RDF star "liberal baseline"

RDF-star liberal baseline (2025.01.08)

ABSTRACT SYNTAX

graph                 ::= triple*
triple                ::= tripleStructure
tripleStructure       ::= subject predicate object
subject               ::= NoLiteralTerm
predicate             ::= iri 
object                ::= term
NoLiteralTerm         ::= NoLiteralAtomicTerm | tripleTerm
NoLiteralAtomicTerm   ::= iri | BlankNode
tripleTerm            ::= tripleStructure
term                  ::= NoLiteralTerm | literal

Notes:

• RDF 1.1 syntax is the above without the tripleTerm category and its dependents.
• A term is denoted by r, a triple by t, and a graph by g.
• Given a triple structure s, we denote the subject, predicate, object of s as s.s, s.p, s.o, respectively.

SIMPLE SEMANTICS

An RDF simple interpretation I is a structure <IR, IP, IS, IL, IEXT, RE> 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 IS from IRIs into IR ⋃ IP, called the interpretation of IRIs.
4. A partial mapping IL from literal into IR, called the interpretation of literals.
5. A mapping IEXT from IP into 2IR x IR, called the extension of properties.
6. ⏩ A partial injective mapping RE from IR x IP x IR into IR, called the interpretation of triple terms. ⏪

A is a mapping from BlankNode to IR.

Given I and A, the function [I+A](.) is defined over terms, triples, and graphs as follows.

• [I+A](r) = IS(r)   if r is a iri
• [I+A](r) = IL(r)   if r is a literal
• ⏩ [I+A](r) = RE([I+A](r.s), [I+A](r.p), [I+A](r.o))   if r is a tripleTerm ⏪️
• [I+A](r) = A(r)   if r is a BlankNode

• [I+A](t) = TRUE   if and only if   <[I+A](t.s), [I+A](t.o)> ∈ IEXT([I+A](t.p))

• [I+A](g) = TRUE   if and only if   ∀ t ∈ g . [I+A](t) = TRUE

A simple interpretation I is a model of a graph g   if and only if   ∃ A . [I+A](g) = TRUE.

The set of all models of a graph g is called models(g).

Simple entailment: g ⊨ g'   if and only if   models(g) ⊆ models(g').

Notes:

• RDF 1.1 simple semantics is the above without the parts within ⏩...⏪ marks.

RDF(S) SEMANTICS

⏩ RDF(S) Interpretations add the following new IRIs with the namespace prefix rdf and rdfs: rdf:reifies, rdfs:Proposition. ⏪️

RDF(S) semantics restricts the function [I+A](.) only with the following additional metamodelling conditions:

• ⏩ <r, [I+A](rdfs:Proposition)> ∈ IEXT([I+A](rdf:type))
            if r ∈ range(RE) or
            if ∃ x,y . RE(x,[I+A](rdf:reifies),r)=y ⏪️
• ⏩ <r, [I+A](rdfs:Resource)> ∈ IEXT([I+A](rdf:type))
            if r ∈ range(RE) or
            if ∃ x,y,z . RE(x,z,r)=y or
            if ∃ x,y,z . RE(r,z,x)=y ⏪️
• ⏩ <r, [I+A](rdf:Property)> ∈ IEXT([I+A](rdf:type))
            if ∃ x,y,z . RE(x,r,z)=y ⏪️

A RDF(S) interpretation I is a RDF(S) model of a graph g   if and only if   ∃ A . [I+A](g) = TRUE.

The set of all RDF(S) models of a graph g is called rdf-models(g).

RDF(S) entailment: g ⊨ g'   if and only if   rdf-models(g) ⊆ rdf-models(g').

⏩ Additional metamodelling sound and complete entailment patterns (informative): ⏪

	if the triple structure appears in S	then S RDF(S) entails
reif1	sss aaa <<(xxx yyy zzz)>>	<<(xxx yyy zzz)>> rdf:type rdfs:Proposition . <<(xxx yyy zzz)>> rdf:type rdfs:Resource . sss rdf:type rdfs:Resource . aaa rdf:type rdf:Property .
reif2	<<(xxx yyy zzz)>> aaa ooo	<<(xxx yyy zzz)>> rdf:type rdfs:Proposition . <<(xxx yyy zzz)>> rdf:type rdfs:Resource . ooo rdf:type rdfs:Resource . aaa rdf:type rdf:Property .
reif3	sss rdf:reifies ooo	ooo rdf:type rdfs:Proposition .

Notes:

• RDF 1.1 RDF semantics is the above without the parts within ⏩...⏪ marks.
• The above entailment pattern are valid only for generalised RDF; for RDF, we have to use a restricted set of sound and incomplete entailment patterns.

BEST PRACTICE (informative)

The RDF-concepts and RDF-primer documents should be conveying the right message 😊, namely reification is sanctioned only if it makes use of the property rdf:reifies or any of its subproperties; the subject of rdf:reifies is called a reifier and the object is called a proposition.

TURTLE SYNTAX (informative)

A triple term is represented in Turtle as follows:

<<( s p o )>>

The following temporary shortcut for a term (a triple term reifier) is added to Turtle, which can be used in arbitrary subject or object position:

<< s p o ~ e >>

which expands in Turtle, following the best practice, as follows:

<< s p o ~ e >> p1 o1 .   ==>   e rdf:reifies <<( s p o )>> .  e p1 o1 .
s1 p1 << s p o ~ e >> .   ==>   e rdf:reifies <<( s p o )>> .  s1 p1 e .