89. In the second case, he got an on-time promotion in 8/89, but recording of the
fact was procrastinated until 11/89. It is obvious that these two cases, a retroactive event and an
on-time event but record-delayed, are not the same. Note that Figure 5(b) and Figure 8(c) are
seemingly identical although they have different semantics.
More dramatic example can be shown as follows. Consider the proactive event of Figure 6 in
which a research grant is awarded in 10/91 and the grant is effective from 7/92. And assume that
we are trying to model the event with valid time and transaction time. If the proactive event
happens to be recorded on 10/91, it might be regarded as a correct recording for the proactive
event. However, if the recording time happens to be 12/91, 7/92, or 10/92, then the recorded event
will be wrongly interpreted as another proactive event, on-time event, or even retroactive event,
respectively. Figure 10 illustrates that situation.
A more general situation is illustrated in Figure 11. We have four events, Ei, Ej, El, and
Ek, which are, respectively, an on-time event, a retroactive event, another on-time event, and a
proactive event. Facts generated by the events are Pi, Pj, Pi, and Pk, respectively. The events
are recorded by transactions, Ti, Tj, Ti, and Tk, respectively. Integers represent time points. Note
that Ti is an error correction that adds a missing event El, which happened prior to Ek, to the
database.8
8As mentioned before, we regard all the events as being the same kind, such as events of promotion. Especially,
El should not be misunderstood as an event of error-discovery. Such an event, bearing a different semantics, should
have occurred somewhere between time points, 10 and 15.









3 5 10 15
I I I .
P, .P P1 ... Pk

3 7 10 12..''
i '1 i '
E, E, "'E. ", Ek

3 '".9 ':-3. 15
1i 1 "1 >
T, T, Tk 1


Valid Time


Event Time



Transaction Time


Figure 11: A general situation of event-occurring and its recording


Reference Time concept Representing sort of validity
[7] Event any sort of validity
Transaction registration-based validity
[18] Logical any sort of validity
Physical none (reference time for logical time)
[26] Valid any sort of validity
Transaction in rollback DB registration-based validity
Transaction in temporal DB none (reference time for valid time)
User-defined any sort of validity


Figure 12: Classification of time concepts


4.4 Classification and maximal set of time concepts

Based on the discussion so far, we can classify the previously proposed time concepts using the
notion of sort of validity and the supplemental time concepts needed to preserve multiple past
states.
As shown in Figure 12, the event time of [7], the logical, valid, and user-defined times are in
essence the same time in the sense that they all represent an arbitrary single sort of validity; the
sort of validity is determined by characteristic functions, or more practically, by users' needs.
Here, the meaning of user-defined time in question needs to be bound to a chronological time,
which is used as time-stamps to represent the temporal validity of certain data objects. Using
time-stamps as data (attribute) values may be unavoidable due to the limitations of underlying
temporal database models. In the temporal relational database, for example, birth-date of a faculty
member need be included as an attribute of a relation TEACHERS, because the birth-date is not
directly related to the validity of a faculty member but to that of a person, which is in a more
generic object set (i.e., entity superclass) of TEACHERS. It should be noted that, however, the
user-defined time has the same nature with the valid time in terms of representing validity.
The transaction time of [7] and that of [26] used in rollback databases represent the registration-










based validity discussed in section 2. The definition of characteristic functions for that validity is
perhaps the simplest. For any object e and any property Pi, VP,(e,t) will be 1 if t is equal to the
time the recording is carried out, and will keep the value until recording of a different property Pj is
performed. In a real situation, this characteristic function is easily implemented by automatically
setting the start point of valid period of a data value to the system clock. Note that the registration-
based validity is a special case of a single sort validity represented by a valid time.
From the foregoing arguments we can conclude that the valid time alone (we have chosen this
terminology) is enough to represent the temporal validity of data objects unless the preservation of
multiple past states is a concern. If an application requires a multisort validity, that requirement
can be easily fulfilled by providing multiple valid times in a temporal database system.
On the other hand, the physical time and the transaction time of [26] used in their temporal
databases do not represent a sort of validity by themselves. They act as a reference time for the
valid (or logical) time to measure the time difference we discussed in this section. Employing the
two-dimensional time, consisting of valid and transaction times, we are able to preserve multiple
past states produced by error corrections and delayed updates (not by retroactive updates). Never-
theless, it should be noted that only one sort of validity is represented by the two-dimensional time.
For example, Figure 9 illustrates a sort of validity. In the figure, (d) depicts that validity most
precisely, while (b) and (c) show stale ones. As a consequence, supporting multisort validity and
preserving all database states by multidimensional time are two independent things. Accordingly,
it is possible that we have an one-dimensional time and a two-dimensional time to support two
sorts of validity, if necessary.
Using the time notions introduced in this paper, multidimensional time may be configured in
three ways:

