A Survey on how Description Logic Ontologies Benefit from Formal Concept Analysis

A Surv ey on ho w Description Logic On tologies Benefit from F ormal Concept Analysis Barı¸ s Sertk a ya SAP Researc h C enter Dres den, German y baris.sert kaya@sap.com Abstract. Although the notion of a concept as a collection of ob jects sharing certain properties, and the notion o f a conceptual hierarc hy are fundamental to both F o rmal Concept A nal ysis and Description Logics, the w ays co ncepts are described and obtained differ significantly betw een these tw o researc h areas. Despite these d ifferences, there hav e b een sev- eral attemp ts to bridge the gap b etw een these tw o formalis ms, and at- tempts to apply metho ds from one field in the other. The present w ork aims to give an ov erview on the researc h done in combining Description Logics and F ormal Concept Analysis. 1 In tro duction F ormal Concept Analysis (FCA) [28] is a field of a pplied mathematics tha t aims to formalize the notio ns of a concept and a conceptua l hierar ch y by means of mathematical t o ols. On the other hand D escriptio n Logics (DLs) [3 ] are a class of logic-bas ed k nowledge representation fo rmalisms tha t ar e used to repr esent the conceptual knowledge of an a pplication doma in in a str uctured wa y . Although the notion of a co ncept as a collec tion of ob jects s haring cer tain prop erties, a nd the notion of a conceptual hierarch y a re fundamen tal to b oth FCA and DLs, the ways concepts are descr ibed and o btained differ significantly b etw een these t wo r esearch ar eas. In DLs, the relev ant concepts of the applica tion domain are formalized by s o-called concept descriptions, which are expressio ns built from unary predicates (that are called atomic concepts), and binary predica tes (that are ca lled atomic roles) with the help of the conc ept co nstructors provided by the DL language. Then in a second step, these co ncept descriptions a re used to des crib e prop erties o f individuals o ccurring in the domain, a nd the roles are used to describ e relations betw een these individuals. On the other hand, in F CA, one starts with a so-called formal context, which in its simplest for m is a wa y of sp ecifying which attributes are sa tisfied by which ob jects. A for mal concept o f such a cont ext is a pair co nsisting o f a set of ob jects called extent, and a set of attributes called inten t such that the inten t c onsists of exactly those a ttributes that the ob jects in the exten t ha ve in common, and the exten t consists of exactly those ob jects that s hare all attributes in the inten t. There are several differe nces b etw een these approaches. Fir st, in F CA one starts with a purely extensio nal descr iption of the application domain, and then derives the formal concepts of this sp ecific doma in, which provide a useful struc- turing. In a w ay , in FCA the intensional knowledge is o btained from the e xten- sional part of the kno wledge. On the other hand, in DLs t he in tensional definition of a concept is given indep endent ly of a sp ecific domain (in terpretation), and the description of the individuals is only pa rtial. Seco nd, in FCA the pr op erties are atomic, and the intensional descr iption of a formal co ncept (by its inten t) is just a conjunction o f such prop erties . DLs usua lly provide a r icher languag e for the intensional definition of co ncepts, which ca n b e seen as an expres sive, yet decidable sublanguag e of first-order pr edicate logic. Despite these differences, there have been several attempts to bridge the gap betw een these tw o forma lisms, and a ttempts to apply metho ds fro m o ne field to the other. F o r example, th ere ha ve bee n efforts to enrich FCA with more complex prop erties similar to concept constructor s in DLs [60,45,44,23,46]. On the other hand, DL r esearch has b enefited fro m FCA metho ds to s olve some pro blems en- countered in kno wledge repre sentation using DLs [1,55,10,12,48,14,49,52 ,16,7,53,50,4,5,13]. The present work aims to give an ov erview on thes e works done for bridging the gap b etw een the t wo forma lisms. In Sectio n 2 w e g ive a shor t introduction to DLs without go ing into technical details. W e assume that the reader is familiar with FCA. W e do no t introduce F CA, we r efer the reader to [2 8] for details . In Section 3 we summariz e the existing work done by other r esearchers in the field. In Section 4 we summarize our own contributions to the field, and conclude with Section 5. 2 Description Logics Description Logics (DLs) [3] ar e a cla ss of knowledge repre sentation for malisms that are used to r epresent the t erminolo gical knowledge of an applicatio n do main in a structured wa y . Since their intro duction, DLs have been used in v a rious ap- plication domains suc h as medical informatics, soft ware engineering , co nfigura- tion of technical systems, na tural languag e pro cessing, databases and web-based information systems. But their most notable success so far is the adoption of the DL-bas ed languag e OWL 1 [34] as the standa rd ontology la nguage for the semantic web [17]. Syn tax In DLs, one formalizes the relev ant notions of an application domain b y c onc ept descriptio ns . A concept description is an expression built fro m ato mic concepts, which a re una ry pr edicates, a nd a tomic r oles, which a re binary pr edi- cates, by using the concept constr uctors provided b y the par ticular DL langua ge in use. DL languages are identified with t he concept co nstructors they allow. F or instance the smallest prop ositionally clo sed langua ge allowing fo r the construc- tors ⊓ (conjunction), ⊔ (disjunction), ¬ (negatio n), ∀ (v alue restrictio n) and ∃ (existential restrictio n) is called ALC . 