RDF Semantics

W3C Recommendation 10 February 2004

PRE-PUBLICATION DRAFT. This document is being prepared for possible publication by the W3C, but it may change, be delayed, or never be published. The "Version" URIs and "Status of this Document" section may reflect its planned location and status, not present reality.

This Version:: http://www.w3.org/2001/sw/RDFCore/TR/WD-rdf-mt-20030117/
Latest Version:: http://www.w3.org/TR/rdf-mt/
Previous Version:: http://www.w3.org/TR/2003/PR-rdf-mt-20031215/
Editor:: Patrick Hayes (IHMC)< phayes@ihmc.us>
Series Editor: Brian McBride (Hewlett Packard Labs)<bwm@hplb.hpl.hp.com>

Please refer to the errata for this document, which may include some normative corrections.

Status of this Document

This document has been reviewed by W3C Members and other interested parties, and it has been endorsed by the Director as a W3C Recommendation. W3C's role in making the Recommendation is to draw attention to the specification and to promote its widespread deployment. This enhances the functionality and interoperability of the Web.

This is one document in a set of six (Primer, Concepts, Syntax, Semantics, Vocabulary, and Test Cases) intended to jointly replace the original Resource Description Framework specifications, RDF Model and Syntax (1999 Recommendation) and RDF Schema (2000 Candidate Recommendation). It has been developed by the RDF Core Working Group as part of the W3C Semantic Web Activity (Activity Statement, Group Charter) for publication on 10 February 2004.

Changes to this document since the Proposed Recommendation Working Draft are detailed in the change log.

The public is invited to send comments to www-rdf-comments@w3.org (archive) and to participate in general discussion of related technology on www-rdf-interest@w3.org (archive).

A list of implementations is available.

The W3C maintains a list of any patent disclosures related to this work.

This section describes the status of this document at the time of its publication. Other documents may supersede this document. A list of current W3C publications and the latest revision of this technical report can be found in the W3C technical reports index at http://www.w3.org/TR/.

0. Introduction

0.1 Specifying a formal semantics: scope and limitations

RDF is an assertional language intended to be used to express propositions using precise formal vocabularies, particularly those specified using RDFS [RDF-VOCABULARY], for access and use over the World Wide Web, and is intended to provide a basic foundation for more advanced assertional languages with a similar purpose. The overall design goals emphasise generality and precision in expressing propositions about any topic, rather than conformity to any particular processing model: see the RDF Concepts document [RDF-CONCEPTS] for more discussion.

Exactly what is considered to be the 'meaning' of an assertion in RDF or RDFS in some broad sense may depend on many factors, including social conventions, comments in natural language or links to other content-bearing documents. Much of this meaning will be inaccessible to machine processing and is mentioned here only to emphasize that the formal semantics described in this document is not intended to provide a full analysis of 'meaning' in this broad sense; that would be a large research topic. The semantics given here restricts itself to a formal notion of meaning which could be characterized as the part that is common to all other accounts of meaning, and can be captured in mechanical inference rules.

This document uses a basic technique called model theory for specifying the semantics of a formal language. Readers unfamiliar with model theory may find the glossary in appendix B helpful; throughout the text, uses of terms in a technical sense are linked to their glossary definitions. Model theory assumes that the language refers to a 'world', and describes the minimal conditions that a world must satisfy in order to assign an appropriate meaning for every expression in the language. A particular world is called an interpretation, so that model theory might be better called 'interpretation theory'. The idea is to provide an abstract, mathematical account of the properties that any such interpretation must have, making as few assumptions as possible about its actual nature or intrinsic structure, thereby retaining as much generality as possible. The chief utility of a formal semantic theory is not to provide any deep analysis of the nature of the things being described by the language or to suggest any particular processing model, but rather to provide a technical way to determine when inference processes are valid, i.e. when they preserve truth. This provides the maximal freedom for implementations while preserving a globally coherent notion of meaning.

Model theory tries to be metaphysically and ontologically neutral. It is typically couched in the language of set theory simply because that is the normal language of mathematics - for example, this semantics assumes that names denote things in a set IR called the 'universe' - but the use of set-theoretic language here is not supposed to imply that the things in the universe are set-theoretic in nature. Model theory is usually most relevant to implementation via the notion of entailment, described later, which makes it possible to define valid inference rules.

An alternative way to specify a semantics is to give a translation from RDF into a formal logic with a model theory already attached, as it were. This 'axiomatic semantics' approach has been suggested and used previously with various alternative versions of the target logical language [Conen&Klapsing] [Marchiori&Saarela] [McGuinness&al]. Such a translation for RDF and RDFS is also given in the L_base specification [LBASE]. The axiomatic semantics style has some advantages for machine processing and may be more readable, but in the event that any axiomatic semantics fails to conform to the model-theoretic semantics described in this document, the model theory should be taken as normative.

There are several aspects of meaning in RDF which are ignored by this semantics; in particular, it treats URI references as simple names, ignoring aspects of meaning encoded in particular URI forms [RFC 2396] and does not provide any analysis of time-varying data or of changes to URI references. It does not provide any analysis of indexical uses of URI references, for example to mean 'this document'. Some parts of the RDF and RDFS vocabularies are not assigned any formal meaning, and in some cases, notably the reification and container vocabularies, it assigns less meaning than one might expect. These cases are noted in the text and the limitations discussed in more detail. RDF is an assertional logic, in which each triple expresses a simple proposition. This imposes a fairly strict monotonic discipline on the language, so that it cannot express closed-world assumptions, local default preferences, and several other commonly used non-monotonic constructs.

