In this paper we present BR-Explorer, an FCA-based algorithm that addresses the problem of retrieving the relevant objects for a given query. Initially, a formal context representing the relation between a set of objects and the corresponding set of attributes is given, and the associated concept lattice is built. BR-Explorer starts by generating a formal concept representing the considered query, and classifies this query concept in the concept lattice. Then, BR-Explorer tries to locate the so-called ``pivot'' concept in the concept lattice, for building step by step the query result (considering the pivot superconcepts in the concept lattice). Finally, BR-Explorer returns a set of objects ranked with respect to their relevance w.r.t. the query

Devignes, Marie-Dominique

Messai, Nizar

Napoli, Amedeo

Smaïl-Tabbone, Malika

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HAL Id: inria-00103913https://hal.inria.fr/inria-00103913Submitted on 5 Oct 2006HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.BR-Explorer: An FCA-based algorithm for InformationRetrievalNizar Messai, Marie-Dominique Devignes, Amedeo Napoli, MalikaSmaïl-TabboneTo cite this version:Nizar Messai, Marie-Dominique Devignes, Amedeo Napoli, Malika Smaïl-Tabbone. BR-Explorer: AnFCA-based algorithm for Information Retrieval. Fourth International Conference On Concept Latticesand Their Applications - CLA 2006, Oct 2006, Hammamet/Tunisia. ￿inria-00103913￿BR-Explorer: An FCA-based algorithm forInformation RetrievalNizar Messai, Marie-Dominique Devignes, Amedeo Napoli, and MalikaSmäıl-TabboneUMR 7503 LORIA, BP 239, 54506 Vandœuvre-lès-Nancy, FRANCE{messai,devignes,napoli,smail}@loria.frhttp://www.loria.fr/∼messaiAbstract. In this paper we present BR-Explorer, an FCA-based algo-rithm that addresses the problem of retrieving the relevant objects for agiven query. Initially, a formal context representing the relation betweena set of objects and the corresponding set of attributes is given, andthe associated concept lattice is built. BR-Explorer starts by generatinga formal concept representing the considered query, and classifies thisquery concept in the concept lattice. Then, BR-Explorer tries to locatethe so-called “pivot” concept in the concept lattice, for building step bystep the query result (considering the pivot superconcepts in the con-cept lattice). Finally, BR-Explorer returns a set of objects ranked withrespect to their relevance w.r.t. the query.1 IntroductionInformation Retrieval (IR) has always been a major concern in Formal ConceptAnalysis (FCA) [4,1]. Indeed, an obvious analogy exists between object-attributeand document-term tables. Accordingly, formal concepts of a concept lattice maybe seen as a pair (answer, query) where the query corresponds to the intentof the concept while the answer corresponds to the extent of the concept. Thesubsumption relation between formal concepts can be considered as a special-ization/generalization relation between such queries. Moreover, the way formalconcepts are classified in a concept lattice allows an easy browsing (navigation)of the lattice structure and hence provides a second way for using concept lat-tices in IR, namely IR by navigation. The two forms of IR using concept lattices(by querying and by browsing) can easily be combined. Such a combination pro-vides more precise results retrieved in a flexible way. In fact, a query can first besubmitted to a lattice-based IR system to locate the formal concept containingthe most precise answer. Once the answer concept is identified, additional resultscan be identified by browsing the concept lattice.This paper details an FCA-based IR algorithm called BR-Explorer. BR-Explorer exceeds the classical document-term field to deal with a more spe-cialized one, namely bioinformatic data bases retrieval [3,5]. This paper givesa formal description and generalization of the research work presented in [3]showing that it may be generalized to IR based on FCA principles.2 Formal definitionsIn the following, we suppose that there exists a formal context K = (G,M, I),where G is a set of objects, M a set of attributes, and I is an incidence relation(on G × M). The set of concepts that may be built from the formal contextK = (G,M, I) is denoted by B(G,M, I), and the resulting concept lattice byB(G, M, I) [2]. Figure 1 represents an example of formal context and its corre-sponding concept lattice (drown with the ConExp 1 system). BR-Explorer triesFig. 