1 W eb Ontology Lan gu age. See http://www.w3 .org/TR/owl- features Typically , a DL know le dge b ase consists o f a terminolo gic al b ox ( TBox ), which defines the ter minology of an applica tion domain, and an assertional b ox ( ABox ), which contains facts ab out a specific world. In its simplest form, a TBox is a set of c onc ept definitions of the form A ≡ C that a ssigns the co ncept name A to the concept descriptio n C . W e call a finite set o f gener al c onc ept inclusion (GCI) axioms a gener al TBox . A GCI is an expressio n of the fo rm C ⊑ D , w here C and D are tw o p ossibly complex co ncept des criptions. I t sta tes a sub concept/sup erco ncept r elationship b etw een the tw o concept descriptions. An ABox is a set of c onc ept assertions of the form C ( a ), which means that the individual a is an instance of the concept C , a nd r ole assertions of the form R ( a, b ), which means that the individua l a is in R -relatio n with individual b . F or instance the following TBo x contains t he definitio n of a landlo ck ed coun- try , which is a country that only has b or ders on land, and the definition of an o cean country that has a bo rder to an o cea n. T := { Landl o ckedCountry ≡ Country ⊓ ∀ has Bor derT o . Land OceanCountry ≡ Co untry ⊓ ∃ hasBorderT o . Ocean } The following ABox states the fa cts ab out the individuals P ortug al , Austr ia , and Atla ntic O cean . A := { Landlo ckedCountry ( Austr ia ) , Country ( P or tug al ) , Ocean ( Atlantic Oc e an ) , hasBorderT o ( P or tug al , Atl antic O cea n ) } Seman tics The meaning of DL conce pts is given b y means o f a n interpr etation I , which is a tuple co nsisting of a domain ∆ I and an interpr etation fun ction · I . The interpretation function maps every concept o ccurring in the T Box to a subset of the domain, every role to a binary relation on the domain, and every individua l name o c curring in the ABo x to a n element o f the domain. The meaning of complex concept descriptio ns is g iven inductively based o n the constructors used in the concept description. F or instance, the concept descr iption Cou ntry ⊓ ∃ has Bor derT o . Ocean is inter- preted as the intersection of the set of count ries and the set of elements of the domain that hav e a bo rder to an o cean. W e s ay that an in terpretatio n I is a mo del of a TBox T if it satisfies a ll concept definitions in T , i.e., for ev ery con- cept definition A ≡ C in T , it maps A and C to the sa me s ubset of the domain. Similarly , we s ay that I is a mo del of an ABox A , if it satisfies all concept and role assertions in A , i.e., for every co ncept asser tion A ( a ) in A , the interpretation of a is an ele men t of the interpretation of A , and for every role asser tion r ( a, b ) the in terpreta tion o f r co nt ains the pair consisting of the interpretations of a and b . The semantics of DL ABoxes is the op en-world semantics , i.e., a bsence o f information ab out a n individual is not interpreted as negative information, but it only indicates lack of k nowledge ab out that indiv idual. Inferences In an application, once we get a description of the application do- main using DLs a s describ ed ab ov e, w e can make inferences, i.e., deduce implicit consequences fro m the explicitly repres ent ed knowledge. The ba sic inference on concept descriptions is subsumption . Given tw o concept descriptions C a nd D , the subsumption problem C ⊑ D is the problem of chec king whether the concept description D is more genera l than the concept descr iption C . In other words, it is the pro blem of determining whether the fir st co ncept a lwa ys, i.e., in every int erpre tation denotes a subset of the set denoted by the second one. W e say that C is s ubsumed by D w.r.t. a T Box T , if in every mo del of T , D is more general than C , i.e., the in terpre tation of C is a subset of the interpretation of D . W e denote this as C ⊑ T D . F or instance, in the exa mple ab ove, the concepts Landlo ckedCountry and OceanCountry are both trivially subsumed by the concept Country . The t ypical inference pro blem for ABoxes is instanc e che cking , which is the problem o f deciding whether the interpretation of a given individual is an element of the interpretation of a g iven concept in every co mmon mo del o f the TBox and the ABox. F or instance, from T and A given ab ove it follows that P or tug al is a n o cean country , altho ugh A do es not contain the assertion OceanCountry ( P or tu g al ). Mo dern DL systems like F aCT++ [57], Racer [2 9], Pellet [54], KAON2 [4 0], Hermit [4 1] and CEL [9] pr ovide their users with in- ference ser vices that solve these inference pr oblems, which are also known as standar d infer enc es . 3 Existing work on DLs and F CA The ex isting w ork done by other resear chers tow ards bridging the gap b etw een F CA und DLs, a nd a ttempts to apply metho ds from o ne field to the other ca n roughly b e collected under tw o categ ories: – efforts to enrich the language of FCA by b or rowing constructo rs from DL languages [60,45,44,23,46] – efforts to employ FCA metho ds in the solution of problems encountered in knowledge represe n tation with DLs [1,5 5,10,48,49,50,4,21,20,5] Below we ar e go ing to discuss some of these efforts briefly . 3.1 Enric hing FCA with DL cons tructors Theory-driv en logi cal scaling In [45], Prediger and Stumme have used DLs in Conc eptual In formation Systems , which a re data analys is tools ba sed on FCA. They can be used to ex tract data fro m a r elational database and to store it in a for mal context b y using so-called c onc eptual sc ales . Pr ediger and Stumme hav e combined DLs with attribute exploratio n in order to define a new kind of conceptual s cale. In this appro ach, DLs provide a rich language to sp ecify which F CA a ttributes cannot o ccur together , and a DL r easoner is used during the attribute exploratio n pro cess as an exp er t to answer the implication questions, and to provide a count erex ample whenever the implica tion do es not hold. T ermino logical attribute logic In [44], Predig er has worked on intro ducing logical constructor s int o FCA . She has enriched F CA with relatio ns, existen- tial and universal