Ontology and Context. Isabel Cafezeiro Departamento de Ciência da Computação Universidade Federal Fluminense Niterói - RJ, Brazil

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1 Ontology and Contxt Isabl Cafziro Dpartamnto d Ciência da Computação Univrsidad Fdral Fluminns Nitrói - RJ, Brazil isabl@dcc.ic.uff.br dward Hrmann Hauslr, Alxandr Radmakr Dpartamnto d Informática Pontifícia Univrsidad Católica Rio d Janiro, Brazil hrmann,radmakr@inf.puc-rio.br Abstract This papr considrs a formal framwork to contxtualiz ontologis and prsnts a formal algbra to manipulat ths ntitis providing svral ways of composing ontologis, contxts or both. Th algbra givs th flxibility that is rquird to modl applications whr th maning of an ntity dpnds on nvironmnt constraints or whr dynamic changs in th nvironmnt must b considrd. Ontologis ar usd in computr scinc to dscrib ral world things by hirarchically organizing concpts and nriching this hirarchy with rlationships among concpts. A ral word ntity to b rprsntd is always rlatd to a contxt: a smantically consistnt body of information in which th ntity maks sns. Th nd of contxts is growing with th adoption of nw paradigms of computation whr information concrning ithr physical or computational nvironmnt is rquird, as in Computr-Awar Mobil Applications, whr th nvironmnt suffr dynamic r-configurations [8]. In this papr w propos an algbra for manipulat Contxtualizd Ontologis. W adopt an homognous dscription of ntitis and contxts and dfin maps that consistntly link ntitis and contxts. This framwork givs th rquird flxibility, making possibl th combination of ntitis or contxts in svral ways, th changing and inhritanc of contxt by an ntity, and othr usful oprations. W considr a fw basic formal concpts what maks it accssibl vn for thos that ar not familiar with formal mthods. Th formal approach for th proposd algbra not only givs a rigorous dfinition of Contxtualizd Ontologis, but also givs insights to sound implmntation of nvironmnts. In addition, it is abstract nough to mak possibl th rplacmnt of ontologis by othr kind of tchniqu for rprsnting knowldg allowing th us of th sam algbra in a larg class of domains. This papr is organizd as follows: Sction givs an ovrviw of th approach. Sction prsnts th complt algbra. Sction 3 prsnts th formal basis and justifis th adoptd formal mthod. Sction 4 concluds th papr. Contxtualizd ntitis ntitis ar dscribd by thr parts: th ntity itslf, a contxt, and a link btwn th ntity and its contxt. As both ntity and contxt ar ontologis, an ntity can b th contxt of othr ntity. Th contxt, givs gnral information about th ntity or about th nvironmnt whrin th ntity oprats. Any contxt is linkd to a (mta)contxt. Th link btwn th ntity and its contxt, plays th rol of nsuring that th ntity and its contxt ar cohrnt, that is, th contxt rspcts (prsrvs) information concrning th natur of th ntity. For xampl, suppos that an ntity has parts, rlatd by f. Thn a link F from to its contxt C must rspcts th parts and intrnal rlationships of. Thus, F : C is such that F (f(, )) = F (f)[f ( ),F( )]. In ordr to avoid violating intrnal constitution of ntitis, fw constraints ar statd about links: (i) any ntity must hav an idntity link, that maps th ntity to itslf, and thus th ntity may b viwd as a (non-informativ) contxt of itslf; (ii) an ntity is calld domain of a link, whil a contxt is calld codomain of a link; (iii) links can b composd in an associativ way if th codomain of th first is th domain of th scond. A tripl (ntity, link, contxt), also rprsntd by c, will b namd contxtualizd ntity. If th ntity, or contxt, is rprsntd by ontologis, w us th nam contxtualizd ontology. W will us th symbol to dnot composition of contxtualizd ntitis. In th squl w prsnt modular constructs that can b applid to contxtualizd ntitis, in ordr to cohrntly combin ntitis, contxts or both. W divid th oprations in thr classs: ntity Intgration, Contxt Intgration and Combind Intgration.

