Coloring RDF Triples to Capture Provenance

Transcription

1 Coloring RDF Triples to Capture Provenance G. Flouris 1, I. Funulaki 1, P. Peiaitis 1,2, Y. Theoharis 1,2, an V. Christophies 1,2 1 Institute of Computer Science, FORTH, Greece an 2 Computer Science Department, University of Crete, Greece {fgeo, funul, ppe, theohari, christop}@ics.forth.gr bstract. Recently, the W3C Linking Open Data effort has booste the publication an inter-linkage of large amounts of RDF atasets on the Semantic Web. Various ontologies an knowlege bases with millions of RDF triples from Wikipeia an other sources, mostly in e-science, have been create an are publicly available. Recoring provenance information of RDF triples aggregate from ifferent heterogeneous sources is crucial in orer to effectively support trust mechanisms, igital rights an privacy policies. Managing provenance becomes even more important when we consier not only explicitly state but also implicit triples (through RDFS inference rules) in conjunction with eclarative languages for querying an upating RDF graphs. In this paper we rely on colore RDF triples represente as quaruples to capture an manipulate explicit provenance information. 1 Introuction Recently, the W3C Linking Open Data [29] effort has booste the publication an interlinkage of large amounts of RDF atasets on the Semantic Web [1]. Various ontologies an knowlege bases with millions of RDF triples from Wikipeia [26] an other sources have been create an are available online [25]. In aition, numerous ata sources in e-science are publishe nowaays as RDF graphs, most notably in the area of life sciences [27], to facilitate community annotation an interlinkage of both scientific an scholarly ata of interest. Finally, Web 2.0 platforms are consiering RDF an RDFS as non-proprietary exchange formats for the construction of information mashups [16, 28]. In this context, it is of paramount importance to be able to store the provenance of a piece of ata in orer to effectively support trust mechanisms, igital rights an privacy policies. Provenance means origin or source an refers to from where an how the piece of ata was obtaine [33]. In the context of scientific communities, provenance information can be use in the proof of the correctness of results an in general etermines their quality. In some cases, provenance of ata is consiere more important than the result itself. The popularity of the RDF ata moel [8] an RDF Schema language (RDFS) [2] is ue to the flexible an extensible representation of information, inepenently of the existence or absence of a schema, uner the form of triples. n RDF triple, (subject,- property,object), asserts the fact that subject is associate with object through property. RDFS is use to a semantics to RDF triples, by imposing inference rules [15] 193

2 2 G. Flouris, I. Funulaki, P. Peiaitis, Y. Theoharis, an V. Christophies (mainly relate to the transitivity of subsumption relationships) which can be use to entail new implicit triples (i.e., facts) that are not explicitly asserte. Currently, there is no aequate support for managing (i.e., querying an upating) provenance information of implicit RDF triples. There are two ifferent ways in which such implicit knowlege can be viewe an this affects the semantics of upate operations only. Uner the coherence semantics [10], implicit knowlege oes not epen on the explicit one but has a value on its own; therefore, there is no nee for explicit support of some triple. Uner this viewpoint, implicit triples are first-class citizens, i.e., consiere of equal value as explicit ones. On the other han, uner the founational semantics [10] each implicit triple epens on the existence of the explicit triple(s) that imply it: implicit knowlege is only vali as long as the supporting explicit knowlege exists. It shoul be emphasize that the selecte viewpoint is irrelevant as far as stanar implication an querying is consiere, but it affects the way upates shoul be performe; in particular, the coherence semantics correspons to belief set changes, whereas founational semantics correspons to belief base changes, in the belief revision terminology [10]. In this paper we propose the use of colors (as in [4]) in orer to capture the provenance of RDF ata an schema triples. Intuitively, the color of an implicit an explicit triple represents the source from which the triple was obtaine. We recor the color of an RDF triple as a fourth column, hence obtaining an RDF quaruple as in [7, 17]. s p o c a 1 p b 1 b 1 q 1 b 1 t 1 1 r b 2 2 r b 1 c 3 c 3 s p o a 1 p b 1 b 1 q 1 b 1 t 1 1 r b 2 2 r b 1 To provie the intuition of the granularity levels of provenance for RDF Fig. 1. Granularity Levels of Provenance atasets that we capture with this work, we compare it with the representation of provenance information in the relational context. For instance, authors in [4] use colors to capture the provenance of relational tables, tuples an attributes. If we consier that a relational tuple of the form [a 1 :v 1,..., a k :v k ] with tuple ientifier ti correspons to a set of triples (ti, a j, v j ), j = 1,... k, then (see Fig. 1): a color assigne to (a) a single triple captures provenance at the level of an attribute of the relational tuple; (b) a collection of triples sharing the same subject captures provenance at the level of the relational tuple an finally (c) a set of triples whose subjects are instances of the same schema class, captures provenance of the relational table. The quaruples use to represent colore RDF/S triples leverage the syntax of RDF Name Graphs [5]: an RDF name graph can be moele by arbitrary sets of triples sharing the same color. The main contributions of this work are: We rely on the notion of colors to capture provenance information of explicit an implicit RDF triples. In particular, we employ a semigroup structure, efine by a set of colors to recor an a binary operation + to reason over the provenance of RDF triples. (a) (b) (c) 194