1. The formal context K = (G, M, I) and its corresponding concept latticeB(G, M, I)answer a query Q = ({x}, {x}′) where {x}′ is a set of given attributes describingthe constraints that must be satisfied by objects to be retrieved.Definition 1 (Query). A query Q is a pair ({x}, {x}′) where {x}′ is a set ofattributes and x is a “dummy object” satisfying the constraints expressed by theattributes in {x}′.As in the well-known FCA-based IR algorithms [1], BR-Explorer retrieves objectsby classifying the query in a concept lattice organizing the considered objects.The insertion of the query in the concept lattice can be considered as the additionof a new entry in the initial formal context. Consider as an example the queryQ = ({x}, {x}′), where {x}′ = {m4,m6,m7}. The addition of this query to theformal context K = (G,M, I) yields the formal context KQ = (GQ,MQ, IQ). Toallow this extension of formal context, we define the operator ⊕.Definition 2 (Extension of a formal context). For a formal context K =(G,M, I) and a query Q = ({x}, {x}′) we define the addition operator ⊕ asfollows:(G,M, I) ⊕ ({x}, {x}′) = (G ∪ {x},M ∪ {x}′, I ∪ ({x}, {x}′))1 http://sourceforge.net/projects/conexpIn this way, two alternatives are possible: computing the new concept latticefrom scratch or using an incremental classification algorithm such as [6]. Thesecond alternative has been chosen in the present research work. The conceptlattice B(GQ,MQ, IQ) associated to the formal context KQ = (GQ,MQ, IQ) isshown in figure 2.Before starting the retrieval of relevant objects for the considered query, twothings must be defined: (1) the relevance criterion allowing to decide whether anobject is relevant to the query or not and (2) the retrieval starting point in theconcept lattice B(GQ,MQ, IQ) allowing to avoid the whole concept lattice scan.Definition 3 (Relevance criterion). Consider an entry ({a}, {a}′) in a for-mal context K = (G, M, I), and a query Q = ({x}, {x}′). The object a is relevantwith respect to Q if and only if {a}′∩{x}′ 6= Ø, i.e. there is at least one attributein {x}′ shared with the object a.The retrieval starting point is the formal concept representing the query in theconcept lattice B(GQ,MQ, IQ). Depending on whether {x} is closed [2] in GQ ornot, this concept may be different from the query Q. In all the cases this conceptis called the pivot concept, denoted Pand defined as follows.Definition 4 (Pivot concept). Consider K = (G,M, I) a formal context andQ = ({x}, {x}′) a query. The pivot concept in the concept lattice B(GQ,MQ, IQ)of the formal context KQ = (GQ,MQ, IQ) is the concept P = ({x}′′, {x}′).In the example introduced above, the pivot concept in B(GQ,MQ, IQ) is P =({g7, x}, {m4,m6,m7}) (figure 2).Considering the relevance criterion defined above, the following propositioncan be stated.Proposition 1. Consider a formal context K = (G,M, I) and a query Q =({x}, {x}′). All the relevant objects with respect to Q in G are in the extent ofthe pivot concept P = ({x}′′, {x}′), namely {x}′′, and the extents of the pivotsuperconcepts in the concept lattice B(GQ,MQ, IQ).Proof. Consider the objects in {x}′′, the extent of the pivot concept. Accordingto the definition of the pivot concept P = ({x}′′, {x}′) (i.e. definition 4) and thedefinition of relevance (i.e. definition 3), all the objects in {x}′′ are relevant withrespect to the query Q = ({x}, {x}′) since they share all the attributes in {x}′,the query intent. For the case of the pivot superconcepts, consider C = (A,B) asuperconcept of P in B(GQ,MQ, IQ), i.e. P = ({x}′′, {x}′) ⊑ C = (A,B). Then,by definition of the lattice ordering, B ⊆ {x}′, meaning that each object in Ashares at least an element with {x}′, and hence is relevant.Based on the subsumption relation, the so-called upper cover defined hereafterallows to scan only the interesting parts of the concept lattice for retrieving therelevant objects of the considered query.Definition 5 (upper cover). (1) Consider a formal context K = (G,M, I),the set of formal concepts B(G,M, I) and the concept lattice B(G,M, I). Theupper cover of a formal concept Y ∈ B(G,M, I) is the set of all direct upperneighbors [2] of Y in B(G,M, I):upper−cover(Y ) = {C ∈ B(G,M, I) |Y ⊑ C and ∄ Z ∈ B(G,M, I) |Y ⊑ Z ⊑ C}(2) Given a set {Cj}j∈J of formal concepts in B(G,M, I), the upper cover ofthe set {Cj}j∈J (J a set of elements in N) is defined as the union of the uppercover of each concept Cj:upper−cover({Cj}j∈J) =⋃j∈Jupper−cover(Cj)3 The BR-Explorer algorithmConsider a query Q = ({x}, {x}′), a formal context K = (G,M, I) and theconcept lattice B(G, M, I). BR-Explorer proceeds as follows. Firstly, the queryAlgorithm 1 BR-ExplorerRequire: K = (G,M,I), B(G,M,I) and Q = ({x},{x}′)Ensure: Robjects1: Insert Q into B(G,M,I)2: P = ({x}′′,{x}′) := Locate Pivot(B(GQ,MQ,IQ),Q)3: n := 1 /* n is the level in B(GQ,MQ,IQ) from P */4: SUBSn−1 := {P}5: rank := 16: if {x}′′ 6= {x} then7: Rrank := {x}′′ \ {x}8: Robjects := (rank,Rrank)9: rank := rank + 110: end if11: while SUBSn−1 6= Ø do12: SUBSn := upper-covers(SUBSn−1)13: Rrank := Ø14: for all C = (A,B) ∈ SUBSn such that B 6= Ø do15: Rrank := Rrank ∪ A16: end for17: EmergingObjects := Rrank\ ({x} ∪ R1,R2,...,Rrank−1)18: Robjects := Robjects ∪ (rank,EmergingObjects)19: n := n + 120: rank := rank + 121: end whileQ = ({x}, {x}′) is classified and inserted in the lattice B(G,M, I) (Algorithm1 line 1). This classification yields a new concept lattice B(GQ,MQ, IQ) and apivot concept P = ({x}′′, {x}′) (line 2; P is given by the procedure Locate P ivot:algorithm 2). The set of objects that are in {x}′′ and in the extents of theAlgorithm 2 Locate PivotRequire: B(GQ,MQ,IQ) and Q = ({x},{x}′)Ensure: P = ({x}′′,{x}′)1: found := false2: SUBS := {⊥} /* ⊥ is the bottom concept in B(GQ,MQ,IQ) */3: while ! found do4: for each C = (A,B) ∈ SUBS do5: if {x}′ = B then6: P := C7: found := true8: break9: else if X′ ⊂ B then10: SUBS := upper-cover(SUBS)11: break12: end if13: end for14: end whilesuperconcepts of P are assigned to the result set Robjects (lines 8 and 18) as apair (rank, set of objects) (line 18). This pair is interpreted as: the objects in setof objects have the rank rank in the final result for the considered query. Thisform of Robjects allows the memorization of the rank of each object in the finalresult during the objects insertion in the result.The result construction starts by considering the set SUBS0 containing onlythe concept P (SUBS0 = {P}). At this step, if {x}′′\{x} 6= ∅ then the objectsin {x}′′\{x} are added to Robjects with the appropriate rank (first rank in thiscase). The next step consists in considering SUBS1 = upper-cover(SUBS0).The set of objects in the extents of the concepts in SUBS1 and not already inthe result (the emerging objects) are added to Robjects with the correspondingrank. The algorithm proceeds in the same way for SUBS2, SUBS3 etc untilan empty set SUBSn is reached. At each step i, if the concept ⊤ appears inthe set of concepts SUBSi and if the intent of ⊤ is the empty set, then theobjects in its extent are ignored. Figure 2 shows a running example of BR-Explorer where the formal context considered is K = (G,M, I) given in figure1 and the query is Q = ({x}, {m4,m6,m7}). The pivot concept returned bythe procedure Locate P ivot is P = ({g7, x}, {m4,m6,m7}) and the result isRObjects = {(1, {g7}), (2, {g6}), (3, {g4, g5})}.The way BR-Explorer proceeds to retrieve relevant objects for a given queryallows this algorithm to achieve high performances in term of recall and precision.In fact, BR-Explorer involves in part an operation considered as query refine-ment in [1] by looking for relevant objects in the pivot superconcepts. In this way,BR-Explorer increases the recall without decreasing the precision since as statedin proposition 1, the objects that are in the extents of the pivot superconceptsare also relevant w.r.t. the query (proposition 1).Fig. 2. Steps of the BR-Explorer execution on the concept lattice B(GQ, MQ, IQ)4 ConclusionThe algorithm BR-Explorer presented in this paper is aimed at IR and queryanswering in a concept lattice. It has been successfully applied in biology [3] andmay be used in other application domains, that can be formalized using a set ofobjects and a set of corresponding attributes. One original aspect characterizingBR-Explorer is the way objects are retrieved and the way the result is progres-sively built. This gives to BR-Explorer a different behavior, contrasting other IRapproaches in the field of FCA, such as those presented in [1] and [4].References1. Claudio Carpineto and Giovanni Romano. Concept Data Analysis: Theory andApplications. John Wiley & Sons, 2004.2. Bernhard Ganter and Rudolf Wille. Formal Concept Analysis. Springer, mathemat-ical foundations edition, 1999.3. Nizar Messai, Marie-Dominique Devignes, Amedeo Napoli, and Malika Smail-Tabbone. Querying a bioinformatic data sources registry with concept lattices.In Proceedings of ICCS 2005, Kassel, Germany, July 18-22, 2005, pages 323–336.4. Uta Priss. Lattice-based Information Retrieval. Knowledge Organization, 27(3):132–142, 2000.5. Malika Smail-Tabbone, Shazia Osman, Nizar Messai, Amedeo Napoli, and Marie-Dominique Devignes. Bioregistry: a structured metadata repository for bioinfor-matic databases. In Proceedings of CompLife’05, Konstanz, Germany, September25-27, 2005.6. Dean van der Merwe, Sergei A. Obiedkov, and Derrick G. Kourie. AddIntent: ANew Incremental Algorithm for Constructing Concept Lattices. In Proceedings ofICFCA 2004, Sydney, Australia, February 23-26, 2004, pages 372–385.

BR-Explorer: An FCA-based algorithm for Information Retrieval

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