Particular uses of RDF, including as a basis for more expressive languages such as DAML+OIL [DAML] and OWL [OWL], may impose further semantic conditions in addition to those described here, and such extra semantic conditions can also be imposed on the meanings of terms in particular RDF vocabularies. Extensions or dialects of RDF which are obtained by imposing such extra semantic conditions may be referred to as semantic extensions of RDF. Semantic extensions of RDF are constrained in this recommendation using the keywords MUST , MUST NOT, SHOULD and MAY of [RFC 2119]. Semantic extensions of RDF MUST conform to the semantic conditions for simple interpretations described in sections 1.3 and 1.4 and 1.5 and those for RDF interpretations described in section 3.1 of this document. Any name for entailment in a semantic extension SHOULD be indicated by the use of a vocabulary entailment term. The semantic conditions imposed on an RDF semantic extension MUST define a notion of vocabulary entailment which is valid according to the model-theoretic semantics described in the normative parts of this document; except that if the semantic extension is defined on some syntactically restricted subset of RDF graphs, then the semantic conditions need only apply to this subset. Specifications of such syntactically restricted semantic extensions MUST include a specification of their syntactic conditions which are sufficient to enable software to distinguish unambiguously those RDF graphs to which the extended semantic conditions apply. Applications based on such syntactically restricted semantic extensions MAY treat RDF graphs which do not conform to the required syntactic restrictions as syntax errors.

An example of a semantic extension of RDF is RDF Schema [RDF-VOCABULARY], abbreviated as RDFS, the semantics of which are defined in later parts of this document. RDF Schema imposes no extra syntactic restrictions.

0.2 Graph Syntax

This document uses the N-Triples syntax described in the RDF test cases document [RDF-TESTS] to describe RDF graphs. This notation uses a node identifier (nodeID) convention to indicate blank nodes in the triples of a graph. While node identifiers such as '_:xxx' serve to identify blank nodes in the surface syntax, these expressions are not considered to be the label of the graph node they identify; they are not names, and do not occur in the actual graph. In particular, the RDF graphs described by two N-Triples documents which differ only by re-naming their node identifiers will be understood to be equivalent . This re-naming convention should be understood as applying only to whole documents, since re-naming the node identifiers in part of a document may result in a document describing a different RDF graph.

The N-Triples syntax requires that URI references be given in full, enclosed in angle brackets. In the interests of brevity, the imaginary URI scheme 'ex:' is used to provide illustrative examples. To obtain a more realistic view of the normal appearance of the N-Triples syntax, the reader should imagine this replaced with something like 'http://www.example.org/rdf/mt/artificial-example/'. The QName prefixes rdf:, rdfs: and xsd: are defined as follows:

Since QName syntax is not legal N-Triples syntax, and in the interests of brevity and readability, examples use the convention whereby a QName is used without surrounding angle brackets to indicate the corresponding URI reference enclosed in angle brackets, e.g. the triple

<ex:a>
    <http://www.w3.org/1999/02/22-rdf-syntax-ns#type>
    <http://www.w3.org/2000/01/rdf-schema#Class> .

In stating general semantic conditions, single characters or character sequences without a colon indicate an arbitrary name, blank node, character string and so on. The exact meaning will be specified in context.

0.3 Graph Definitions

A set of names is referred to as a vocabulary. The vocabulary of a graph is the set of names which occur as the subject, predicate or object of any triple in the graph. Note that URI references which occur only inside typed literals are not required to be in the vocabulary of the graph.

A proper instance of a graph is an instance in which a blank node has been replaced by a name, or two blank nodes in the graph have been mapped into the same node in the instance.

Any instance of a graph in which a blank node is mapped to a new blank node not in the original graph is an instance of the original and also has it as an instance, and this process can be iterated so that any 1:1 mapping between blank nodes defines an instance of a graph which has the original graph as an instance. Two such graphs, each an instance of the other but neither a proper instance, which differ only in the identity of their blank nodes, are considered to be equivalent. We will treat such equivalent graphs as identical; this allows us to ignore some issues which arise from 're-naming' nodeIDs, and is in conformance with the convention that blank nodes have no label. Equivalent graphs are mutual instances with an invertible instance mapping.

A merge of a set of RDF graphs is defined as follows. If the graphs in the set have no blank nodes in common, then the union of the graphs is a merge; if they do share blank nodes, then it is the union of a set of graphs that is obtained by replacing the graphs in the set by equivalent graphs that share no blank nodes. This is often described by saying that the blank nodes have been 'standardized apart'. It is easy to see that any two merges are equivalent, so we will refer to the merge, following the convention on equivalent graphs. Using the convention on equivalent graphs and identity, any graph in the original set is considered to be a subgraph of the merge.

One does not, in general, obtain the merge of a set of graphs by concatenating their corresponding N-Triples documents and constructing the graph described by the merged document. If some of the documents use the same node identifiers, the merged document will describe a graph in which some of the blank nodes have been 'accidentally' identified. To merge N-Triples documents it is necessary to check if the same nodeID is used in two or more documents, and to replace it with a distinct nodeID in each of them, before merging the documents. Similar cautions apply to merging graphs described by RDF/XML documents which contain nodeIDs, see RDF/XML Syntax Specification (Revised) [RDF-SYNTAX].