quant ifiers, and negation, obtaining a la nguage like the DL ALC , which s he ha s called terminolo gische Merkmalslo gik ( t erminolo gic al at- tribute lo gic 2 ). In the s ame work she has a lso presented applications of her ap- proach in enriching formal cont exts with new knowledge, applications in many v alued formal contexts, and a pplications for so-called sc ales , which are fo rmal contexts that ar e used to obta in a standar d formal context from a man y v alued formal context. Relational concept analysis In [46], Roua ne et al. have pres ent ed a com- bination of F CA and DLs that is called r elational c onc ept analysis . It is an adaptation of FCA that is intended for analyzing ob jects describ ed by relational attributes in data mining. The approach is based on a collection of formal con- texts called r elational c ontext family and relations b etw een these contexts. The relations b etw een the contexts are binary rela tions be t ween pairs o f ob ject sets that b elong to tw o different contexts. Pro cess ing these contexts and relations with r elational concept analysis metho ds yields a set o f conce pt lattices (o ne for each input context) such tha t the formal concepts in different lattices ar e linked by re lational a ttributes, which ar e similar to r oles in DLs, or ass o ciations in UML. One dis tinguishing fea ture of this approach from the other efforts that int ro duce relations into F CA is that the formal concepts and relations b etw een formal concepts of differ ent contexts can be ma pped into concept descriptions in a subla nguage of ALE , which is called F L − E in [46]. F L − E a llows for conjunc- tion, v alue restric tion, existential restriction, and to p and b ottom conc epts. In this a pproach, after the fo rmal concepts and rela tions hav e bee n o btained and mapp ed into F L − E co ncept descriptio ns, DL reaso ning is used to c lassify a nd chec k the consistency of these descr iptions. 3.2 Applying FCA metho ds in DLs Subsumption hierarc h y of conjunctions of DL concepts In [1], Baa der has used FCA for an efficient computation of an extended subsumption hierarch y of a set of DL concepts. More precisely , he used attribute ex ploration for comput- ing the subs umption hier arch y of all conjunctions o f a set of DL concepts. The main motiv a tion for this work was to determine the interaction b etw een defined concepts, which migh t not easily b e seen by just lo ok ing at the s ubsumption hierarch y of defined co ncepts. In o rder to explain this, the following exa mple has b een given: assume tha t the defined concept N oDaughter stands for those peo ple who hav e no daughters, NoSon stands for tho se p eople who have no sons, and NoSmallChild s tands for th ose p eople who hav e no small c hildren. Obviously , there is no subsumption r elationship b etw een these three concepts. On the other hand, the conjunction NoDaug hter ⊓ No Son is subsumed b y NoSmallChild , i.e., 2 This translation is ours. if an individual a b elong s to NoSo n and N oDaughter , it also b elong s to NoS- mallChild . How ever, this ca nnot b e derived from the informa tion that a b elong s to NoSon and NoDaug hter by just lo oking at the s ubsumption hier arch y . This small example demonstrates that runtime inferences co ncerning individuals ca n be made faster by preco mputing the subsumption hierarchy not only for defined concepts, but also for all co njunctions of defined co ncepts. T o this purp ose, Baader defined a for mal context who se attributes were the defined DL conce pts, and whose ob jects were all p ossible counterexamples to subsumption relationships, i.e., in terpretatio ns together with an elemen t of the int erpre tation domain. This formal context has the pro p erty that its co ncept lattice is is omorphic to the required subsumption hierarch y , namely the sub- sumption hier arch y of conjunctions o f the defined DL co ncepts. How ever, this formal context has the disadv antage that a standard subsumption alg orithm can not b e used as exp ert for this cont ext within attribute explo ration. In o rder to ov ercome this problem, the appro ach was reconside red in [12] and a new for - mal context that has the same pr op erties but for whic h a usual subsumption algorithm could b e us ed a s exp ert was intro duced. Subsumption hierarc h y of conjunctions and disjunctions of DL con- cepts In [55], Stumme has extended the ab ovemen tioned subsumption hierarch y further with disjunctions of DL concepts. More pr ecisely , he presented how the complete lattice of all p ossible combinations of co njunctions and disjunctions of the co ncepts in a DL TBox can be computed by using FCA. T o this aim, he use d another knowledge a cquisition to ol of FCA instead of attribute explo- ration, na mely distribut ive c onc ept explor ation [56]. In the lattice computed by this metho d, the supr emum of t wo DL concepts in the la ttice cor resp onds to the disjunction of these co ncepts. Subsumption hie rarc hy of least common s ubsumers In [10] Baader and Molitor hav e used FCA f or supp orting b ottom-up co nstruction of DL knowledge bases. In the b ottom-up a pproach, the knowledge engineer do es not directly de- fine the concepts of her application domain, but she gives typical ex amples o f a concept, and the system comes up with a co ncept description f or these examples. The pro ce ss of co mputing such a co ncept description cons ists of first computing the most sp e cific c onc epts that the given ex amples b elong to, and then comput- ing the le ast c ommon subsumer of these concepts. Here the choice of exa mples is crucia l for the quality of the resulting concept description. If the examples are to o similar , the resulting concept descr iption will b e to o sp ecific; conv ersely , if they a re to o distinct, the resulting concept description will b e to o gener al. In order to o vercome this, Baader and Molitor have used a ttribute explora tion for computing the subsumption