2 Th Algbra of Contxtualizd ntitis. ntity Intgration Oprations in this class hav th purpos of intgrating ntitis that shar th sam contxt. As ntitis ar cohrnt with rspct to thir contxt, an opration of this class is guidd by th contxt and rsults a nw ntity that is also attachd to that contxt. Figur, part (a) shows ntitis and that shar th contxt C. Th right diagram of part (a) shows th producd ntity, namd. (a) C C (b) C C C C C Figur. (a) Intgration of ntitis sharing th sam contxt. (b) Intgration of contxts of a sam ntity. Th original ntitis play th rol of contxt to th producd ntity. By transitivity, th original contxt is also a contxt for th producd ntity. whol plant growth stag rproductiv stag flowring fruit formation lif stags activ innativ frtil unproductiv pr-frtil rproductiv matur Figur. Whol Plant Growth Stag contxtualizd by Lif Stags. In ordr to corrctly dfin, w considr that right diagram of part (a) of Figur, has th following two proprtis: (i) Th diagram is commutativ, that is, from ach componnt of, th sam componnt in C is rachd by following th path or th path in th diagram. This proprty nsurs th cohrnc of, and with rspct of th contxt. (ii) Th bottom ntity is th mor complt ntity that maks th diagram commut. By mor complt w man that all componnts of and that ar linkd to th sam lmnt in C hav a corrsponding in, and nothing mor than this is prsnt in. Thr is a uniqu ntity constructd in this way that maks th diagram commuts (s Sction 3). This uniqu ntity is th smantic intrsction of and with rspct to th contxt. Thus, if and giv diffrnt approachs about a subjct C, thn xprss thir agrmnt with rspct to C. human gstativ stag pos-partum first ag scond ag third ag lif stags innativ activ frtil pr-frtil rproductiv matur unproductiv Figur 3. Human contxtualizd by Lif Stags. xampl: Figurs, 3 and 4 show contxtualizd ntitis: at lft, th ntity, at right, th contxt. Figur 4 shows th smantic intrsction of contxtualizd ntitis of Figurs (about plant [7]) and 3 (about human bing). Both ar contxtualizd by an ontology that dscribs lif stags. Th rsult of th intgration (Figur 4) mbodis th smantic intrsction of plant and human. Both plant and human ontologis could also b viwd as contxt to th rsulting ntity. As statd by (i), componnts of th ntity in Figur 4 corrspond to thos of plant and human that ar linkd to th sam componnt of lif stag. As statd by (ii), all th componnts that satisfis (i) ar prsnt in th ntity of Figur 4. In this way, hirarchy is prsrvd by th rsulting links. Not also that it is th nt formd by th links that givs th maning of ach ntity, as it stablishs th gnral contxt of ntitis. Dfinition (ntity Intgration) Givn two contxtualizd ntitis sharing th sam contxt : C and : C, th intgration of and with rspct to C is th contxtualizd ntity C, such that, (i) Thr xists : and : such that =, and, (ii) For any othr othr ntity, with links : and : thr xists a uniqu link! : with! = and! = Th xistnc of links : and : nsurs th smantic rlationship of all componnts of with thos of and. Th commutativ condition is (i). Condition (ii) xprsss that mbody th complt intrsction, and not part of it: if thr xists anothr ntity ( ) that could also b put in th plac of, thn it could b smantically mappd in. Following, w prsnt an algorithm that implmnts th opration. Th first loop constructs th st of concpts C of th rsulting ontology, rspcting th partial ordr (H C )of, and C. Th scond loop constructs th st