1. Combination of valid and transaction times.

2. Combination of valid and event times.

3. Combination of valid, event, and transaction times.

Using (1), we can preserve multiple histories generated by error corrections and delayed updates.
But we lose incomplete pasts generated by retroactive updates. Of course, the database does not
keep any information concerning when events happened, much less the types of events. Figure 13
shows the roles of valid time and transaction time, which is a simplified view of Figure 11. As we
can see, no information about events exists in Figure 13. However, from the transaction time it is
possible to infer that transaction T1 was an error correction.
Using (2), we can preserve multiple histories generated by retroactive updates. But erroneous
pasts generated by error corrections and delayed updates cannot be preserved. Figure 14 illustrates
the situation simplified from Figure 11 by employing only the valid time and the event time. In









3 5 10 15
|I I I >
.P, '*?, "P ... / Pk
3 ''. 9 '13'. 15
I 1 I "I >
T, T, Tk i

Figure 13: Combination of valid and trans;

3 5 10 15

SPP, PI Pk
3 '.7 10 12..'
I i F"
E, E, El Ek


Valid Time



Transaction Time


action times


Valid Time


Event Time


Figure 14: Combination of valid and event times


the figure we don't have any information regarding transactions. As a result, it is not possible to
infer that recoding of El was done by an error correction TI.
Lastly, using all the three times (case (3)), all multiple pasts generated by those operations can
be preserved. This approach will give us the most general interpretation of temporal databases.
However, the generality needs to be weighed against the increased complexity in interpretation
from users' viewpoint. While the implementation of three-dimensional time is possible, we do not
believe that four- (or higher) dimensional time is manageable and that benefits from its generality
would offset the far increasing intricacy of interpretation. From this practical standpoint, we claim
that the valid, event, and transaction times are a maximal set of times, which are needed and can
be implemented in temporal databases.


5 One-dimensional interpretation

The significance of temporal and sort of validity defined earlier is that they lead to a formal
representation of the evolution of data values, which we call the history of an object. Before we can
define validity and preservation of multiple pasts using two or three time concepts (leading to two-
or three-dimensional representations, respectively), we need to establish the history of an object in
terms of its evolution. This section formally defines the history of an object and relevant concepts,
and shows how the history is constructed from data manipulation operations.

5.1 History of an object

A temporal data value possessed by an object is represented by a pair, (P, r), where P is a property
and r is a valid period of P for the object. Usually, an object may have several properties during its











i1 Ti Ti2
II II'
Pil P, Pi2


Figure 15: History of an object

lifetime. From the definition of sort of validity, it is guaranteed that valid period of each property
does not overlap one another. Then, temporal data values which an object has had during its
lifetime can be grouped together. We call such a group the history of an object. The history of an
object reflects its evolution along the time dimension. A formal definition of the history is given
below.


Definition: Given an object set A and its property set P and given a sort of validity Sp, a history
H of object e in A is a set of temporal data values,

H = {(P, 7,r) |V' P(e, r) = 1, VP, E Sp}

where property Pi 7, r, is the valid period of Pi, and V'jP, is the extended characteristic function
of Vjp,.
A history can be depicted as shown in Figure 15. In the figure, the valid period of the property
Pi (indicated in the figure as P i and Pi2) is the union of r7, and Ti2.