hierarch y of all least common s ubsumers of a given set of concepts. In this hierarchy one can easily see the p osition of the least co n- cept description that the chosen examples belong to, and decide whether these examples are appropriate for obtaining the intended concept descr iption. How- ever, t here may be exp onentially man y least common subsumers, a nd depe nding on the DL in use, b oth the lea st common subsumer computation a nd subsump- tion test can b e e xp ensive o per ations. The use of attribute explo ration provides us with c omplete informa tion on how this hie rarch y lo oks like without explicitly computing all leas t co mmon subsumers and class ifying them. Relational explo ration In his Ph.D thesis [49], Rudolph ha s combined DLs and FCA for acquiring complete relationa l knowledge ab out an application do- main. In his approach, which he calls r elational explor ation , he uses DLs for defining FCA attributes, and FCA for r efining DL knowledge bases. More pre- cisely , DLs makes use of the interactive knowledge acquisition metho d o f FCA, and F CA b enefits from DLs in terms of expr essing relational knowledge. In [48,49], Rudolph use s the DL F LE for this purpo se, which is the DL that allows for the constr uctors conjunction, existential restric tion, a nd v a lue restriction. In his previo us work [4 7], he uses the DL E L , which allows for the constructors conjunction and existential re striction. In b oth cases, he defines the semantics by means of a s pec ial pair o f for mal contexts c alled binary p ower c ont ext family , which are used for expressing rela tions in FCA. Binary p ow er context families hav e a lso been used for g iving semantics to c onc eptual gr aphs . In order to collect information a bo ut the for mulae expressible in F LE , in [48,49] he defines a forma l co nt ext called F LE - c ontext . The attributes of this formal context are F LE - concept descr iptions, and the ob jects are the elements o f the domain ov er which these concept descr iptions are interpreted. In this context, an ob ject g is in relation with a n attribute m if and only if g is in the in terpre- tation of m . Thus, a n implication holds in this formal context if and only if in the given mo del the concept description resulting from the conjunction of the attributes in the pre mise of the implication is subsumed by the concept descrip- tion formed from the conclusion. This is how implicatio ns in F LE -contexts give rise to subsumption r elationships b etw een F LE concept descriptions. In order to o btain c omplete knowledge ab out the subsumption rela tionships in the given model b etw een arbitrar y F LE concepts, Rudolph g ives a m ulti-step exploratio n algorithm. In the first step of the algo rithm, he starts with an F LE - context whose attributes are the atomic co ncepts occ urring in a knowledge base. In explora tion step i + 1, he defines the set of attr ibutes a s the union of the set of a ttributes from the first step and the set of concept descriptions formed by universally qua n tifying all a ttributes of the cont ext a t step i w.r.t. all ato mic roles, and the set of concept descriptions for med by existentially quantifying all concept in tents of the context at step i w.r .t all atomic roles. Rudolph points out that, at an ex ploration step, there ca n be some concept descr iptions in the at- tribute set that a re equiv alent, i.e., attributes that can b e re duced. T o this aim, he introduces a metho d that he calls empiric att ribut e r e duction . In pr inciple, it is p oss ible to ca rry out infinitely many ex ploration steps, which means that the algorithm will not terminate. In order to guarantee termination, Rudolph re- stricts the num ber of explo ration steps. After carr ying out i steps o f exploration, it is then p o ssible to decide subsumption (w.r.t. the given mo del) b et ween a ny F LE concept descriptio ns up to role depth i just by using the implication bases obtained as a result of the e xploration steps. In addition, he als o c harac terizes the cases where finitely man y s teps are sufficient to acquire co mplete inf orma tion for deciding subsumption b et ween F LE concept descr iptions with ar bitrary r ole depth. Rudolph ar gues that his metho d can b e use d to supp ort the knowledge engineers in designing, building and r efining DL ontologies. This metho d has bee n implemented in the too l Re lexo. 3 Exploring Finite M o dels in the DL E L g f p In [4] Baader and Distel hav e extended classical F CA in order to provide supp or t for analyz ing rela tional structures by us ing efficient FCA algorithms. In this appr oach the ato mic a t- tributes a re r eplaced by complex formulae in some log ical lang uage, and da ta is repr esented using r elational structures rather than just formal contexts. This extension is later instantiated with atrr ibutes defined in the DL E L , a nd with relational structures defined over a signature of unary and binary pr edicates, i.e., mo dels for E L . In this setting an implicatio n corr esp onds to a GCI in E L . This approach a t the first sight seems to b e very close to the approa ch int ro duced in [48,49]. One of the main diff erences b etw een these approaches is that in [4] the authors use o ne context with infinitely many complex attributes, where as in [49] Rudolph uses a n infinite family of contexts, each ha ving finitely ma n y attributes that a re obtained b y restricting the role depth of co ncepts. In [4] the authors additionally show that for the DLs E L and E L gf p , which extends E L with cyc lic concept definitions interpreted with gr e atest fix p oint semantics, the s et o f GCIs holding in a finite mo del a lwa ys ha s a finite ba sis. That is, ther e is a lwa ys a finite subset of the infinitely many GCIs from which the rest follows. Later in [5] the authors have shown how to compute this basis efficiently by using methods from FCA. In a follow-up pa per [22], Distel has describ ed ho w this metho d can be mo dified to allow ABox individuals as counterexamples to GCIs. 4 Con tributions to com bining DLs and FCA Our co nt ribution to the DL resear ch by means of FCA metho ds falls mainly under tw o to pics: 1) supp orting b ottom- up co nstruction of DL k nowledge bases, 2) completing DL knowledge bases. In Section 4.1 we briefly desc rib e the use of F CA in the former, and in Section 4.2 we briefly descr ib e the us e of F CA in the latter contribution. 