Capturing Provenance of RDF Triples through Colors

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1 Capturing Provenance of RDF Triples through Colors G. Flouris FORTH-ICS, Greece I. Funulaki FORTH-ICS, Greece Y. Theoharis FORTH-ICS, Greece University of Crete, Greece P. Peiaitis FORTH-ICS, Greece University of Crete, Greece V. Christophies FORTH-ICS, Greece University of Crete, Greece STRCT 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. INTRODUCTION 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 Permission to make igital or har copies of all or part of this work for personal or classroom use is grante without fee provie that copies are not mae or istribute for profit or commercial avantage an that copies bear this notice an the full citation on the first page. To copy otherwise, to republish, to post on servers or to reistribute to lists, requires prior specific permission an/or a fee. Copyright 200X CM X-XXXXX-XX-X/XX/XX...$5.00. 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] (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 provenance information of implicit RDF triples affecting the semantics of query an upate RDF languages. There are two ifferent ways in which such implicit knowlege can be viewe an this affects the assignment of provenance information to implicit triples, as well as the semantics of the upate operations. 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], implicit knowlege is only vali as long as the supporting explicit knowlege is there. Therefore, each implicit triple epens on the existence of the explicit triple(s) that imply it. In this paper we propose the use of colors (as in [4]) in orer to capture the provenance of RDF ata an schema triples. We recor the color of an RDF triple as a fourth column, hence obtaining an RDF quaruple as in [7, 17]. To provie the intuition of the granularity levels of provenance for RDF atasets that we capture with this work, we compare it with provenance in the relational worl. 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 Figure 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 RDFS class, captures provenance

2 of the relational table. n RDF Name Graph [5] can be moele by arbitrary sets of triples sharing the same color. 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 s p o a 1 p b 1 b 1 q 1 b 1 t 1 1 r b 2 2 r b 1 Figure 1: Granularity Levels of Provenance 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 an abelian group, efine by a set of colors to recor an a binary operation + to reason over the provenance of RDF triples. 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 when colore RDF triples are represente as quaruples. In aition we iscuss atomic upate operations (i.e., inserts an eletes) for sets of RDF quaruples uner the coherence semantics [10]. In this work, we consier coherence semantics, because we believe that implicit knowlege is equally important to the explicit one, i.e., the semantics of the RDF/S graph shoul be etermine by its logical closure rather than its syntactic formulation (i.e., explicit triples). The paper is structure as follows: in Section 2 we present the motivating example that we will use in 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 MOTIVTING EXMPLE 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. Figure 2 shows relation Q(s, p, o, c) that stores both schema an ata triples where s, p an o 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). We will 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). (a) (b) (c) s p o c q 1 &NYT enorses &. Obama q 2 &NYT rf:type Newspaper c 4 q 3 Newspaper rf:type rfs:class q 4 Newspaper rfs:subclassof Mass Meia 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 Mass Meia Newspaper Figure 2: Relation Q(s, p, o, c) Meia c 5 c 4 &NYT rfs:class c rf:property 5 c 5 c c5 1 supports enorses Person c 1 enorses enorses Caniate c 5 &. Obama rf:type rfs:subclassof rfs:subpropertyof rfs:omain rfs:range Figure 3: Graph representation for relation Q(s, p, o, c) Note that we can assign ifferent colors to the same triple (s, p, o) (quaruples q 1 an q 16). In 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 [2], are represente with boxes, an ovals respectively. Instances of classes contain their URI reference an to istinguish between iniviual resources an classes, we prefix the URI of the former with the & symbol. RDFS built-in properties [2] rfs:subclassof, rf:type an rfs:subpropertyof are represente by ashe, otte an otte-ashe arrows respectively. Figure 3 shows the graph obtaine from the quaruples in Q(s, p, o, c) of Figure 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). 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 RDFS 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 col-

