Formalization

Considering an entailment regime E, a dataset-interpretation with respect to E is an E-interpretation. Given an interpretation I and a dataset D having default graph G and named graphs NG, I(D) is true if and only if I(G) is true.

Examples of entailment and non-entailments

Consider the following dataset:

{ ex:s  ex:p  ex:o . }
ex:g1 { ex:a  ex:b  ex:c }

does not dataset-entail:

{ ex:s  ex:p  ex:o .
  ex:a  ex:b  ex:c .}

but dataset-entails:

{}  # empty default graph
ex:g2 { ex:x  ex:y  ex:z }

Since graph names are not particularly constrained, one can use them in triples, for instance:

{ ex:g1  ex:author  ex:Bob .
  ex:g1  ex:created  "2013-09-17"^^xsd:date .}
ex:g1 { ex:a  ex:b  ex:c }

but it would dataset-entail:

{ ex:g1  ex:author  ex:Bob .
  ex:g1  ex:created  "2013-09-17"^^xsd:date .}
ex:g1 { ex:x  ex:y  ex:z }

Properties of this dataset semantics

Assuming this semantics is convenient since it merely ignores named graphs in a dataset for any reasoning task. As a result, datasets can be simply treated as regular RDF graphs by extracting the default graph. Named graphs can still be used to preserve useful information, but it bears no more meaning than a commentary in a program source code.

The obvious disadvantage is that, since named graphs are completely disregarded in terms of meaning, there is no guarantee that any information intended to be conveyed by the named graphs is preserved by inference.

Formalization: first version

We define a dataset-interpretation with respect to an entailment regime E as an E-interpretation. Given a dataset-interpretation I and a dataset D having default graph G and named graphs NG, I(D) is true if and only if I(G) is true and for all ng in NG, I(ng) is true.

This is equivalent to I(D) is true if I(H) is true where H is the merge of all the RDF graphs, named or default, appearing in D.

Formalization: second version

We define a dataset-interpretation with respect to an entailment regime E as an E-interpretation. Given a dataset-interpretation I and a dataset D having default graph G and named graphs NG, I(D) is true if and only if I(H) is true where H is the union of all the RDF graphs, named or default, appearing in D.

An alternative presentation of this variant is the following: define I+A to be an extended interpretation which is like I except that it uses A to give the interpretation of blank nodes; define blank(D) to be the set of blank nodes in D. Then I(D) is true if and only if [I+A](D) is true for some mapping A from blank(D) to the set of resources in I.

Examples

Consider the following dataset:

{ ex:s  ex:p  ex:o . }  # default graph
ex:g1 { ex:a  ex:b  ex:c }

dataset-entails:

{ ex:s  ex:p  ex:o .
  ex:a  ex:b  ex:c .}

If the entailment regime E is RDFS with the recognized datatype xsd:integer, then the following RDF dataset is RDFS-dataset-inconsistent:

{ }  # empty default graph
ex:g1 { ex:age  rdfs:range  xsd:integer . }
ex:g2 { ex:bob  ex:age  "twenty" .}

Properties of this dataset semantics

This semantics allows one to partition the triples of an RDF graph into multiple named graphs for easier data management, yet retaining the meaning of the overall RDF graph. Note that this choice of semantics does not impact the way graph names are interpreted: it is possible to further constrain the graph names to denote the RDF graph associated with it, or other possible constraints. The possible interpretations of graph names, and their consequences, are presented in the next sections.

This semantics is implicitly assumed by existing graph store implementations. The OWLIM RDF database management system implements reasoning techniques over RDF datasets that materialize inferred statements into the database [[citation needed]]. This is done by taking the union of the graphs in the named graphs, applying standard entailment regimes over this RDF graph and putting the inferred triples into the default graph.

This dataset semantics makes all triples in the named graphs contribute to a global knowledge, thus making the whole dataset inconsistent whenever two graphs are mutually contradictory. In situations where named graphs are used to store RDF graphs obtained from various sources on the open Web, inconsistencies or contradictions can easily occur. Notably, Web crawlers of search engines harvest all RDF documents, and it is known as a fact that the Web contains documents serializing inconsistent RDF graphs as well as documents that are mutually contradicting yet consistent on their own. In this case, this semantics can be seen as problematic.

