An Interval Algebra for Indeterminate Time

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1 From AAAI- Proceedings. Copyright 2, AAAI ( All rights resered. An Interal Algebra for Indeterminate Time Wes Cowley Department of Computer Science and Engineering Uniersity of South Florida P.O. Box 213 Tampa, FL Dimitris Plexousakis Department of Computer Science Uniersity of Crete & ICS-FORTH P.O. Box 22, Heraklion Greece 7149 Abstract Temporal indeterminacy is an inherent problem which arises when capturing and manipulating temporal data in many application areas. As such, representation and manipulation of timestamps with indeterminacy is a requirement for these applications. We present an extension of Allen s thirteen interal relationships to indeterminate temporal interals based on a noel representation for indeterminate timestamps. The timestamps can be deried from and translated to interal constraints. We proide a set of simple and useful operators for manipulating both conex and non-conex indeterminate interals represented by these timestamps. Introduction Temporal indeterminacy exists when one does not know precisely either the location of an eent on the time line or its duration. The temporal knowledge about the eent is indefinite or incomplete. This is an inherent problem when reasoning in many application domains. Planning systems are sometimes faced with the infeasibility of completely modeling all aspects of a situation. Knowledge representation and reasoning systems may be faced with insufficient information to completely determine a desired time interal. The knowledge about the time interal is indeterminate one may know a period of time the interal definitely coers but there may also be a period of time into which the interal may extend. The prealent method for handling temporal indeterminacy in the temporal reasoning community is to use a temporal constraint network (TCN) on interals (Allen 193). The TCN allows the reasoner to arrie at a disjunction of possible relationships between pairs of interals. For example, Anger, et al. describe a temporal knowledge base using this approach (Anger et al. 19). The knowledge of the location and duration of an eent on the time line is encoded in the relationships between that eent and others in the network. We examine a noel representation for indeterminate interals based on the endpoints of the interals determinate and indeterminate regions and define a set of operators, which are both useful and non-complex, for manipu- Copyright cfl 2, American Association for Artificial Intelligence ( All rights resered. lating indeterminate timestamps. Conersions between interal constraints in the form of the thirteen interal relationships (Allen 193) and the indeterminate timestamps is straightforward. We use the indeterminate interals to extend the traditional interal relationships to allow for indeterminacy. To the best of our knowledge, this is the first effort in this direction. Our approach differs from TCNs in that the knowledge of an interal s position and duration is encoded in the interal s representation rather than in relationships between interal, allowing for non-complex manipulation of the knowledge. One application of this is in temporal integrity constraints for knowledge bases such as Telos (Plexousakis 1995). Another method associates characteristic functions with interals to represent the leel of possibility that any gien point is actually contained in the interal (Bouzid & Mouaddib 199). In this approach, the characteristic function of interals are modified as more knowledge is gained and schedules eole. A similar method, in the temporal database field, describes a timestamp with a probability function (Dyreson & Snodgrass 199). The probability and characteristic function approach is both more flexible and more complex than that described here as it allows for reasoning oer the likelihood that a point is in the interal or not. In our method a point is either definitely in the interal or possibly in the interal. In another approach, Gadia, et al propose a set oriented representation which supports both indeterminacy and incompleteness with a three-alued logic (Gadia, Nair, & Poon 1992). Their work is the closest to the indeterminate interals described here, but proides a different set of operators. They do not describe a method for translating between interal constraint notation and their representation, complexity of the operators, or algorithms. Our work does not examine temporally missing alues as Gadia does. Koubarakis examines indeterminacy in the context of temporal constraint databases, where temporal indeterminacy is described by local and global constraints on ariables representing the end points of the interals (Koubarakis 1994; 1997). This is similar to the interal oriented TCN approach in that knowledge of a time interal is encoded in the constraints. The purpose in this case is less oriented toward deduction and more toward finite representation of

