NdProperties: Encoding Contexts in RDF Predicates with Inference Preservation

Abstract

Annotating RDF triples within a context (such as provenance) is becoming more and more important in the recent years. A number of approaches exist to represent this annotations, but how to distinguish the information of each context and be able to reason with it is still a challenge. In this work, we present formalizations of the necessary transformations in the RDF graph to encode the contextual information as contextualization function in RDF, and describe the existing approaches according to this formalization. We then present NdProperties, an OWL-DL ontology inspired by NdFluents and singleton properties, and present alternative contextualization functions for it.

Keywords

• RDF

-

OWL

Contexts Reasoning Rei cation

Introduction

In the recent years, the necessity of annotating RDF statements with contextual information have been increasing. However, a triple can only represent a binary relations between a subject and an object. To make explicit the relationship between the statement and its context, either the information must be encoded outside of the RDF syntax, or the statement and its context must be split into multiple triples, possibly resulting in the subject and the object being disconnected.

With most common approaches for representing contextual information, inferences that were following from the original statement are not preserved. Recent approaches allow for some inferences to be preserved. However, the result of the inference has no connection with the context. This makes impossible to separate the knowledge in di erent context, and can lead to data that is inconsistent or that does not correspond with the ground truth, due to some undesirable inference results.

In this work we present NdProperties, an OWL-DL ontology to use triple predicates to relate statements to their context. This approach is inspired by Singleton Properties [ 8 ] (which also use the predicates to represent the statement) Copyright c 2018 for this paper by its authors. Copying permitted for private and academic purposes. and NdFluents [ 5 ] (which is able to better separate the contexts), NdProperties can be seen as a concrete implementation of a generic approach where a subset of the terms is contextualized, explored also for all the terms in NdTerms [ 10 ]. The NdProperties ontology can be implemented in di erent ways, in order to allow more or less inferences.

The rest of the document is organized as follows: Section 2 presents some definitions, necessary to understand the rest of the work; Section 3 present current approaches to annotate statements with additional data; Section 4 describes the NdProperties ontology, with Section 5 studying its inference capabilities; nally, we provide some discussion possible future work in Section 6. 2

Preliminaries

In order to go from a set of triples, plus the desired contextual annotations, to a set of contextualized triples (i.e., a set of rei ed or modi ed triples, plus additional triples that express the annotations), it is necessary to perform a transformation. In this section we introduce the notation we will use for the rest of the paper, formalize this transformation as a contextualization function, and present some desired properties that such a function should have. This section summarizes and updates previous work in the topic [ 5, 10 ].

We assume in nite disjoint sets I (IRIs), B (blank nodes), and L (literals). An RDF triple is a tuple (s; p; o) 2 (I [ B) I (I [ B [ L), where s is called the subject, p is the predicate and o is the object. We write T the in nite set of triples. An RDF graph G is a set of RDF triples. For the sake of simplicity, we will assume that a contextual annotation is a single IRI. We assume an in nite set of contexts C I.

De nition 1 (Contextualization function). A contextualization function in RDF is any function f : C 2T ! 2T .

Usually, contextualization functions need to create fresh terms that will be used to reify the statements. This is often done by renaming existing terms using one or more injective functions ren, which may or may not depend on the context or the type or term being renamed. This function is in practice usually arbitrary. The idea of a contextualization function is that it expresses that the knowledge of the original graph holds in the context. However, many meaningless contextualization functions are possible that do not necessarily encode that knowledge. In order to identify useful contextualization functions, we propose the following property: De nition 2 (Inference preservation). A contextualization function f preserves inferences wrt a set of graphs G i for all c 2 C and all graphs G; G0 2 G, i G j= G0 then f (c; G) j= f (c; G0).

Inference preservation expresses the idea that knowledge that can be inferred in the original graph should also be inferred within the context. It is also possible that the contextualization function preserves some inferences, but it is not able to encapsulate the inferred knowledge in the context. We call this property Non-contextual Inference preservation. Although this property can be useful if the goal of contextualization function is to simply annotate triples that are considered universally true, it can lead to undesirable inferences if that is not the case, as we will see in Section 5.

De nition 3 (Non-contextual Inference preservation). A contextualization function f preserves entailments wrt a set of ontologies O i for all c 2 C and all ontologies O; O0 2 O, i O j= O0 then f (c; O) j= O.

This de nitions will be used to study existing rei cation approaches, as well as the newly de ned NdProperties. 3

Existing approaches for representing context

In this section we present and describe the most relevant rei cation approaches (namely RDF rei cation, n-ary relations, the singleton and companion properties, and NdFluents), and formalize their contextualization functions. In the following de nitions, we will use t as the parameter of the ren function when the fresh term does not depend on a concrete term of the triple. Note that for RDF rei cation and n-ary relations more than one contextualization function could be possible. For the sake of simplicity, we consider an elementary transformation were the added term will be used to represent the context.

RDF rei cation [2, Sec. 5.3] is the standard W3C model to represent information about a statement. A contextualization function for rei cation can be de ned as follows.