1. Interpretations

1.1 Technical note (Informative)

RDF does not impose any logical restrictions on the domains and ranges of properties; in particular, a property may be applied to itself. When classes are introduced in RDFS, they may contain themselves. Such 'membership loops' might seem to violate the axiom of foundation, one of the axioms of standard (Zermelo-Fraenkel) set theory, which forbids infinitely descending chains of membership. However, the semantic model given here distinguishes properties and classes considered as objects from their extensions - the sets of object-value pairs which satisfy the property, or things that are 'in' the class - thereby allowing the extension of a property or class to contain the property or class itself without violating the axiom of foundation. In particular, this use of a class extension mapping allows classes to contain themselves. For example, it is quite OK for (the extension of) a 'universal' class to contain the class itself as a member, a convention that is often adopted at the top of a classification hierarchy. (If an extension contained itself then the axiom would be violated, but that case never arises.) The technique is described more fully in [Hayes&Menzel].

In this respect, RDFS differs from many conventional ontology frameworks such as UML which assume a more structured hierarchy of individuals, sets of individuals, etc., or which draw a sharp distinction between data and meta-data. However, while RDFS does not assume the existence of such structure, it does not prohibit it. RDF allows membership loops, but it does not mandate their use for all parts of a user vocabulary. If this aspect of RDFS is found worrying, then it is possible to restrict oneself to a subset of RDF graphs which do not contain any such 'loops' of class membership or property application while retaining much of the expressive power of RDFS for many practical purposes, and semantic extensions may impose syntactic conditions which forbid such looped constructions.

The use of the explicit extension mapping also makes it possible for two properties to have exactly the same values, or two classes to contain the same instances, and still be distinct entities. This means that RDFS classes can be considered to be rather more than simple sets; they can be thought of as 'classifications' or 'concepts' which have a robust notion of identity which goes beyond a simple extensional correspondence. This property of the model theory has significant consequences in more expressive languages built on top of RDF, such as OWL [OWL], which are capable of expressing identity between properties and classes directly. This 'intensional' nature of classes and properties is sometimes claimed to be a useful property of a descriptive language, but a full discussion of this issue is beyond the scope of this document.

Notice that the question of whether or not a class contains itself as a member is quite different from the question of whether or not it is a subclass of itself. All classes are subclasses of themselves.

Readers who are familiar with conventional logical semantics may find it useful to think of RDF as a version of existential binary relational logic in which relations are first-class entities in the universe of quantification. Such a logic can be obtained by encoding the relational atom R(a,b) into a conventional logical syntax, using a notional three-place relation Triple(a,R,b); the basic semantics described here can be reconstructed from this intuition by defining the extension of y as the set {<x,z> : Triple(x,y,z)} and noting that this would be precisely the denotation of R in the conventional Tarskian model theory of the original form R(a,b) of the relational atom. This construction can also be traced in the semantics of the L_base axiomatic description [LBASE].

1.2 URI references, Resources and Literals.

This document does not take any position on the way that URI references may be composed from other expressions, e.g. from relative URIs or QNames; the semantics simply assumes that such lexical issues have been resolved in some way that is globally coherent, so that a single URI reference can be taken to have the same meaning wherever it occurs. Similarly, the semantics has no special provision for tracking temporal changes. It assumes, implicitly, that URI references have the same meaning whenever they occur. To provide an adequate semantics which would be sensitive to temporal changes is a research problem which is beyond the scope of this document.

The semantics does not assume any particular relationship between the denotation of a URI reference and a document or Web resource which can be retrieved by using that URI reference in an HTTP transfer protocol, or any entity which is considered to be the source of such documents. Such a requirement could be added as a semantic extension, but the formal semantics described here makes no assumptions about any connection between the denotations of URI references and the uses of those URI references in other protocols.

The semantics treats all RDF names as expressions which denote. The things denoted are called 'resources', following [RFC 2396], but no assumptions are made here about the nature of resources; 'resource' is treated here as synonymous with 'entity', i.e. as a generic term for anything in the universe of discourse.

The different syntactic forms of names are treated in particular ways. URI references are treated simply as logical constants. Plain literals are considered to denote themselves, so have a fixed meaning. The denotation of a typed literal is the value mapped from its enclosed character string by the datatype associated with its enclosed type. RDF assigns a particular meaning to literals typed with rdf:XMLLiteral, described in section 3.

1.3 Interpretations

The basic intuition of model-theoretic semantics is that asserting a sentence makes a claim about the world: it is another way of saying that the world is, in fact, so arranged as to be an interpretation which makes the sentence true. In other words, an assertion amounts to stating a constraint on the possible ways the world might be. Notice that there is no presumption here that any assertion contains enough information to specify a single unique interpretation. It is usually impossible to assert enough in any language to completely constrain the interpretations to a single possible world, so there is no such thing as 'the' unique interpretation of an RDF graph. In general, the larger an RDF graph is - the more it says about the world - then the smaller the set of interpretations that an assertion of the graph allows to be true - the fewer the ways the world could be, while making the asserted graph true of it.

