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Preprint0(2001)?–? 1 Approximate Qualitative Temporal Reasoning ThomasBittner QualitativeReasoningGroup,DepartmentofComputerScience,NorthwesternUniversity E-mail:[email protected] Wepartitionthetime-lineindifferentways, forexample, intominutes,hours, days, etc. When reasoning about relations between events and processes we often reason about their (cid:0) (cid:1) locationwithin such partitions. Forexample, happenedyesterdayand happenedtoday, (cid:0) (cid:1) consequently and are disjoint. Reasoning about thesetemporal granularities so farhas focussedontemporalunits(relationsbetweenminute,hourslots). Ishallargueinthispaper thatinourrepresentationsandreasoningproceduresweneedintoaccountthateventsandpro- cessesoftenlieskewtothecellsofourpartitions(Forexample,‘happenedyesterday’doesnot (cid:0) meanthat startedat12a.m.andended0p.m.)Thishastheconsequencethatourdescrip- tionsoftemporallocationofeventsandprocessesareoftenapproximateandroughinnature ratherthanexactandcrisp.InthispaperIdescriberepresentationandreasoningmethodsthat taketheapproximatecharacterofourdescriptionsandtheresultinglimits(granularity)ofour knowledgeexplicitlyintoaccount. Keywords:ApproximateReasoning,QualitativeReasoning,TemporalRelations,Granularity, Ontology 1. Introduction Formal systems that support reasoning about calendar units and clock units are called temporal granularities [3]. Such systems have been the subject of intensive re- search in recent years, e.g., [2,27,12]. These systems provide foundations for task and processmanagement[19,13,18],forworkondatabasesystems[20],on(geographic)information systems [23], and they are relevant also in many other domains. Essentially, temporal granularities describe ways of partitioning the time-line and methods for reasoning about cells within the partitions which result. Examples of partitions are: the partition of the time-line into fifteen minute slots produced by your favorite calendar application, or the partition of the time-line created by the succession of update operations of some data-base system. Partitionsof the time-line can be rough. Consider, for example, the partition with cells labeled ‘before World War 2’, ‘during World War 2’, Report Documentation Page Form Approved OMB No. 0704-0188 Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. 1. REPORT DATE 3. DATES COVERED 2001 2. REPORT TYPE 00-00-2001 to 00-00-2001 4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER Approximate Qualitative Temporal Reasoning 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION Northwestern University ,Qualitative Reasoning Group, Department of REPORT NUMBER Computer Science,1890 Maple Avenue,Evanston,IL,60201 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution unlimited 13. SUPPLEMENTARY NOTES 14. ABSTRACT 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF ABSTRACT OF PAGES RESPONSIBLE PERSON a. REPORT b. ABSTRACT c. THIS PAGE 49 unclassified unclassified unclassified Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 2 T.Bittner/ApproximateQualitativeTemporalReasoning ‘after World War 2’. (In what follows I use the terms calendar-partition, db-partition, andWW2-partitioninordertorefertotheseexamples.) Acriticalassumptionunderlyingmostformalsystemsdealingwithtemporalgran- ularitiesisthattheboundaries(beginningsandendings)oftheeventsandprocesseswe want to represent are made to coincide with the boundaries between the partition cells withintherepresentation. Forexample,ifweplanameetingwithinthepartitioncreated byourcalendarapplication,thenthebeginningandendingofthemeetingmustcoincide with thebeginning and ending of theavailable fifteenminute slots. I shall argue below that this assumption masks a deep-running problem, which cannot be resolved merely bychoosingafinerresolution(e.g., fiveminuteslots). Rather, onemustgive uptheas- sumptionthatboundariesofeventsorprocessesneedtocoincidewiththeboundariesof partitioncells1. Thisresultsinapproximateratherthanexactrepresentationsofthetem- poral locations of events andprocesses. In this paper I will