De nition 4 (RDF rei cation contextualization function). Let c 2 C and G 2 2T be a context and a graph respectively. We de ne the contextualization function fR such that fR(c; G) = f(renc(t); type; Statement); (renc(t); subject; s); (renc(t); predicate; p); (renc(t); object; o) j (s; p; o) = t 2 Gg.

Techniques for representing n-ary relations in RDF [ 9 ] were published in 2006 as a W3C note. A contextualization function for them can be de ned as follows. De nition 5 (N-ary relations contextualization function). Let c 2 C and G 2 2T be a context and a graph respectively. We de ne the contextualization function fnary such that fnary(c; G) = f(s; rencs(p); rencr(p)); (rencr(p); rencv(p); o) j (s; p; o) = t 2 Gg.

Singleton properties [ 8 ] are a recent proposal based on creating a unique property for each triple and using it to reify the triple. It extends the RDF semantics in order to make each singleton property unique and to include its extension in the extension of the original property. However, these semantics can be emulated with the following contextualization function.

De nition 6 (Singleton Property contextualization function). Let c 2 C and G 2 2T be a context and a graph respectively. We de ne the contextualization function fsp such that fsp(c; G) = f(s; ren(p); o); (ren(p); type; SingletonP roperty); (ren(p); singletonP ropertyOf; p); (ren(p); type; F unctionalP roperty); (ren(p); type; InverseF unctionalP roperty) j (s; p; o) = t 2 Gg [ ((singletonP ropertyOf; subP ropertyOf; subP ropertyOf )).

The Companion Property [ 4 ] is an attempt to reduce the number of unique properties that are generated by singleton properties. Its contextualization function can be de ned as follows.

De nition 7 (Companion Property contextualization function). Let c 2 C and G 2 2T be a context and a graph respectively. We de ne the contextualization function fcp such that fcp(c; G) = f(s; ren1(p); o); (ren1(p); companionP ropertyOf; p); (s; ren2(p); ren3(t)); (ren3(t); idP ropertyOf; ren1(p)) j (s; p; o) = t 2 Gg [ (companionP ropertyOf; subP ropertyOf; subP ropertyOf ).

NdFluents [ 5 ] was created with the purpose of improving inference preservation. It consists in considering individuals as separate entities that exist according to each context. Its contextualization function can be de ned as follows. De nition 8 (NdFluents contextualization function). Let c 2 C and G 2 2T be a context and a graph respectively. We de ne the contextualization function fndf such that fndf (c; G) = f(renc(s); p; renc(o)); (renc(s); type; renc(contextualP art)); (renc(o); type; renc(contextualP art)); (renc(ContextualP art); subClassOf; ContextualP art); (renc(s); renc(contextualP artOf ); s); (renc(o); renc(contextualP artOf ); o); (renc(s); renc(contextualExtent); (renc(o); renc(contextualExtent); renc(t)); (renc(t); type; renc(Context)); j (s; p; o) = t 2 Gg [ f(renc(Context); subClassOf; Context); (renc(contextualExtent); subP ropertyOf; contextualExtent); (renc(contextualP artOf ); subP ropertyOf; contextualP artOf )g.

NdTerms [ 10 ] is a generalization of NdFluents that has been speci cally studied in the context of Description Logics. It extends the approach to every term in the ontology instead of just the individuals. If the initial ontology is in OWL, then it can be represented in RDF, and the contextualization could be expressed according to the above Def. 1. However, it also introduces several notions that are required to ensure other desirable properties, that are fairly long to give in detail here. So we do not formalize the related contextualization function.

In addition to these approaches, there are diverse ways to express context with extensions of the syntax of RDF. Most prominently Named graphs [ 3 ], RDF* [ 6 ], and Notation 3 [ 1 ] are three such cases. Named graphs extend the syntax of RDF by adding a fourth element to the triples. RDF* extends RDF with the possibility to use an RDF* triple in subject and object positions of another RDF* triple. Notation-3 extends this further with more constructs, such as rule de nitions, variable quanti cations. These approaches do not conform to the de nition of contextualization function, since the result of their transformation is not an RDF graph. Hence, they will be kept out of scope for comparisons in the rest of the paper. 4

The NdProperties Ontology and Contextualization

Function

NdFluents [ 5 ] and NdTerms [ 10 ] are based on the idea if creating contextual terms (individuals for the former, all terms for the latter) that mirror the terms of the graph or ontology. They can be seen as concrete instantiations of a general approach in which a subset of the terms is contextualized. NdProperties is a new instantiation where the relations are contextualized. It creates a di erent property for each existing property in the graph and each di erent context we want to use to annotate the data. The ontology can be seen down below, and the contextualization function is shown in De nition 9. )

While the ontology can be used \as is", in order to represent more than one context it is necessary to be extended for each context (in a similar fashion as NdFluents [ 5 ]). In 1 we show the representation of a statement with temporal annotations using this ontology. The non-dashed parts belong to the temporal extension, while the dashed parts correspond to the NdProperties ontology plus other possible extensions. Other dimensions, such as provenance, can be modeled similarly to the temporal dimension by replacing the appropriate classes and properties. We will use the function ren in order to generate these terms in De nition 9.

