Ieee Transactions On Fuzzy Systems, Vol. 16, No. 2, April 2008 517 .
IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 2, APRIL 2008517Fuzzifying Allen’s Temporal Interval RelationsSteven Schockaert, Martine De Cock, and Etienne E. KerreAbstract—When the time span of an event is imprecise, it canbe represented by a fuzzy set, called a fuzzy time interval. In thispaper, we propose a framework to represent, compute, and reasonabout temporal relationships between such events. Since our modelis based on fuzzy orderings of time points, it is not only suitable toexpress precise relationships between imprecise events (“Rooseveltdied before the beginning of the Cold War”) but also imprecise relationships (“Roosevelt died just before the beginning of the ColdWar”). We show that, unlike previous models, our model is a generalization that preserves many of the properties of the 13 relationsAllen introduced for crisp time intervals. Furthermore, we showhow our model can be used for efficient fuzzy temporal reasoningby means of a transitivity table. Finally, we illustrate its use in thecontext of question answering systems.TABLE IALLEN’S TEMPORAL INTERVAL RELATIONS BETWEEN INTERVALSA [a ; a ] AND B [b ; b ]Index Terms—Fuzzy ordering, fuzzy relation, interval algebra,question answering, temporal reasoning.I. INTRODUCTIONEMPORAL representation and reasoning is an importantfacet in the design of many intelligent systems. For example, question answering systems require at least some basictemporal representation scheme to answer simple temporalquestions such as “When was Franklin Roosevelt born?” Toenable question answering systems to answer more complextemporal questions, considerable effort has been made to extracttemporal information from natural language texts (e.g., [1],[12], [15], [16], and [23]–[25]) and to analyze complex temporal questions (e.g., [22]). However, temporal relationshipsexpressed in natural language are often vague, e.g., “Rooseveltdied just before the end of the Second World War.” Moreover,historic time periods are more often than not characterizedby a gradual beginning and/or ending [17]. The traditionaltemporal reasoning formalisms need to be extended to copewith this kind of vagueness, which is inherently associated withreal–world temporal information.One of those well-known formalisms is Allen’s temporal interval algebra [3]. Allen defined a set of 13 qualitative relationsthat may hold between two compact intervalsand. Table I shows how Allen expressed these preciserelations by means of constraints on the boundaries of the crispintervals involved. In this paper, we extend Allen’s work to amore general formalism that can handle precise as well as imprecise relationships between crisp and fuzzy intervals.TManuscript received December 6, 2006; revised January3, 2007. The work ofS. Schockaert was supported by the Research Foundation—Flanders.S. Schockaert is with the Department of Applied Mathematics and ComputerScience, Ghent University, 9000 Gent, Belgiumj, and also with the ResearchFoundation—Flanders, Belgium (e-mail: Steven.Schockaert@UGent.be).M. De Cock and E. E. Kerre are with the Department of Applied Mathematicsand Computer Science, Ghent University, 9000 Gent, Belgium (e-mail: Martine.DeCock@UGent.be; Etienne.Kerre@UGent.be).Digital Object Identifier 10.1109/TFUZZ.2007.895960Our first concern is generalizing the definitions of the qualitative relations of Table I to make them applicable to fuzzy intervals as opposed to only crisp intervals. Indeed, when an event ischaracterized by a gradual beginning and/or ending, it is naturalto represent the corresponding time span as a fuzzy set, whichwe call a fuzzy (time) interval. Depending on the intended application, this fuzzy set can either be defined by an expert (e.g.,[17] and [20]) or constructed automatically (e.g., [27]). Sincewe cannot refer to the gradual beginning and endings of a fuzzyinterval in the same way we refer to the boundaries of a crispinterval, we first have to provide a way to express that, for instance, the beginning of a fuzzy interval is before the begin(as needed in the definition of thening of a fuzzy intervalqualitative relation “overlaps”). We suggest to do this by measuring the highest extent to which there exists a time point inthat occurs before all the time points in . In general, in our approach, qualitative relations between fuzzy intervals are definedin terms of the ordering of the gradual beginning and endings ofthese intervals, which in turn are defined in terms of the orderingof the time points belonging to these intervals. The resultingqualitative relations between the fuzzy intervals are gradual, i.e.,they may hold to some degree only; hence the name fuzzy temporal interval relations. When and are crisp, our approachreduces to Allen’s work.Our second goal is providing a means to model impreciserelations to be able to express that, for instance, event tookplace just before event , or that occurred long after . Although these kind of relations are not considered in Allen’s original model, in our approach we arrive at them quite elegantly byusing imprecise orderings of time points in the model sketchedabove. The resulting approach is applicable again to both crispand fuzzy time intervals.This paper is organized as follows. In the next section, we review related work concerning fuzzifications of Allen’s intervalrelations. In Section III, we show how imprecise relationships1063-6706/ 25.00 2008 IEEE
518IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 2, APRIL 2008between time points can be modelled by using fuzzy orderings.In Section IV, we rely on relatedness measures for fuzzy sets tolift these imprecise orderings of time points into relationshipsbetween fuzzy time intervals [29]. This results in a generalization of Allen’s 13 interval relations that are also applicable whenthe time intervals are fuzzy. Furthermore, this framework ispowerful enough to additionally express imprecise relationshipsthat are not considered in Allen’s original model. We show thatour model preserves many important properties regarding (ir)reflexivity, (a)symmetry, and transitivity, and that our generalizeddefinitions remain mutual exclusive and exhaustive. Moreover,in Section V, we discuss fuzzy temporal reasoning and introduce a transitivity table to derive new temporal knowledge inan efficient way. This transitivity table is a generalization of thetransitivity table that was introduced by Freksa in [13], whichshows that no transitivity properties are lost in our generalizedframework. Section VI illustrates the usefulness of our approachwithin the context of question answering systems. Sections IIIand IV contain many new results that require a mathematicalproof. To preserve the continuity of the main text, we presentthese proofs in the Appendixes.Nagypál and Motik [17] and of Ohlbach [20], which areconcerned with generalizing Allen’s interval relations whenthe time span of an event is represented as a fuzzy set. However, these approaches suffer from a number of importantdisadvantages. For example, the relation “equals” defined in[17] is not reflexive in general; for a continuous fuzzy setin ,,while, taking into account Allen’s intended meaningandof these relations, one would expect. Moreover, imprecisetemporal relations cannot be expressed. An approach similar to[17] was suggested in [8] within the context of ranking fuzzynumbers. Ohlbach [20] suggests an alternative approach thatallows one to model imprecise temporal relations such as “more or less finishes ” based on measures of overlap forfuzzy sets. However, this approach cannot handle imprecisetemporal relations such as “ was long before .” Moreover,as pointed out in [20], many of the (ir)reflexivity, (a)symmetry,and transitivity properties of the original temporal relationsare lost in this approach; hence it is not suitable for temporalreasoning. Imprecise temporal relations are also consideredin [10]; however, only crisp intervals are considered in thisapproach.II. RELATED WORKMost fuzzifications of Allen’s interval algebra deal with uncertainty rather than imprecision (e.g., [4], [9]–[11], and [14]).These approaches assume that—in the face of complete knowledge—the time span of an event can always be modelled as acrisp (time) interval. For example, Dubois and Prade [9] represent a time interval as a pair of possibility distributions thatdefine the possible values of the endpoints of the crisp interval.Using possibility theory, the possibility and necessity of each ofthe interval relations can then be calculated. This approach alsoallows to model imprecise relations such as “ was long before.” In a different approach adopted by Dutta [11], time intervals are abstract entities and the possibility, for each intervaland each event , that occurs in is defined. In [4], uncertaintyregarding the temporal relations that hold between crisp time intervals is considered in order to reason with statements such asholds is 0.6.” Guesgen et al. [14]“the possibility thatproposed a similar approach based on the notion of a conceptualneighborhood, a notion originally introduced in [13].Temporal information is expressed with respect to a certainlevel of granularity (e.g., years, days, seconds, etc.), which partitions the timeline. In [7], it is argued that the time span of eventsoften skews to the cells of this partitioning. Therefore, a roughset approach is adopted in which the time span of an event is represented by a lower approximation consisting of 1) the cells ofthe partitioning that are fully included in this time span and 2) anupper approximation consisting of the cells of the partitioningthat at least partially overlap with this time span. The temporalinterval relations are redefined, using a directed variant of the region connection calculus (RCC) [21], to cope with these “roughtime intervals.”In [6], it is suggested to represent time intervals as fuzzysets, but no definitions of the interval relations are given.Most