presentformal methods for theapproximate representationof events and processeswith respect to partitions of the time-line. Ialsopresent formalmeansto derive relationsbetweeneventsandprocesses capturedinapproximaterepresentations. The paper is structured as follows. It starts with a discussion of relationships between events and processes and partitions of the time-line. In Section 3 qualitative relations between temporal regions are defined. These relations are then generalized to relationsbetweenapproximationsofeventsandprocesseswithrespecttoanunderlying partition in Sections 4-8. In Section 9 the relationships between the notions of approx- imationand granularity willbe discussedas wellas limitsand potentialapplicationsof theproposedformalism. 2. Referenceandapproximation Partitions of the time-line are used both as frames of reference and as the basis forapproximations. Inordertounderstandtherelationshipsbetweenthesetwouseswe needtounderstandtherelationshipsbetween eventsandprocesseson theonehandand ourpartitionsofthetime-lineontheother. 2.1. Temporalgranularities Temporal granularities describe ways of partitioning the time-line and ways in whichsuchpartitionscanbeusedasframesofreference. Examplesofpartitionsdealing (cid:2) Similarpointsweremade,forexample,in[17]and[11]fromadifferentperspective. T.Bittner/ApproximateQualitativeTemporalReasoning 3 asframesofreferencearethecalendar-partition,thedb-partition,andtheWW2-partition mentioned above. We use them in expressions like ‘We will meet on Monday morning at8a.m.’ or‘Themeetingwilltakeonehour’,or‘Therearewereseveralchangessince thelastupdate’,or‘BerlinwastheculturalcenterofEuropebetweentheWorldWars’. (cid:3)(cid:5)(cid:4)(cid:7)(cid:6)(cid:9)(cid:8)(cid:11)(cid:10)(cid:13)(cid:12) The time-line, , itself is conceived as the totally ordered class of time points, i.e., the class of all possible boundaries of temporal regions, forming a directed one-dimensional space [33,25]. Usually it is assumed that the time-line is isomorphic to the real numbers, reflecting the intuition that boundaries can be located anywhere. Given the point-based view of the time-line, temporal intervals (topologically simple one-dimensional regions) can be represented by ordered pairs of boundary points [24]. Temporal intervals, in general, are such that each interval has proper parts. In those domainstherearenoatomictemporalintervals. A partition of the time-line is a set of time intervals (cells) that do not overlap but sum to the whole time-line. In opposition to the time-line itself, partition cells are countable, and so human beings can name them. One way of naming them consists in assigning integer numbers to them in such a way that the ordering of the integers corresponds to the ordering of the underlying time-line: for example, your computer internally counts the seconds that have passed by since January 1st 1970 in order to give you the time. Partitionshave different granularities and they can be hierarchically organizedinvirtueofthefactthatsomepartitionsincludeothers. Thisoccurswhenever thecellsofafinerpartitionsubsumethecellsofpartitionsatacoarserlevel. Forexample, afifteenminuteslotinyourcalendarmightbesubsumedbythreefiveminuteslots. (For a detailed discussion see [41].) Every partition has minimal cells, i.e., cells that do not have (proper)subcells. Forexampleminimalcells inyour calendarmay befiveminute slots. This, however, does not mean that there do not exist events that are shorter than fiveminutes. Thisdoesonlymeanthatyourcalendar‘doesnotcare’aboutthoseevents. These intuitions about the time-line and its partitions were formalized in the granularity-modelproposedin[3]. Moreover,thismodeltakesspecialkindsofpartitions intoaccount,includingpartitionswithholesorgaps,partitionsdeterminedbyattributes suchasworkingdays,holidays,andsoon,andpartitionswithcellsofdifferentsizes. 2.2. Occurrentsandpartitions In this paper we consider events and processes such as ‘Your meeting with your boss on Mondaymorning from 8a.m. to 9 a.m. in heroffice’, ‘My childhood’, ‘World War 2’, ‘AAAI-2000’. Following [38] I call such spatio-temporal objects occurrents. 