ContextualProperty TemporalProperty

ProvenanceProperty

TrustProperty temporalPropertyOf contextualPropertyOf Paris

France

Context capitalOf temporalExtent

Interval contextualExtent

t1

Fig. 1: Temporal Extension of NdProperties De nition 9 (NdP{ contextualization function). Let c 2 C and G 2 2T be a context and a graph respectively. We de ne the contextualization function fNdP such that fNdP (c; G) = f(s; renc(p); o) j (s; p; o) = t 2 Gg[ f(renc(p); type; renc(ContextualP roperty)) j p 2 t 2 Gg[ f(renc(p); renc(contexualP ropertyOf ); p) j p 2 t 2 Gg[ frenc(p); renc(contextualExtent); renc(t)g f(renc(ContextualP roperty); subClassOf; ContextualP roperty)g[ frenc(contextualP ropertyOf ); subP ropertyOf; contextualP ropertyOf g[ frenccontextualExtent; subP ropertyOf; contextualExtentg[ frenc(t); type; renc(Context)g[ frenc(Context); subClassOf; Contextg

Note that the semantics of the property p are lost in fNdP (e.g., it's domain and range, or as a functional property, transitive property, etc.). However, it is also possible to have contextualization functions that retain totally or partially the semantics of p. For the sake of comparison, we will de ne a contextualization function fNdP + , in which we the contextual properties are described in an identical way as the original ones.

De nition 10 (NdP+ contextualization function). Let c 2 C and G 2 2T be a context and a graph respectively. We de ne the contextualization function fNdP + such that fNdP + (c; G) = fNdP (c; G) [ f(renc(p); q; r) j (p; q; r) = t 2 Gg

In the following section we will compare the inference preservation of the two NdProperties contextualization functions against the other approaches. 5

Reasoning with NdProperties

Following a similar approach as for NdFluents [ 5 ], we analyze for which rules from the pD* fragment of OWL [ 7 ] the contextualization functions preserve inferences. For each rule, we check if the contextualization function for each approach has Inference Preservation or Non-Contextual Inference Preservation wrt a graph that contains the corresponding triples. Following NdFluents evaluation, we apply the contextualization function only on triples that do not include

Constraint Conclusion Reif. N-Ary S.P. NdF NdP{ NdP+ RDF, RDFS, or OWL vocabularies. The intuition behind this is that rei cation approaches are usually used to annotate data on relations between resources. Table 1 shows the D* (modi ed RDFS) entailment rules and rule preservations for each one of the approaches, whereas Table 2 presents the same information for P entailments (modi ed subset of OWL). Note that we remove those rows where both condition and conclusion include only triples with RDF, RDFS, or OWL vocabularies. A symbol P indicates that there is rule preservation for the corresponding approach, while a symbol PNC denotes non-contextual rule preservation. A symbol ! means that it is possible to have undesirable inferences This happens when the contextualization function annotates at least one triple of the condition, and either we have non-contextual inference preservation, or we have inference preservation but the function does not modify the triple in the conclusion.

For other approaches the conclusions of NdFluents [ 5 ] hold: rei cation and n-ry relations show poor preservation of rules, where most of those rules could be considered tautologies. The Singleton Property provides a mixture of inference preservation and non-contextual inference preservation for all the rules, that can be useful when we want to annotate universally true facts, but it is not usable when we want to have contextual information that is not universally true. NdFluents, by contrast, has neither non-contextual rule preservation nor rule preservation that can lead to undesirable inferences for any rule. As for NdProperties, we can observe that NdP{ behaves slightly better than rei cation and n-ary relations, having inference preservation for tow additional rules: rdf1 and rdfp11. NdP+ shows the potential of this approach: adding axioms to the contextual property leads to having inference preservation for the associated rules (which was not possible for NdFluents). However, in four out of eight cases it can lead to undesirable inferences, which means that what axioms should be added to the contextualization function needs to be carefully considered. 6

Conclusion and Future Work

NdProperties can be seen as a concrete instantiation of a general approach of contextualizing a subset of the terms in an ontology. Other instantiations that follow this approach are NdFluents [ 5 ] (where the individuals are contextualized) and NdTerms [ 10 ] (where all the terms are contextualized). In this paper, we have present an ontology to represent NdProperties in RDF and showed two possible contextualization function among many. We compare these contextualization functions against existing rei cation approaches in terms of inference preservation and see that it preserves more desirable entailments, with the additional advantage that can be tuned to preserve only a subset of desired inferences.

In the future, we plan to continue our research in the general approach of contextualizing terms, studying their instantiations and properties and how to combine and relate di erent contexts, and performing a systematic comparison of the di erent instantiations of the approach.

Acknowledgements: This work is supported by H2020 Marie Sklodowska-Curie ITN No 642795. Authors would like to thank Prof. Pierre Maret for his comments.

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