2 potentially infinite domains. In the remainder of this paper we will first describe our assumed temporal structure and chosen representation for indeterminate interals followed by a discussion of an extension of the interal relationships to handle indeterminacy. Finally, we will gie our conclusions and aenues for future work. Valid Interal Stamps Motiation Indeterminate time may arise in a number of ways and is nearly unaoidable when dealing with the alid time of facts in a knowledge base. We are interested in associating a alid time period with facts in order to support a temporal integrity constraint implementation, such as in, e.g., (Plexousakis 1995; Cowley 1999) in which constraints include combinations of the thirteen interal relationships (Allen 193). As such, it is appropriate that the alid time periods are also specified using interal relationships. Because most of these relationships do not precisely constrain the time periods inoled, the resulting time stamps are indeterminate. Thus, we must be able to derie indeterminate alid time stamps from interal constraints, manipulate those indeterminate interals, and translate the results back to interal constraints. Points We assume a linear discrete time line (Benthem 1991) bounded by the range of the underlying integer type in which temporal points are represented. We further assume distinct elements 1, 1, andnil. With the exception of the incomparable element nil, which represents a non-specified alue, points are totally ordered by. The relationships, =, and» between two points carry the expected meaning. More formally, we assume a temporal structure T which is comprised of a set of points P = f ;p 1 ;p ;p 1 ;g and a relation. The set P = P [ f 1; 1; nilg is isomorphic to the set Z = Z [ f 1; 1; nilg. We define a one-to-one and onto mapping function, P!Z, and its inerse as follows ρ p if p = 1; 1; or nil (p) = i for p i ;i2z ρ 1 i if i = 1; 1; or nil (i) = for i 2Z p i The relation between members of the set P is defined as p 1 p 2, (p 1 ) (p 2 ) and can be extended to the special elements 1 and 1 by obsering that p 2P 1p 1. Finally, is undefined if either operand is nil. Similar translations gie the meanings for = and». Note that from the definition of, p i = p j ) i = j. The following functions permit us to determine which of a pair of points occurs earlier or later on the time line. Definition 1. We define The functions min p, max p, min p, and max p P P!P as follows min p (p 1 ;p 2 )= max p (p 1 ;p 2 )= min p (p 1 ;p 2 )= max p (p 1 ;p 2 )= nil if p 1 = nil _ p 2 = nil p 1 if p 1 p 2 p 2 otherwise nil if p 1 = nil _ p 2 = nil p 1 if p 2 p 1 p 2 otherwise nil if p 1 = nil ^ p 2 = nil p 1 if p 2 = nil _ p 1 p 2 p 2 otherwise nil if p 1 = nil ^ p 2 = nil p 1 if p 2 = nil _ p 2 p 1 otherwise p 2 min p (max p ) chooses the earliest (latest) operand if both are not nil. Where exactly one operand is nil, min p and max p return the other point. Gien a point p, we need to refer to its next and preious points on the time line. Definition 2. The functions preious and next, each with signature P!P, are defined as follows ρ p if p 2 f 1; 1; nilg preious(p) = 1 ( (p) 1) otherwise ρ p if p 2 f 1; 1; nilg next(p) = 1 ( (p) +1) otherwise Determinate Interals In order to discuss periods of time, as opposed to instants, we use interals. Definition 3. A conex interal is a set of consecutie points. An interal I can be represented by its lowest and highest points hi s ;I e i;i s» I e, I s = nil, I e = nil. We say that p 2 I iff I s» p» I e. The empty interal, which contains no point, is represented by hnil; nili or ;. Wesay that a conex interal is infinite if one or both endpoints is 1 or 1. I refers to the set of all conex interals. Points may be implicitly conerted to interals by the function interal P!I, definedasinteral(p) =hp; pi. We will use the 13 basic interal relationships (Allen 193) for comparing interals for ordering and inclusion. The concept of intersection of conex interals is important. This carries the same meaning as in set theory, namely that there is at least one point in common. Definition 4. Two conex interals I 1 and I 2 intersect if I 1s» I 2e ^ I 2s» I 1e. Note that the empty interal, hnil; nili does not intersect with any interal due to the incomparability of nil. The infinite interal, h 1; 1i intersects with all non-empty interals. We also need the concept of adjacency of conex interals, which is defined next.