The following definition of an interpretation is couched in mathematical language, but what it amounts to intuitively is that an interpretation provides just enough information about a possible way the world might be - a 'possible world' - in order to fix the truth-value (true or false) of any ground RDF triple. It does this by specifying for each URI reference, what it is supposed to be a name of; and also, if it is used to indicate a property, what values that property has for each thing in the universe; and if it is used to indicate a datatype, that the datatype defines a mapping between lexical forms and datatype values. This is just enough information to fix the truth-value of any ground triple, and hence any ground RDF graph. (Non-ground graphs are considered in the following section.) Note that if any of this information were omitted, it would be possible for some well-formed triple to be left without a determinate value; and also that any other information - such as the exact nature of the things in the universe - would, regardless of its intrinsic interest, be irrelevant to the actual truth-values of any triple.

All interpretations will be relative to a set of names, called the vocabulary of the interpretation; so that one should speak, strictly, of an interpretation of an RDF vocabulary, rather than of RDF itself. Some interpretations may assign special meanings to the symbols in a particular vocabulary. Interpretations which share the special meaning of a particular vocabulary will be named for that vocabulary, e.g. 'rdf-interpretations', 'rdfs-interpretations', etc. An interpretation with no particular extra conditions on a vocabulary (including the RDF vocabulary itself) will be called a simple interpretation, or simply an interpretation.

RDF uses several forms of literal. The chief semantic characteristic of literals is that their meaning is largely determined by the form of the string they contain. Plain literals, without an embedded type URI reference, are always interpreted as referring to themselves: either a character string or a pair consisting of a character string and a language tag; in either case, the character string is referred to as the "literal character string". In the case of typed literals, however, the full specification of the meaning depends on being able to access datatype information which is external to RDF itself. A full discussion of the meaning of typed literals is described in section 5 , where a special notion of datatype interpretation is introduced. Each interpretation defines a mapping IL from typed literals to their interpretations. Stronger conditions on IL will be defined as the notion of 'interpretation' is extended in later sections.

Throughout this document, precise semantic conditions will be set out in tables which state semantic conditions, tables containing true assertions and valid inference rules, and tables listing syntax, which are distinguished by background color. These tables, taken together, amount to a formal summary of the entire semantics. Note that the semantics of RDF does not depend on that of RDFS. The full semantics of RDF is defined in sections 1 and 3 ; the full semantics of RDFS in sections 1, 3 and 4.

IEXT(x), called the extension of x, is a set of pairs which identify the arguments for which the property is true, that is, a binary relational extension. This trick of distinguishing a relation as an object from its relational extension allows a property to occur in its own extension, as noted earlier.

The assumption that LV is a subset of IR amounts to saying that literal values are thought of as real entities that 'exist'. This amounts to saying that literal values are resources. However, this does not imply that literals should be identified with URI references. Note that LV may contain other items in addition to plain literals. There is a technical reason why the range of IL is IR rather than restricted to LV. When interpretations take account of datatype information, it is syntactically possible for a typed literal to be internally inconsistent, and such ill-typed literals are required to denote a non-literal value, as explained in section 5.

The next sections define how an interpretation of a vocabulary determines the truth-values of any RDF graph, by a recursive definition of the denotation - the semantic "value" - of any RDF expression in terms of those of its immediate subexpressions. These apply to all subsequent semantic extensions. RDF has two kinds of denotation: names denote things in the universe, and sets of triples denote truth-values.

1.4 Denotations of Ground Graphs

The denotation of a ground RDF graph in I is given recursively by the following rules, which extend the interpretation mapping I from names to ground graphs. These rules (and extensions of them given later) work by defining the denotation of any piece of RDF syntax E in terms of the denotations of the immediate syntactic constituents of E, hence allowing the denotation of any piece of RDF to be determined by a kind of syntactic recursion.

In this table, and throughout this document, the equality sign = indicates identity and angle brackets <x,y> are used to indicate an ordered pair of x and y. RDF graph syntax is indicated using the notational conventions of the N-Triples syntax described in the RDF test cases document [RDF-TESTS]: literal strings are encloded within double quote marks, language tags indicated by the use of the @ sign, and triples terminate with a 'code dot' . .

If the vocabulary of an RDF graph contains names that are not in the vocabulary of an interpretation I - that is, if I simply does not give a semantic value to some name that is used in the graph - then these truth-conditions will always yield the value false for some triple in the graph, and hence for the graph itself. Turned around, this means that any assertion of a graph implicitly asserts that all the names in the graph actually refer to something in the world. The final condition implies that an empty graph (an empty set of triples) is trivially true.

Note that the denotation of plain literals is always in LV; and that those of the subject and object of any true triple must be in IR; so any URI reference which occurs in a graph both as a predicate and as a subject or object must denote something in the intersection of IR and IP in any interpretation which satisfies the graph.

As an illustrative example, the following is a small interpretation for the artificial vocabulary {ex:a, ex:b, ex:c, "whatever", "whatever"^^ex:b}. Integers are used to indicate the non-literal 'things' in the universe. This is not meant to imply that interpretations should be interpreted as being about arithmetic, but more to emphasize that the exact nature of the things in the universe is irrelevant. LV can be any set satisfying the semantic conditions. (In this and subsequent examples the greater-than and less-than symbols are used in several ways: following mathematical usage to indicate abstract pairs and n-tuples; following N-Triples syntax to enclose URI references, and also as arrowheads when indicating mappings.)

For example, I(

<ex:a> <ex:b> <ex:c>
    .

) = true if <I(ex:a),I(ex:c)> is in IEXT(I(<ex:b>)), i.e. if <1,2> is in IEXT(1), which is {<1,2>,<2,1>} and so does contain <1,2> and so I(<ex:a <ex:b> ex:c>) is true.