Formalization

In order to be consistent with RDF model theory, blank nodes used as graph names are treated like existential variables. Consequently, their semantics is formalized according to the same notation presented in [[RDF11-MT]]:

Suppose I is an interpretation and A is a mapping from a set of blank nodes to the universe IR of I. Define the mapping [I+A] to be I on names, and A on blank nodes on the set: [I+A](x)=I(x) when x is a name and [I+A](x)=A(x) when x is a blank node; and extend this mapping to triples and RDF graphs using the rules given above for ground graphs.

A dataset-interpretation I with respect to an entailment regime E is an E-interpretation extended to named graphs and datasets as follows:

• if (n,g) is a named graph where the graph name is an IRI, then I(n,g) is true if and only if I(n) = (n,g).
• if D is a dataset comprising default graph DG and named graphs NG, then I(D) is true if and only if there exists a mapping from blank nodes to the universe IR of I such that [I+A](DG) is true and for all named graph (n,g) in NG, [I+A](n) = (n,g).

Examples

Consider the following dataset:

{ }  # empty default graph
ex:g1 { ex:a  ex:b  ex:c }
ex:g2 { ex:x  ex:y  ex:z }

dataset-entails:

{ }
_:b { ex:a  ex:b  ex:c }
ex:g2 { ex:x  ex:y  ex:z }

but does not dataset-entail:

{ }
ex:g1 { []  ex:b  ex:c }
ex:g2 { ex:x  ex:y  ex:z }

nor:

{ }
ex:g1 {  }

If the entailment regime E is RDFS with the recognized datatype xsd:integer, then the following RDF dataset is RDFS-dataset-inconsistent:

{ ex:age  rdfs:range  xsd:integer .
  ex:me  ex:age  ex:g1 . }  # default graph
ex:g1 { ex:s  ex:p  ex:o }

The graph name can be used in triples to attached metadata (here ex:hasNextVersion is a custom term that does not enforce a formal constraint, so it is up to the implementation to decide how to treat it):

{ ex:g1  ex:published  "2013-08-26"^^xsd:date .
  ex:g1  ex:hasNextVersion  ex:g2 .}
ex:g1 { ex:s1  ex:p1  ex:o1 .
        ex:s2  ex:p2  ex:o2 }
ex:g2 { ex:s1  ex:p1  ex:o1 }

Properties of this dataset semantics

There are important implications with this semantics. In this case, a named graph pair can only entail itself or a graph that is structurally equivalent if the graph name is a blank node. Graph names have to be handled almost like literals. Unlike other IRIs or blank nodes, their denotation is strictly fixed, like literals are. This means that graph IRIs may possibly clash with constraints on datatypes, as in the example above.

A variant of this dataset semantics imposes that the graph name denotes the RDF graph itself, rather than the pair. This means that two occurrences of the same graph in different named graph pairs actually identify the same thing. Thus, the graph names associated with the same RDF graphs are interchangeable in any triple in this case.

Formalization

There are several possible formalizations of this leading to similar entailments. One way is to interpret the graph name as denoting a graph, and a named graph pair is true if this graph entails the graph inside the pair. In this case, a dataset-interpretation with respect to an entailment regime E is an E-interpretation such that:

• given a mapping A from blank nodes to the univers IR and a named graph pair ng = (n,G), [I+A](ng) is true if [I+A](n) is an RDF graph and E-entails G;
• for a dataset D = (DG,NG), I(D) is true if there exists a mapping A from blank nodes to the universe IR such that [I+A](DG) is true and for all named graph ng in NG, [I+A](ng) is true;
• I(D) is false otherwise.

Examples

Consider the following dataset:

{ }  # empty default graph
ex:g1 { ex:YoutubeEmployee  rdfs:subClassOf  ex:GoogleEmployee .
        ex:steveChen  rdf:type  ex:YoutubeEmployee . }
ex:g2 { ex:chadHurley  rdf:type  ex:YoutubeEmployee }

RDFS-dataset-entails:

{ }
ex:g1 { ex:steveChen  rdf:type  ex:GoogleEmployee }

but does not RDFS-dataset-entail:

{ }
ex:g2 { ex:chadHurley  rdf:type  ex:GoogleEmployee }

Graph names used in triples that express metadata do not necessarily generate inconsistency:

{ ex:g1  ex:validAfter  "2006"^^xsd:gYear .
  ex:g1  ex:published  "2013-08-26"^^xsd:date .
  ex:g2  ex:validAt  "2005"^^xsd:gYear .}
ex:g1 { ex:YoutubeEmployee  rdfs:subClassOf  ex:GoogleEmployee .
        ex:steveChen  rdf:type  ex:YoutubeEmployee . }
ex:g2 { ex:chadHurley  rdf:type  ex:YoutubeEmployee }

(here, ex:validAfter and ex:validAt are custom terms that do not enforce a formal constraint, but may be used internally for, e.g., checking the temporal validity of triples in the named graph).