3 Coloring RDF Triples to Capture Provenance 3 We exten the RDFS inference rules [15] for computing the composite provenance of implicit RDF triples. In this respect, we evise an algorithm for etermining on the fly the provenance of non-materialize implicit RDF triples. We stuy the semantics of provenance propagation an querying for the subclass, subproperty an type RDF hierarchies when colore RDF triples are represente as quaruples. In aition, we iscuss atomic upate operations (i.e., inserts an eletes) of RDF quaruples; upates are consiere uner the coherence semantics which allow us to preserve more knowlege uring upate operations [10]. The paper is structure as follows: in Section 2 we present the motivating example that we will use throughout this paper. Section 3 presents the basic RDF an RDFS notions. In Section 4 we introuce the notions of color an quaruple an iscuss inference rules for sets of RDF quaruples. Section 5 iscusses simple queries an atomic upates. We present relate work in Section 6 an conclue in Section 7. 2 Motivating Example We will use, for illustration purposes, an example taken from the News application omain. The RDFS schema of our News example captures information relate to newspapers an politicians as well as the relationships between them an is contribute by ifferent information sources. s p o c q 1 &NYT enorses &. Obama q 2 &NYT rf:type Newspaper c 4 q 3 Newspaper rf:type rfs:class c 3 q 4 Newspaper rfs:subclassof Mass Meia c 3 q 5 Mass Meia rfs:subclassof Meia c 5 q 6 Caniate rf:type rfs:class c 5 q 7 &. Obama rf:type Caniate c 5 q 8 enorses rf:type rf:property q 9 enorses rfs:omain Newspaper q 10 enorses rfs:range Caniate q 11 Caniate rfs:subclassof Person q 12 supports rf:type rf:property q 13 supports rfs:omain MassMeia q 14 supports rfs:range Person q 15 enorses rfs:subpropertyof supports q 16 &NYT enorses &. Obama q 17 Meia rf:type rfs:class c 5 Fig. 2. Relation Q(s, p, o, c) Relation Q(s,p,o,c) that stores both schema an ata triples is shown in Fig. 2. For Q(s, p, o, c), the s, p an o columns stan for the subject, preicate an object of an RDF triple. Column c is use to store the provenance (color) of a triple (s, p, o), which correspons to the source from which this triple originates from an is represente by a URI (the URI of the source). We say that a triple (s, p, o) is colore c iff (s, p, o, c) is in Q(s, p, o, c). The set of triples (s, p, o) can be obtaine by projecting on columns s, p an o of Q(s, p, o, c). In Q(s, p, o, c), a triple (s, p, o) can be assigne ifferent colors (e.g., quaruples q 1 an q 16 ); this way, we can capture ata integration scenarios in which the same piece of information originates from ifferent sources. Since RDF/S graphs can be seen as a kin of noe an ege labele irecte graphs, we use the following graphical notation: classes an properties (binary relations between classes) of an RDFS vocabulary, are represente with boxes, an ovals respectively. Instances of classes contain their URI reference an to istinguish between in- 195

4 4 G. Flouris, I. Funulaki, P. Peiaitis, Y. Theoharis, an V. Christophies iviual resources an classes, we prefix the URI of the former with the & symbol. RDFS built-in properties rfs:subclassof, rf:type an rfs:subpropertyof are represente by ashe, otte an otte-ashe arrows respectively. Fig. 3 shows the graph obtaine from the quaruples in Q(s, p, o, c) of Fig. 2. For (s, p, o, c), we color the ege with c an we raw s p c o (except if p is one of the built-in RDF/S properties where we raw s c o). Meia c 5 Mass Meia rfs:class c rf:property 5 c 5 c 3 c c5 1 supports Person rf:type rfs:subclassof rfs:subpropertyof c 3 c 1 Newspaper enorses Caniate rfs:omain rfs:range c 4 &NYT enorses enorses c 5 &. Obama Fig. 3. Graph representation of Q(s, p, o, c) great part of the information capture by a set of RDF triples can be inferre [15] by the transitivity of class an property subsumption relationships (rfs:subclassof an rfs:subpropertyof respectively) state in the associate schemas. For instance, although not explicitly asserte in Q(s, p, o, c), we can infer that Newspaper is a subclass of Meia because (i) Newspaper is a subclass of Mass Meia (quaruple q 4 ) an (ii) Mass Meia is a subclass of Meia (quaruple q 5 ). The question raise here is the following: what is the color of the implicit RDF triple? : the triple cannot be colore either c 3 or c 5 (colors of quaruples q 4 an q 5 respectively). In terms of provenance, we view the origin of this triple as composite: this triple shoul be colore c 3 an c 5. possible way to represent this fact is to a quaruples (Newspaper, rfs:subclassof, Meia, c 3 ) an (Newspaper, rfs:subclassof, - Meia, c 5 ) in Q(s, p, o, c). ut, in this case, the query return the triples colore c 3 woul falsely return (Newspaper, rfs:subclassof, Meia). Hence, composite origin cannot be capture by associating with the implicit triple separately the colors of its implying triples. Instea, we capture the provenance of implicit triples using colors efine by applying the special operation + on the colors of its implying triples. For instance, we color triple (Newspaper, rfs:subclassof, Meia) with the color c (3,5) = c 3 + c 5. Color c (3,5) can be seen as a new, composite source of information: it is a new URI an is assigne to the triples which are implie by triples colore c 3 an c 5 (as in the above example). Note that in this paper, we only consier where provenance [33], i.e., we recor the sources that contribute in generating a particular triple, not the processes 196