3 of rlations R, rspcting th rlationships rl of, and C. For concpts, th algorithm considrs concpts x in C that ar linkd to th sam concpt in C C as a concpt x in C, that is: f (x )=f (x ). Th imag of f (x ) (or f (x )) will compos th st of concpts C of th nw ntity. Th componnts and o and ar dfind for ach concpt addd to C linking th addd concpt to th invrs imag of it by f in C,or by f in C. For rlations, th algorithm acts in a similar way. Th algorithm rturns th composit (could b ), which is th link C. Algorithm. (ntity Intgration) Input: : C and : C Output: C Notation: x i ar variabls for concpts of ntitis and y i ar variabls for rlations of ntitis. (C,R,H C,rl ) idntify th componnts of an ntity. f is componnt f of a link and g is componnt g of a link. Thsymbol dnots th association by a function of th lmnt at th lft to th lmnt at th right of th symbol. Initial conditions: C, R ar mpty sts and, ar mpty functions. For all x C If thr is x C with f (x )=f (x ) C := C f (x ) := (f (x ) C ) x := f (f (x ) C ) x For all y R If thr is y R with g (y )=g (y ) R := R g (y ) g := g (g (y ) C ) y g := g (g (y ) C ) y rturn (f,g ) ( ) innativ rproductiv matur lif stags lif stags activ innativ frtil unproductiv pr-frtil rproductiv matur Figur 4. Th smantic intrsction btwn contxtualizd ntitis of Figurs and 3.. Contxt Intgration In Sction. w showd an opration that rsults in an ntity with mor than on contxt. This situation happns not only as a consqunc of that opration, but also in modling many ral world aspcts, whr a singl ntity can b viwd in diffrnt ways. This motivats th dfinition of a class of oprations that act in th contxts of a singl ntity. A nw contxt is producd as a rsult of combining and intgrating givn contxts. Th rsulting contxt must b cohrnt with rspct to th corrsponding ntity. Part (b) of Figur shows ntity with two contxts C and C, maning that all componnts of that ntity can b undrstood both as concpts of C or as concpts of C. Th right diagram of part (b) of Figur shows th producd contxt C. All componnts of th original contxts will b rprsnt in th nw contxt, as th rsulting links hav th original contxts as domain. Thus this opration is a summation (amalgamation) of contxts. Th commutativity of th diagram nsurs that it is a cohrnt sum with rspct to th ntity: a componnt of th ntity is mappd into (diffrnt) concpts in C and C, and thn, mappd to th sam concpt in C. Th top contxt C is th lss informativ contxt that maks th diagram commut. By lss informativ w man that all lmnts of C and C ar rprsnt in C, and nothing mor than this. It can b provd that thr is a uniqu contxt constructd in this way that maks th diagram commut. whol plant growth stag rproductiv stag whol plant growth stag rproductiv stag grmination part-of part-of imbibition sding growth part-of part-of radicl shoot mrgnc mrgnc Figur 5. Part of Whol Plant Growth Stag ontology. xampl: A plant growth stag ontology is composd by two sparat parts: (Figur 5) and rproductiv stag (Figur 6). Ths parts can b dvlopd in sparat and glud latr to form th complt plant growth stag. In th glu procss part of th ontology (th glu points) must b contxtualizd by th ontologis containing th nw parts. Th intgration of ths contxts rsults th whol plant growth stag ontology. Dfinition (Contxt Intgration) Givn two contxtualizations of th sam ntity : C and : C, th contxt intgration of C and C with rspct to is th contxtualizd ntity C, such that, (i) Thr xists : C C and : C C such that =, and, (ii) For any othr othr contxt C, with maps : C C and : C C thr xists a uniqu map!:c C with! = and! =. Th xistnc of maps : C C and : C C nsurs that all componnts of contxts C and C ar smantically mappd to th nw contxt C, and thus, C is a kind of smantic union of C and C. Th commutativ proprty is (i). Condition (ii) xprsss that C is th lss