The truth of the fourth triple is a consequence of the rather idiosyncratic interpretation chosen here for typed literals.

In this interpretation IP is a subset of IR; this will be typical of RDF semantic interpretations, but is not required.

For example, I(<ex:a> <ex:c> <ex:b> .) = true if <I(ex:a), I(<ex:b>)>, i.e.<1,1>, is in IEXT(I(ex:c)); but I(ex:c)=2 which is not in IP, so IEXT is not defined on 2, so the condition fails and I(

<ex:a> 
  <ex:c> <ex:b> .

) = false.

It also makes all triples containing a plain literal false, since the property extension does not have any pairs containing a plain literal.

To emphasize; this is only one possible interpretation of this vocabulary; there are (infinitely) many others. For example, if this interpretation were modified by attaching the property extension to 2 instead of 1, none of the above triples would be true.

This example illustrates that any interpretation which maps any URI reference which occurs in the predicate position of a triple in a graph to something not in IP will make the graph false.

1.5. Blank Nodes as Existential Variables

Blank nodes are treated as simply indicating the existence of a thing, without using, or saying anything about, the name of that thing. (This is not the same as assuming that the blank node indicates an 'unknown' URI reference; for example, it does not assume that there is any URI reference which refers to the thing. The discussion of Skolemization in appendix A is relevant to this point.)

An interpretation can specify the truth-value of a graph containing blank nodes. This will require some definitions, as the theory so far provides no meaning for blank nodes. Suppose I is an interpretation and A is a mapping from some set of blank nodes to the universe IR of I, and 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(E) to be the set of blank nodes in E. Then the above rules can be extended to include the two new cases that are introduced when blank nodes occur in the graph:

Notice that this does not change the definition of an interpretation; it still consists of the same values IR, IP, IEXT, IS, LV and IL. It simply extends the rules for defining denotations under an interpretation, so that the same interpretation that provides a truth-value for ground graphs also assigns truth-values to graphs with blank nodes, even though it provides no denotation for the blank nodes themselves. Notice also that the blank nodes themselves are perfectly well-defined entities; they differ from other nodes only in not being assigned a denotation by an interpretation, reflecting the intuition that they have no 'global' meaning (i.e. outside the graph in which they occur).

For example, the graph defined by the following triples is false in the interpretation shown in figure 1:

since if A' maps the blank node to 1 then the first triple is false in I+A', and if it maps it to 2 then the second triple is false.

Note that each of these triples, if thought of as a single graph, would be true in I, but the whole graph is not; and that if a different nodeID were used in the two triples, indicating that the RDF graph had two blank nodes instead of one, then A' could map one node to 2 and the other to 1, and the resulting graph would be true under the interpretation I.

This effectively treats all blank nodes as having the same meaning as existentially quantified variables in the RDF graph in which they occur, and which have the scope of the entire graph. In terms of the N-Triples syntax, this amounts to the convention that would place the quantifiers just outside, or at the outer edge of, the N-Triples document corresponding to the graph. This in turn means that there is a subtle but important distinction in meaning between the operation of forming the union of two graphs and that of forming the merge. The simple union of two graphs corresponds to the conjunction ( 'and' ) of all the triples in the graphs, maintaining the identity of any blank nodes which occur in both graphs. This is appropriate when the information in the graphs comes from a single source, or where one is derived from the other by means of some valid inference process, as for example when applying an inference rule to add a triple to a graph. Merging two graphs treats the blank nodes in each graph as being existentially quantified in that graph, so that no blank node from one graph is allowed to stray into the scope of the other graph's surrounding quantifier. This is appropriate when the graphs come from different sources and there is no justification for assuming that a blank node in one refers to the same entity as any blank node in the other.

2. Simple Entailment between RDF graphs

Following conventional terminology, I satisfies E if I(E)=true, and a set S of RDF graphs (simply) entails a graph E if every interpretation which satisfies every member of S also satisfies E. In later sections these notions will be adapted to other classes of interpretations, but throughout this section 'entailment' should be interpreted as meaning simple entailment.

Entailment is the key idea which connects model-theoretic semantics to real-world applications. As noted earlier, making an assertion amounts to claiming that the world is an interpretation which assigns the value true to the assertion. If A entails B, then any interpretation that makes A true also makes B true, so that an assertion of A already contains the same "meaning" as an assertion of B; one could say that the meaning of B is somehow contained in, or subsumed by, that of A. If A and B entail each other, then they both "mean" the same thing, in the sense that asserting either of them makes the same claim about the world. The interest of this observation arises most vividly when A and B are different expressions, since then the relation of entailment is exactly the appropriate semantic license to justify an application inferring or generating one of them from the other. Through the notions of satisfaction, entailment and validity, formal semantics gives a rigorous definition to a notion of "meaning" that can be related directly to computable methods of determining whether or not meaning is preserved by some transformation on a representation of knowledge.

This section gives a few basic results about simple entailment and valid inference. Simple entailment can be recognized by relatively simple syntactic comparisons. The two basic forms of simply valid inference in RDF are, in logical terms, the inference from (P and Q) to P, and the inference from foo(baz) to (exists (?x) foo(?x)).

These results apply only to simple entailment, not to the extended notions of entailment introduced in later sections. Proofs, all of which are straightforward, are given in appendix A, which also describes some other properties of entailment which may be of interest.