3 ore either or c 5 (colors of quaruples q 4 an q 5 resp.). In terms of provenance, we view the origin of this triple as composite: this triple shoul be colore an c 5. possible solution is to a quaruples (Newspaper, rfs:subclassof, Meia, ) an (Newspaper, rfs:subclassof, Meia, c 5) in Q(s, p, o, c). ut, in this case, query return the triples colore 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 which creates a new color. This color can then be assigne to explicit an not just to implicit triples. For instance, we color triple (Newspaper, rfs:subclassof, Meia) with the color c (3,5) = + c 5. In Section 4 we iscuss the properties of operation + in etail. s with annotate atabases [4, 11, 12], we aim at supporting both provenance propagation an provenance querying for RDF ata. 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,) 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 an more specifically inserts an eletes. One can (a) insert an (b) elete an explicit or an implicit quaruple. For our upate operations we ahere to the coherence semantics [10] an we consier reunant-free graphs. ccoring to the coherence semantics, we must explicitly retain all the implicit triples that will no longer be implie ue to the eletion of an explicit triple. Reunant-free graphs have been chosen, because they offer a number of avantages in the case of transaction management for concurrent upates an queries. Consier for instance, the eletion of quaruple (&. 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 [10], the implicit triple woul be lost. Note that it woul not be correct to remove the implicit knowlege in this example, because the removal of (&. Obama, rf:type, Person, c (1,5) ) was not ictate by the user s upate itself, an shoul therefore be avoie. Furthermore, when reunancy elimination is applie, there is no way to know whether (&. Obama, rf:type, Person, c (1,5) ) was explicitly asserte or not an consequently, we on t know whether it shoul be remove. In this respect, coherence semantics is more aequate when consiering reunant-free RDF/S graphs since they allow one to retain as much information as possible in the presence of upates. 3. PRELIMINRIES 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 builtin 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) 1 coul be use as objects of triples escribing class an property types. Furthermore, one can assert instance of relationships of resources with the RDF preicate rf:type (type), while subsumption relationships among classes an properties are expresse with the RDFS rfs:subclassof (sc) an rfs:sub- PropertyOf (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 a 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). It shoul be finally stresse that RDFS schemas are essentially escriptive an not prescriptive, esigne to represent ata. We believe that this flexibility in representing schema-relaxable (or schema-less) information, is the main reason for RDF an RDFS popularity. Using the uniform formalism of RDF triples, we are able to represent in a flexible way both schema an instances in the form of RDF/S graphs. It shoul be note that RDF/S graphs are not classical irecte labele graphs, because, for example, an RDFS preicate (e.g., rfs:subpropertyof) may relate other preicates (e.g., enorses an supports). Thus, the resulting structure is not a graph in the strict mathematical sense. 4. PROVENNCE FOR RDF/S DT 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 an abelian group where I U is the set of colors. We efine the neutral color ε I, an for each color c I we efine its inverse enote by c; + is a binary operation with the following properties: + ε = (Neutral) + ( ) = ε (Inverse) + = (Iempotence) + = + (Commutativity) + ( + ) = ( + ) + (ssociativity) 1 In parenthesis are the terms we will use in the paper to refer to the RDFS built-in classes an properties.

4 inary operation + is efine to capture the provenance of implicit RDF triples. 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 its implying triples an not by the orer of application of the inference rules (commutativity an associativity). For an implicit RDF triple, its implying triples are those use to obtain it through the application of the inference rules that we will iscuss in Section 4.1. We shoul note here that the inverse color oes not introuce negative knowlege: if a triple is colore (the inverse of color ) the intuition is that the triple can have any color except. The neutral color is a color that is absorbe by any other color. We enote with the color obtaine by applying operation + on colors an : = +. We say that color c k is a efining color of,2,...,n an write c k,2,...,n iff k {1, 2,..., n}. 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 Dataset 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 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 that are useful to recor an reason about provenance information in the presence of upates. 4.1 Inference In a similar manner as for RDF graphs (i.e., sets of triples) we efine a consequence operator that abstracts a set of inference rules which compute the closure of an RDF ataset. Let be an RDF Dataset. We write Cn() to refer to the closure of. Entailment for RDF atasets is efine as for RDF graphs: an RDF ataset 1 entails RDF ataset 2 iff the closure of 2 is a subset of the closure of 1 moulo the colors. We say that a ataset entails a quaruple q, an write q, iff q belongs to the closure of. We say that a triple t = (s, p, o) is colore c iff there exists a quaruple (s, p, o, c) in 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 the implicit triple, using the colors of its implying triples. For instance, 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, Reflexivity of sc I (1) : Transitivity of sc I (2) : Reflexivity of sp I (3) : Transitivity of sp I (4) : Transitivity of I (5) : class instantiation Transitivity of I (6) : property instantiation (C, type, class, ) (C, sc, C, ) (C 1, sc, C 2, ), (C 2, sc, C 3, ) (C 1, sc, C 3, ) (P, type, prop, ) (P, sp, P, ) (P 1, sp, P 2, ), (P 2, sp, P 3, ) (P 1, sp, P 3, ) (x, type, C 1, ), (C 1, sc, C 2, ) (x, type, C 2, ) (P 1, sp, P 2, ), (x 1, P 1, x 2, ) (x 1, P 2, x 2, ) Table 1: Inference Rules for RDF Datasets ) an (Mass Meia, rfs:subclassof, Meia, c 5). We overloa the closure operator Cn in orer to capture the closure of an RDF Dataset, compute using the inference rules of Table 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 those 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. 5. QUERYING ND UPDTING RDF DTSETS 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;