5 Coloring RDF Triples to Capture Provenance 5 (e.g., the particular inference rules) that le to its generation (how provenance) [33]. In Section 4 we iscuss the properties of operation + in more etail. s with annotate atabases [11, 12], we aim at supporting both provenance propagation an provenance querying over the subclass, subproperty an type RDF hierarchies. In the case of provenance propagation, queries are targete to the original ata an provenance is propagate to the query results. The result of these queries are explicit an implicit colore RDF triples. For instance, one can ask the query return all instances of class Newspaper. The result of the query will be (&NYT, rf:type, - Newspaper, c 4 ). Query return all subclasses of Meia will return (Newspaper, rfs:subclassof, Meia, c (3,5) ) an (Mass Meia, rfs:subclassof, Meia, c 5 ). Note that (Newspaper, rfs:subclassof, Meia, c (3,5) ) is an implicit quaruple obtaine by applying the transitive rfs:subclassof subsumption relationship on quaruples (Mass Meia, rfs:subclassof, Meia, c 5 ) an (Newspaper, rfs:subclassof, Mass Meia,c 3 ) as previously iscusse. On the other han, in the case of provenance querying, queries are explicitly targete at provenance information. Examples falling in this category are queries asking for the color of a given triple, or queries that filter triples given a color. For instance, the query return the color of (Newspaper, rfs:subclassof, Meia), will return c (3,5). In this work, we support atomic upates; more specifically, one can either (a) insert a quaruple, or (b) elete an explicit or implicit quaruple. s previously state, for our upate operations we ahere to the coherence semantics [10], accoring to which we must explicitly retain all the implicit triples that will no longer be implie after the eletion of an explicit triple; we must also ensure that the resulting set of quaruples is reunant-free. s an example, consier the eletion of (&. Obama, rf:type, Caniate, c 5 ). ccoring to the coherence semantics we must retain the implicit information (&. Obama, rf:type, Person, c (1,5) ) implie by quaruples (&. Obama, rf:type, Caniate, c 5 ) an (Caniate, rf:subclassof, Person, ). Ha we followe the founational semantics, the implicit triple (an all the information it carries) woul be lost, which woul correspon to an unnecessary loss of information. 3 Preliminaries In this work, we ignore non-universally ientifie resources, calle unname or blank noes [14]. In this respect, we consier two isjoint an infinite sets U, L, enoting the URIs an literals respectively. Definition 1 n RDF triple (subject, preicate, object) is any element of the set T = U U (U L). The RDF Schema (RDFS) language [2] provies a built-in vocabulary for asserting user-efine schemas in the RDF ata moel. For instance, RDFS names rfs:resource (res), rfs:class (class) an rf:property (prop) 3 coul be use as objects of triples escribing class an property types. Furthermore, one can assert instance of relationships 3 In parenthesis are the terms we use to refer to the RDFS built-in classes an properties. 197