5.1.1 Special time points in history

Suppose that property Pi of object e has valid period ri which is a set of disjoint time intervals,
{{ ti, t + 1, ., tj 1}, {tk, tk + 1, .. tl 1}, *. {tm, tm + 1, t 1}}.

Acquisition and release points Now we are interested in two kinds of special time points
over the valid period r,: the initial points ti,tk, ,tm of the time intervals, in which an object
(say e) acquires a property (say P;), and the time points tj, tL, t,, immediately after the final
points of the time intervals, where the object e loses that property. We call these acquisition
and release points, respectively. The valid period ri of the property Pi can also be denoted as
{(tL ~ tj 1), (t ~ tl 1) 1), . (t t 1)}.
A valid period may contain an indefinite time interval, being the latest one in the valid period.
An indefinite time interval, as a part of a valid period, is produced if there is no matching release
point to an acquisition point. When a valid period contains an indefinite time interval, the valid
period is called indefinite valid period.











ti tj tk tl

P; P, P;


Figure 16: Special time points in history


Figure 16 illustrates acquisition and release points. Property Pi has two acquisition points ti
and tl and a release point tj, while property Pj has an acquisition point tj and a release point tk.
The valid period of property Pi is an indefinite one.


Birth and death points Among the release points of a history, the one which is not an acquisi-
tion point of another property is termed a death point. In Figure 16, the release point tk is a death
point. Similarly, among the acquisition points, the first one in a history or the first one after a
death point is termed a birth point. In Figure 16, the acquisition points ti and tl are birth points.
Note that a history may have multiple birth and death points. If the history for an object has
more than one birth point (which implies at least one death point), as the one shown in Figure 16,
we say that the object has been reincarnated.


5.1.2 Lifespan

A lifespan of an object is defined to be a time interval between its birth point and its closest death
point. The death point itself is not included in the lifespan. To be more specific, let's assume that
tb is a birth point and td is its closest death point. Then, the relevant lifespan is a time interval,

{tb, tb + 1, ..., td 1}. If there is no matching death point for a birth point, then the relevant
lifespan becomes indefinite. An indefinite lifespan is denoted by (tb oc)9, in the same manner
an indefinite time interval or valid period is denoted. In Figure 16, the relevant object has two
lifespans. One is (ti ~ tk 1), and the other is an indefinite lifespan, (ti ~ oo). The reincarnation
of an object can be stated in another way: if an object has multiple lifespans, the object is said to
be reincarnated.
For the sake of simplicity, we assume that if an object has subordinate objects, all objects
including the main object have the same lifespan. This assumption seems to be reasonable as
insert and delete operations determining lifespan of an object (by specifying the birth and death
points, respectively) usually take effect on an object as a whole1.

9The denotation might have to be (tb ~ oo 1). However, it does not make any practical difference, although the
two are not strictly equal.
10Sometimes we may confront with situations in which this assumption is too restrictive. For instance, as part of
the schema evolution, some attributes may be dropped from (or added to) an existing relation. Then, those affected
attributes may have to be treated as having different lifespan.











Snapshot


ti tj tk tl

Pi Pj P,



Figure 17: Snapshot


5.1.3 Snapshot

Since valid periods of properties in a history are disjoint, it is obvious that given a time point at most
one property becomes valid. By moving the time point along the time axis we can get properties
of an object at various moments. We, informally (somewhat imprecisely), call the property (data
value) obtained at a time point a snapshot. Figure 17 depicts a snapshot. In the figure, the snapshot
at the given time point is property Pj. A formal definition of snapshot is given below.


Definition: Given a property set P and a history H, the snapshot is a function,

sn : T P

given by

sn Pi if t E and (Pi, i) EH
e otherwise

where t is a time point in the (valid) time domain T, Pi is a property in P and c is the empty
property.
Analogous to the valid period, the snapshot provides us with information directly extractable
from a history. While the valid period gives information regarding time periods) an object pos-
sessed a given data value, the snapshot renders information about what data value an object
possessed at a given time point.

