The efficiency of SPARQL query evaluation against Linked Open Data may benefit from schema-based indexing. However, many data items come with incomplete schema information or lack schema descriptions entirely. In this position paper, we outline an approach to an indexing of linked data graphs based on schemata induced through Formal Concept Analysis. We show how to map queries onto RDF graphs based on such derived schema information. We sketch next steps for realizing and optimizing the suggested approach

Brosius, Dominik

Staab, Steffen

Southampton (e-Prints Soton)

Linked Data Querying through FCA-basedSchema IndexingDominik Brosius and Steffen StaabInstitute for Web Science and Technologies, University of Koblenz-Landau,{dbrosius, staab}@uni-koblenz.deAbstract. The efficiency of SPARQL query evaluation against LinkedOpen Data may benefit from schema-based indexing. However, many dataitems come with incomplete schema information or lack schema descrip-tions entirely. In this position paper, we outline an approach to an in-dexing of linked data graphs based on schemata induced through FormalConcept Analysis. We show how to map queries onto RDF graphs basedon such derived schema information. We sketch next steps for realizingand optimizing the suggested approach.Keywords: Linked Data, Formal Concept Analysis, Schema Indexing1 IntroductionThe ease of consuming Linked Open Data (LOD) depends on the availability ofschema information. This holds for tasks such as browsing, where a user may notknow the structure of the data found in a dataset, or querying, where schemainformation could be used for query optimization. In the LOD setting a querycan be evaluated against the original LOD datasets through query federation orby evaluating it centrally against a LOD crawl. In both scenarios, the efficiencyof query evaluation may be improved by identifying those datasets schematicallymatching the query – maybe partially – while leaving out others.In the LOD cloud schema information is sparse, as many data items comewith incomplete schema information or lack schema descriptions entirely. Meth-ods for schema induction and ontology learning have been studied to remedythis problem ([4], cf. section 2). A method, that has successfully been used inthis context, is Formal Concept Analysis (FCA) (cf. [2], [3], [12]).To our knowledge, FCA has not been used for constructing indices, thatallow a schema-oriented query-to-graph mapping. Particularly, we are not awareof previous work on mapping queries to formal contexts. We expect that FCAwill benefit this kind of indexing through the formally sound construction ofschemata that are free of external heuristics and concise at the same time. Inthis paper, we outline an approach for such a schema index and point to futurework on an implementation.2 Dominik Brosius and Steffen Staab2 Related WorkFor our approach to schema indexing, we propose a property-based schema in-duction using FCA. Property-based type clustering is discussed in literatureas a remedy for the schema sparsity in LOD datasets. The feasibility of suchapproaches has been analyzed in [5]. The authors show that the properties of re-sources within LOD datasets can be used for deducing resource types. [10] arguefor a property-based data access for programming with Linked Data in order todeal with a lack of type descriptions. [6] further corroborate this position, as theyargue for property-based concept definitions. In order to ensure quality of suchdefinitions, they propose a system for incorporating interactive user feedback.Schema induction, the learning of schema information from data, is a way tohandle schema sparsity. [11] describes a statistical approach to schema inductionthrough association rule mining on transaction tables. A recent approach ispresented in [7]. It combines the mining of property-based entity descriptionsand type clustering over these using DBSCAN. An overview over the field ofschema/ontology learning is given in [4].FCA has been applied to problems related to ontology learning. In [3] it hasbeen used for learning taxonomies from natural language text. The authors of[2] and [12] use attribute exploration from FCA for semi-automatic approachesto ontology completion. Lately, [1] used FCA-based association rule mining forcompleting type definitions given in DBPedia data through further implied in-formation.We combine the induction of schema with building and providing an indexfor query-to-graph mapping. An index