6 6 G. Flouris, I. Funulaki, P. Peiaitis, Y. Theoharis, an V. Christophies of resources with the RDF preicate rf:type (type), whereas subsumption relationships among classes an properties are expresse with the RDFS rfs:subclassof (sc) an rfs:subpropertyof (sp) preicates respectively. In aition, RDFS rfs:omain (omain) an rfs:range (range) preicates allow one to specify the omain an range of the properties in an RDFS vocabulary. In the rest of this paper, we consier two isjoint an infinite sets of URIs of classes (C U) an property types (P U). 4 Provenance for RDF/S Data In the same spirit as in [4] for relational atabases, we assign a color to an RDF triple in orer to capture its provenance. Definition 1. Structure (I, + ) is a commutative semigroup where: I U is the set of colors, isjoint from the sets of class C an property types P. + is a binary operation with the following properties:,, c 3 I + = (Iempotence) + = + (Commutativity) + ( + c 3 ) = ( + ) + c 3 (ssociativity) inary operation + is efine to capture the provenance of implicit RDF triples (see Table 1) an returns a composite color which is also in I, i.e., if a triple t is implie by triple t 1 (whose color is ) an t 2 (whose color is ), then the color of t shoul be + an is enote by c (1,2). In fact, c (1,2) is a new URI in I whose exact form is an implementation etail that we on t aress in this paper. Note that, for n colors, there are O(2 n ) possible composite colors, so the storage of a composite color woul require O(n) bits in an efficient implementation. For an implicit RDF triple, its implying triples are those use to obtain it through the application of the inference rules that will be iscusse in Section 4.1. We say that color c k is a efining color of c (1,2,...,n) an write c k,2,...,n iff k {1, 2,..., n}, i.e., if c (1,2,...,n) can be obtaine by an operation of the form c (1,2,...,n) = c k + c for some color c. In other wors, if a triple t is colore c (1,2,...,n) an c k is a efining color of c (1,2,...,n), then t is an implicit triple which has been inferre using some triple colore c k. The intuition behin the properties of the + operation is the following: (a) an implicit RDF triple obtaine from triples of the same color inherits the color of its implying triples (iempotence) (b) the color of an implicit triple is uniquely etermine by the colors of the triples that imply it an not by the orer of application of the inference rules (commutativity an associativity). We shoul also note that + is a new color unless =. In this manner, we keep a clear separation of what is given (explicit) an what can be implie (implicit). Moreover, the choice of iempotence is ue to the semantics we give to colors as sources of information an the type of provenance we support: we nee to know which sources participate in the creation of a new triple irrespective of which triples contribute to its implication. On the other han, commutativity an associativity stem from the fact that the set of triples that are implie by a given triple set is the same irrespective of the orer in which the inference rules are applie. 198

7 Coloring RDF Triples to Capture Provenance 7 Definition 2 n RDF quaruple (subject, preicate, object, color) is any element of the set D = U U (U L) I. Using this efinition, we can efine the notion of an RDF ataset featuring triples associate with their provenance information as follows: Definition 3 n RDF Dataset is a finite set of quaruples in D ( D). Note that none of the existing approaches in the literature combine intentional an extensional assignment of triples to provenance information (colors). In [30] RDF Name Graphs, capturing the provenance of RDF triples, are efine intentionally through SPRQL [31] views an o not support the explicit assignment of triples to such graphs, whereas in [5] a purely extensional efinition is followe. The notion of colors as introuce in this paper allows us to capture both the intentional an extensional aspects of RDF graphs (i.e., sets of RDF triples) that are useful to recor an reason about provenance information in the presence of upates. 4.1 Inference Similarly to RDF graphs (i.e., sets of triples) we efine a consequence operator that abstracts a set of inference rules. These rules compute the closure of an RDF ataset, enote by Cn(), which is obtaine using the inference rules of Table 1. We say that a ataset entails a quaruple q, an write q, iff q Cn(). We say that a triple t = (s, p, o) is colore c iff (s, p, o, c) Cn(). The inference rules shown in Table 1 exten those specifie in [15] in a straightforwar manner to take into account colors. They compute the color of an implicit triple, using the colors of its implying triples. For instance, for our motivating example of Section 2, quaruple (Newspaper, rfs:subclassof, Meia, c (3,5) ) is obtaine by applying the Transitivity of sc I (2) rule on quaruples (Newspaper, rfs:subclassof, Mass Meia, c 3 ) an (Mass Meia, rfs:subclassof, Meia, c 5 ). 4.2 Reunancy Elimination In our work we consier that the RDF atasets are reunant free. n RDF ataset is reunant free if there oes not exist a quaruple that can be implie by others when applying inference rules I (1) I (6) of Table 1. The etection an removal of reunancies is straightforwar using these rules. In the sequel, we assume that queries an upates are performe upon reunant free RDF atasets. In effect, this means that reunancies are etecte (an remove) at upate time rather than at query time. This choice was mae because we believe that in real scale Semantic Web systems, query performance shoul prevail over upate performance. Reunant free RDF atasets were chosen because they offer a number of avantages in the case of transaction management for concurrent upates an queries. 199