5 [[type]] = {(x, y, c) (2) {I [[sc]] = {(x, y, c) (1) {I [[sp]] = {(x, y, c) (3) {I,I(5),I(2),I(4) } (x, type, y, c)} } (x, sc, y, c)} } (x, sp, y, c)} [[omain]] = {(x, y, c) (x, omain, y, c)} [[range]] = {(x, y, c) (x, range, y, c)} [[p]] = {(x, y, c) (4) {I,I(6) } (x, p, y, c)} Table 2: [[r]] for properties type, sc, sp, omain, range an 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 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 in the same spirit as in [21] as follows: a mapping µ is a partial function µ : (V C) U I. The omain of µ (om(µ)) is the subset of V C where µ is efine. Let µ enote a mapping an?v a variable, then 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 Dataset, enote by [[r]]. Given an RDF Dataset, [[r]] for properties type, sc, sp, omain, range an 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. 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. 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 (a an b are constant URIs an c is a color ientifier in 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 the implicit triples are consiere of equal value as the explicit ones an hence we nee to retain information that woul be lost in the case of a triple eletion. Moreover, we shoul enforce that the resulting RDF atasets will remain vali with respect to the employe RDFS 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 RDF 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].

6 insert(, type,, ) (, type,, ) is inserte (, type,, ) is elete since it coul be inferre by existing quaruples (a) Class Instance Insertion (1) insert(, type,, c1 ) (, type,, ) is inserte (, type,, ) is reunant an elete (b) Class Instance Insertion (2) 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 non-reunant 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. The algorithms presente here iffer from the algorithms that hanle inserts an eletes in the absence of provenance information: when computing the triples to be inserte (or elete) as a sieeffect of the upate operation, we must compute the colors of the non-materialize implicit triples, whereas in the original algorithms no such computation is necessary INSERT Operation primitive insert operation is of the form: insert(s, p, o, i) where s, p U, o U L, i I lgorithm 1: Class Instance Insertion lgorithm 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 ; - insert(, type,, ) (, type,, ) is not inserte since it is reunant per the transitivity of class instantiation inference rule an because + ( - ) = ( + )+ ( - ) = (c) Class Instance Insertion (3) - Figure 4: Examples of Class Instance Insertions 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). Finally, the quaruple is inserte (line 7). Examples of class instance insertions are shown in Figure 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 the quaruple 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 (recall that we follow the coherence semantics) an (2) elete the quaruples that if retaine woul imply the quaruple we wish to elete (lines 2 7). In orer to ensure that the RDF ataset is still vali after the upates, we must remove all properties originating from

7 lgorithm 2: Class Instance Deletion lgorithm 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 ; C D elete(, type,, ) (, type, C, ) is inserte (, type, D, ) is not inserte since quaruple (, type,, ) woul still be inferre (, type,, ) is elete because otherwise (, type,, woul still be inferre. (a) Class Instance Deletion (1) elete(, type,, ) (, type,, ) is elete (, type,, ) is inserte because it was initially inferre C D (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 5 an 6. Figure 6 shows a class instance eletion with the necessary property instance eletions to ensure a vali RDF ataset. 5.3 Complexity nalysis When working with colore RDF triples, the basic kins of queries that we nee to answer are: what is the color of a triple an which triples have a given color. oth kins of queries are classifie as provenance querying problems. In orer to answer them over an RDF ataset we consier that a triple t = (s, p, o) is colore c iff there exists a quaruple (s, p, o, c) in Cn(). In our work we consier reunant-free RDF atasets: implicit triples are not materialize. In this section we iscuss an algorithm that computes the color of non-materialize implicit RDF triples of an RDF ataset without materializing the closure Cn() of an iscuss its complexity. For this analysis, we consier to be an RDF ataset. We enote by M t the set of triples of : M t = {(s, p, o) c I, (s, p, o, c) a quaruple in } an with M c the set of colors in. In orer to etermine whether a triple t is colore with a color c in an RDF ataset, we must consier all possible combinations of triples in M t that involve at least one triple that is colore by one of the efining colors of c. For (b) Class Instance Deletion (2) - elete(, type,, ) (, type,, ) is elete an (, type,, ) is inserte. It is inferre using the transitivity of class instantiation inference rule an since + (- ) = ( + )+ (- ) = (c) Class Instance Deletion (3) - Figure 5: Examples of Class Instance Deletions

