TODO

Introduction

Background and Motivation

TODO, citing [[RDF-STAR-FOUNDATION]]

Overview

The RDF data model lets you state facts in three-part subject-predicate-object statements known as triples. For example, with a single RDF triple you can say that employee38 has a familyName of "Smith". A triple's predicate is a property specified with an IRI (an Internationalized version of a URI) to identify the namespace of the property name. A triple's subject and object can each be an IRI referencing any entity, and the object can also be a literal value such as "Smith" or data of other types such as dates, numbers, or Boolean values.

The subject and object of a triple can themselves reference triples. In the statement "employee22 claims that employee38 has a jobTitle of 'Assistant Designer'", the object of the triple that has employee22 as its subject references the statement "employee38 has a jobTitle of 'Assistant Designer'". This use of a triple as the subject or object resource of another triple so that we can say things about that triple is known as reification.

The concept of reification has always been part of RDF, but expressing it in RDF concrete syntaxes such as Turtle, N-Triples, and RDF/XML has been verbose and cumbersome. This specification describes a new, more compact conceptual data model and Turtle concrete syntax for reification known as RDF* (pronounced "RDF star") and Turtle* (pronounced "Turtle star"), respectively. This model and syntax enable the creation of concise triples that reference other triples as subject and object resources.

Triples that include a triple as a subject or an object are known as RDF* triples. The following dataset shows the example RDF* triples from above using the Turtle* syntax, which uses double angle brackets to enclose a triple serving as a subject or object resource:

        
      

After declaring a prefix so that IRIs can be abbreviated, the first triple in this example asserts that employee38 has a familyName of "Smith". Note that this dataset does not assert that employee38 has a jobTitle of "Assistant Designer"; it says that employee22 has made that claim. In other words, the triple "employee38 has a jobTitle of 'Assistant Designer'" is not what we call an asserted triple, like "employee38 has a familyName of 'Smith'" above; it is known as an embedded triple. (If we added the triple `:employee38 :jobTitle "Assistant Designer"` below the triple about employee22's claim in the example above, then this triple about employee38's jobTitle would be both an embedded triple and an asserted one.)

This specification also describes an extension to the SPARQL Protocol and Query Language known as SPARQL* (pronounced "SPARQL star") for the querying of RDF* triples. For example, the following SPARQL* query asks "who has made any claims about employee38?"

                

SPARQL query triple patterns that include a triple pattern as a subject or object are known as SPARQL* triple patterns.

For the remainder of this document, examples will assume that the following prefixes have been declared to represent the IRIs shown with them here:

`:``<http://www.example.org/>`
`rdfs:``<http://www.w3.org/2000/01/rdf-schema#>`
`owl:``<http://www.w3.org/2002/07/owl#>`
`prov:``<http://www.w3.org/ns/prov#>`
`dc:``<http://purl.org/dc/elements/1.1/>`
`dct:``<http://purl.org/dc/terms/>`

Concepts and Abstract Syntax

In the following, we we introduce a number of SPARQL*-specific definitions, which rely on the following notions (extending some of them) defined in [[[RDF11-CONCEPTS]]] [[RDF11-CONCEPTS]]: blank node, default graph, graph name, IRI, literal, named graphs, object, predicate, RDF dataset, RDF graph, RDF triple, and subject

An RDF* graph is a set of RDF* triples.

An RDF* triple is a 3-tuple defined recursively as follows:

As for RDF triples, we call the 3 components of an RDF* triple its subject, predicate and object, respectively. From the definitions above, it follows that any RDF graph is also an RDF* graph. Note also that, by definition, an RDF* triple cannot contain itself and cannot be nested infinitely.

IRIs, literals, blank nodes and RDF* triples are collectively known as RDF* terms.

For every RDF* triple t, we define its constituent terms (or simply constituents) as the set containing its subject, its predicate, its object, plus all the constituent terms of its subject and/or its object if they are themselves RDF* triples. By extension, we define the constituent terms of an RDF* graph to be the union set of the constituent terms of all its triples.