relevant to our approach are definitely the work ofIII. FUZZY ORDERING OF TIME POINTSA. DefinitionsThe fuzzy temporal interval relations that we will define inthe next section are based on orderings between the time pointscontained in the intervals. Throughout this paper, we representtime points as real numbers. A real number can, for example, beinterpreted as the number of milliseconds since January 1, 1970,or the number of years since 1900. Because we want to modelimprecise temporal relations, we need a way to express for twotime points and that is long before , that is just before, and that and occur at approximately the same time.Letand. Then the extent to which is) can be expressed by thelong before (with respect toin defined as [9]fuzzy relationififotherwise(1)for all and in . The partial mappingis depictedin Fig. 1(a). The parameters and define how the concept“long before” should be interpreted. For a time point to be longbefore to degree one, the time gap with should be at least. If the time gap with is smaller than , the time point islong before to degree zero. In between there is a gradual transition. Although it seems natural to impose that is positive, for. Moreover, as pointedtechnical reasons we only requireout by Ohlbach [20], in some applications it may be desirableto express that is (long) before to a (small)for somestrictly positive degree, which is only possible in our approach ifis a generalization of thewe allow negative values of .crisp strict ordering relation . Indeed, imposing,
SCHOCKAERT et al.: FUZZIFYING ALLEN’S TEMPORAL INTERVAL RELATIONS519Fig. 1. Fuzzy ordering of time points. (a) L(:; b): fuzzy set of time points long before b. (b) L(:; b): fuzzy set of time points before or at approximately(:; b): fuzzy set of time points at approximately the same time as b. (d) J(:; b): fuzzy set of time points just before b. (e)the same time as b. (c) EOverview of fuzzy relations between time points a and b.we obtainwise.The fuzzy relationifinandother-is defined as [9]Example 1: Assume that a time point corresponds to theand,number of years since January 1900. Usingwe obtain, for example(2)represents the extent to whichfor all and in .is not “long before” (with respect to), in other words,the extent to which is before or at approximately the same timeas . It holds thatififotherwiseexpressing that 20 occurred long before 23 to a low degree, that20 occurred just before 23 to a high degree, etc. On the otherhand, we also have(3)ifandotherMoreover,is a generalization of the crisp ordering .wise, i.e.,As will become clear in Section IV, we only need the fuzzyrelationsandto model imprecise temporal intervalrelations. The degreeto which occurs at approxito whichmately the same time as , and the degreeis just before , can easily be expressed usingand,i.e.,(4)(5)An overview of the four fuzzy relations between time points isgiven in Fig. 1.In other words, although 23 is not considered to be long beforeor just before 20, it is still considered to be before or at approximately the same time as 20 to a high degree.B. PropertiesThe fuzzy relationsandbehave as can be intuitively expected from orderings. First recall that, in general, afuzzy relation in a universe is called, for an arbitrary triangular norm :for all in ;1) reflexive ifffor all in ;2) irreflexive ifffor all and in ;3) symmetric iff4) –asymmetric ifffor all andin ;for all5) –transitive iff, and in .
520IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 2, APRIL 2008Furthermore, if a fuzzy relationinis reflexive, symis called a fuzzy –equivalencemetric, and –transitive,relation. A fuzzy – –ordering relation [8] is then defined asa –transitive fuzzy relation in , which is:-reflexive, i.e.,for all and in1);2) - -antisymmetric, i.e.,for all and inTransitivity of the fuzzy orderings is of particular importancefor temporal reasoning. Many interesting properties regardingandfollow from an importantthe transitivity ofcharacterization of their composition. Recall that, in general, thesup– composition of two fuzzy relations and in is thein defined for each and in byfuzzy relation(6)Throughout this paper, we use–norm, i.e.,to denote the ŁukasiewiczTABLE IIRELATION BETWEEN THE BOUNDARIES OF THE CRISP INTERVALS [a ; aAND [b ; b ], AND THE FUZZY INTERVALS A AND B]Corollary 3:is a fuzzy––ordering.The following proposition is a generalization of the trichotomy law, stating that if is long before , and cannotbe at approximately the same time and cannot be before .and, it holds thatProposition 2: For(15)for all and in [0, 1].Proposition 1 (Composition): Let; it holds thatfor alland(7)(8)(9)(10)and.whereFor, we obtain the following interesting corollary.and , , and inCorollary 1: For(11)(12)(13)(14)Equation (11) expresses thatis–transitive while (12)is–transitive. Furthermore, (13) and (14)says thatand, generexpress a mixed transitivity betweenalizing that fromand, it follows that, andand, it follows that.similarly, that fromCorollary 1 and hence Proposition 1 do not