4 T.Bittner/ApproximateQualitativeTemporalReasoning (cid:14) (cid:15) Every event or process, , is located at a region of time, , bounded by the beginning (cid:3) (cid:12) (cid:15)(cid:17)(cid:16)(cid:19)(cid:18) (cid:14) and the end of its existence, i.e., . Occurrents have temporal parts, which are occurrents themselves, and which are located at parts of the temporal regions at which theirwholesarelocated. Consider now occurrents and their temporal location with respect to partitions of thetime-line. Thisrelationshipcanbedescribedintermsofrelationsbetweentheexact (cid:3) (cid:12) (cid:18) (cid:14) temporalregion ofagiven occurrent, , andthecells ofthecorrespondingpartition. For example, we can describe the temporal location of your meeting with your boss, relative to the above-mentioned partition of the time-line into fifteen minute slots, by (cid:3) (cid:12) (cid:18) (cid:14) saying that the temporal location of this occurrent, , is identical to the sum of the fourconsecutive cellsbetween8a.m. and9a.m. Considerourthreeexamplepartitions: calendar-partition,db-partition,andWW2- partition. For each of these partitions we have a number of occurrents whose temporal (cid:3) (cid:12) (cid:18) (cid:14) locations, , can be exactly represented with respect to one of these partitions (As inthecaseofyourmeetingwithyourboss). Exactrepresentationinthiscontext means thatthe boundaries of these events coincidewith boundaries of corresponding partition (cid:3) (cid:12) (cid:18) (cid:14) cells. More precisely, we can say that the temporal location, , of such occurrents is identical to some mereological sum of partition cells. The majority of occurrents, however, cannot be represented exactly with respect to partitions in this way. This is because their beginnings and endings do not coincide with the beginnings and endings of partition cells. Consider, for example, the location of the beginning of the German carnivalseasonwhichoccurseveryyearonNovember11that11o’clockand11minutes. Thisboundaryliesskewtotheboundariesofthetime-slotsofyourcalendardividedinto 15-minute slots that start at each full hour. Consequently, the location of the occurrent ‘Carnival season 2001’ cannot be represented exactly with respect to the partitions of suchcalendars. Onecaneasilyseethatmostoccurrentsandmostpartitionsstandinthis kind of relationship. This is because most events andprocesses occur independentlyof ourpartitioningactivity. (Thisholds,too,ofmostmeetingswithyourboss.) We have two ways of dealing with this issue: (1) Whenever we want to use a partitionasaframeofreferencewecanconstructanadhocpartitionofsuchasortthat the occurrents we refer to are located exactly at some corresponding sums of partition cells. Thiscanbeachieved, forexample,bychoosingpartitionswhicharesuchthatthe occurrents of interest are parts of the partition, e.g., ‘before the occurrent of interest’, ‘during the occurrent of interest’, ‘after the occurrent of interest’, etc 2. Another way (cid:20) MorecomplexformsofpartitionsofthissortareoftenusedinGeographicInformationSystems. Parti- T.Bittner/ApproximateQualitativeTemporalReasoning 5 is to refinepartitionsuntil theoccurrents of interest canbe representedexactly, e.g.,by switching from hours to minutes, from seconds to nano-seconds, and so on. Or (2) we use an approximation theoretic approach and describe temporal location and extension approximatelyratherthanexactly. Forexample,wesaythattheGermancarnivalseason beginsatsometimebetween11a.m. and11.15a.m.,i.e.,thatitoccupiesonlyapartof thecorrespondingtimeslot. Obviously(1)ispreferable. Unfortunatelyitisnotalwayspossibletoconstructor tousepartitionsinthatsense: (a)Animportantadvantageoftheuseoffamiliarcalendar- like partitionsas framesof referenceis thatsuch partitionsdo notchange and thatthey canbere-usedindifferentcontextsinvolvinginindependenteventsandprocesseswhich may need to be synchronized. Frames of reference are, by definition, relatively stable over time. Constructing partitions on the fly as occurrents occur is thus