Consider the following RDF* triple (represented in Turtle*):
        
      
Its set of constituent terms comprises the IRIs `:name`, `:statedBy`, `:bob`, the blank node `_:a`, the literal `"Alice"`, and the triple `<< _:a :name "Alice" >>`.

An RDF* triple used as the subject or object of another RDF* triple is called an embedded triple. An RDF* triple that is an element of an RDF* graph is called an asserted triple. Note that, in a given RDF* graph, the same triple MAY be both embedded and asserted.

An RDF* dataset is a collection of RDF* graphs, and comprises:

Again, this definition is an extension of the notion of RDF dataset, hence it follows that any RDF dataset is also an RDF* dataset.

Turtle*

In this section, we present Turtle*, an extension of the Turtle format [[TURTLE]] allowing the representation of RDF* graphs. For the sake of conciseness, we only describe here the differences between Turtle* and Turtle.

Grammar

Turtle* is defined to follow the same grammar as Turtle, except for the EBNF productions specified below, which replace the productions having the same number (if any) in the original grammar.

[8] `objectList` ::= object annotation`?` `(` ',' object annotation`?` `)*`
[10] `subject` ::= iri `|` BlankNode `|` collection `|` embTriple
[12] `object` ::= iri `|` BlankNode `|` collection `|` blankNodePropertyList `|` literal `|` embTriple
[27] `embTriple` ::= '<<' embSubject verb embObject '>>'
[28] `embSubject` ::= iri `|` BlankNode `|` embTriple
[29] `embObject` ::= iri `|` BlankNode `|` literal `|` embTriple
[31] `annotation` ::= '{|' predicateObjectList '|}'

The changes are that `subject` and `object` productions have been extended to accept embedded triples, which are described by the new productions 27 to 29. Note that embedded triples accept a more restricted range of subject and object expressions than asserted triples. Additionally, the `objectList` production now accepts an optional annotation after each object.

Parsing

A Turtle* parser is similar to a Turtle parser as defined in Section 7 of the Turtle specification [[TURTLE]], with an additional item in its state :

Additionally, the |curSubject| can be bound to any RDF* term (including an embedded triple).

A Turtle* document defines an RDF* graph composed of a set of RDF* triples. The `subject` and `embSubject` productions set the |curSubject|. The `verb` production sets the |curPredicate|. The `object` and `embObject` productions set the |curObject|. Finishing the `object` production, an RDF* triple |curSubject| |curPredicate| |curObject| is generated and added to the RDF* graph.

Beginning the `embTriple` production records the |curSubject| and |curPredicate|. Finishing the `embTriple` production yields the RDF* triple |curSubject| |curPredicate| |curObject| and restores the recorded values of |curSubject| and |curPredicate|.

Beginning the `annotation` production records the |curSubject| and |curPredicate|, and sets the |curSubject| to the RDF* triple |curSubject| |curPredicate| |curObject|. Finishing the `annotation` production restores the recorded values of |curSubject| and |curPredicate|.

All other productions MUST be handled as specified by Section 7 of the Turtle specification [[TURTLE]], while still applying the changes above recursively.

Other Concrete Syntaxes

While this document specifies only one concrete syntax, nothing prevents other concrete syntaxes of RDF* from being proposed. In particular, other existing concrete syntaxes for RDF, such as RDF/XML [[RDF-SYNTAX-GRAMMAR]], could be extended to support RDF*. In particular, the N-Triples syntax [[N-TRIPLES]] being a subset of Turtle, an appropriate subset of Turtle* could be defined to extend N-Triples accordingly.

SPARQL* Query Language

This Section introduces SPARQL*, which is an RDF*-aware extension of the RDF query language SPARQL [[SPARQL11-QUERY]]; i.e., SPARQL* can be used to query RDF* graphs.