3 Definition 5. We say that two interals I 1 and I 2 are adjacent if next(i 1e )=I 2s _ next(i 2e )=I 1s. We will also need the concept of non-conex interals to talk about sets of non-consecutie points. Definition 6. A non-conex interal I is a set of conex interals I i which neither intersect nor are adjacent. Indeterminate Interals A alid interal stamp (VIS) is an extension of the interal concept. A VIS describes the set of points which are definitely in the interal as well as those which may be in the interal. Through this mechanism we can represent the indeterminacy inherent in interal constraints. Definition 7. A conex alid interal stamp (CVIS) is a 4- tuple V = hi s ;D s ;D e ;I e i associated with a fact in a knowledge base. The points are restricted so that D s = nil, D e = nil and D s = nil ) (I s = nil, I e = nil). D s ;D e, if not nil, are time stamps marking the end points of the determinate interal D; that period of time during which the associated fact is true. I s and I e, if not nil,represent the extended period of time before and after the determinate interal, respectiely, during which the associated fact may be true. The indeterminate interal is the non-conex interal I = fi L ;I H g where I L and I H are ; If I s = nil I L = hvi s ;VI e i If D = ; hvi s ; preious(vd s )i otherwise ρ ; If Ie = nil _ D = ; I H = hnext(vd e );VI e i otherwise A CVIS for which both I s and I e are nil is called fully determinate, while one with both D s and D e being nil is called fully indeterminate. The empty VIS is represented as hnil; nil; nil; nili or ;. We will use CV to refer to the set of all CVISs. An interal I can be conerted implicitly to a CVIS with the function cis I!CV,definedascis(I) = hnil;i s ;I e ; nili Example 1. Figure1ashowstheCVISh1; 2; 4; 6i. This contains the points f2; 3; 4g determinately and f1; 5; 6g indeterminately. Figure 1b shows the same period coered by the fully indeterminate CVIS h1; nil; nil; 6i. Finally, Figure 1c shows the left infinite CVIS h 1; 4; 6; nili. We will often need to obtain the earliest and latest points in a CVIS independently of whether that point is determinate or not. Definition. We define the functions upper and lower as upper(v ) = max p (VD e ;VI e ) and lower(v ) = min p (VD s ;VI s ), both with signature CV! P. We will also need to know the latest point at which a CVIS V can start and the earliest point at which it can end. To understand these functions, obsere that if a CVIS V has a determinate region then the interal will start no later than (a) (b) 1 3 (c) 4 6 Figure 1 Examples of conex alid interal stamps VD s and end no earlier than VD e. On the other hand, if V is fully indeterminate then it may start as late as VI e and end as earlier as VI s. Definition 9. We define two functions, maxlo and minup, with signature CV!P as maxlo(v ) = min p (VD s ;VI e ) and minup(v ) = max p (VI s ;VD e ). The following extends the definition of membership in an interal. Definition 1. We denote membership of a point p in a CVIS V as p 2 V and define that membership as p 2 V, lower(v )» p» upper(v ). We will also use p 2 VD to represent p s determinate inclusion in V and p 2 VI to represent p s indeterminate inclusion in V. Next we extend the concept of intersection to conex alid interal stamps. Definition 11. Two CVISs, V 1 and V 2, intersect if lower(v 1 )» upper(v 2 ) ^ lower(v 2 )» upper(v 1 ).Wesay V 1 and V 2 determinately intersect if V 1 D 6= ;^V 2 D 6= ;^V 1 D intersects V 2 D. The concept of adjacency can be extended to conex alid interal stamps as follows. Definition 