4 whol plant growth stag rproductiv stag whol plant growth stag rproductiv stag flowring fruit formation Figur 6. Anothr part of Whol Plant Growth Stag ontology. informativ contxt that can rprsnt th union of C or C. Algorithm. (Contxt Intgration) Input: : C and : C Output: C Notation: similar of ntity Intgration. Initial conditions: C is th mpty st and,f ar mpty functions. (i) For all x C C C := C C x := (f (x) C C ) (x C C) := (f (x) C C ) (x C C) (ii) For all x C that is not in th imag of f C C := C C x := (x C C ) (x C C) (iii) For all x C that is not in th imag of f C C := C C x := (x C C ) (x C C) rturn f Th algorithm shows how to construct th st of concpts and th partial ordr of concpts (H C ) of th rsulting ontology. Th st rlations R and th function rl is constructd in th sam way and is not shown. Th first stp (loop (i)) adds concpts from in C. Th imag of ths concpts by f i is collapsd in C. Th loops (ii) and (iii) add C C th concpts of C and C that ar not in th imag of or. Thn or ar dfind for ths concpts. Th compositions ( ) (f,g ) nsur that ach concpt of C is ithr a collapsd lmnt from C and C or an lmnt of C and C that dos not com from..3 Combind Intgration W dfind ways of oprating contxtualizd ntitis by combining ntitis (.) or contxts (.). In this sction w show how to oprat contxtualizd ntitis as a whol, considring both ntity and contxt. W prsnt th motivation for th oprations and th formal dfinitions. W do not prsnt th algorithms du to limitation of spac. c Th commutativ squar C C c C dfins a map btwn contxtualizd ntitis nsuring th cohrnc of ntitis and contxts. Following th lft way (c ), w nsur that C is cohrnt with C, and thus, is also a contxt for. Following th right way ( ), w nsur that C is a contxt for, and as is also a contxt for, C is a contxt for. Dfinition 3 (Map Btwn Contxtualizd ntitis) Givn two contxtualizd ntitis C and C, a pair c contxtualizd ntitis (C C, ) is a map from c to if C = C is a contxtualizd ntity..3. Th Rlativ Intrsction Th rlativ intrsction givs commonalitis among ntitis with diffrnt contxts. It is th cohrnt intrsction of two givn contxtualizd ntitis with rspct to a third contxtualizd ntity, and is prformd in thr contxtualizd ntitis, as in th diagram C C C. Th rsult is a contxtualizd ntity dfind by th commutativ diagram of Figur 7 (a): a contxtualizd ntity (C ) mappd into contxtualizd ntitis (C,C ) which ar mappd into th uppr contxtualizd ntity (C). Thus, C is th mor informativ contxtualizd ntity that is cohrnt with C and C with rspct to C. Th rlativ intrsction is a paralll opration acting both on contxts and ntitis, as shows Figur 7 (b). Th vrtical arrows ar contxtualizd ntitis. Th pairs of links btwn ntitis (down squar) and btwn contxts (uppr squar) ar maps btwn contxtualizd ntitis. By dfinition of maps btwn contxtualizd ntitis, all th latral squars of th cub in Figur 7(b) commut. Th rlativ intrsction will b dfind in such a way that nsurs that th bottom and top squars of th cub also commut. (a) C m C C m m m C (b) C C C Figur 7. (a, b) Rlativ Intrsction. xampl. C (c) Univrsity Graduation Spcialization UFF Computr C Scinc Prog.Lang. (c) An Dfinition 4 (Rlativ Intrsction) Givn two maps btwn contxtualizd ntitis m : C C and m : C C, th rlativ intrsction of C and C with rspct to C is

5 th contxtualizd ntity C, with maps m : C C and m : C C such that, (i) m m = m m, and, (ii) For any othr othr contxtualizd ntity C, with maps m : C C and m : C C thr xists a uniqu map!:c C with m! =m and m! =m..3. Th Collapsing Union Th collapsing union of contxtualizd ntitis acts both in contxt and ntity of contxtualizd ntitis and rsults th union of thm, possibly collapsing som componnts. In a dual way of subsction.3., it is prformd in thr contxtualizd ntitis forming a diagram as C C C. Th rsult is a contxtualizd ntity dfind by th commutativ diagram of Figur 8 (b): a contxtualizd ntity C that is th targt of links of both contxtualizd ntitis C and C, thus any concpt in C or C is mappd a componnt in C. Morovr, th concpts of C ar mappd to th sam concpt of C via links through C or C. Thus, C is th lss informativ contxtualizd ntity that is cohrnt with C and C. Collapsd componnts of C ar dfind by C. As wll as th rlativ intrsction, th collapsing union is a paralll opration acting both on contxts and ntitis (Figur 8 (b). (a) C m m C C m m C (b) C C C C (c) C Graduation Spcialization FirstYar Computr Scinc