Initial Definitions

In the following, we introduce a number of SPARQL*-specific definitions, which rely on the following notions, defined in [[[SPARQL11-QUERY]]] [[SPARQL11-QUERY]]: RDF term, query variable, triple pattern, property path pattern, property path expression, and solution mapping.

A SPARQL* triple pattern is a 3-tuple that is defined recursively as follows:

  1. Every SPARQL triple pattern is a SPARQL* triple pattern;
  2. If |t| and t' are SPARQL* triple patterns, |x| is an RDF term or a query variable, and |p| is an IRI or a query variable, then (|t|, |p|, |x|), (|x|, |p|, |t|), and (|t|, |p|, t') are SPARQL* triple patterns.

As for RDF* triples, a SPARQL* triple pattern MUST NOT contain itself.

A SPARQL* basic graph pattern (BGP*) is a set of SPARQL* triple patterns.

A SPARQL* property path pattern is a 3-tuple (|s|,|p|,|o|) where

A SPARQL* solution mapping μ is a partial function from the set of all query variables to the set of all RDF* terms. The domain of μ, denoted by dom(μ), is the set of query variables for which μ is defined.

The notion of a SPARQL* solution mapping extends the notion of a standard SPARQL solution mapping; that is, every SPARQL solution mapping is a SPARQL* solution mapping. However, in contrast to SPARQL solution mappings, SPARQL* solution mappings may map variables also to RDF* triples.

All notions related to SPARQL solution mappings carry over naturally to SPARQL* solution mappings. In particular, the definition of compatibility extends naturally to SPARQL* solution mappings: two SPARQL* solution mappings μ1 and μ2 are compatible if, for every variable v that is both in dom(μ1) and in dom(μ2), μ1(v) and μ2(v) are the same RDF* term. In this case, μ1 ∪ μ2 is also a SPARQL* solution mapping. Moreover, for any SPARQL* solution mapping μ we write card[Ω](μ) to denote the cardinality of μ in a multiset Ω of such mappings. Finally, given a BGP* |B| and a SPARQL* solution mapping μ, we write μ(|B|) to denote the result of replacing every variable |v| in |B| for which μ is defined with μ(v).

Next, we aim to carry over the notion of solutions for BGPs to BGP*. To this end, we first define an auxiliary concept that carries over the notion of an RDF instance mapping [[RDF11-MT]] to RDF*.

An RDF* instance mapping σ is a partial function from the set of all blank nodes to the set of all RDF* terms. The domain of σ, denoted by dom(σ), is the set of blank nodes for which σ is defined.

Similar to the corresponding notation for solution mappings, for an RDF* instance mapping σ and a BGP* |B| we write σ(|B|) to denote the result of replacing every blank node |b| in |B| for which σ is defined with σ(b).

Now we are ready to define the notion of solution for BGP*.

Given a BGP* |B| and an RDF* graph |G|, a SPARQL* solution mapping μ is a solution for the BGP* |B| over |G| if it has the following two properties

Grammar

SPARQL* is defined to follow the same grammar as SPARQL, except for the EBNF productions specified below, which replace the productions having the same number (if any) in the original grammar.

[60] `Bind` ::= 'BIND' '(' ( Expression `|` EmbTP ) 'AS' Var ')'
[75] `TriplesSameSubject` ::= VarOrTermOrEmbTP PropertyListNotEmpty `|` TriplesNode PropertyList
[80] `Object` ::= GraphNode `|` EmbTP
[81] `TriplesSameSubjectPath` ::= VarOrTermOrEmbTP PropertyListPathNotEmpty `|` TriplesNode PropertyListPath
[105] `GraphNodePath` ::= VarOrTermOrEmbTP `|` TriplesNodePath `|`
[174] `EmbTP` ::= '<<' EmbSubjectOrObject Verb EmbSubjectOrObject '>>'
[175] `EmbSubjectOrObject` ::= Var `|` BlankNode `|` iri `|` RDFLiteral `|` NumericLiteral `|` BooleanLiteral `|` EmbTP
[176] `VarOrTermOrEmbTP` ::= Var `|` GraphTerm `|` EmbTP

This introduces a notation for embedded triple patterns (productions [174] and following), which is similar to the one defined for embedded triples in , but accepting also variables. These embedded triple patterns are allowed in the subject ([75], [81]) and object ([80], [105]) positions of SPARQL* triple patterns, as well as in BIND statements ([60]).