12. We say that two CVISs, V 1 and V 2,areadjacent if next(upper(v 1 )) = lower(v 2 )_next(upper(v 2 )) = lower(v 1 ). We say that V 1 and V 2 are determinately adjacent if V 1 D 6= ;^V 2 D 6= ;^V 1 D adjacent V 2 D. Note that two CVISs which are determinately adjacent may intersect but will not determinately intersect. The work of (Cowley 1999) depicts how to translate from interal constraints of the form R I, wherer is an interal relationship and I is a determinate interal, to a conex alid interal stamp. We will also need to consider non-conex VISs. Definition 13. A non-conex alid interal stamp (NVIS) is a finite set of CVISs which meet two conditions. 1) No two CVISs may intersect. 2) If any two CVISs are adjacent, then both must hae a definite interal and they must not be determinately adjacent. NV refers to the set of all NVISs. A CVIS can be implicitly conerted to an NVIS with the

4 function nis CV! NV, definedasnis(c) =fcg. Definition 14. V = CV [ NV is the set of all VISs. Operators on Valid Interal Stamps There are a number of operators on VISs which correspond to the similarly named ones on Boolean expressions and sets. Specifically, we will define conjunction, intersection, union, and difference of VISs below. To aoid confusion, we will annotate the usual operators to emphasize that they carry different semantics than the familiar ones. For example, we use [ for the union of VISs. When operands are known to be conex (non-conex) VISs, we will use c [ ( n [). When the familiar semantics from logic or set theory are sufficient, we will use the unannotated operators. Conjunction There are times when a single interal constraint may not adequately express the knowledge one has about a fact s alid time. For that we must conjoin multiple constraints. We will do this by way of the VIS conjunction operator 1 ^. The effect of ^ is to produce a determinate region which includes the determinate region of both operands and an indeterminate region which includes only points in the indeterminate region of both operands. The operator uses the knowledge expressed by two VISs in order to increase the accuracy to which the alid time of a fact is known. In order to compute the resulting VIS, V r, from the conjunction of two existing CVISs, V 1 and V 2,wemustfirst ensure that the conjunction is satisfiable. For example, there are no dates which satisfy the condition contains Jan-9 and during Mar-9. On the other hand, contains Jan-9 and after No-97 can be satisfied. Each operand restricts the range of the other. Definition 15. Two CVISs, V 1 and V 2,areconjunction compatible, denoted V 1 ^comp V 2,if1) V 1 intersects V 2,2) V 2 D 6= ; ) (lower(v 1 ) = V 2 D s ^ V 2 D e = upper(v 1 )), and 3) V 1 D 6= ;)(lower(v 2 ) = V 1 D s ^ V 1 D e = upper(v 2 )). We extend this definition to NVISs by saying that two NVISs are conjunction compatible if V 1i 2 V 1 9V 2j 2 V 2 such that V 1i ^comp V 2j and V 2j 2 V 2 9V 1i 2 V 1 such that V 2j ^comp V 1i. Conjunction compatibility ensures that the indefinite region of each interal encloses the definite region of the other. Otherwise, there would be a point which is in the definite region of one operand but not in the other operand at all. That is the condition which results from inconsistent interal constraints. Definition 16 ( ^). If V 1 and V 2 are VISs such that V 1^comp V 2 we define their conjunction, denoted by V 1 ^ V2,asthe 1 The term conjunction refers to the operation on interal constraints which ^ supports. It might be more appropriate to think of the effect on a VIS as strengthens. VIS V