inappropriate. (b)Itisoftennotpossibletorefinepartitionsasneeded;ourmeasuringinstrumentshave onlyafiniteresolution. (c)Oftenwedonotknowwhencertaineventsexactlyoccur;for example, I know that my boss came to talk to me during my lunch break, but I do not knowexactlywhenshearrivedorwhensheleft. Assumingtherelative stabilityofthe sortsof partitions of thetime-lineweuse as frames of reference, we can distinguish two different classes of occurrents according to the ways they behave with respect to such stable partitions: bona fide occurrents on the one hand, and fiat occurrents, on the other. Consider again your meeting with your boss. ItwasscheduledforMondaymorningandhadacorresponding,neatentryinyour calendar. Theplannedmeetingstartsandendsinexactlythewayinwhichitisenteredin yourcalendar. Theactualmeeting,ontheotherhand,startsatatimewhichdependson when peopleactuallyshow up andwhen theboss decidestoendit. Itisveryimportant to notice that we have two distinct occurrents here: (1) the occurrent ‘Planned meeting with the boss’ that is at home in the realm of calendars and scheduling applications, and(2)theactualmeetinginvolvingactualpeopleandtheiractivitiesofdrinkingcoffee, standingabout,rollingtheireyes,etc. The planned meeting is a fiat occurrent. This means that it is defined by its fiat boundaries, which are the result of human conceptualization [39,40]. Human beings have completecontrolover thetemporallocation ofsuch fiatoccurrents; they can, pro- vided they act early enough, postpone and cancel them at will. In particular such fiat occurrentscanbescheduledinsuchawaythattheirboundariescoincidewiththebound- tionsarerefinedinstepwisefashionbyaddingmoreandmoreoccurrents. Iftwoobjectsoverlap,then theiroverlapformsaseparatecell. Thisprocessofconstructingapartitioniscalledspatialenforcement [31]. 6 T.Bittner/ApproximateQualitativeTemporalReasoning ariesofcellsofsomepartitionofthetime-line3: theplannedmeetingstartsexactlyat8 a.m. andendsexactlyat9a.m. Theactualmeeting,incontrast,isabonafideoccurrent. Thismeansthatitexiststo a significant degree independently of human conceptualization [39,40]. This occurrent involvespeopleandtheiractions. Youcangetfiredduringameeting,youcanarrivetoo lateforameeting,oryoumayhavetoleaveitearly. Noticethatthedistinctionbetweenbonafideandfiatoccurrentsdoesnotimplythat all fiats coincide with the boundaries of our partitions and that all bona fide occurrents lie skew to them. Often bona fide occurrents are themselves cells of our partitions (for example in case of the WW2-partition). On the other hand there are fiat occurrents that lie skew to the boundaries of our partitions, for example, the fiat occurrent ‘the second five minutes of the planned meeting with the boss’ lies skew to the boundaries of your calendar partition which consists of fifteen minute slots. The point is that in the fiat domain we often have the freedom to place occurrents nicely since we are in charge of creating and placing them (as in the case of a planned meeting). We often can adjust the fiat occurrents to our reference partitions. On the other hand we often adjust or create our partitions with respect to bona fide occurrents if we are able to do so (as in the case of the WW2-partition). Otherwise we represent bona fide occurrents approximately by describing their relations (e.g., full overlap or partial overlap) to the cellsofsomereferencepartition. Goingdeeperintothetheoryofbonafideandfiatoccurrentsgoesbeyondthescope of this paper. See [39,40]4 for details. The important point however is that the two kinds of occurrents behave differently with respect to those partitions of the time-line which we human beings construct. Fiat occurrents can be scheduled in such a way that they can be represented exactly, i.e., their boundaries can be placed in such a way that they coincide with the cell boundaries of the appropriate partition of the time-line. (Think of the way you plan your meetings.) Bona fide occurrents, on the other hand, behavedifferently, sincetheyareindependentofourhumanconceptualization. Theyare affectedbyclimateandthemoodoftheparticipants,bythepunctualityofthetransport (cid:21) Fiatoccurrentscanofcoursebescheduledalsoinsuchawaythattheirboundariesdonotcoincidewith regularpartitionsofthetime-lineasinthecaseofthecarnivalseasonorinthecaseofraceswhichbegin (cid:22) withtheshootingofthestartingpistole. Smith’s theory of bona fide and fiat objects shows that the formerly contrary positions of realism and idealismcanbecombinedonthebasisoftheviewthatmanyentitiesinrealityenjoyindependentexistence but have boundaries which depend on our human demarcations. In this context his work represents a continuationofthatofBrentano,Husserl,andIngarden. T.Bittner/ApproximateQualitativeTemporalReasoning 7 system, and by other, internal and external factors not under our control. That is why theyusuallydonotexactlyfitintoourpartitions. Inordertoestablisharelationshipbetweenbonafideoccurrentsandcalendarpar- titionsweneedtodealwithapproximationsofthetemporallocationsofbonafideoccur- rents, rather than their exact locations (which may in any case not be capable of being exactly determined). This provides the motivation for extending the granularity model proposedby[3]totakeapproximationsintoaccount. 2.3. Approximation InthissectionIshowthattheviewofpartitionsasframesofreferenceisaspecial caseofapproximation. IntheremainderofthepaperIthenconcentrateonthenotionof approximation. 2.3.1.Roughapproximations The notion of approximation is based on the definition of an indiscernibility relation with respect to a partition of some domain. Rough set theory [35] provides the formal foundations. Rough approximation is based on approximating subsets of a set when the set is equipped with an equivalence relation that partitions the set as a whole (cid:23) (cid:24)(cid:25)(cid:16)(cid:27)(cid:26)(cid:29)(cid:28)(cid:31)(cid:30)! #"(cid:11)$&%(cid:11)’ into equivalence classes. Given aset with apartition , anarbitrary (cid:6) (cid:6) (*)+(cid:23) ,.-(cid:17)/0%+12(cid:26) ’ subset canbe approximated by a function fo po no . The value of (cid:3) (cid:12) ,.- " (cid:28) (cid:30) )(cid:19)( (cid:28) (cid:30)43 (5(cid:16)76 is defined to be fo if , it is no if , and otherwise the value is po. Intuitively fo, po and no are interpreted as ‘full overlap’, ‘partial overlap’ and ‘no overlap’. Usingthenotion ofapproximation wethen candefinean equivalence (indiscerni- (cid:6) (cid:23) (cid:28) (8)9(cid:23) /:(cid:28)<;=(?>@,.AB(cid:16)@,.- bility) relation in the domain of subsets of : . The ( ( approximation of a subset is exact if and only if is identical to a union of elements (cid:28)(cid:31)(cid:30)C$D(cid:24) ofpartitionelements. Formallywecanwrite: (cid:3) (cid:12) (cid:3) (cid:12) (cid:6) ,.- >FEG":/H,.- " $B(cid:26) ’ exact fo no IJ(LK Otherwisetheapproximationiscalledrough. Intheexactcasetheequivalenceclass containsasingleelement. Thereisobviouslyarelationbetween thegranularity(resolution)ofagivenparti- tionandthenumberofsetswecanrepresentexactlyandthe‘roughness’oftheapprox- imationoftheothersets. ThiswillbediscussedinSection9. 8 T.Bittner/ApproximateQualitativeTemporalReasoning 2.3.2.Approximationandtemporalgranularities In [9] and [43] the technique of approximation of subsets of a set was applied to regions of space by interpreting the three values fo, po, and no as different degree of spatial overlap. On the spatial interpretation these values measure the extent to which ( (cid:23) a region overlaps with the cells of the partition of a spatial region . In this paper I considertheapproximationofregionsinthetemporaldomainwiththeobviousinterpre- tationoffo,po,andnoinaone-dimensionalspace. Partitionsofthetime-linewerehitherousedmainlytoprovideaframeofreference with respect to which temporal location and extension were described. From the point of view of approximations the time-granularities used in [3] are approximations that are exact in the sense that occurrents have a temporal location that is identical to the (cid:3) (cid:12) (cid:6) EM":/H,.- " $8(cid:26) ’ mereologicalsumofcorrespondingpartitioncells,i.e., fo no . , A ,4- Giventheexactnessoftheapproximations, and ,onecaneasilyderivequal- (cid:28) ( N itativerelationsbetweenthe(non-empty)intervals and (assumingfo no): (cid:3) (cid:12) (cid:3) (cid:12) (cid:3) (cid:6) (cid:12) EG":/(cid:31),.A " (cid:16)O,.