Translation to the Algebra

Based on the SPARQL grammar, the SPARQL specification defines the process of converting graph patterns and solution modifiers in a SPARQL query string into a SPARQL algebra expression [SPARQL11-QUERY, Section 18.2]. This process must be adjusted to consider the extended grammar introduced above. In the following, any step of the conversion process that requires adjustment is discussed.

Variable Scope

As a basis of the translation, the SPARQL specification introduces a notion of in-scope variables. To cover the new syntax elements introduced in this notion MUST be extended as follows.

  • A variable is in-scope of a BGP* |B| if the variable occurs in |B|, which includes an occurrence in any embedded triple pattern in |B| (independent of the level of nesting).
  • A variable is in-scope of a property path pattern if the variable occurs in that pattern, which includes an occurrence in any embedded triple pattern in the pattern (independent of the level of nesting).
  • A variable is in-scope of a BIND clause of the form `BIND ( T AS v )` (where |T| is an embedded triple pattern) if the variable is variable |v| or the variable occurs in the embedded triple pattern |T|. As for standard BIND clauses with expressions, variable |v| must not [be] in-scope from the preceding elements in the group graph pattern in which [the BIND clause] is used [SPARQL11-QUERY, Section 18.2.1]].

Expand Syntax Forms

The translation process starts with expanding abbreviations for IRIs and triple patterns [SPARQL11-QUERY, Section 18.2.2.1]. This step MUST be extended in two ways:

  1. Abbreviations for triple patterns with embedded triple patterns MUST be expanded as if each embedded triple pattern was a variable (or an RDF term).

    For instance, the following syntax expression:
                    
                  
    must be expanded to
                    
                  
  2. Abbreviations for IRIs in all embedded triple patterns MUST be expanded.

    For instance, the embedded triple pattern
                    
                  
    must be expanded to
                    
                  

Translate Property Path Patterns

The translation of property path patterns has to be adjusted because the extended grammar allows for SPARQL* property path patterns whose subject or object is a SPARQL* triple pattern.

The translation as specified in the W3C specification distinguishes four cases. The first three of these cases do not require adjustment because they are taken care of either by recursion or by the adjusted translation of basic graph patterns (as defined in below). However, the fourth case MUST be adjusted as follows.

Let |X| |P| |Y| be a string that corresponds to the fourth case in [SPARQL11-QUERY, Section 18.2.2.4]. Given the grammar introduced in , |X| and |Y| may be an RDF term, a variable, or an embedded triple pattern, respectively (and |P| is a property path expression). The string |X| |P| |Y| is translated to the algebra expression `Path`(X’,|P|,Y’) where X’ and Y’ are the result of calling a function named `Lift` for |X| and |Y|, respectively. For some input string |Z| (such as |X| or |Y|) that can be an RDF term, a variable, or an embedded triple pattern, the function `Lift` is defined recursively as follows:

  1. If |Z| is an embedded triple pattern <<|S|,|P|,|O|>> then return the SPARQL* triple pattern (`Lift`(|S|), |P|, `Lift`(|O|));
  2. Otherwise, return |Z|.
The purpose of this translation step is to convert any property path pattern as can be written based on the extended grammar for SPARQL* (cf. ) into a SPARQL* property path pattern as considered in the algebra. To this end, the function `Lift` translates every embedded triple pattern as can be written in the SPARQL* syntax into a SPARQL* triple pattern.