4.1 Supp orting b ottom-up construction of DL Ontologies T raditionally , DL k nowledge bases ar e built in a top- down manner , in the sense that first the relev ant no tions of the domain ar e formalized by c oncept descr ip- tions, and then these co ncept descriptions a re used to sp ecify pr op erties of the individuals o ccur ring in the domain. How ever, this top-down a pproach is not 3 http://rel exo.ontoware.o rg alwa ys adequate. On the one hand, it might not alwa ys b e intuitiv e which no- tions of the doma in a re the relev a nt ones for a pa rticular a pplication. On the other hand, even if this is intuitiv e, it might not alwa ys be ea sy to come up with a clea r for mal des cription of these notions, esp ecially for a do main ex pe rt who is not an exp ert in knowledge eng ineering. In o rder to overcome this, in [8 ] a new appro ach, called “botto m-up a pproach”, was introduced for co nstructing DL knowledge bas es. In this approach, instead of directly defining a new co n- cept, the domain exp er t introduces several typical examples as ob jects, which are then automatically genera lized into a concept des cription by the system. This description is then offer ed to the doma in ex per t as a p oss ible c andidate for a definition o f the co ncept. The tas k of co mputing such a concept descriptio n can b e split into tw o subtasks : – computing the mos t sp ecific concepts of the given o b jects, – and then computing the le ast common subsumer of these concepts. The most sp e cific c onc ept (msc) o f an o b ject o is the most specific co ncept de- scription C expr essible in the given DL la nguage that has o as an instance . The le ast c ommon su bsu mer (lcs) of concept descriptions C 1 , . . . , C n is the most sp e- cific concept description C expr essible in the g iven DL langua ge that subsumes C 1 , . . . , C n . The problem of computing the lcs and (to a more limited extent) the msc has a lready b een inv estigated in the litera ture [8,39,2]. The metho ds for computing the least c ommon subsumer a re re stricted to rather inexpr essive descriptions logics no t allowing for disjunction (and thus not allowing for full neg ation). In fact, for langua ges with disjunction, the lcs of a collection of co ncepts is just their disjunction, and nothing new can b e lea rned from building it. In co nt ras t, for languages without disjunction, the lcs extracts the “commona lities” of the g iven collection of co ncepts. Mo de rn DL systems like F a CT++ [33,57], Racer [2 9], Pellet [54], and Hermit [41] ar e based on very expressive DLs, and there exist larg e knowledge bases that use this ex pressive power and can b e pro cessed by these sys tems. In order to allow the user to re-use concepts defined in such existing knowledge bases and still supp ort the user in defining new concepts with the b ottom-up appro ach sketc hed above, in [15,14,16] we hav e pro po sed the following extende d b ottom-up appr o ach : as sume that there is a fixed b ackgr ound terminolo gy defined in an expr essive DL; e .g., a large on tology written b y experts, whic h the user has bought from some on tology provider. The user then wan ts to extend this terminolo gy in order to a dapt it to the needs of a particula r applica tion domain. How ever, s ince the user is not a DL expe rt, he employs a less e xpressive DL a nd needs suppo rt thr ough the bo ttom-up appr oach when building this user -sp ecific extension o f the ba ckground terminology . There a re several r easons for the user to employ a restr icted DL in this s etting: first, such a r estricted DL may be ea sier to comprehend and use for a non-ex pe rt; second, it may allow for a mor e intuitiv e g raphical or frame-like user in terface; t hird, to use the b ottom-up approach, the lcs m ust exist and ma ke sense, and it must b e pos sible to compute it with reasona ble effort. T o make this mor e precis e, c onsider a background terminolo gy (TBox) T defined in an expressive DL L 2 . When defining new concepts, the user employs only a sublanguag e L 1 of L 2 , for which computing the lcs makes sense. How ever, in addition to primitive concepts and roles, the co ncept descriptions wr itten in the DL L 1 may a lso con tain names of co ncepts defined in T . Let us call suc h con- cept descriptio ns L 1 ( T )-conce pt desc riptions. Given L 1 ( T )-concept des criptions C 1 , . . . , C n , we wan t to compute their lcs in L 1 ( T ), i.e ., the lea st L 1 ( T )-concept description that subsumes C 1 , . . . , C n w.r.t. T . In [14,16] we ha ve considered the case where L 1 is the DL ALE and L 2 is the DL ALC , and shown the following result: – If T is an acyclic ALC -TBox, then the lcs w.r .t. T of ALE ( T )-concept de- scriptions always exis ts. Unfortunately , the pr o of of this result do es not yield a practical alg orithm. Due to this, in [14,16,53] we hav e developed a more practical appro ach. Assume that L 1 is a DL for which leas t common subsumers (without background TBox) alwa ys exist. Given L 1 ( T )-conce pt descriptions C 1 , . . . , C n , one can compute a common subsumer w.r.t. T by just ignoring T , i.e., by treating the defined names in C 1 , . . . , C n as primitive and co mputing the lcs of C 1 , . . . , C n in L 1 . How ever, the common subsumer obtained this wa y will usually b e to o g eneral. In [14,16,53], work we presented a metho d for computing “go o d” common subsumer s w.r.t. background TBoxes, which may not b e the le ast common subsumers, but w hich are better tha n the commo