Empty Graph Lemma. The empty set of triples is entailed by any graph, and does not entail any graph except itself. [Proof]

The relationship between merging and entailment is simple, and obvious from the definitions:

Merging lemma. The merge of a set S of RDF graphs is entailed by S, and entails every member of S. [Proof]

This means that a set of graphs can be treated as equivalent to its merge, i.e. a single graph, as far as the model theory is concerned. This can be used to simplify the terminology somewhat: for example, the definition of S entails E, above, can be paraphrased by saying that S entails E when every interpretation which satisfies S also satisfies E.

The example given in section 1.5 shows that it is not the case, in general, that the simple union of a set of graphs is entailed by the set.

The interpolation lemma completely characterizes simple RDF entailment in syntactic terms. To tell whether a set of RDF graphs entails another, check that there is some instance of the entailed graph which is a subset of the merge of the original set of graphs. Of course, there is no need to actually construct the merge. If working backwards from the consequent E, an efficient technique would be to treat blank nodes as variables in a process of subgraph-matching, allowing them to bind to 'matching' names in the antecedent graph(s) in S, i.e. those which may entail the consequent graph. The interpolation lemma shows that this process is valid, and is also complete if the subgraph-matching algorithm is. The existence of complete subgraph-checking algorithms also shows that RDF entailment is decidable, i.e. there is a terminating algorithm which will determine for any finite set S and any graph E, whether or not S entails E.

Such a variable-binding process would only be appropriate when applied to the conclusion of a proposed entailment. This corresponds to using the document as a goal or a query, in contrast to asserting it, i.e. claiming it to be true. If an RDF document is asserted, then it would be invalid to bind new values to any of its blank nodes, since the resulting graph might not be entailed by the assertion.

The interpolation lemma has an immediate consequence a criterion for non-entailment:

Note again that this applies only to simple entailment, not to the vocabulary entailment relationships defined in rest of the document.

Several basic properties of entailment follow directly from the above definitions and results but are stated here for completeness sake:

Monotonicity Lemma. Suppose S is a subgraph of S' and S entails E. Then S' entails E.[Proof]

The property of finite expressions always being derivable from a finite set of antecedents is called compactness. Semantic theories which support non-compact notions of entailment do not have corresponding computable inference systems.

Compactness Lemma. If S entails E and E is a finite graph, then some finite subset S' of S entails E.

2.1 Vocabulary interpretations and vocabulary entailment

Simple interpretations and simple entailment capture the semantics of RDF graphs when no attention is paid to the particular meaning of any of the names in the graph. To obtain the full meaning of an RDF graph written using a particular vocabulary, it is usually necessary to add further semantic conditions which attach stronger meanings to particular URI references and typed literals in the graph. Interpretations which are required to satisfy extra semantic conditions on a particular vocabulary will be generically referred to as vocabulary interpretations. Vocabulary entailment means entailment with respect to such vocabulary interpretations. These stronger notions of interpretation and entailment are indicated by the use of a namespace prefix, so that we will refer to rdf-entailment, rdfs-entailment and so on in what follows. In each case, the vocabulary whose meaning is being restricted, and the exact conditions associated with that vocabulary, are spelled out in detail.

3. Interpreting the RDF Vocabulary

3.1 RDF Interpretations

RDF vocabulary

rdf:type

rdf:Property 
          rdf:XMLLiteral rdf:nil rdf:List rdf:Statement rdf:subject rdf:predicate 
          rdf:object rdf:first rdf:rest rdf:Seq rdf:Bag rdf:Alt rdf:_1 rdf:_2 
          ... rdf:value

rdf-interpretations impose extra semantic conditions on rdfV and on typed literals with the type rdf:XMLLiteral, which is referred to as the RDF built-in datatype. This datatype is fully described in the RDF Concepts and Abstract Syntax document [RDF-CONCEPTS]. Any character string sss which satisfies the conditions for being in the lexical space of rdf:XMLLiteral will be called a well-typed XML literal string. The corresponding value will be called the XML value of the literal. Note that the XML values of well-typed XML literals are in precise 1:1 correspondence with the XML literal strings of such literals, but are not themselves character strings. An XML literal whose literal string is well-typed will be called a well-typed XML literal; other XML literals will be called ill-typed.

An rdf-interpretation of a vocabulary V is a simple interpretation I of (V union rdfV) which satisfies the extra conditions described in the following table and all the triples in the subsequent table. These triples are called the rdf axiomatic triples.

The first condition could be regarded as defining IP to be the set of resources in the universe of the interpretation which have the value I(rdf:Property) of the property I(rdf:type). Such subsets of the universe will be central in interpretations of RDFS. Note that this condition requires IP to be a subset of IR. The third condition requires that ill-typed XML literals denote something other than a literal value: this will be the standard way of handling ill-formed typed literals.

The rdfs-interpretations described in section 4 below assign further semantic conditions (range and domain conditions) to the properties used in the RDF vocabulary, and other semantic extensions MAY impose further conditions so as to further restrict their meanings, provided that such conditions MUST be compatible with the conditions described in this section.

For example, the following rdf-interpretation extends the simple interpretation in figure 1 to the case where V contains rdfV. For simplicity, we ignore XML literals in this example.