Translate Basic Graph Patterns

After translating property path patterns, the translation process collects any adjacent triple patterns [...] to form a basic graph pattern [SPARQL11-QUERY, Section 18.2.2.5]. This step has to be adjusted because triple patterns in the extended syntax may have an embedded triple pattern in their subject position or in their object position (or in both). To ensure that every result of this step is a BGP*, before adding a triple pattern to its corresponding collection, its subject and object MUST be replaced by the result of calling function `Lift` for the subject and the object, respectively.

Translate BIND Clauses with an Embedded Triple Pattern

The extended grammar in allows for BIND clauses with an embedded triple pattern. The translation of such a BIND clause to a SPARQL algebra expression requires a new algebra symbol:

Then, any string of the form `BIND( T AS v )` with |T| being an embedded triple pattern (i.e., not a standard BIND expression) is translated to the algebra expression `TR`(T’, |v|) where T’ is the result of the function `Lift` for |T|.

Notice, the translation of BIND clauses with an embedded triple pattern as defined in this section is used during the translation of group graph patterns. The case of BIND clauses with an embedded triple pattern is covered in this translation of group graph patterns by the last, “catch all other” `IF` statement (i.e., the `IF` statement with the condition `E is any other form`) and not by the `IF` statement for BIND clauses with an expression.

Evaluation Semantics

The SPARQL specification defines a function eval(|D|(|G|), algebra expression) as the evaluation of an algebra expression with respect to a dataset |D| having active graph |G| [SPARQL11-QUERY, Section 18.6]. Recall that the dataset |D| in the context of SPARQL* is an RDF* dataset and, thus, the active graph |G| is an RDF* graph, and so is any other graph in dataset |D|. The definition of the eval function is recursive; the two base cases of this definition for SPARQL* are given as follows:

  1. For every BGP* |B|, eval(|D|(|G|), |B|) is a multiset Ω that consists of all SPARQL* solution mappings that are a solution for the BGP* |B| over |G|. For every such mapping μ, card[Ω](μ) is the number of distinct RDF* instance mappings σ such that dom(σ) is equivalent to the set of blank nodes in |B| and μ(σ(|B|)) is a subgraph of |G|. (For any SPARQL* solution mapping μ' that is not a solution for |B| over |G|, we have that card[Ω](μ')=0; i.e., μ' is not in Ω.)
  2. For any algebra expression |E| of the form TR(|tp|, ?v) where |tp| is a SPARQL* triple pattern and ?v is a variable (as introduced in ), eval(|D|(|G|), |E|) is a multiset Ω that consists of as many SPARQL* solution mappings as there are solution mappings in Ω', where Ω'=eval(|D|(|G|),{|tp|}), such that for every μ' in Ω' there exists a μ in Ω that has the following four properties:
    1. dom(μ) = dom(μ') ∪ {?v}
    2. μ and μ' are compatible
    3. μ(?v) = μ'(|tp|)
    4. card[Ω](μ) = card[Ω'](μ')

For any other algebra expression, the SPARQL specification defines algebra operators [[SPARQL11-QUERY]]. These definitions can be extended naturally to operate over multisets of SPARQL* solution mappings (instead of ordinary solution mappings). Given this extension, the recursive steps of the definition of the eval function for SPARQL* are the same as in the SPARQL specification.

Query Result Formats

In SPARQL, queries can take four forms: SELECT, CONSTRUCT, DESCRIBE, and ASK - see SPARQL1.1 Query, Section 16 [[SPARQL11-QUERY]]. The first of these returns a sequence of solution mappings that contain variable bindings. The second and third both return an RDF graph, and the last returns a boolean value.

The result of the ASK query form is not changed by the introduction of RDF*, and the result of the CONSTRUCT and DESCRIBE forms can be represented by Turtle*. However, since the SELECT form deals with returning individual RDF terms, the specific serialization formats for representing such query results need to be extended so that the new embedded triple RDF term can be represented. In this section, we propose extensions for the two most common formats for this purpose: [[[sparql11-results-json]]], and [[[rdf-sparql-XMLres]]].