Prog.Lang. Func.Ling. Figur 8. (a, b) Th Collapsing Union. (c) An xampl. Dfinition 5 (Collapsing Union) Givn two maps btwn contxtualizd ntitis m : C C and m : C C, th collapsing union of C and C with rspct to C is th contxtualizd ntity C, with maps m : C C and m : C C such that, (i). m m = m m, and, (ii). for any othr othr contxtualizd ntity C, with maps m : C C and m : C C thr xists a uniqu map!:c C with! m = m and! m = m. xampls: Considr that C,C and C modl Univrsitis and Courss. C has as contxt an ontology modling Univrsitis and as ntity, th instanc of univrsity namd UFF. C has as contxt an ontology modling Graduation Courss and as ntity, th instanc of Graduation Cours Computr Scinc. Finally, C has as contxt an ontology modling Spcialization Courss and, as ntity, th instanc Programming Languag. Figur 7 (c) shows th diagram that sktchs th rlativ intrsction of C and C with rspct to C. InC, th rsulting contxtualizd ntity, C xprsss what th concpts Graduation and Spcialization hav in common, undr th scop of Univrsity. xprsss th commonalitis of Computr Scinc and Programming Languag whil courss of th Univrsity UFF. Th commutativity of th whol diagram nsurs that Computr Scinc, Programming Languag, UFF and thir intrsction ar cohrnt with Graduations, Spcializations and Univrsitis. In Figur 8 (c), C idntifis Functional Languags as a first yar disciplin, and th mappings C C and C C indicat that it occurs in both Graduation in Computr Scinc and in Spcialization in Programming Languags. Th rsult of th collapsing union on this diagram producs a nw cours of graduation with mphasis in Programming Languags, whr quivalnt disciplins of both courss (as functional languags) ar not duplicatd. 3 Th Formal Approach: Catgory Thory Th algbra of contxtualizd ontologis is an application of Catgory Thory. W avoidd making xplicit us of catgorical trms to mak th papr mor radabl for thos who do not know this thory. In this sction, w justify th us of Catgory Thory and show th corrspondnc btwn th concpts prsntd hr and catgorical concpts. Th radr can find in th appropriatd litratur (for xampl, [6]) th proofs that th oprations prsntd hr ar wll dfind. To complt th formal approach, in [5] th radr can s in that th adoptd mchanism to structurally compos and dcompos ontologis is accompanid by a modl-thortic and syntactic mchanism. Catgory Thory provids a grat powr of intgration and introprability of htrognous ntitis bcaus of th focus that is put in rlationship (catgorical morphisms) and not in th things bing rprsntd (catgorical objcts). Things ar dscribd abstractly, accordingly to thir intractions. It is possibl to dfin svral catgoris (according to th kinds of things to b dscribd) that can b rlatd by rlationships btwn catgoris (catgorical functors). Functors that prsrv proprtis of catgoris making possibl th co-xistnc of htrognous things. Catgory Thory offrs a st of ways of combining ntitis, som of which (as colimits) ar traditionally usd to intgration. It has bn succssfully usd in situations whr introprability is a crucial point, as in formal spcification of systms [], softwar architctur [3, 4], and, mor rcntly, in ontology intgration [, 0, ]. In Contxt Awar Applications, Catgory Thory contributs not only formalizing th objct of study, but also giving good dirctions to oprationaliz th concpt and facilitating implmntations. Lt C b a catgory with objcts o and morphisms m. W cal C a domain of knowldg. Th objcts of C ar n-