IS: ex:a=>1, ex:b=>1, ex:c=> 2, rdf:type=>T, rdf:Property=>P, rdf:nil=>1, rdf:List=>P, rdf:Statement=>P

,
    rdf:subject=>

1, rdf:predicate=>1, rdf:object=>1

,
    rdf:first=>

1, rdf:rest=>1, rdf:Seq=>P

,
    rdf:Bag=>

P, rdf:Alt=>P

, rdf:_1, rdf:_2, ...
    =>

This is not the smallest rdf-interpretation which extends the earlier example, since one could have made IEXT(T) be {<1,2>,<T,2>}, and managed without having P in the universe. In general, a given entity in an interpretation may play several 'roles' at the same time, as long as this can be done without violating any of the required semantic conditions. The above interpretation identifies properties with lists, for example; of course, other interpretations might not make such an identification.

Every rdf-interpretation is also a simple interpretation. The 'extra' structure does not prevent it acting in the simpler role.

3.2. RDF entailment

S rdf-entails E when every rdf-interpretation which satisfies every member of S also satisfies E. This follows the wording of the definition of simple entailment in Section 2, but refers only to rdf-interpretations instead of all simple interpretations. Rdf-entailment is an example of vocabulary entailment.

It is easy to see that the lemmas in Section 2 do not all apply to rdf-entailment: for example, the triple

is true in every rdf-interpretation, so is rdf-entailed by the empty graph, contradicting the interpolation lemma for rdf-entailment. Section 7.2 describes exact conditions for detecting RDF entailment.

3.3. Reification, Containers, Collections and rdf:value

The RDF semantic conditions impose significant formal constraints on the meaning only of the central RDF vocabulary, so the notions of rdf-entailment and rdf-interpretation apply to the remainder of the vocabulary without further change. This includes vocabulary which is intended for use in describing containers and bounded collections, and a reification vocabulary to enable an RDF graph to describe, as well as exhibit, triples. In this section we review the intended meanings of this vocabulary, and note some intuitive consequences which are not supported by the formal model theory. Semantic extensions MAY limit the formal interpretations of these vocabularies to conform to these intended meanings.

The omission of these conditions from the formal semantics is a design decision to accomodate variations in existing RDF usage and to make it easier to implement processes to check formal RDF entailment. For example, implementations may decide to use special procedural techniques to implement the RDF collection vocabulary.

3.3.1 Reification

Semantic extensions MAY limit the interpretation of these so that a triple of the form

is true in I just when I(aaa) is a token of an RDF triple in some RDF document, and the three properties, when applied to such a denoted triple, have the same values as the respective components of that triple.

This may be illustrated by considering the following two RDF graphs, the first of which consists of a single triple.

_:xxx rdf:type rdf:Statement .

     _:xxx rdf:subject <ex:a> .

     _:xxx rdf:predicate <ex:b> .

     _:xxx rdf:object <ex:c> .

The second graph is called a reification of the triple in the first graph, and the node which is intended to refer to the first triple - the blank node in the second graph - is called, rather confusingly, a reified triple. (This can be a blank node or a URI reference.) In the intended interpretation of the reification vocabulary, the second graph would be made true in an interpretation I by interpreting the reified triple to refer to a token of the triple in the first graph in some concrete RDF document, considering that token to be valid RDF syntax, and then using I to interpret the syntactic triple which the token instantiates, so that the subject, predicate and object of that triple are interpreted in the same way in the reification as in the triple described by the reification. This could be stated formally as follows: <x,y> is in IEXT(I(rdf:subject)) just when x is a token of an RDF triple of the form

and y is I(aaa); similarly for predicate and object. Notice that the value of the rdf:subject property is not the subject URI reference itself but its interpretation, and so this condition involves a two-stage interpretation process: one has to interpret the reified node - the subject of the triples in the reification - to refer to another triple, then treat that triple as RDF syntax and apply the interpretation mapping again to get to the referent of its subject. This requires triple tokens to exist as first-class entities in the universe IR of an interpretation. In sum: the meaning of the reification is that a document exists containing a triple token which means whatever the first graph means.Note that this way of understanding the reification vocabulary does not interpret reification as a form of quotation. Rather, the reification describes the relationship between a token of a triple and the resources that triple refers to. The reification can be read intuitively as saying "'this piece of RDF talks about these things" rather than "this piece of RDF has this form".

The semantic extension described here requires the reified triple that the reification describes - I(_:xxx) in the above example - to be a particular token or instance of a triple in a (real or notional) RDF document, rather than an 'abstract' triple considered as a grammatical form. There could be several such entities which have the same subject, predicate and object properties. Although a graph is defined as a set of triples, several such tokens with the same triple structure might occur in different documents. Thus, it would be meaningful to claim that the blank node in the second graph above does not refer to the triple in the first graph, but to some other triple with the same structure. This particular interpretation of reification was chosen on the basis of use cases where properties such as dates of composition or provenance information have been applied to the reified triple, which are meaningful only when thought of as referring to a particular instance or token of a triple.

Although RDF applications may use reification to refer to triple tokens in RDF documents, the connection between the document and its reification must be maintained by some means external to the RDF graph syntax. (In the RDF/XML syntax described in RDF/XML Syntax Specification (Revised) [RDF-SYNTAX], the rdf:ID attribute can be used in the description of a triple to create a reification of that triple in which the reified triple is a URI constructed from the baseURI of the XML document and the value of rdf:ID as a fragment.) Since an assertion of a reification of a triple does not implicitly assert the triple itself, this means that there are no entailment relationships which hold between a triple and a reification of it. Thus the reification vocabulary has no effective semantic constraints on it, other than those that apply to an rdf-interpretation.