r such that V r D = fpjp 2 V 1 D _ p 2 V 2 Dg and V r I = fpjp 2 V 1 I ^ p 2 V 2 Ig. If(V 1 ^comp V 2 ) then V 1 ^ V2 = ;. Because the order of the interal constraints in a alid time specification should not matter, it is important that the order of VISs with respect to ^ not matter. Hence, this theorem, which is straightforward to show. Theorem 1. (V; ^) forms a commutatie monoid with identity ^ = h 1; nil; nil; 1i. Intersection The intersection of two CVISs is defined similarly to conjunction. The precondition is not as restrictie. It is simply necessary that the interals intersect by definition 11. On the other hand, the result of intersection is more restrictie than that of conjunction. Specifically, the determinate region produced by conjunction includes the determinate regions of both operands. If a point is determinate in only one of the operands, it is determinate in the result. For intersection, the determinate region includes only those points determinately in both operands. If a point is determinately in one operand but indeterminately in the other, the intersection contains that point indeterminately. Definition 17 ( ). We define the intersection of two VISs, V 1 and V 2, denoted by V 1 V2,astheVIS V r such that V r D = fpjp 2 V 1 D ^ p 2 V 2 Dg and V r I = fpjp 2 V 1 ^ p 2 V 2 ^ (p 2 V 1 I _ p 2 V 2 I)g. As with ^, the following is straightforward to proe. Theorem 2. (V; ) forms a commutatie monoid with identity = hnil; 1; 1; nili Union When an insert operation is performed for a fact whose non-temporal attributes are the same as an existing fact one of two things must happen. Either the operation is treated as a new insert, resulting in a pair of facts which differ only in their alid interal stamps; or the VIS of the existing fact is updated to include the alid time specified for the insert. We proide the union operator to support the latter semantics. Definition 1 ( [). We define the union of two VISs, V 1 and V 2, denoted by V 1 [ V2,astheVISV r such that V r D = fpjp 2 V 1 D _ p 2 V 2 Dg and V r I = fpjp 62 V r D ^ (p 2 V 1 I _ p 2 V 2 I)g. Again, the following is straightforward. Theorem 3. (V; [) forms a commutatie monoid with identity [ = hnil; nil; nil; nili. Difference When performing a deletion or update operation where the alid time specified for the operation is only a

5 portion of the alid time for the affected facts, we will need to produce the difference of two alid interal stamps. In general, such a difference results in a non-conex interal. Definition 19 ( ). We define the difference of two VISs, V 1 and V 2, denoted by V 1 V2 as the VIS V r such that V r D = fpjp 2 V 1 D ^ p 62 V 2 g and V r I = fpj(p 2 V 1 D ^ p 2 V 2 I) _ (p 2 V 1 I ^ p 62 V 2 D)g. Note that is neither commutatie nor associatie. The identity element, denoted by, is the empty interal. Complexity of VIS Operators The basic operators on CVISs ^comp, c^, c, c [,and c, are all constant time. This should be clear from their definitions, which depend solely on the alues of the end points. For an NVIS V, we will use jv j to indicate the number of CVISs V i 2 V. Based on an analysis of the algorithms deeloped to implement the operators, we hae the results in table 1 for NVISs. Those algorithms and proofs of these results are omitted due to lack of space. Operator Complexity ^comp O(jV 1 jjv 2 j) n ^ O(jV1 jjv 2 j) n O(jV1 jjv 2 j) n [ O((jV1 jjv 2 j) 2 ) n O(jV 2 j 4 jv 1 j) Table 1 Complexity of Operators oer NVISs Indeterminate Interal Relationships Now we turn to comparing the indeterminate interals by examining how the thirteen