6 titis ovr which th domain of knowldg is constructd, and th morphisms of C ar th mappings btwn ths ntitis. Whn modling an application, o is th concpt ovr which th application is about and m xprss ways of rlating this concpt. A contxtualizd ntity is an objct of th catgory C, that is, th catgory that has as objcts morphisms of C and as morphisms pairs of C morphisms (m, m ) that dfins commutativ squars on C, as in Dfinition 3. A map btwn contxtualizd ntitis is a morphism in C. Th opration ntity intgration (Dfinition ) is a pullback in C. In ordr to nsur that any diagram of th form o o o has a pullback it is provd that C is finitly complt[5]. Th opration contxt intgration (Dfinition ) is a pushout in C. In ordr to nsur that any diagram of th form o o o has a pushout it is provd that C is finitly cocomplt[5]. Th combind oprations ar prformd in C, which is also finitly complt and cocomplt, as a consqunc of compltnss and cocompltnss of C [6]. Th rlativ intrsction (Dfinition 4) and collapsing union (Dfinition 5) ar rspctivly th pullback and pushout in C. 4 Conclusion Considring that contxts ar ssntial to clarify th maning of ntitis, and that applications that considr dynamic changs of th nvironmnt rquir nw forms of rprsnting th contxt, w prsnt in this papr an algbra to support th rprsntation of contxts and to mphasiz its rlationships with ntitis. Abstraction, modularity and rus ar important proprtis that guid this approach, which ar achivd by th uniform rprsntation of ntitis and contxts and by th compositional dfinitions of oprations. Th rol of an objct (as ntity or contxt) is givn by th nt of links from or to this objct. Thus, th maning of an ntity is dpndnt of its rlationships. This abstract framwork givs a sound basis for implmnting systms whr contxts play a cntral rol. It also rachs th 5 proprtis raisd in [9] for a formal modl of contxtawarnss. It is xpansiv as th transitiv composition of links maks possibl to xprss how a distant ntity can affct an agnt s bhavior. Spcificity is nsurd by th flxibility in th combination of contxt. Th notion of contxt is xplicit and sparatd from th ntity what nsurs control ovr its contxt. Maintnanc of contxt is transparnt to th ntity, that is fr of knowing intrnal dtails of th contxt. W add to ths proprtis introprability of htrognous dscriptions. This framwork support th coxistnc of diffrnt kinds of catgoris dfind ovr diffrnt tchniqus of knowldg rprsntation, as long as ach catgory is finitly complt and co-complt. Rfrncs [] T. Bnch-Capon and G. Malcolm, Formalizing Ontologis and thir Rlations, in: K. V. Andrsn, J. K. Dbnham, R. Wagnr, ds., Procdings of 6th Intrnational Confrnc on Databas and xprt Systms Applications (999). [] I. Cafziro and. H. Hauslr, Smantic Introprability via Catgory Thory, Confrncs in Rsarch and Practic in Information Tchnology, Vol. 83. J. Grundy, S. Hartmann, A. H. F. Landr, L. Maciaszk and J. F. Roddick, ds. 6th Intrnational Confrnc on Concptual Modling, R 007, Auckland, Nw Zaland. [3] I. Cafziro and. H. Hauslr, Catgorical Limits and Rus of Algbraic Spcifications, in Advancs in Logic, Artificial Intllignc and Robotics, ds J. MihoroAb and J. Silva, IOS Prss, Amstrdan, (00) [4] I. Cafziro and. H. Hauslr, Algbraic Framwork for Rvrs nginring on Spcifications, to appar in Procdings of th Th 6th Congrss of Logic Applid to Tchnology, LAPTC007, SP, Brazil. [5] I. Cafziro and. H. Hauslr, Dscription Logics as Institutions, Univrsidad Fdral Fluminns, Nitroi, Brasil. (007). [6] S. MacLan, Catgoris for th Working Matmatician, Brlin: Springr Vrlag(997). [7] A. Pujar and othrs, Whol-Plant Growth Stag Ontology for Angiosprms and Its Application in Plant Biology, Plant Physiology - Amrican Socity of Plant Biologists, Avaiabl at (006) [8] A. Ranganathan, R.. McGrath, R. H. Campbll and M. D. Mikunas, Th Us of Ontologis in a Prvasiv Computing nvironmnt, Th knowldg ngnring Rviw, 8:3 (004) [9] G. C. Roman, C. Julin, J. Payton, A Formal Tratmnt of Contxt Awarnss, Procdings of FAS 04 Vol. 98-, LNSC. (004). [0] M. Schorlmmr, and Y. Kalfoglou, Progrssiv Ontology Alignmnt for Maning Coordination: An Information-Thortic Foundation, in Procdings of th 4th AAMAS 05, Utrcht, Holland. [] A. Tarlcki and D. Sannlla, Algbraic prliminaris, in:. Artsiano, d., Algbraic Foundations of Systms Spcification, (Brlin, 999) 3-30.