SPARQL* Query Results JSON Format

The result of a SPARQL SELECT query is serialized in JSON as defined in [[[sparql11-results-json]]], which specifies a JSON representation of variable bindings to RDF terms (see [sparql11-results-json, Section 3.2]). To accommodate the new RDF term for embedded triples that RDF* introduces, the table of RDF term JSON representations in sparql11-results-json, Section 3.2.2 is extended with the following entry:

An embedded triple with subject RDF term `S`, predicate RDF term `P` and object RDF term `O`
                {
                  "type": "triple",
                  "value": {
                     "subject": S,
                     "predicate": P,
                     "object": O
                  }
                }
              
where `S`, `P` and `O` are encoded using the same format, recursively.
Consider the following RDF term, an embedded triple in Turtle* syntax:
            
          
This term is represented in JSON as follows:
                {
                  "type": "triple",
                  "value": {
                     "subject": {
                        "type": "uri",
                        "value" "http://example.org/alice"
                     },
                     "predicate": {
                        "type": "uri",
                        "value" "http://example.org/name"
                     },
                     "object": {
                        "type": "literal",
                        "value" "Alice",
                        "datatype": "http://www.w3.org/2001/XMLSchema#string"
                     },
                  }
                }
          

SPARQL* Query Results XML Format

The result of a SPARQL SELECT query is serialized in XML as defined in [[[rdf-sparql-XMLres]]]. This format proposes an XML representation of variable bindings to RDF terms.

To accommodate the new RDF term for embedded triples that RDF* introduces, the list of RDF terms and their XML representations in [rdf-sparql-XMLres, Section 2.3.1] is extended as follows:

An embedded triple with subject term `S`, predicate term `P`, and object term `O`
              
            
where `S`, `P` and `O` are encoded recursively, using the same format, without the enclosing `<binding>` tag.
Consider the following RDF term, an embedded triple in Turtle* syntax:
            
          
This term is represented in XML as follows:
            
          

SPARQL* Update

RDF* Semantics

In this section, we provide a model-theoretic semantics for RDF*, by extending the one defined in [[[RDF11-MT]]] [[RDF11-MT]].

Definitions

An RDF* triple is said to be ground if it has no blank node in its constituent terms. An RDF* graph is ground if all its triples are ground. This definition generalizes the notion of ground RDF graph. IRIs, literals and ground RDF* terms are collectively known as ground RDF* terms.

An RDF* simple interpretation |I| is a structure consisting of:

  1. A non-empty set |IR| of resources, called the domain or universe of |I|.
  2. A set |IP|, called the set of properties of |I|.
  3. A mapping |IEXT| from |IP| into the powerset of |IR|×|IR| i.e. the set of sets of pairs (|x|,|y|) with |x| and |y| in |IR|.
  4. A mapping |IS| from IRIs into |IR|∪|IP|.
  5. A partial mapping |IL| from literals into |IR|.
  6. A partial mapping |IT| from ground RDF* triples into |IR|, such that for any two triples (|s1|,|p1|,|o1|) and (|s2|,|p2|,|o2|), |IT|(|s1|,|p1|,|o1|)=|IT|(|s2|,|p2|,|o2|) implies |IG|(|s1|)=|IG|(|s2|), |IG|(|p1|)=|IG|(|p2|) and |IG|(|o1|)=|IG|(|o2|), where |IG|=|IS|∪|IL|∪|IT|.

This definition is identical to the definition of simple interpretation [[RDF11-MT]] up to item 5 included. Item 6 extends it to support RDF* triples. Any RDF simple interpretation can be considered as an RDF* simple interpretation with |IT|=∅.