hold for an arbitrarytriangular norm in general.For,is reflexive andis irreflexive. Thefollowing corollary results from the obvious reflexivity andsymmetry ofand Corollary 1.is a fuzzy–equivalence relation.Corollary 2:––antisymmetry of.From (4), we obtain theCombined with the reflexivity of, we establish yet another interesting corollary.andin .IV. FUZZY TEMPORAL INTERVAL RELATIONSA. Ordering of Vague BoundariesWe define a fuzzy time period as a normalized fuzzy setin , which is interpreted as the time span of some event. Recall that a fuzzy set is called normalized if there exists ain such that. Furthermore, a fuzzy (time) intervalis a convex and upper semicontinuous normalized fuzzy setin , i.e., for each in ]0,1], the –level setis a closed interval.1 If andare fuzzy time intervals, theboundaries of and can be gradual. Hence, we cannot referto these boundaries in the same way we refer to the boundariesof crisp intervals to define temporal relations in the manner ofTable I. Nonetheless, we can use the fuzzy orderings betweentime points defined in the previous section. One possibility tomeasure, for example, the extent to which the beginning of afuzzy time interval is long before the beginning of a fuzzytime interval is to look at the highest extent to which thereexists a time point in that occurs long before all time pointsin . Similarly, for instance, to express the degree to which thebeginning of is before or at the same time as the ending of ,we can use the highest extent to which there exists a time pointin that occurs before or at the same time as some time pointin . This can be accomplished by using relatedness measures,as shown in Table II. For an arbitrary fuzzy relation in , and1All the properties of the fuzzy temporal interval relations in this paper arevalid for arbitrary fuzzy time periods. Hence from a syntactic point of view, neither convexity nor upper semicontinuity is required. However, from a semanticpoint of view, it seems natural to consider only temporal interval relations between fuzzy time intervals since the convexity condition is needed to adequatelygeneralize the notion of an interval, while the upper semicontinuity conditionreflects the fact that time intervals are closed intervals.
SCHOCKAERT et al.: FUZZIFYING ALLEN’S TEMPORAL INTERVAL RELATIONS521TABLE IIITRANSITIVITY TABLE FOR RELATEDNESS MEASURESfuzzy setsas [28]andin , these relatedness measures are definedProposition 4 (Reflexivity) [28]: If(28)(29)(16)(17)(18)(19)(20)(21)where is a left-continuous –norm andcator, defined for all and in [0, 1] asits residual impli-These definitions are closely related to the sup– compositionof fuzzy relations and to the subproduct and superproduct offuzzy relations [5]. In the remainder of this paper, we willand its residual implimainly use the Łukasiewicz –normcatorin the definition of the relatedness measures, i.e.,for all and in [0, 1]. Whenand, we omitthe subscripts of , , and in (16)–(21). In the remainder ofthis section, let and be fuzzy relations in , and , , andfuzzy sets in . We recall the following three propositionsfrom [28].Proposition 3 [28]: If and are normalized, then(22)(23)(24)(25)(26)(27)is reflexive, thenProposition 5 (Irreflexivity) [28]: Ifis irreflexive, then(30)(31)Substituting fuzzy time intervals for and and eitherorfor in the propositions above shows that our approach for modeling relations between the vague boundaries offuzzy time intervals is sound. For example, from (23) and (27),we deriveHence the degree to which the beginning of is long before theend of is at least as high as the degree to which the beginningof is long before the beginning of . Furthermore, ifto ensure the reflexivity of, from (28) and (29), we deriveHence the ending of is less than or approximately equal to theending of to degree one and the beginning of is less thanor approximately equal to the beginning of to degree one. Inthe same way, from (30) and (31), we obtainIn other words, the ending of is not “long before” the endingof and the beginning of is not “long before” the beginningof . Finally, as a result of the following important proposition,the transitivity behavior of the ordering of the interval boundaries is preserved.Proposition 6 (Transitivity): For normalized fuzzy sets , ,and , the relatedness measures exhibit the transitivity properbe the entry in thisties displayed in Table III. Lettable on the row corresponding with the relatedness measure