5.2 Effect of data manipulation on special time points

In addition to the fact that the special time points are delimiters of a valid period, practically
they are usually the only time points preserved in the database with respect to the validity of data
values. And only data manipulation operations play the role of affecting the temporal information
stored in a database. The information conveyed by data manipulation operations is defined as
follows (object identification is assumed and is not explicitly shown).










I P (t, {tj})
u P (t, {tj})
D c (ti)
c : P (tl, {t})

where I, U, D, and C denote insert, update, delete, and error correction operations, respectively,
and P is a property (data value), and ti and tj are acquisition and release points, respectively. In
the definition, items in braces ({}) are optional.
For insert operation, ti is the acquisition point of P and the birth point of the relevant object.
Usually only ti will be specified considering that in most situations the release point of a property
is not known at the time of insertion. In that case, tj is assumed to be oo, resulting in an indefinite
valid period.
For update operation, ti is the acquisition point of P. In addition, ti also should be the release
point of the previous property because at most one property can be valid at a given time. tj will
not be specified unless release point of P is also known. If tj is omitted, it is assumed to be oo.
For delete operation, the empty property (c) is recorded. That is, the relevant object will lose its
property and cease to exist.11 The acquisition point of the empty property, ti, effectively becomes
the release point of the previous property and eventually a death point of the relevant object. Note
that only ti can be given in delete operation. The release point, say tj, of empty property can be
given only by insert operation (i.e., reincarnation).
The error correction operation is quite general that it operationally encompasses all the other
operations. If a corrective property P is given, it is operationally similar to insert or update
operation. If P is not given, it is operationally a delete operation. However, the error correction
operation needs to be semantically differentiated from the rest of the operations considering user's
intention for its use. That is, insert, update, and delete operations contribute to the normal
evolution of data objects, whereas correction operation is used to rectify errors occurred on the
record of an evolution.
A correction operation can be easily identified by the acquisition point ti which will be less than
or equal to the last special point in a history. For example, suppose that Pk is the current (or the
last) property in a history and tk is the acquisition point of the property, being the final special
point in the history. If an operation adds a property Pi with acquisition point ti being less than
tk, then the operation is actually an error correction operation.
1In section 3 we assumed that if an object has no property the object does not exist.











Insert
ti


( }


Pi
Update
ti tj
] I ]


(b)


Pi i
Delete
ti tj tk
(c) I i I
Pi P


Figure 18: A developing history


5.3 Construction of history

Typically, the history of an object is created by an insert operation, developed by update operations,
and eventually terminated by a delete operation. Once terminated, an object can be reincarnated by
a new insert operation (thereby creating another lifespan). Often, the data manipulation operations
contribute only one special point and that too periodically. Thus, we need a rationale to make up
a complete valid period (regardless of whether it is a definite or an indefinite one) using the values
given as part of the data manipulation operations. The following assumption, called isochronal
recording, is used for that purpose.


Assumption 1 (Isochronal recording) If and only if a property of an object changes, the change
is recorded in the database.12


The above assumption implies that if a change is not recorded (by users), the current property
of the object is assumed to be unchanged and will continue to hold. In other words, if an object
has property Pi at ti, then the object is regarded as still retaining property Pi at ti + 1, unless
a new property Pj is recorded at ti + 1. Based on the assumption, the validity of a property at
intermediate points between an acquisition point and a release point can be derived; that is, the
property of the acquisition point is retained up to its release point.13
Figure 18 shows a history based on the following data manipulation operations:

12Strictly, a change cannot be recorded simultaneously, causing temporal discrepancies between the modeled world
and the recorded world. We will discuss this issue in the next section.
13This behavior of a property corresponds to that of "step-wise constant" time sequences described in [23]. In fact,
there are properties that do not display this behavior. For example, a property describing an act, such as "shipping
a product", is valid for a short period of time, not until the next act. Nonetheless, the isochronal assumption still
works. That is, if a property changes soon after its occurrence, such a change also need be recorded simultaneously
according to the assumption.










I : Pg (ti)
S: P, (tj)

D C(tk).