Semantic condition for ground graphs

The denotation of a ground RDF* graph in an RDF* simple interpretation |I| is then given by the following rules, where the interpretation is also treated as a function from expressions (terms, triples and graphs) to elements of the universe and truth values:

  • if |E| is a literal then |I|(|E|) = |IL|(|E|)
  • if |E| is an IRI then |I|(|E|) = |IS|(|E|)
  • if |E| is a ground RDF* triple then |I|(|E|) = |IT|(|E|)
  • if |E| is an RDF* graph then |I|(|E|) =
    • true if for every asserted triple (|s|,|p|,|o|) ∈ |E|, (|I|(|s|),|I|(|o|)) ∈ |IEXT|(|I|(|p|))
    • false otherwise

Since |IL| and |IT| are partial mappings, |I|(|E|) may be undefined for some literal or triple |E|. In that case, |E| has no semantic value in |I|, so any asserted triple having E as subject or object it will fail to satisfy the condition above, hence any graph containing such asserted triple will be false.

In the original condition for simple entailment [[RDF11-MT]], the denotation of an RDF triple is a boolean, while above we define the denotation of an RDF* triple to be an element of the universe. However, the denotation of any RDF graph is always the same under the original condition and the condition above. Therefore, the condition above can be considered as an extension of the original one to support any RDF* graph.

Semantic condition with blank nodes

Given an RDF* graph |E|, we call the embedded blank nodes of |E| the set of blank nodes appearing in subject or object position of some embedded triple in |E|; we call the open blank nodes of |E| all the other blank nodes appearing in |E|.

A mapping from a set blank nodes into a set of ground RDF* terms is called a grounding function. We define the extended application of a grounding function Γ to other RDF* terms and to RDF* graphs as follows:

  • if |E| is a blank node then Γ*(|E|) = Γ(|E|) if it is defined, |E| otherwise
  • if |E| is an IRI or a literal, then Γ*(|E|) = |E|
  • if |E| is an RDF* triple (|s|,|p|,|o|), then Γ*(|E|) = (Γ*(|s|),Γ*(|p|),Γ*(|o|))
  • if |E| is an RDF* graph, then Γ*(|E|) = {Γ*(|t|) for all |t| ∈ |E|}

Suppose |I| is an RDF* simple interpretation and |A| is a mapping from a set of blank nodes to the universe |IR| of |I|. Define the mapping [|I|+|A|] of RDF* terms into |IR| to be |A| on blank nodes of the set, and |I| on any other term; and extend this mapping to RDF* triples and RDF* graphs using the rules given above for ground graphs. Then the denotation of any RDF* graph in |I| is given by:

For the special case of a standard RDF graph and an RDF simple interpretation, the condition above is equivalent to the one defined in [[[RDF11-MT]]], which only requires a mapping |A|, but no grounding function. Indeed, since RDF graphs have no embedded triples, the empty mapping is the only possible choice for Γ, and Γ* maps the graph to itself.

Entailment

Following [[[RDF11-MT]]], we extend the notions of satisfiability and entailment. An RDF* simple interpretation satisfies |E| when |I|(|E|)=true. |E| is (simply) satisfiable when an RDF* simple interpretation exists which satisfies it, otherwise (simply) unsatisfiable. An RDF* graph |G| simply entails an RDF* graph H when every interpretation which satisfies |G| also satisfies |H|. If two RDF* graphs |G| and |H| each entail the other then they are logically equivalent.

Any semantic extension of RDF MAY be extended to RDF* by replacing the semantic conditions, the notion of satisfiability and the notion of entailment, defined in [[[RDF11-MT]]], by their corresponding extension defined above. This is notably the case for Datatype entailment and RDFS entailment.

Design Rationale

In this section, we discuss a number of desired features of RDF* semantics in order to shed light on the design choices made in the previous section

Quoting without asserting

RDF* must be able to quote a triple without asserting it, so that we can represent peoples' beliefs or claims without endorsing them, or represent facts that are no longer or not yet true. This is ensured by the fact that only asserted triples are considered to determine if the denotation of a graph is true or false.

For example, the following graph:
          
        
does not entail `:alice foaf:knows :bob`, and the SPARQL* query below executed against the graph above would return no result.
          