for subgraph querying is presented in[14]. There the authors make use of a precomputed lattice of subgraphs in orderto narrow down the candidates for subgraph queries. An approach similar toours is demonstrated in [8]. The authors present a schema-based index that isconsisting of three layers, each supporting different types of queries. The indexis constructed through clustering of RDF type information in a stream-basedfashion. The approaches presented in [7], [8] and [12] either introduce heuristicsor human oversight or require case specific parameter tuning. We sketch a FCA-based schema index that is free of such external factors and general enough tomap the diverse data found in the LOD cloud.3 PreliminariesIn the following we give a short introduction to Formal Concept Analysis (FCA),a method for conceptual knowledge discovery and representation, as well as thebasics of RDF and SPARQL.We lend the following definitions 1, 2, 3 from [13]:Definition 1 (Formal Context). A formal context K := (G,M, I) is a triplecomprised of a set of objects G, a set of attributes M and an indicidence relationI ⊆ G×M encoding that ”g has attribute m” iff gIm.Linked Data Querying through FCA-based Schema Indexing 3Definition 2 (Formal Concept). For A ⊆ G and B ⊆ M [13] define A′ :={m ∈M |∀g ∈ A : gIm}, B′ := {g ∈ G|∀m ∈ B : gIm}.A formal concept is a pair (A,B) with A′ = B and B′ = A. A is the extent,B is the intent of the concept. As in [13], we abbreviate the set of all formalconcepts for a formal context as L(G,M, I).Formal concepts fall naturally into a hierarchy, called a concept lattice, bya subconcept-superconcept-relation defined via the subset relations over G andM , respectively.Definition 3 (≤). For two concepts (A1, B1), (A2, B2) [13] define (A1, B1) ≤(A2, B2) ⇔ A1 ⊆ A2 (equivalently, ⇔ B1 ⊇ B2). The pair (L(G,M, I),≤) iscalled concept lattice.RDF Datasources of the LOD cloud contain RDF1 data. For now, we abstractfrom the possibility of blank nodes in RDF data.Definition 4 (RDF Graph). Given the set of RDF resource identifiers U andthe set of literals L and having the set of RDF terms T := U ∪L: An RDF graphis a set of RDF triples D ⊆ {<s, p, o> |s ∈ U, p ∈ U, o ∈ T}. We further definethe set of all known RDF graphs D := {D|D is a known graph}.SPARQL A language for formulating graph oriented queries against RDFdatasets is SPARQL2. For this paper, we will focus on Basic Graph Patternsthat are used for formalizing graph pattern matching and, thus, are fundamen-tal to SPARQL(cf. [9]). Here, we further assume that queries are only evaluatedagainst the default graph of a datasource.Definition 5 (Basic Graph Pattern). Given a set V of variable names, aTriple Pattern tp is a triple (s, p, o) ∈ (U∪V )×(U∪V )×(T ∪V ). A Basic GraphPattern (BGP) of a query q is a set containing a number of Triple Patterns. Weabbreviate the set of all queries only containing a single BGP as BGP .With this restriction to single-BGP queries, we define the solution to a queryq through matching its BGP against the RDF graph D as q(D) (cf. [9]). Wefurther define a solution to q against a set of graphs D as q(D) := q(⋃D∈DD).An example for a BGP and a possible solution is given in fig. 1 b), c).4 Schema IndexIn a query federation system a central query processor accepts and processesqueries by dispatching (parts of) the queries to datasets (i.e., computing nodesserving them). In the case of LOD these datasets are graphs. The schema in-dex maps queries onto minimal sets of graphs required for evaluating the given1 Resource Description Framework; http://www.w3.org/TR/rdf-concepts/2 http://www.w3.org/TR/sparql11-overview/4 Dominik Brosius and Steffen Staab(a)(b)album titleex:abbey:road ”Abbey Road”ex:quadrophenia ”Quadrophenia”... ...(c)Fig. 1: Examples of (a) RDF graphs, (b) a BGP and (c) a solution to the BGPqueries. The optimal schema index minimizes the subset of all known RDFgraphs to be used for returning the complete solution of a query q:Di(q) = argminDj⊆D∧q(Dj)=q(D)|Dj |. (1)The improvement of efficiency of query evaluation then depends on the num-ber of graphs left out ofDi(q). Judging a graph to be relevant for query evaluationmay be based on a schematic match of a query and the graph.In our approach the matching will be informed by an underlying lattice ofproperty type clusters induced from the graph data via FCA. With our ap-proach, the schema index constructs and maintains a schema lattice coveringevery graph encountered. We build this lattice by