interal relationships used in Allen s interal algebra (Allen 193) can be extended to the indeterminate interals represented by CVISs. We will need to define both definite and potential satisfaction of the relationships when applied to CVISs, resulting in a total of 26 interal relationships. Two principles are used in reaching these definitions. Principle 1. For V 1 definitely R V 2 to hold, where R is an interal relationship, we must ;V 2 such thatv 1 ^comp V1 ^ V 2 ^comp V2 ^ ((V 1 1 )D R (V 2 2 )D) If this principle is met, it is not possible to conjoin any compatible interal constraint to either V 1 or V 2 such that R will not be satisfied. Principle 2. For V 1 potentially R V 2 to hold, where R is an interal relationship, we must hae 9V 1 ;V 2 such thatv 1 ^comp V 1 ^ V 2 ^comp V 2 ^ (V 1 1 ) definitely R (V 2 2 ) This ensures that it is possible, ia the conjunction of additional interal constraints with V 1 and V 2, to arrie at a pair of interals which definitely satisfy R. Definition 2. For two CVISs, V 1 and V 2,wesay ffl V 1 definitely equals V 2 if V 1 I = ;^V 2 I = ;^V 1 D = V 2 D ffl V 1 potentially equals V 2 if lower(v 1 )» maxlo(v 2 ) ^ lower(v 2 )» maxlo(v 1 ) ^ minup(v 2 )» upper(v 1 ) ^ minup(v 1 )» upper(v 2 ) ffl V 1 definitely before V 2 if ffl V 1 potentially before V 2 if ffl V 1 definitely meets V 2 if upper(v 1 ) lower(v 2 ) minup(v 1 ) maxlo(v 2 ) V 1 I e = nil ^ V 2 I s = nil ^ V 1 D e = V 2 D s ^ V 1 D s V 2 D s ^ V 1 D e V 2 D e ffl V 1 potentially meets V 2 if lower(v 1 ) maxlo(v 2 ) ^ minup(v 1 ) upper(v 2 ) ^ lower(v 2 )» upper(v 1 ) ^ minup(v 1 )» maxlo(v 2 ) ffl V 1 definitely oerlaps V 2 if V 1 D s lower(v 2 ) ^ upper(v 1 ) V 2 D e ^ V 2 D s V 1 D e ffl V 1 potentially oerlaps V 2 if lower(v 1 ) maxlo(v 2 ) ^ minup(v 1 ) upper(v 2 ) ^ lower(v 2 ) upper(v 1 ) ^ lower(v 1 ) preious(upper(v 1 )) ^ lower(v 2 ) preious(upper(v 2 )) ffl V 1 definitely during V 2 if V 2 D s lower(v 1 ) ^ upper(v 1 ) V 2 D e ffl V 1 potentially during V 2 if lower(v 2 ) maxlo(v 1 ) ^ minup(v 1 ) upper(v 2 ) ^ lower(v 2 ) preious(upper(v 2 )) ffl V 1 definitely starts V 2 if V 1 I s = nil ^ V 2 I s = nil ^ V 1 D s = V 2 D s ^ upper(v 1 ) minup(v 2 ) ffl V 1 potentially starts V 2 if lower(v 1 )» maxlo(v 2 ) ^ lower(v 2 )» maxlo(v 1 ) ^ minup(v 1 ) upper(v 2 ) ffl V 1 definitely finishes V 2 if V 1 I e = nil ^ V 2 I e = nil ^ V 1 D e = V 2 D e ^ V 2 D s lower(v 1 )

6 ffl V 1 potentially finishes V 2 if minup(v 2 )» upper(v 1 ) ^ minup(v 1 )» upper(v 2 ) ^ lower(v 2 ) maxlo(v 1 ) We can make some interesting obserations about the compatibility of the arious interal relationships. First, we note that the interal relationships between determinate interals are mutually exclusie. Lemma 1. Let I 1, I 2 be interals and R 1, R 2 be determinate interal relationships. Then, I 1 R 1 I 2 )(I 1 R 2 I 2 ). Proof. This follows directly from the definitions of the relationships. This obseration extends to the definite relationships between alid interal stamps. Theorem 4. Let V 1 and V 2 be alid interal stamps and let R 1 and R 2 be determinate interal relationships, R 1 6= R 2. Then, V 1 definitely R 1 V 2 )(V 1 definitely R 2 V 2 ). Proof. Assume that V 1 definitely R 1 V 2. Then 9V1 ;V 2 such that (V 1 1 )D R 1 (V 2 2 )D. By lemma 1, we know that ((V 1 1 )D R 2 (V 2 2 )D). Principle 1 then requires that (V 1 definitely R 2 V 2 ). We cannot make the same conclusion regarding potential relationships, howeer. Theorem 5. Let V 1 and V 2 be alid interal stamps and let R 1 and R 2 be determinate interal relationships, R 1 6= R 2. Then, V 1 potentially R 1 V 2 6) (V 1 potentially R 2 V 2 ). Proof. By example Let p 1 and p 2 be points, p 1 p 2. Let V 1 = V 2 = hp 1 ; nil; nil;p 2 i. Choose V1 = V 2 = hnil;p 1 ;p 2 ; nili. Then, (V 1 1 )D equals (V 2 2 )D. By principle 2 V 1 potentially equals V 2. Now, choose V1 = hnil;p 1 ;p 1 ; nili and V2 = hnil;p 2 ;p 2 ; nili. So, (V 1 1 )D before (V 2 V 1 potentially before V 2. 