n subsumers computed by ig noring the TBox. In the present work we do not give the g cs algor ithm in detail. W e only demo nstrate it on an exa mple. The algorithm is describ ed in de tail in [16]. Example 1. As a simple example, consider the ALC -TBox T : NoSon ≡ ∀ has -child . F emale , NoDaughter ≡ ∀ ha s-child . ¬ F emale , SonRichDo ctor ≡ ∀ has-child . ( F emale ⊔ ( Do ctor ⊓ Ri ch )) , DaughterHappyDoctor ≡ ∀ has-child . ( ¬ F emale ⊔ ( Do ctor ⊓ Happy )) , ChildrenDo ctor ≡ ∀ ha s-child . Doct or , and the ALE -c oncept descriptions C := ∃ has-child . ( NoSon ⊓ Daug hterHappyDoctor ) , D := ∃ has-chi ld . ( NoDaughter ⊓ SonRichDo ctor ) . By ignor ing the T Box, we obtain the ALE ( T )-c oncept description ∃ has-chi ld . ⊤ as a co mmon subsumer of C , D . Ho wev er, if we take into account that bo th NoSon ⊓ DaughterHappyDoctor and N oDaughter ⊓ SonRichDo ct or are subsumed by the co ncept Chil drenDo ctor , then we obtain the more sp ecific common sub- sumer ∃ has-child . ChildrenDo ctor . The gcs o f C, D is even mo re specific. In fact, the least co njunction of (negated) co ncept names subsuming b oth NoSon ⊓ DaughterHappyDoctor and NoDa ughter ⊓ SonRichDo c tor is ChildrenDo ctor ⊓ Daug hterHappyDoctor ⊓ Son RichDo ctor , and th us the gcs of C, D is ∃ has-child . ( ChildrenDo ctor ⊓ D aughterHappyDoctor ⊓ SonRichDo ctor ) . The co njunct ChildrenDo ctor is actually redundant since it is implied by the remainder of the co njunction. ⋄ In order to implemen t the gcs alg orithm, we must b e able to compute the smallest co njunction of (negated) concept na mes that subsumes tw o such con- junctions C 1 and C 2 w.r.t. T . In principle, one ca n co mpute this smallest con- junction by testing, for every (negated) concept name whether it subsumes b oth C 1 and C 2 w.r.t. T , and then take the co njunction of those (neg ated) concept names for which the test was pos itive. Howev er, this r esults in a large num ber of (po ssibly ex pe nsive) calls to the subsumption algor ithm for L 2 w.r.t. (general or (a)cyclic) TBoxes. Since, in our a pplication scenar io (b ottom-up co nstruction of DL knowledge bas es w.r.t. a given ba ckground terminolog y), the TBox T is assumed to b e fixed, it ma kes sense to preco mpute this information. This is wher e FCA comes into play . By using the a ttribute ex ploration metho d [2 4] (p ossibly with ba ckground knowledge [2 5,26,27]), we compute the ab ov ementioned smallest conjunction, which is r equired for computing a gcs. T o this purpo se we define a formal context who se co ncept lattice is isomor phic to the subsumption hierarch y we are in terested in. In general, the subsumption relation induces a pa rtial order , and not a la ttice structur e o n concepts. How ev er, in the case of conjunctions of (neg ated) concept names, all infima exist, a nd thus a lso all suprema, i.e ., this hiera rch y is a complete lattice. The e xp erimental results in [1 6] have shown that the use o f this hier arch y and its use in gcs co mputation are indeed quite efficient. 4.2 Completing DL Ontologies The sta ndardization of OWL [34] as the ontology langua ge for the seman tic web [1 7] led to the fact that several ontology editors lik e P rot´ eg´ e [3 8], and Swoop [37] now supp or t OWL, and ontologies written in OWL are employ ed in more and mor e applica tions. As the size of these ontologies g rows, to ols that suppo rt improving their qualit y b eco me mor e impo rtant. The to ols a v ailable un til now us e DL reaso ning to detect inconsistencies and to infer co nsequences, i.e., implicit knowledge that can b e deduced from the ex plicitly repres ented knowledge. There are also promising appr oaches that allow to pinp oint the rea- sons for inconsistencie s a nd for certain c onsequences, and that help the ontol- ogy engineer to resolve incons istencies and to remov e un wan ted consequences [51,36,35,32,11,43]. These approaches addres s the qua lit y dimension of sound- ness o f an ont olog y , bo th within itself (consistency) and w.r.t. the intended application domain (no unw an ted co nsequences). In [6,7] we hav e considered a different quality dimension: c ompleteness . W e hav e provided a bas is for forma lly well-founded techniques and too ls that support the o nt olog y engine er in c hecking whether an o ntology contains a ll the re lev ant infor mation ab out the application domain, and to extend the ontology appro priately if this is not the case. As alrea dy mentioned, a DL knowledge base (now adays often calle d ontol- ogy) usually co nsists of tw o parts, the terminolog ical part (TBo x), which defines concepts and a lso states additional co nstraints (GCIs) on the interpretation of these concepts, and the assertional part (ABo x), whic h describ es individuals and their relationship to each other and to c oncepts. Given an application domain and a DL knowledge bas e de scribing it, we ca n ask whether the k nowledge base contains all the r elev a nt infor mation ab out the domain: – Are all the rele v ant cons traints that hold b etw een concepts in the domain captured by the TBox? – Are all the relev ant individuals existing in the do main repr esented in the ABox? As a n example, consider the OWL on tolog y for human protein pho sphatases that has b een describ ed and used in [59]. This ontology was develope d ba sed on information from peer- reviewed publications. The human protein phosphatase family has been well ch ar acterised exp erimentally , and detailed kno wledge about different class es o f such pr oteins is av ailable. This k nowledge is represented in the terminologica l part of the on tology . Moreo ver, a large set of hum an phosphatases has b een identified and do cumented by exp ert biolo gists. The se ar e describ ed as individuals in the as sertional part of the ontology . One can now ask whether the information ab out pr otein phospha tases c ontained in this ontology is co mplete: are all the r elationships that hold amo ng the intro