- " (cid:28) ( implies equal (cid:3) (cid:12) (cid:3) (cid:12) (cid:3) (cid:12)(cid:17)R (cid:3) (cid:12) (cid:3) (cid:6) (cid:12) EG"(cid:13)/(cid:31),.- " (cid:16) 1P,.A " (cid:16) QH":/(cid:31),.A " (cid:16)O,.- " (cid:28) ( fo fo and implies contains (cid:3) (cid:12) (cid:3) (cid:12) (cid:3) (cid:12)(cid:17)R (cid:3) (cid:12) (cid:3) (cid:6) (cid:12) EG"(cid:13)/(cid:31),.- " (cid:16) 1P, A " (cid:16) QH":/(cid:31), A " (cid:16)O,.- " ( (cid:28) fo fo and implies containedBy (cid:3) (cid:3) (cid:12)(cid:17)R (cid:3) (cid:12)(cid:17)R (cid:12) (cid:3) (cid:12)(cid:17)R (cid:3) (cid:12) (cid:3) (cid:6) (cid:12) EG"(cid:13)/ ,4A " (cid:16) ,.- " (cid:16) 1S,.A " (cid:16)O,.- " (cid:28) ( noor no implies disjoint (cid:6)UTV(cid:6)LW (cid:3) (cid:12) (cid:3) (cid:12) QH" /H,.A " (cid:16)O,.- " (cid:16) fo and (cid:3)XT(cid:31)(cid:12) (cid:3)XT(cid:31)(cid:12) (cid:3)YWZ(cid:12)\[ (cid:3)YWZ(cid:12) (cid:3) (cid:6) (cid:12) ,4A NF,.- ,4A ,.- (cid:28) ( and implies partiallyoverlap These follow immediately from the definitions. As already mentioned, things become more complicated when multiple frames of reference are involved [3]. In this paper I concentrate on a single frame of reference and I consider the derivation of relations betweentemporalregionsgiventheirroughapproximation. 3. Qualitativerelationsbetweenregions In this section I define qualitative relations between one-dimensional regions. Thesedefinitions arebased exclusively on themeet operation and provide thebasis for thedefinition of correspondingrelationsbetween approximations later on in thispaper. The meet operation is interpreted as the overlap of regions. Two regions have a non- (cid:3)^]‘_DaFR (cid:12) (cid:16)cb empty meet if and only if they share parts (or interior points in point-set topologicalterms). It is important to stressthat the same or similar relationshave been definedalsoelsewhere, e.g.,[1,21,36,24,15]. IusethenotationofRCCfromtheregion T.Bittner/ApproximateQualitativeTemporalReasoning 9 connection calculus in order to stress thecorrespondence between therelations defined inthispaperandrelationsdefinedbyCohnandhisco-workers. Correspondenceinthis context means thatIamtalking aboutregular regions thatsatisfytheRCC-axioms [36] and that similar relations could be defined or have been defined in terms of RCC, e.g., de [36,16,15]. Iusesub-andsuperscripts(e.g.,RCC )wherethesuperscriptreferstothe number of relations in the denoted set and the subscript refers to the dimension of the regionsandtheembeddingspace. The contribution of this section is the specific style of definition that allows us to generalize relations between temporal regions to relations between approximations of temporal regions. This methodology was originally proposed in [8] for the definition of relations between approximations of spatial regions. The usage of the outcome of ]‘_fa ]‘_Da ] ]‘_fa a (cid:16)gb (cid:16) (cid:16) the meet operation ( , , ) in order to define relations is ]h_Ca (cid:16)Ob somewhatsimilartothetechniqueEgenhoferusedinhisintersectionmatrices( ]*_&aiR (cid:16)Ob and ),e.g.,[21]. 3.1. Boundaryinsensitiverelations 3.1.1.RCC5relations ] a Giventworegions and , eachboundary insensitive topologicalrelation (RCC5 relation)betweenthemcanbedeterminedbyconsideringthefollowingtripleofBoolean truthvalues[8]: (cid:3)^]j_‘a8R (cid:6)C]j_‘a ]4(cid:6):]j_‘a ak(cid:12)ml (cid:16)Ob (cid:16) (cid:16) The correspondence between such triples and boundary insensitive relations between regionsonanundirectedlineisgiveninthetableinFigure1[8]. nporqjtvs u nporq t n nporq t q PP(x,y) RCC5 F F F DR T F F PO T T F PP T F T PPi T T T EQ DR(x,y) PO(x,y) PPi(x,y) EQ(x,y) disjoint partial overlap proper part(y,x) equal Figure 1. Definition of the RCC5 relations and the corresponding RCC5 lattice. (The bold boundary (cid:0) (cid:1) encloses andthedashedboundaryencloses .) As a set the relations defined in the table in Figure 1 are jointly exhaustive and