8 E E q D elete(, type, E, ) q D c 4 p c 4 p c 5 q C &o a. (, type,, c1) is elete since otherwise (, type, y, ) ) woul still be inferre b. (, p, &o,c 5) is elete since is no longer an instance of the omain of property p (class ) c. (, p, &o, c (5,4) ) is inserte since it was inferre in the initial graph. c 4 p c 4 q c (5,4) q C &o &o' &o' Figure 6: Class Instance an Property Instance Deletion 4 c 9 1 c 4 c 4 2 c9 5 3 Figure 7: sc Hierarchy instance, consier the class subsumption hierarchy shown in Figure 7. To compute the set of triples colore c (2,3,9) = + + c 9, we nee to consier the combination of all triples colore with at least one of colors, an c 9. brute force approach woul be to compute Cn() in orer to compute the color of a given triple. However, there is a much faster approach which avois the computation of Cn(), but, instea, computes irectly the set of triples from which t can be implie. Following this approach, we can etermine whether a triple t is colore c in O( M t log( M t )) time. Determining the color of a triple not implie by others is trivial: these are all the triples that o not involve the RDFS type, sc an sp relationships (the inference rules in Table 1 consier only those). This is achieve in O(log( M t )) by a simple search for t in. To etermine the color of implicit triples 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 triples from which triple (, sc, ), for instance, 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 triples, we are going to keep the colors of each triple. Recall that 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 triples (i.e., eges) that we have alreay examine. lgorithm 3: Traverse Subsumption Graph Data: Classes source an target, RDF Dataset, Set of colors S; Result: Set of colors S; 1 if source=target then 2 return S; 3 S ; 4 foreach class x, where x is a superclass of source an subclass of target o 5 Let (source, sc, x, c) ; 6 Let S x ; 7 forall colors c i in S o 8 S x S x {c i + c}; 9 en 10 Traverse Subsumption Graph(x,target,,S x); 11 S S S x 12 en 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. For instance, consier the subclass hierarchy shown in Figure 7. 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 9} since the triples that etermine that is a subclass of 1, 2 are etermine by triples colore {}, {, c 9}

9 respectively. For each of the superclasses of, we perform exactly the same process an we will be extening the sets of colors that we have obtaine. The result of this process is the set of colors: {c (3,4), c (9,4), c (2,4), c (2,9,3) } where a color c (1,...,n) = c n. Given the fact that we prune subsumption paths that are not use to imply the triple in question using the labeling scheme, an that no cycles are allowe, the escribe process will, in the worst case, consier each triple in the RDF ataset once. Given that each triple access requires a search in, the cost of each access is O(log( M t )) (using an aequate inexing system). Therefore, the total complexity is O( M t log( M t )). 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 triples of the form (x, Q, y) such that Q is an explicit or implicit subproperty of P. The rest of the recursion steps are as in the above cases. 6. RELTED WORK So far, research on recoring provenance for RDF ata has focuse on either associating triples with an RDF name graph [5] 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 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]. Note that none of the existing approaches 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 that are useful to recor an reason about provenance information in the presence of upates. On the other sie of the spectrum, a significant amount of work on the issue has been one for relational an treestructure atabases [3, 4, 13, 12]. In [3, 4] authors iscuss explicit provenance recoring uner copy-paste semantics where all operations are a sequence of elete-insert-copypaste operations. In that work, new ientifiers are introuce in the case in which the same object is elete an then re-inserte, whereas in our case we are able to recognize the corresponing triple, an consequently preserve provenance information. In [13], fine-graine where an how provenance for relational atabases is capture; however, upates are not consiere in that work. Finally, in [12] authors consier a colore algebra to annotate columns an rows of relational tables at a coarse graine level which bares similarities to our color-base approach to recor provenance of RDF triples. 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. This problem has been overlooke in the majority of approaches that eal with managing provenance information for RDF graphs. s a future work we will stuy a more general algebraic structure as in [13] to capture the provenance of triples obtaine by the SPRQL operators [23]. 8. REFERENCES [1] T. erners-lee, J. Henler, an O. Lassila. The Semantic Web. Scientific merican-merican Eition, [2] D. rickley an R.V. Guha. RDF Vocabulary Description Language 1.0: RDF Schema [3] P. uneman,. P. Chapman, an J. Cheney. Provenance Management in Curate Databases. In SIGMOD, 2006.

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