8 8 G. Flouris, I. Funulaki, P. Peiaitis, Y. Theoharis, an V. Christophies Reflexivity of sc Transitivity of sc (C, type, class, ) (C 1, sc, C 2, ), (C 2, sc, C 3, ) I (1) : (C, sc, C, ) I (2) : (C 1, sc, C 3, c (1,2) ) Reflexivity of sp Transitivity of sp (P, type, prop, ) (P 1, sp, P 2, ), (P 2, sp, P 3, ) I (3) : (P, sp, P, ) I (4) : (P 1, sp, P 3, c (1,2) ) Transitivity of class instantiation Transitivity of property instantiation (x, type, C 1, ), (C 1, sc, C 2, ) (P 1, sp, P 2, ), (x 1, P 1, x 2, ) I (5) : (x, type, C 2, c (1,2) ) I (6) : (x 1, P 2, x 2, c (1,2) ) Table 1. subset of the RDFS Inference Rules for RDF Datasets 5 Querying an Upating RDF Datasets 5.1 Querying RDF Datasets In this section we iscuss a simple class of queries that allow one to express queries on the subclass, subproperty an type hierarchies of an RDF ataset. We consier V, C to be two sets of variables for resources an colors respectively; V, C, U (URIs) an L (literals) are mutually isjoint sets. We efine a simple form of a quaruple pattern, calle q-pattern which is an element from (U V) {type sc sp} (U V) (I C). In our context, a query is of the form (H,, C) where H (hea) is a q-pattern, (boy) is an expression efine after the anteceent of the inference rules presente in Section 4.1, an C (constraints) is a conjunction of atomic preicates. Each atomic preicate has the form: (1) v = const for v V; (2) v v for v, v C; (3) c = c k where c C an c i I. ccoring to the above efinition, one can express constraints on resources (1), on colors (2), as well as to specify that a color consiere in the query is efine by a set of other colors (3). In aition, we require that all variables that appear in the hea of the query (H) appear in the query s boy (). This restriction is impose in orer to have computationally esirable properties. We enote variables with?x,?y,... for resources an?,?,... for colors. To efine the query semantics, we use the notion of mapping (as in [21]) as follows: a mapping µ is a partial function µ : (V C) U. The omain of µ (om(µ)) is the subset of V C where µ is efine. For a variable?v, µ(?v) enotes the resource or color to which?v is mappe through µ. To efine the semantics of a q-pattern we must efine first the semantics of a property r over an RDF ataset, enote by [[r]]. Given an RDF ataset, [[r]] for properties type, sc, sp, omain, range an user efine property p is given in Table 2. We write S q to enote that ataset entails quaruple q when the inference rules in S are applie on. 200

9 [[type]] = {(x, y, c) {I (2) [[sc]] = {(x, y, c) {I (1) Coloring RDF Triples to Capture Provenance 9,I(5),I(2) } (x, type, y, c)} } (x, sc, y, c)} [[sp]] = {(x, y, c) (3) {I,I(4) [[omain]] = {(x, y, c) (x, omain, y, c)} } (x, sp, y, c)} [[range]] = {(x, y, c) (x, range, y, c)} [[p]] = {(x, y, c) (4) {I (x, p, y, c)},i(6) } Table 2. [[r]] for type, sc, sp, omain, range an user efine property p [[(a, exp,?y,?c)]] = {µ om(µ) = {?y,?c} an (a, µ(?y), µ(?c)) [[exp]] } [[(?x, exp, a,?c)]] = {µ om(µ) = {?x,?c} an (µ(?x), a, µ(?c)) [[exp]] } [[(?x, exp,?y, c)]] = {µ om(µ) = {?x,?y} an (µ(?x), µ(?y), c) [[exp]] } [[(a, exp, b,?c)]] = {µ om(µ) = {?c} an (a, b, µ(?c)) [[exp]] } [[(a, exp, b, c)]] = {µ om(µ) = an (a, b, c) [[exp]] } Table 3. Semantics of q-patterns We write r to enote the semantics of property r when no inference rule is use. We can now efine the semantics of a q-pattern. Consier an RDF ataset an q = (?X, exp,?y,?c) a q-pattern, where exp is one of sc, sp, type, omain an range. Then the evaluation of q over is efine as follows: [[q]] = {µ om(µ) = {?X,?Y,?c} an (µ(?x), µ(?y ), µ(?c)) [[exp]] } In Table 3 we give the semantics of some q-patterns when URIs an colors are consiere (where a, b U an c I). Finally, given a mapping µ we say that µ satisfies an atomic preicate C, enote by µ C, per the following conitions: µ (?x = const) iff µ(?x) = const, const U,?x om(µ) µ (?x =?y) iff µ(?x) = µ(?y),?x,?y om(µ) µ (?c =?c ) iff µ(?c) = µ(?c ),?c,?c om(µ) µ (?c?c ) iff µ(?c) µ(?c )?c,?c om(µ) µ (?c = c k ) iff µ(?c) = c k,?c om(µ) For a query Q = (H,, C) an RDF ataset an mapping µ, such that µ [[]], an µ C, µ(h) is the quaruple obtaine by replacing every variable?x in om(µ) with µ(?x). The color variable (if any) is replace by the color obtaine by applying the + operator as specifie by the inference rules I (1) to I (6) of Table 1. The answer to Q is the union of the quaruples µ(h) for each such mapping µ. 5.2 Upating RDF Datasets In this section we iscuss atomic upate operations (inserts an eletes). Recall that in our work we follow the coherence semantics [10] accoring to which we nee to retain implicit information that woul be lost uring a triple eletion. Moreover, we enforce that the resulting RDF atasets will remain vali with respect to the employe RDFS 201