522and the column corresponding with the relatedness measure. Then it holds that.For example, the entry on the sixth line and the third columnof Table III should be read asIEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 2, APRIL 2008TABLE IVFUZZY TEMPORAL INTERVAL RELATIONS.; ; ;,;, () () ( ; ; ; ; ( ; ), AND (;;),)Using Proposition 1, we obtain as a special casewhich generalizes the statement that if the beginning of isand the ending ofis before thebefore the beginning ofending of , then the beginning of is before the ending of .This correspondence with the transitivity behavior of the crisprelations and also reveals why, for some entries in Table III,we have no information at all, i.e., the entries that equal one.For example, from the fact that the beginning of is beforethe ending of , and the fact that the beginning of is beforethe ending of , we can conclude nothing about the relativepositioning of and . As a consequence, the entry on the firstrow, first column equals one.All results from this section can easily be generalized to anarbitrary universe and an arbitrary left-continuous –norm. Ourcommitment to the Łukasiewicz –norm is mainly motivated bywithand, as exemplithe rich interactions offied by Propositions 1 and 2.B. Relations Between Fuzzy Time PeriodsUsing the expressions in Table II, it is straightforward to generalize the temporal interval relations from Table I: using theminimum to generalize the conjunctions in Table I, we obtainthe generalized definitions in Table IV. Due to the idempotencyof the minimum, using the minimum to combine the differentconstraints on the vague boundaries in this way seems muchmore natural than, for example, using the Łukasiewicz –norm.Moreover, it turns out that this choice of the minimum is a prerequisite for some desirable properties of the fuzzy temporal interval relations, which will be introduced further on in this section.Note that the definitions in Table IV coincide with Allen’sandequals zero andandoriginal definitions if eachare crisp sets. Quantitative information ( happened at leastfour years after ) and semiquantitative information ( happened long after ) can be expressed using values or different from zero. The (semi)quantitative information we mayhave at our disposal about the relative positioning of the beginnings of and is independent of the semiquantitative information we may have at our disposal concerning the endings ofand ; hence, the fuzzy relationinvolves two different sets of parametersand. Onthe other hand, the two relatedness measures in the definitiontogether express that the ending of is approxofimately equal to the beginning of ; hence the same set of pais used twice. Notice how the notionrametersFig. 2. Fuzzy time intervals A and B .of approximate equality in the definition ofis expressed entirely analogous to the definition ofin (4) by.making use ofExample 2: For the fuzzy time intervals and displayed inand. InFig. 2, it holds thatother words, is considered to be fully before , as the overlapto hold to a degreebetween and is too low forhigher than zero. However, it is clear that also more or lessmeets . By increasing the value of , we apply a stricter definition of “long before” and a more tolerant definition of “approximately at the same time.” Hence the degree to which islong before decreases and the degree to which more or lessmeets increases. We obtainWhen is sufficiently large, the end of is not considered to belong before the beginning of anymore, hence. A similar observation can be made when increasing the valueofOur generalization preserves several interesting properties ofAllen’s original algebra, many of which are lost in other approaches. First, Allen’s temporal interval relations are jointlyexhaustive, which means that between any two time intervals, atleast one of the temporal relations holds. For fuzzy time periods
SCHOCKAERT et al.: FUZZIFYING ALLEN’S TEMPORAL INTERVAL RELATIONS523we obtain a generalization, using the Łukasiewicz –conormdefined for all and in [0, 1], as, andbe deProposition 10 [(Ir)reflexivity]: Let ,,(). Thefined as in Table IV, and let,,,,,,,,,relationsandare irreflexive, i.e., let be one of the aforementionedfuzzy relations and let and be fuzzy time periods. It holdsthatProposition 7 (Exhaustivity): Letriods. It holds thatandbe fuzzy time pe(38)Furthermore, it holds that(39)(32)IfFor nondegenerate time intervals, i.e., time intervalswith, Allen’s relations are mutually exclusive. Thismeans that at most one of the temporal relations holds betweentwo given nondegenerate time intervals, and hence preciselyone. We call a fuzzy time period nondegenerate with respectiff, i.e., if the beginning of istolong before the end of . Again, we obtain a generalization ofthis property.Proposition 8 (Mutual Exclusiveness): Let and be non). Moredegenerate fuzzy time periods with respect to (2over, let and both be one of the 13 fuzzy temporal relaandtions defined in Table IV,. If, it holds that(33)The condition that and should be nondegenerate fuzzy timeor. This isperiods is only needed when or isnot different from the traditional crisp case. For example, usingAllen’s definitions, we have for two crisp intervalsandthatholds. However, if, weholds. Likewise, if, we have thatalso have thatholds.Finally, we obtain generalizations of the (a)symmetry and the(ir)reflexivity properties of Allen’s relations., andbe deProposition 9 [(A)symmetry]: Let ,fined as in Table IV, and let,().,,,,,,,,The relations, andare–asymmetric, i.e., let be one of theaforementioned fuzzy relations and let and be