duced c lasses of phosphatas es captured b y the constra in ts in the T Box, or are there relationships that hold in the do main, but do not follow fro m the TBox? Are all p ossible kinds of hu man pr otein phosphatases repres ented by individuals in the ABox, or are there phosphatases that hav e not yet b een included in the ontology or even not yet hav e bee n identified? Such ques tions cannot b e answered by a n a utomated to o l a lone. Clearly , to chec k whether a g iven relations hip b etw een concepts—which do es not follow from the TBox—holds in t he do main, one needs to ask a domain exp ert, and the same is true for questions reg arding the exis tence of individuals no t describ ed in the ABox. The rˆ ole of the automated to ol is to ensure that the exp ert is asked as few questions as p os sible; in particular, she should not be asked triv ial questions, i.e., questions that could actually b e answered based o n the represe n ted knowl- edge. In the a bove example, answering a non-trivial questio n regarding h uman protein phos phatases may require the biologist to study the relev ant litera ture, query existing protein databas es, or even to carry o ut new exp er iment s. Thus, the exp ert may b e pr ompted to acquir e new biological knowledge. The attribute ex ploration metho d of FCA ha s proved to b e a successful knowledge acquisition method in v arious application domains. One of th e earliest applications of this appro ach is descr ib ed in [58], where the domain is lattice theory , and the go al of the ex ploration pro cess is to find, on the one ha nd, all v alid rela tionships b etw een prop erties of lattices (like b eing distributive), and, on the other hand, to find counterexamples to all the rela tionships that do not hold. T o answer a query whether a ce rtain r elationship holds, the lattice theory exp ert must either c onfirm the re lationship (by using results from the literature or by ca rrying out a new pro of for this fact), or g ive a counterexample (again, by either finding one in the literature or constr ucting a new one). Although this sounds very similar to what is needed in o ur case , we cannot directly use this approa ch. The main r eason is the op en-world semantics of de- scription logic knowledge bases. Consider an individua l i fr om an ABo x A and a co ncept C occ urring in a TBox T . If we canno t deduce from the TBox T and A that i is an instance of C , then we do not assume that i do es not b elong to C . Instead, we only ac cept this as a conseq uence if T a nd A imply that i is an in- stance of ¬ C . Th us, our knowledge ab out the relationships b etw een individua ls and concepts is incomplete: if T and A imply neither C ( i ) nor ¬ C ( i ), then we do not know the rela tionship be t ween i and C . In contrast, classica l F CA and attribute e xploration assume that the knowledge ab out ob jects is complete: a cross in row g and column m of a formal context says that ob ject g has attribute m , and the abs ence of a cro ss is interpreted as saying that g do es not hav e m . There has b een some work on how to extend FCA a nd attribute explora tion from complete knowledge to the cas e o f partial knowledge [25,18,30,31,19,49], and how to ev a luate formulas in formal contexts that do not c ontain complete information [4 2]. How ever, these works are based on assumptions tha t are dif- ferent fr om ours . In particular , they as sume that the exp ert cannot a nswer a ll queries and, as a cons equence, the knowledge o btained after the exploration pro cess may still b e incomplete and the relationships b etw een concepts that a re pro duced in the end fall into tw o categor ies: relatio nships that are v alid no ma t- ter ho w the incomplete part o f the knowledge is completed, and relationships that are v alid only in some completio ns of the incomplete part of the knowledge. In co nt ras t, our inten tion is to complete the knowledge base, i.e., in the end we wan t to have complete knowledge ab out these r elationships. What may b e incomplete is the descr iption of individuals used during the explor ation pr o cess. In [7,53] we hav e intro duced an extension of FCA that can deal with partial knowledge. This extension is based o n the notion of a p artial c ontext that consists of a set of p artial obje ct descriptions ( p o d ). A p o d is a tuple ( A, S ) where A represents the set of attributes that the p o d is known to have, and S repres ents the set of attributes that the p o d is k nown not to hav e. A and S ar e disjoint and their union need no t b e the whole attribute set, i.e., for so me attributes it might be unknown whether the po d has this attribute or not. W e say that a p o d ( A, S ) r efutes an implication L → R if L ⊆ A and R ∩ S 6 = ∅ . W e also say that a partia l context r efutes an implication if t here is a p o d in this partia l con text that refutes this implication. Ba sed on these, we define the notion of an un de cide d implic ation , which is an implication that does not follow from a given set of implications, and that is not refuted by a partial context. Then the a ttribute explora tion metho d for partial co ntexts can be formulated as enumerating undecided implications as efficient a s po ssible. In [7,53] we hav e described a version o f attribute exploration algorithm that works for this s etting, and proved that this algo rithm terminates and it is co rrect. Later w e have shown that g iven a DL knowledge base ( T , A ), any individual in A gives ris e to a p o d, and thus A induces a partial context. This enables us to use our attribute exploration algorithm o n partial contexts for A countr ies Asian EU European G8 Mediterranean Syria + - - - + T urkey + - + - + F rance - + + + + German y - + + + - Switzerland - - + - - USA - - - + - T able 1. The partial context b efore completion finding completing DL knowledge bases. As a result o f running this alg orithm on a DL knowledge base, the knowledge base is complete w.r .t. a n