10 10 G. Flouris, I. Funulaki, P. Peiaitis, Y. Theoharis, an V. Christophies schema. The notion of valiity has been escribe in various fragments of the RDFS language ([18, 32]), an is use to overrule certain triple combinations. In the context of RDF atasets, the valiity constraints are applie (an efine) at the level of the ataset, but the color-relate part of the quaruple is not consiere. The valiity constraints that we consier in this work concern the isjointness between class, property an color names an the acyclicity of rfs:subclassof an rfs:subpropertyof subsumption relationships. n aitional valiity constraint that we consier in our work is that the subject an object of the instance of some property shoul be correctly classifie uner the omain an range of the property respectively. For a full list of the relate valiity constraints, see [19]. The semantics of each atomic upate is specifie by its corresponing effects an sie-effects. The effect of an insert or elete operation consists of the straightforwar insertion/eletion of the requeste quaruples. The sie-effects ensure that the resulting RDF ataset continues to be vali an non-reunant as iscusse in [35]. Upate semantics ahere to the principle of minimal change [9], per which a minimal number of insertions an eletions shoul be performe in orer to restore a vali an nonreunant state of an RDF ataset. The effects an sie-effects of insertions an eletions are etermine by the kin of triple involve, i.e., whether it is a class instance or property instance insertion or eletion. Due to space restrictions we only escribe class instance insertions an eletions. INSERT Operation: primitive insert operation is of the form: insert(s, p, o, i) where s, p U, o U L, i I Data: insert(x, type, y, i), RDF ataset Result: Upate RDF ataset if ( (x, y, i) [[type]] ) then return ; if (y / C) then return ; forall ((x, z, i ) type s.t. (y, z, i ) [[sc]] an i = i + i ) o = \ {(x, type, z, i )}; en = {(x, type, y, i)}; return lgorithm 1. Class Instance Insertion lgorithm formal escription of the insertion of a quaruple (x, type, y, i) in an RDF ataset along with its sie-effects can be foun in lgorithm 1. t line 1 we examine if the quaruple alreay belongs to the semantics of property type. If not, then we ensure that y is a class (lines 2 3). If it is, then we remove all class instantiation quaruples from the RDF ataset which can be implie through the quaruple to be inserte an the class subsumption relationships (lines 4 6), to guarantee that the result is reunant free. Finally, the quaruple is inserte (line 7). n example of a class instance insertion is shown in Figures 4(a) an 4(b). 202

11 Coloring RDF Triples to Capture Provenance 11 Data: elete(x, type, y, i), RDF ataset Result: Upate RDF ataset 1 if ( (x, y, i) [[type]] ) then return ; 2 forall ((x, y, i ) [[type]],(y, y, i ) [[sc]] s.t. i = i + i ) o 3 forall (y, z, k) sc s.t. y! = z o 4 if (z, y, h) [[sc]] then = {(x, type, z, i + k)} 5 en 6 = \ {(x, type, y, i )} 7 en 8 forall (x, o, h) q, s.t. (q, c, i) omain o 9 if (x, c, j) [[type]] then 10 = \ {(x, q, o, h)} ; 11 forall q s.t. (q, q, h ) [[sp]] o 12 if (x, e, k) [[type]] s.t. (q, e, k ) omain then = {(x, q, o, h + h )} 13 en en 16 forall (o, x, h) q, s.t. (q, c, i) range o 17 if (x, c, j) [[type]] then 18 = \ {(o, q, x, h)} ; 19 forall q s.t. (q, q, h ) [[sp]] o 20 if (x, e, k) [[type]] s.t. (q, e, k ) range then = {(o, q, x, h + h )} 21 en en 24 return lgorithm 2. Class Instance Deletion lgorithm DELETE Operation primitive elete operation is of the form: elete(s, p, o, i) where s, p U, o U L, i I. formal escription of the eletion of a quaruple (x, type, y, i) is given in lgorithm 2. t line 1 we examine if (x, type, y, i) belongs to the semantics of property type. If this is the case, then we must (1) insert all the quaruples that are implie by the quaruple that we wish to elete (per the coherence semantics) an (2) elete the quaruples that if retaine woul imply the quaruple we wish to elete (lines 2 7). To ensure that the RDF ataset is still vali after the upates, we must remove all properties originating from (or reaching resp.) x whose omain (or range resp.) is a class that x is no longer an instance of (lines 8 23). Examples of class instance eletions can be foun in Figures 4(c) an 4(). 5.3 Complexity nalysis When working with colore RDF triples, one of the basic kins of queries that we nee to answer is what is the color of a triple. This is a provenance query an essentially 203