fuzzy timeperiods. It holds that(34)Furthermore, it holds that(35)Iftoandare nondegenerate fuzzy time periods with respect, it holds that(36)(37)is a nondegenerate fuzzy time period with respect to, it holds that(40)In Propositions 7–10, fuzzy relations of the formandare used to express the concepts “long before” and “moreor less before.” In principle, more general classes of fuzzy relations could be used to this end, i.e., fuzzy relations that cannotor. However, as can easilybe written as eitherbe seen from their proof in Appendix III, these propositions remain valid for more general classes of fuzzy relations, providedsome weak assumptions are satisfied. For example, let andbe arbitrary fuzzy relations in that are used to express theconcepts “long before” and “more or less before,” respectively.forThen, Proposition 7 remains valid ifall and in . For Propositions 8–10 to hold, we also have,, etc., areto assume, among others, that ,irreflexive.andHowever, using fuzzy relations of the formto express fuzzy orderings of time points has a number of important advantages. As shown in Section III-B, these fuzzy relationscomposisatisfy many desirable properties, and their sup–tion can be conveniently characterized (Proposition 1), whichis important for reasoning with fuzzy temporal relations. Moreover, in [30], we have shown that this choice allows one to evaluate the fuzzy temporal interval relations in an efficient way forpiecewise linear fuzzy intervals, an important prerequisite formost real-world applications.V. FUZZY TEMPORAL REASONINGWhen,, andarecrisp intervals, using Allen’s original definitions, we can deandthatduce, for example, fromholds. Indeed by, we have, and by,; fromand, we concludewe have, or in other words,. When , , and arefuzzy time intervals, we would like to make similar deductions,oreven when the interval relations are imprecise (i.e.,). To this end, we use the Łukasiewicz –normto generalize such deductions. For example, let , , and be fuzzyand. Furthermoretime intervals,,,, andletas before. We obtain the equation shown at the
524IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 16, NO. 2, APRIL 2008bottom of the page. Using Table III, i.e., the transitivity tablefor relatedness measures and (10), we obtainwhere. Inand,particular, when,we havestating how the degree to which is during and the degree towhich more or less meets can be used to compute a lowerbound for the degree to which is long before . This is ageneralization of the statement that if occurs during andmeets , then occurs before .andAs another example, in the crisp case from, one can conclude thatholds, under the assumption that is a nondegenerate interval [3]. In our generalized approach, we obtain for fuzzy time intervals , , andAssuming thatthatis nondegenerate with respect to, i.e.,, and using Table III and (9), we obtainThis deduction process can easily be automated, which is whatwe have done to obtain Table V. In the crisp case, the temporal relation that results from composing two temporal relations is not always fully determined. For example, for crisp inand, we havetervals , , and such that,, ormay hold since we can dethatand. Freksa [13] definedduce only thata set of coarser temporal relations, which he calls conceptualneighborhoods, and provided a transitivity table that is deductively closed for Allen’s original relations as well as the conororceptual neighborhoods. For example,is equivalent to. The definitions of the relevant conceptual neighborhoods are shown in Table VI. Generalizing these definitions to cope with fuzzy time intervals andimprecise temporal relations is straightforward, using again therelatedness measures from Table II. To obtain Table V, we haveassumed that , , and are nondegenerate fuzzy time intervals with respect to (0 ,0). One can verify that when , , andare crisp intervals, Table V corresponds to Freksa’s transitivitytable. As a consequence, by restricting Table V to the first 13rows and the first 13 columns, we obtain a transitivity table thatis a sound generalization of Allen’s transitivity table. Note thatwhile Table III serves to derive knowledge about relationshipsbetween the gradual boundaries of fuzzy intervals, Table V isused to reason about the relationships between fuzzy intervalsthemselves.Table V cannot be used for reasoning with (semi) quantitativeor. It istemporal information, i.e., when somenot feasible to construct a more general transitivity table, whichwould permit this and which is still deductively closed. Instead,or, the transitivity table for relatednesswhenmeasures can be used to make deductions. For example, it holdsthatwhereis defined as in Table IV andand thus, using Table III and (10)This result can only be written aswhich does not hold for arbitraryfor a givenand.if
SCHOCKAERT et al.: FUZZIFYING ALLEN’S TEMPORAL INTERVAL RELATIONS525TABLE V; ,; ; ; ,; ; ; ; ; . LETBETRANSITIVITY TABLE FOR FUZZY TEMPORAL INTERVAL RELATIONS (T
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