Properties of this dataset semantics

This semantics assumes that the truth of named graphs is preserved when replacing the RDF graphs inside named graphs with equivalent graphs. This means in particular, that one can normalize literals and still preserve the truth of a named graph. This means too that standard RDF inferences that can be drawn from the RDF graphs inside named graphs can be added to the graph associated with the graph name without impacting the truth of the RDF dataset.

While this semantics does not guarantee that reasoning with RDF datasets will preserve the exact triples of an original dataset, it is semantically valid to store both the original and any entailed datasets.

An example implementation of such a context-based semantics is Sindice [[DELBRU-ET-AL-2008]].

Variants of this dataset semantics

There are several variants of this type of dataset-semantics

• The default graph is interpreted as universal truth, that is, for a named graph (n,G), I(n) E-entails the default graph.
• The graph name does not denote an RDF graph but a resource associated with an RDF graph.
• Each named graph could be associated with a distinct E-interpretation and impose all interpretations to be true for their corresponding graph, in order for the dataset to be true.

Formalization

Let d be the function that maps an IRI to an RDF graph that can be obtained from dereferencing the IRI. For an IRI u, d(u) is empty when dereferencing returns an error or a document that does not encode an RDF graph.

A dataset-interpretation I with respect to an entailment regime E is an E-interpretation such that:

• for a named graph pair ng = (n,G), I(ng) is true if d(n) equals (respectively, is a subgraph of, is entailed by) G;
• for a dataset D = (DG,NG), I(D) is true if I(DG) is true and for all named graph ng in NG, I(ng) is true;
• I(D) is false otherwise.

Examples

Entailments in this semantics depend not only on the content of a dataset but also on the content of the Web and the ability of a reasoner to accept this content. Moreover, the entailments vary whether the considered relation is “equals”, or “subgraph of”, or “entailed by”.

For instance, if the reasoner is offline, then the dereferencing function d in the previous definition always return an empty graph. In this case, if the relation is “equals” or “subgraph of”, only empty named graphs can be true; if the relation is “entails by”, then only named graphs containing axiomatic triples are true. In general, if the relationship is “equals”, named graph do not provide extra entailments.

Properties of this dataset semantics

The distinguishing characteristic of this dataset semantics is the fact that a single RDF dataset can lead to different entailments, depending on the state of the Web. This can be seen as a feature for systems that need to be in line with what is found online, but is a drawback for systems that must retain consistency even when they go offline.

Formalization

A quad-interpretation is a tuple (IR,IP,IEXT,IS,IL,LV) where IR, IP, IS, IL and LV are defined as in RDF and IEXT is a mapping from IP into the powerset of IR × IR union IR × IR × IR.

Since this option modifies the notion of simple-interpretation, which is the basis for all E-interpretations in any entailment regime E, it is not clear how it can be extended to arbitrary entailment regimes. For instance, does the following quad set:

ex:a  rdf:type  ex:c  ex:x .
ex:c  rdfs:subClassOf  ex:d  ex:x .

RDFS-dataset-entails:

ex:a  rdf:type  ex:d  ex:x .

Properties of this dataset semantics

With this semantics, all inferences that are valid with normal RDF triples are preserved, but it is necessary to extend RDFS in order to accommodate for ternary relations. There are several existing proposals that extend this quad semantics by dealing with a specific “dimension”, such as time, uncertainty, provenance. For instance, temporal RDF [[TEMPORAL-RDF]] uses the fourth element to denote a time frame and thus allow reasoning to be performed per time frame. Special semantic rules allow one to combine triples in overlapping time frames. Fuzzy RDF [[FUZZY-RDF]] extends the semantics to deal with uncertainty. stRDF [[ST-RDF]] extends temporal RDF to deal with spatial information. Annotated RDF [[ANNOTATED-RDF]] generalizes the previous proposals.