By the insert operation, the relevant object acquires a data value Pi at ti, and the resultant history
is shown in Figure 18(a). By the update operation, the object gets Pj at tj, and a new history
is shown in Figure 18(b). Lastly, the object loses its data value at tk by the delete operation,
and Figure 18(c) shows the resultant history. Although we have multiple histories for an object as
shown in Figure 18, only the last history is kept in the temporal database with one-dimensional
time.


6 Language and Implementation issues

In this section, we sketch the language and implementation issues for the formalism proposed in this
paper. We believe that our formalism provides a sound basis for various temporal database concepts
without complicating the implementation. We use the relational data model for concreteness as
well as to compare our results with previous work [4, 25, 19, 20].

6.1 Need for a restriction on history

We impose a restriction on the history to ensure that the valid period of a property consists of
a ',.i, time interval. This restriction precludes the recurring valid periods, such as shown in
Figure l(b), to be a part of a history, and simplifies the implementation of temporal validity. If we
allow the valid period to be a set of time intervals, then whenever an object gets a new temporal
data value (that is, the object is updated), the existing history might have to be searched to see if
the same property already exists. If so, a new time interval would be added to the existing valid
period rather than create a new pair of property and valid period for the new temporal data value.
A more serious problem than the search overhead, perhaps, will be that one cannot anticipate
storage (memory/disk) space needed for time-stamps. This will result in waste of space as well as
performance degradation. A ramification of this restriction is that whenever a data value is added
by an update or an error correction operation, the data value is regarded as a new value regardless
of the previous ones. In Figure 15, for example, even though Pi; and Pi2 are the same properties,
they are treated as different ones having their own valid periods of single time interval, Tr, and Tr2.

6.2 Attribute time-stamped relations

Although both attribute and tuple time-stampings have been proposed in the literature along
with their advantages and disadvantages, we adopt the attribute time-stamping in our work. Our
view of an attribute is that of an object that can evolve independently. A major critique against











TEACHERS


Name Address Rank
John Gainesville Assistant
Smith Gainesville Full
Jane Alachua Associate


Figure 19: A non-temporal relation: TEACHERS


TEACHERS

Name Valid time Address Valid time Rank Valid time
John (8/88 ~ oc) Gainesville (8/88 ~ oc) Instructor (8/88 ~ 12/90)
Assistant (1/91 ~ oc)
Smith (1/81 ~ oc) Gainesville (1/81 ~ 10/85) Assistant (1/81 ~ 12/84)
Jacksonville (11/85 ~ 7/86) Associate (1/85 ~ 7/89)
Gainesville (8/86 ~ oc) Full (8/89 ~ oo)
Jane (8/85 ~ 4/89) Ocala (8/85 ~ 4/89) Assistant (8/85 ~ 4/89)
Jane (8/91 ~ o0) Alachua (8/91 ~ o) Associate (8/91 ~ oo)


Figure 20: A temporal relation: TEACHERS


the attribute time-stamping has been its generation of non-1NF (non-first-normal-form) relations.
However, unless one is building an experimental temporal database system using an existing non-
temporal relational one, the tuple time-stamping has little advantage over the attribute time-
stamping but causes enormous space waste and fragmentation of information pertinent to an entity.
Moreover, regarding attribute time-stamped relations as non-1NF is inappropriate in that multiple
data values which an attribute has during its evolution are different than a set value of non-temporal
databases. In the former case, only one data value may be valid at a given time point, whereas the
whole set is currently valid in the latter case. In this respect, an attribute time-stamped relation
may be thought of as 1NF since multiple data values for an attribute in such a relation are still
atomic.
In order to implement the attribute time-stamping, a pair of time-stamps are associated with
each attribute value, representing a valid period of the value. A set of data values and their
time-stamps belonging to an attribute of tuple represents a history of the attribute.
Figure 19 shows a traditional relation TEACHERS whose key attribute is "Name". The relation
is extended to a temporal one by time-stamping each attribute as shown in Figure 20. Note that
the key attribute value does not change while other non-key attribute values (may) change.
As a tuple is inserted, all its attributes get time-stamped by setting the birth point to a specified
value and the death point to oo. In Figure 20, Smith's rank has been evolved by the following four










operations:


I : Assistant (1/81)
U : Full* (1/',)
C : Associate (1/',)
U : Full (8/89).