12 12 G. Flouris, I. Funulaki, P. Peiaitis, Y. Theoharis, an V. Christophies insert(&x, type,, ) insert(&x, type,, c1) c (1,2) &x (&x, type,, ) is inserte (&x, type,, c (1,2)) is elete since it coul be inferre by existing quaruples &x &x (&x, type,, ) is inserte (&x, type,, ) is reunant an elete &x (a) Class Instance Insertion (1) (b) Class Instance Insertion (2) C c 3 D elete(&x, type,, c (1,2)) c 3 C D elete(&x, type,, ) &x (&x, type, C, c (1,2)) is inserte (&x, type, D, c (1,2)) is not inserte since quaruple (&x, type,, c (1,2)) woul still be inferre (&x, type,, ) is elete because otherwise (&x, type,, c (1,2) woul still be inferre. c (1,2) &x &x (&x, type,, ) is elete (&x, type,, c (1,2)) is inserte because it was initially inferre &x c (1,2) (c) Class Instance Deletion (1) () Class Instance Deletion (2) Fig. 4. Examples of Class Instance Insertions ((a) (b)) an Deletions ((c) ()) boils own to fining, for a given triple (s, p, o) an RDF ataset, all quaruples of the form (s, p, o, c) in Cn(). Given that we work with reunant free RDF atasets (so implicit triples are not materialize), we present here an algorithm that computes the color of RDF triples without materializing Cn(), an iscuss its complexity. 4 c 9 1 c 3 c 4 c 4 2 c Consier an RDF ataset whose size is N. In orer to etermine the color of (s, p, o), without computing Cn(), we nee to fin all the possible ways in which a quaruple of the form (s, p, o, c) (for any c) can be inferre using quaruples from. Using the algorithm below, this can be mae in O(N log(n)) time. c 3 Certain quaruples are not involve in the inference rules in Table 1, so they cannot be implie c9 by others: these are all the quaruples that o not involve the RDFS type, sc an sp relationships. Determining the color of such a quaruple is trivial, as Fig. 5. sc Hierarchy we only nee to check (rather than Cn()), so it can be mae in O(log(N)) time by a simple search in (using an appropriate inex). 204

13 Coloring RDF Triples to Capture Provenance 13 To etermine the color of quaruples that involve the aforementione built-in RDFS properties, we view the sc an sp hierarchies as irecte acyclic graphs. Noes in the graph are classes (properties resp.), an there exists an ege between noes x an y iff x is a subclass (subproperty resp.) of y. The problem of obtaining the sequence of quaruples from which e.g., quaruple (, sc,, c) (for any c), can be implie is equivalent to iscovering the path(s) (i.e., sequences of eges) in the DG from noe to noe. For each of these sequences of eges, we keep the color of each involve quaruple. Recall that an ege may involve several quaruples (i.e., a triple can be colore with ifferent colors). The algorithm uses a epth-first search an terminates when is reache. t each step of the algorithm, we (i) fin the superclasses of the context noe (i.e., the noe that we are currently looking at) that are also subclasses of the target noe (e.g., ) an (ii) store the set of colors of the eges that we have alreay examine Data: Classes source an target, RDF Dataset, Set of colors S; Result: Set of colors S; if source=target then return S; res ; foreach class x, where x is a superclass of source an subclass of target o Let (source, sc, x, c) ; Let S x ; forall colors c i in S o S x S x {c i + c}; en res res Traverse Subsumption Graph(x,target,,S x); en return res lgorithm 3. Traverse Subsumption Graph In lgorithm 3 in orer to obtain all the classes that are superclasses of source an subclasses of target (line 4), we use the labeling scheme introuce in [6] that captures the subsumption relationships between classes an properties an allows us to etermine whether a class (or property) is a subclass (or subproperty) of another in constant time. This is achieve by simply comparing the labels of the classes/properties. The labeling scheme is solely use to prune irrelevant subclass an subproperty paths, thereby limiting our search space significantly, without taking into consieration colors of triples. s an example of application, consier the subclass hierarchy shown in Fig. 5. Suppose that we nee to iscover the color of triple (, sc, ). We start with class an in the first step we will keep classes 1, 2 an sets of colors S 1 = { }, S 2 = {c 3, c 9 } since the eges that etermine that is a subclass of 1, 2 are forme by triples colore { }, {c 3, c 9 } respectively. For each of the superclasses of (i.e., 1, 2 ), we perform exactly the same process an exten the sets of colors that we have obtaine us- 205