        

Referential opacity

Embedded triples are referentially opaque, meaning that triples using different terms can be considered different, even if their terms can be inferred to be synonyms. Although RDF* simple entailment has no means to entail any kind of synonymy, it is possible in some semantic extensions, such as OWL [[OWL2-RDF-BASED-SEMANTICS]].

A well known example is the superman problem:

          
        

Intuitively: this graph states that Superman and Clark Kent are the same person, so if Superman can fly, then it follows that Clark Kent can as well. So, under OWL2-entailment, this graph entails `:clarkKent :can :fly`. However, Lois Lane does not know that Superman and Clark Kent are the same person. So from her point of view, the two triples are not equivalent, and she can believe one without believing the other.

Referential opacity is ensured by differentiating the intension of embedded triples (represented by the |IT| mapping) from their extension (the denotations of their subject, predicate and object). Since |IT| is based solely on the syntax of triples, two syntactically different triples can always have different intentions, even if their subjects, predicates and objects are semantically equivalent.

On the other hand, all triples with the same intension are required to have the same extension. So if two RDF* triples denote the same resource T, their subjects, predicates and objects, respectively, are constrained to also denote the same thing.

Blank node scope

Blank nodes in embedded triples have the same scope as blank nodes used in the subject or object position of asserted triples (usually the whole graph or the whole dataset in which they appear). This means that the same blank node identifier used in different embedded triples, or at different levels of nesting, will refer to the same thing.

For example, in the following graph:

          
        

the three occurrence of `_:x` must refer to the same resource in every interpretation of the graph. In other words, it must be the same resource that Alice knows, that she claims is named "Bob", and that she claims works for ACME.

As a consequence, the following query will return `"Bob"`:

          
        

As another consequence, the following graph does not entail the graph above (because the graph below allows the resource known by Alice to be different from the one about which she makes claims).

          
        

Formally, the second graph is satisfied by an interpretation having:

  • |IS|: `:alice`→A, `:knows`→K, `dc:creator`→C, `:bob`→B
  • |IT|: `<< :bob :name "Bob" >>`→T1, `<< :bob :workingFor :acme >>`→T2
  • |IEXT|(K) = {(A,Y)}
  • |IEXT|(C) = {(T1, A), (T2, A)}

with Γ: `_:x`→`:bob`, and A: `_:y`→Y. But this interpretation can not satisfy the first graph, because it would require a grounding function Γ' such that

  • I(Γ'(`_:x`)) = Y in order to satisfy the first triple,
  • Γ'(`_:x`) = `:bob` in order to satisfy the second and third triple,
  • and both can not be true at the same time in the interpretation above.

Interpolation lemma

The interpolation lemma [[RDF11-MT]] states that an RDF graph G simply entails an RDF graph E if and only if a subgraph of G is an instance of E. Intuitively, this means that all graphs simply entailed by G can be constructed by:

  • removing triples,
  • replacing a term with a fresh blank node,
  • splitting a blank node into two (or more) fresh blank nodes.

A design goal of the RDF* semantics was to preserve that property.

We didn't prove it yet...

Historical remarks

SA-mode and PG-mode

A lot of discussions on the RDF* mailing list and GitHub repository refer to SA-mode and PG-mode. Those abbreviations stand for "Separate Assertion mode" and "Property Graph mode". They originate in the fact that different versions of RDF* have been published over the years, with different designs. In PG-mode, any embedded triple was also considered asserted. SA-mode, on the other hand, allowed the use of embedded triples without those triples being automatically asserted, requiring that they be asserted separately when that was intended. SA-mode was more flexible, but induced redundancy in the use-cases that PG-mode was designed to address.

The notion of annotations in the Turtle* syntax was introduced to remove the need for different modes. Rather than interpret the same syntax differently in each mode, which would have caused interoperability problems and required a switch for those modes, it was decided to provide a different syntax for each use case.