The insert operation set the acquisition point of "Assistant" to 1/81. The rank was wrongly updated
to "Full*" by the first update operation. The update set the release point of Assistant to 1/" -" and
the acquisition point of Full* to 1/"' too. The error was found later and the rank was corrected
to "Associate". By the correction, the effect of the first update was removed from the relation and
the value Associate got the acquisition point, 1/" -" The rank was updated to "Full" by the second
update operation. By the update, the release point of Associate and the acquisition point of Full
were both set to 8/89; thus, the valid period of Associate became (1/' 7/89).
Similarly, Jane's rank has evolved through the following operations:

I : Assistant (8/,-.)
D : (5/89)
I : Associate (8/91).

The operations -i,:-: -1 that Jane left the university in 5/89 and rejoined in 8/91.
Note that insert and delete operations take effect on a whole tuple rather than an attribute,
thereby all attributes in a tuple have the same lifespan. All attributes of tuple Jane, for example,
have lifespan, (8/". ~- 4/89) and (8/91 ~ oc). In other words, the object representing Jane was
reincarnated.

6.3 Temporal query languages

Devising query languages over temporal databases has been a dominant part of the temporal
database research. We already presented two rudimentary constructs of temporal query languages,
the valid period and snapshot. Similar language constructs to the snapshot or more sophisticated
forms are found in all work dealing with the temporal query language. They are usually termed
I,. I," or i ii. --I .. clause. The following temporal query is borrowed from TQuel of [25].

range of F is TEACHERS
retrieve F.Name
where F.Rank = "Associate"
when F.Rank overlap 1/89










The answer to the query is "Smith", since the when clause specifies that only attributes whose
valid periods contain the time point, 1/89 be selected.14
The counterpart to the valid period as a language construct is also found in some researchers'
work, but in more diverse and primitive forms. The syntax of the following temporal query is from
TSQL of [19].

select Rank.time-start, Rank.time-end
from TEACHERS
where Name = "Jane" and Rank = "Associate"
time-slice Now

The suffixes, I,.. --I.ilt" and i i -.. .l" in select-clause return a valid period of an attribute
value, and the i ..--li..." clause is similar to the when clause. Accordingly, the answer should
be (8/91, 00).15 However, a language construct implementing the valid period needs to be more
sophisticated, because although we restricted a valid period to be a single time interval, in reality a
data value may have a valid period composed of several time intervals. Smith's address, Gainesville,
for example, has a valid period composed of two time intervals. For such a case, the constructs
time-start and time-end do not work properly.


7 Conclusions

In this paper, we have formally defined the notions of temporal and sort of validity. Use of a char-
acteristic function provides an interpretation based mechanism for defining the notion of validity.
This is analogous to the interpretation of clauses in first-order logic which is essentially based on
the domain as well as the interpretation assigned to constants and functions.
With the notions defined in this paper, we have been able to dispel the confusion among various
time concepts. Also we have shown that retroactive and proactive updates cannot be modeled using
only valid and transaction times. In order to resolve this problem, we have proposed the event time.
Furthermore, we have clarified the combinations of multidimensional times needed for preserving
multiple pasts generated by retroactive updates, error corrections, and delayed updates. In addition,
the notion of history has been defined formally and its implementation using the relational data
model has been sketched.
Handling multidimensional times in temporal databases requires a systematic interpretation
methodology. We have extended this work to two-dimensional representations of time which corre-
spond to the combinations of valid and transaction times as well as valid and event times [16]. In
1The original TQuel is based on the tuple time-stamping. Thus, the query shown above and its interpretation do
not precisely coincide with those of TQuel.
1Again, TSQL is based on the tuple time-stamping. The original TSQL will not allow such a query on attribute
time-stamped tables.










the two-dimensional cases, the notion of valid period is extended to valid pattern and the notion of
history to multi-history. Currently, we are investigating the three-dimensional representation using
all the three time concepts.


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