14 14 G. Flouris, I. Funulaki, P. Peiaitis, Y. Theoharis, an V. Christophies ing operation +. Upon return of the recursion, these colors are place into the set res to be returne. The output of the algorithm are the colors: c (3,4), c (9,4), c (2,4), c (2,9,3), where c (1,...,n) = c n. Given the fact that we prune subsumption paths that are not relevant in line 4 (using the labeling scheme), an that no cycles are allowe, the escribe process will, in the worst case, consier each quaruple in the RDF ataset once. Given that each access requires a search in, the cost of each access is O(log(N)) (using an aequate inexing system). Therefore, the total complexity is O(N log(n)). For the other triples that may appear as implicit ones, the algorithm is similar. In particular, if t is of the form (P, sp, Q), then the process is ientical to the above, except that properties an subproperty relationships are consiere instea of class an subclass relationships. If t is of the form (, type, ) then the process is almost ientical, except that, in the first step of the recursion, we search for all classes whose explicit instance is an are also (implicit or explicit) subclasses of ; the rest is the same. Finally, if t is of the form (x, P, y), then, again, the process is ientical, except that, in the first step of the recursion, we search for all explicit quaruples of the form (x, Q, y, c) such that Q is an explicit or implicit subproperty of P. The rest of the recursion steps are as in the above cases. 6 Relate Work So far, research on recoring provenance for RDF ata has focuse on either associating triples with an RDF name graph [5, 34] or by extening an RDF triple to a quaruple where the fourth element is a URI, a blank noe or an ientifier [7, 17]. These works vary in the semantics of the fourth element which is use to represent provenance, context an access control information. RDF Name graphs have been propose in [5, 34] to capture explicit provenance information by allowing users to refer to specific parts of RDF/S graphs in orer to ecie how creible is, or how evolves a piece of information. n RDF name graph is a set of triples to which a URI has been assigne an can be reference by other graphs as a normal resource; in this manner, one can assign explicit provenance information to this collection of triples. Currently, there is no aequate support on how to manage provenance of implicit an explicit triples in the presence of queries an upates. uthors in [5] o not iscuss RDFS inference, queries an upates in the presence of RDF name graphs. Unfortunately, existing eclarative languages for querying an upating RDF triples have been extene either with RDF name graphs (such as SPRQL [23] an SPRQL Upate [31]) or with RDFS inference support [22, 24], but not with both. In this paper, we attempt to fill this gap by proposing a framework base on the use of colors to capture provenance of RDF triples an reason about the provenance of implicit triples for simple queries an atomic upates. In a previous work [20], we have introuce the notion of RDF/S graphsets which buils upon an extens the notion of RDF name graphs. In that paper we showe that the mechanism of RDF name graphs cannot capture the provenance of implicit RDF triples, an propose RDF graphsets as a solution to this problem. In this paper, we use colors as an elegant an uniform way to capture the provenance of both explicit an implicit RDF triples. Colors are a generalization of 206

15 Coloring RDF Triples to Capture Provenance 15 RDF name graphs [5]: a set of triples colore with a single color can be consiere as belonging to the RDF name graph whose URI is the color (recall that colors are URIs). Colors obtaine by applying the + operation on other colors simulate graphsets [20]. The notion of colors as introuce in this paper allows us to capture both the intentional an extensional aspects of RDF graphs that are useful to recor an reason about provenance information in the presence of upates whereas none of the existing approaches [5, 30] combine intentional an extensional assignment of triples to provenance information. On the other sie of the spectrum, a significant amount of work on the issue has been one for relational an tree-structure atabases [3, 4, 13, 12]. Unlike these works, we consier both recursion an upates, whereas [13] oes not consier upates, [3, 4] supports upates but not recursion an [12] consiers neither recursion or upates. Moreover, we o not focus on provenance propagation through relational operators but through inference rules that can be translate to relational unions an joins with boune recursion. However, inference rules compute on the fly implicit triples without the nee to materialize the provenance of intermeiate results (per recursion step). 7 Conclusion In this paper we propose the use of colors to capture the provenance of RDF ata an schema triples. We use the logical representation of quaruples to store the color of an RDF triple. The use of colors allows us to capture provenance at several granularity levels an can be consiere as a generalization of RDF Name Graphs. One of the main contributions of the paper is the extension of RDFS inference rules to etermine the provenance of implicit RDF triples, an extension which is not possible uner the RDF name graphs approach an has been overlooke in the majority of approaches that eal with managing provenance information for RDF graphs. Note that the extene inference rules o not entail any aitional complexity or scalability concerns to those alreay involving RDFS reasoning (see complexity analysis). s a future work we will stuy a more general algebraic structure (as in [13]) to capture the provenance of triples an mappings obtaine by the SPRQL operators [23]. In aition, we plan to stuy the insertion an eletion of colors where the latter can be expresse as a batch eletion of quaruples sharing the same color. cknowlegements This work was partially supporte by the EU project CSPR (FP IST ). References 1. T. erners-lee, J. Henler, an O. Lassila. The Semantic Web. Scientific merican- merican Eition, D. rickley an R.V. Guha. RDF Vocabulary Description Language 1.0: RDF Schema P. uneman,. P. Chapman, an J. Cheney. Provenance Management in Curate Databases. In SIGMOD, P. uneman, J. Cheney, an S. Vansummeren. On the Expressiveness of Implicit Provenance in Query an Upate Languages. In ICDT,

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