2 )D and Finally, we can see that, as expected, a definite relationship implies a potential one. Theorem 6. Let V 1 and V 2 be alid interal stamps and let R be a determinate interal relationship. Then, V 1 definitely R V 2 ) V 1 potentially R V 2. Proof. Assume V 1 definitely R V 2. Choose V 1 = V 2 = ^. Then V 1 1 = V 1 and V 2 2 = V 2, so (V 1 1 ) definitely R (V 2 2 ). This meets the conditions for principle 2. Therefore, V 1 potentially R V 2. Conclusions and Future Work We hae shown a representation for both conex and nonconex indeterminate interals and proided a useful set of operators on those interals. Further, we hae shown that it is possible to extend the relationships used in Allen s interal algebra (Allen 193) to the indeterminate interals represented by conex alid interal stamps. These extensions are interesting in their own right because of the large amount of prior work in both the artificial intelligence and database fields that is based on the interal algebra and its ariants. One specific application area is in allowing temporal integrity constraints to take into account indeterminism. An obious extension of this current work would be to reformulate Allen s constraint propagation algorithms to use indeterminate interals and inestigate resulting differences in complexity and reasoning power. It would also be interesting to reformulate the potential and definite relationships presented here using interals treated as primitie constructs instead of based on end points. For determinate interals, all thirteen relationships can be defined in terms of either meets or before. This would proide the ability to use dense or continuous timelines as well as discrete. Acknowledgements The authors would like to thank Dr. Manolis Koubarakis and the anonymous referees for their comments. References Allen, J. F Maintaining knowledge about temporal interals. Communications of the ACM 26(11) Anger, F. D.; Mata-Toledo, R. A.; Morris, R. A.; and Rodriguez, R. V. 19. A relational knowledge base with temporal reasoning. In Proceedings of the First Florida Artificial Intelligence Research Symposium, Benthem, J The Logic of Time. Dordrecht, Holland Kluwer Academic Publishers. Bouzid, M., and Mouaddib, A.-I Uncertain temporal reasoning for the distributed transportation scheduling problem. In Proceedings Fifth International Workshop on Temporal Representation and Reasoning, Cowley, W Temporal integrity constraints with temporal indeterminacy. Master s thesis, Uniersity of South Florida. Dyreson, C. E., and Snodgrass, R. T Supporting alid-time indeterminacy. Transactions on Database Systems 23(1)1 57. Gadia, S. K.; Nair, S. S.; and Poon, Y.-C Incomplete information in relational temporal databases. In Proceedings of the 1th International Conference on Very Large Data Bases, Koubarakis, M Database models for infinite and indefinite temporal information. Information Systems 19(2) Koubarakis, M The complexity of query ealuation in indefinite temporal constraint databases. Theoretical Computer Science 171(1 2)25 6. Plexousakis, D Compilation and simplification of temporal integrity constraints. In Rules in Database Systems Second International Workshop,