intended int erpre ation, i.e., if an implicatio n holds in this interpreation then it also follo ws from the TBox, and if not then the ABox contains a counterexample to this implication. F or details of the attribute explor ation o n partial cont exts and its application to DL ontologies we refer the reader to [7,53], and demonstrate on a small example how it works. Example 2. Let our TBox T countr ies contain the following co ncept definitions: AsianCountry ≡ Co untry ⊓ ∃ hasT erritoryIn . { Asia } EUmemb er ≡ Co untry ⊓ ∃ memb erOf . { E U } Europ eanCountry ≡ Co untry ⊓ ∃ hasT erritoryIn . { E ur ope } G8memb er ≡ Cou ntry ⊓ ∃ memb erOf . { G 8 } IslandCountry ≡ Country ⊓ ¬∃ hasT erritoryIn . Continent MediterrenaenCountry ≡ Co untry ⊓ ∃ hasBorderT o . { M e di ter r enae nS ea } Moreov er, let our ABo x A countr ies contain the individuals Syria , T urkey, F r anc e, Germany, Switzerland, US A and ass ume w e are interested in the subsumption relationships b etw een the concept names AsianCountry , EUmemb er , Europ ean- Country , G8member and MediterreneanCountry . T a ble 1 sho ws the partial cont ext induced by A countr ies , a nd T able 2 shows the questions asked by the completion algorithm and the a nswers given to these questions. In or der to sav e s pace, the names of the concepts ar e shortened in bo th tables. The questions with po sitive answers result in ex tension of the TBox with the fo llowing GCIs: G8memb er ⊓ M editerraneanCountry ⊑ EUm emb er ⊓ Europ eanCo untry EUmemb er ⊓ G8memb er ⊑ Europ eanCou ntry AsianCountry ⊓ EUme mb er ⊑ Medite rraneanCountry AsianCountry ⊓ EUme mb er ⊓ Europ eanCountry ⊓ MediterraneanCountry ⊑ G8memb er Moreov er, the questions with neg ative ans wers result in extension of the ABox with the individuals Russia, Cyprus, Sp ain and J ap an . The partial context induced by the resulting ABox A ′ countr ies is s hown in T able 3. The resulting Question Answer Counterex. { G8, Mediterranean } → { EU, Europ ean } ? yes - { European, G8 } → { EU } ? no Ru ssia { EU } → { Europ ean, G8 } ? no Cyprus { EU, G8 } → { Europ ean } ? yes - { EU, Europ ean } → { G8 } ? no Spain { Asian, G8 } → { Europ ean } ? no Ja pan { Asian, EU } → { Mediterranean } ? yes - { Asian, EU, Europ ean, Mediterranean } → { G8 } ? yes - T able 2. Execution of the on tology completion algorithm ( T countr ies , A countr ies ) A ′ countr ies Asian EU European G8 Mediterranean Syria + - - - + T urkey + - + - + F rance - + + + + German y - + + + - Switzerland - - + - - USA - - - + - Russia + - + + - Cyprus + + - - + Spain - + + - + Japan + - - + - T able 3. The par tial context after co mpletion knowledge base ( T ′ countr ies , A ′ countr ies ) is complete w.r.t. the initially s elected concept names. ⋄ Based o n on the descr ibe d appr oach, we implemented a first exp er iment al version of a DL knowledge ba se completio n too l a s an extensio n for the Swoop ontology editor using Pellet as the underlying r easoner . A first ev aluation of this to ol on the OWL o ntology for h uman protein phosphatases with biolog ists as exp erts, was quite promising, but also show ed that the to ol must be improv ed in order to be useful in practice. In particular, w e hav e observed that the exp erts sometimes make error s when a nswering queries. Thus, the to ol should supp ort the exp er t in detecting such err ors, and als o make it p ossible to co rrect err ors without ha ving to restart the completion pro cess from scratch. Another usability issue o n the wish lis t of our e xpe rts was to a llow the p ostp onement of a nswering certain q uestions, while c ontin uing the co mpletion pro ces s with other q uestions. In a follow-up pap er [13] we have addr essed these usability issues. W e hav e improv ed the metho d in such a way that at any time during completio n the exp ert can pause the pro cess, see all of her previous answers or changes to the knowledge base, ’undo’ so me of thos e changes, and co nt inue co mpletion. Here we of co urse paid attention that the exp ert do es not hav e to answer the same questions she ha s a nswered befo re pausing the pro cess. W e hav e achiev ed this by saving previous answers, and using them as background k nowledge when the exp ert co n tinues completion. The other wish o f o ur exp erts, namely p ostp oning questions was so lved pausing completion, changing the or der of attributes, a nd restarting the completion with previous ans wers a s background knowledge. In theory , this method might not p ostp one a question, thus the exper t might be asked the la st question again. How ever, in prac tice the metho d turned o ut to be useful in m any c ases when the ex per t was not able to answer a pa rticular question and wan ted to get another one. W e hav e implement ed our ontology completion metho d together with thes e us ability issues as a plug in for the P rot´ eg´ e on tolog y editor under the name OntoCom P . 4 5 Conclusion W e have summa rized the work done in co mbinin g DLs and FCA . The research done in this field mainly falls under t wo categor ies: 1) efforts to enrich the la n- guage of FCA by b orrowing cons tructors from DL langua ges, and 2) effor ts to employ FCA methods in the s olution of problems encountered in kno wledge rep- resentation with DLs. F or e ach of these catego ries we hav e g iven p o in ters and shortly descr ibe d the relev ant w ork in the litera ture. W e have also describ ed our own contributions, which a re mainly under the second categor y . Recent developmen ts in information technologies like so cial netw orks, W eb 2.0 applications a nd semantic w eb applications ar e bringing up new challenges for r epresenting v ast amounts of knowledge and analyz ing h uge a mounts of data rapidly g enerated by these applicatio ns. The tw o resea rch ar eas we hav e dis- cussed here, na mely DLs and FCA, are lying at the core of repre senting knowl- edge, and analyzing data, resp ectively . 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