applying FCA on a formalcontext extracted from an RDF graph.Definition 6 (Resource-Feature Map). We define a feature domain F asthe set of all known predicates p encountered in D. A resource-feature map f :D × U → 2F assigns an RDF resource r the set of describing features found ina graph: f(D, r) = {p ∈ F |∃o ∈ T :< r, p, o >∈ D}.Example 1. f(gex, ex:quadrophenia) = {ex:title, ex:tracks}There are other possible manifestations of such a resource-feature map. For ex-ample, one could (also) map the types assigned to resources through rdf:type.Definition 7 (Schema Lattice). We define a graph-covering formal contextK := (U,F, If ) with the set of resources U , the feature domain F and If :=Linked Data Querying through FCA-based Schema Indexing 5{(r, p)|∃D ∈ D,∃r ∈ U : p ∈ f(D, r)}. The schema lattice LS = (L(U,F, If ),≤)is the lattice constructed from K via FCA.While extracting features given in the graphs, the schema index will alsostore the set of graphs contributing individual features to the subsequent latticeconstruction.Definition 8 (Feature-contributing Graph). For an RDF resource r and afeature p ∈ F , we define a set of feature-contributing graphs as γ(r, p) = {D|p ∈f(D, r)}. For the features of an intent B, we define the set of contributing graphsas γ′(B) = {D|∃p ∈ B, ∃r ∈ U : D ∈ γ(r, p)}.We conceive the schema index as a pair (LS , Γ ) of the schema lattice LS anda function Γ mapping queries to indexed graphs. In order to look up graphs thata query should be evaluated against, the schema index then derives the queriestype information through the Query-Type Map function:Definition 9 (Query-Type Map). The Query-Type Map Θ is a function Θ :BGP → 2F that maps a query q to a set of intents Θ(q).For a query q the schema index returns the set of schematically appropriategraphs as follows:Γ (q) = {D|D ∈ γ′(B) for some B ∈ Θ(q)}.For our running example (cf. fig. 1), the graph gex would be returned forthe BGP, as here {ex:title, ex:tracks} constitutes an intent reflected in thequery.5 Outlook and ConclusionIn this position paper, we sketch the idea of a schema index for a query-to-graphmapping. For constructing the index we aim at an automatic schema inductionprocess through FCA. We hypothesize that FCA is a good method for this use.One, it is independent of external factors, such as expert involvement as in [12]or a task specific parameter tuning as done in [7]. Two, by design it is a methodtargeting property-based entity descriptions as argued for in [6], [5].Our proposition is that the schema index returns the graphs necessary for acomplete query evaluation:q(Γ (q)) = q(D) (for every query q) (2)However, Γ (q) may return graphs that might be disregarded for evaluating BGP.Hence, the proposed schema index is not necessarily optimal. For a future im-plementation we plan to approximate the optimal case (cf. (1)). This must onlybe done to a degree that is sensible, since potential savings in execution timeare being countered by costs for building and looking up the index.6 Dominik Brosius and Steffen StaabTowards a realization of the proposed schema index, we expect to face ques-tions such as for the size of formal contexts as given through the feature domainsof a graph. Next steps for our research will also involve empirical analysis ofruntime behaviour and scalability of FCA in the LOD use scenario. We have in-dications that in spite of exponential worst-case complexity, the property-basedclustering of data items may lead to empirically acceptable runtime behaviour.A further question in this context is how to efficiently update existing lattices,when, e.g., encountering further features of an object while still ingesting a graphin a stream-based fashion. For such a scenario, we will also look into what lat-tice construction algorithm to choose. Finally, we plan to address the problemsof reducing the size of formal contexts and lattices through appropriate datapreparation and cleansing lattices of ”noisy” concepts.References1. M. Alam, A. Buzmakov, V. Codocedo, and A. Napoli. Mining definitions fromRDF annotations using formal concept analysis. Proc. of IJCAI, pages 823–829,2015.2. F. Baader, B. Ganter, and U. Sattler. Completing description logic knowledgebases using formal concept analysis. Proc. of IJCAI, pages 230–235, 2007.3. P. Cimiano, A. Hotho, and S. Staab. 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Linked data querying through FCA-based schema indexing

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