A reification of a triple does not entail the triple, and is not entailed by it. (The reification only says that the triple token exists and what it is about, not that it is true. The second non-entailment is a consequence of the fact that asserting a triple does not automatically assert that any triple tokens exist in the universe being described by the triple. For example, the triple might be part of an ontology describing animals, which could be satisfied by an interpretation in which the universe contained only animals, and in which a reification of it was therefore false.)

Since the relation between triples and reifications of triples in any RDF graph or graphs need not be one-to-one, asserting a property about some entity described by a reification need not entail that the same property holds of another such entity, even if it has the same components. For example,

_:xxx rdf:type rdf:Statement .

     _:xxx rdf:subject <ex:subject> .

     _:xxx rdf:predicate <ex:predicate> .

     _:xxx rdf:object <ex:object> .

     _:yyy rdf:type rdf:Statement .

     _:yyy rdf:subject <ex:subject> .

     _:yyy rdf:predicate <ex:predicate> .

     _:yyy rdf:object <ex:object> .

     _:xxx <ex:property> <ex:foo> .

3.3.2 RDF containers

RDF provides vocabularies for describing three classes of containers. Containers have a type, and their members can be enumerated by using a fixed set of container membership properties. These properties are indexed by integers to provide a way to distinguish the members from each other, but these indices should not necessarily be thought of as defining an ordering of the container itself; some containers are considered to be unordered.

The RDFS vocabulary, described below, adds a generic membership property which holds regardless of position, and classes containing all the containers and all the membership properties.

One should understand this RDF vocabulary as describing containers, rather than as a vocabulary for constructing them, as would typically be supplied by a programming language. On this view, the actual containers are entities in the semantic universe, and RDF graphs which use the vocabulary simply provide very basic information about these entities, enabling an RDF graph to characterize the container type and give partial information about the members of a container. Since the RDF container vocabulary is so limited, many 'natural' assumptions concerning RDF containers are not formally sanctioned by the RDF model theory. This should not be taken as meaning that these assumptions are false, but only that RDF does not formally entail that they must be true.

There are no special semantic conditions on the container vocabulary: the only 'structure' which RDF presumes its containers to have is what can be inferred from the use of this vocabulary and the general RDF semantic conditions. In general, this amounts to knowing the type of a container, and having a partial enumeration of the items in the container. The intended mode of use is that things of type rdf:Bag are considered to be unordered but to allow duplicates; things of type rdf:Seq are considered to be ordered, and things of type rdf:Alt are considered to represent a collection of alternatives, possibly with a preference ordering. The ordering of items in an ordered container is intended to be indicated by the numerical ordering of the container membership properties, which are assumed to be single-valued. However, these informal interpretations are not reflected in any formal RDF entailments.

RDF does not support any entailments which could arise from enumerating the elements of an rdf:Bag in a different order. For example,

Notice that if this conclusion were valid, then the result of conjoining it to the original graph would also be a valid entailment, which would assert that both elements were in both positions. This is a consequence of the fact that RDF is a purely assertional language.

There is no assumption that a property of a container applies to any of the elements of the container, or vice versa.

There is no formal requirement that the three container classes are disjoint, so that for example something can be asserted to be both an rdf:Bag and an rdf:Seq. There is no assumption that containers are gap-free, so that for example

There is no way in RDF to 'close' a container, i.e. to assert that it contains only a fixed number of members. This is a reflection of the fact that it is always consistent to add a triple to a graph asserting a membership property of any container. And finally, there is no built-in assumption that an RDF container has only finitely many members.

3.3.3 RDF collections

RDF provides a vocabulary for describing collections, i.e.'list structures', in terms of head-tail links. Collections differ from containers in allowing branching structure and in having an explicit terminator, allowing applications to determine the exact set of items in the collection.

As with containers, no special semantic conditions are imposed on this vocabulary other than the type of rdf:nil being rdf:List. It is intended for use typically in a context where a container is described using blank nodes to connect a 'well-formed' sequence of items, each described by two triples of the form



  

  _:c1 rdf:first aaa .

  _:c1 rdf:rest _:c2

where the final item is indicated by the use of rdf:nil as the value of the property rdf:rest. In a familiar convention, rdf:nil can be thought of as the empty collection. Any such graph amounts to an assertion that the collection exists, and since the members of the collection can be determined by inspection, this is often sufficient to enable applications to determine what is meant. Note however that the semantics does not require any collections to exist other than those mentioned explicitly in a graph (and the empty collection). For example, the existence of a collection containing two items does not automatically guarantee that the similar collection with the items permuted also exists:



  

  _:c1 rdf:first <ex:aaa> .

  _:c1 rdf:rest _:c2 .

   _:c2 rdf:first <ex:bbb> .

  _:c2 rdf:rest rdf:nil .

_:c3 rdf:first <ex:bbb> .

  _:c3 rdf:rest _:c4 .

  _:c4 rdf:first <ex:aaa> .

     _:c4 rdf:rest rdf:nil .

Also, RDF imposes no 'well-formedness' conditions on the use of this vocabulary, so that it is possible to write RDF graphs which assert the existence of highly peculiar objects such as lists with forked or non-list tails, or multiple heads:

_:666 rdf:first <ex:aaa> .

     _:666 rdf:first <ex:bbb> .

     _:666 rdf:rest <ex:ccc> .

  _:666 rdf:rest rdf:nil .

It is also possible to write a set of triples which underspecify a collection by failing to specify its rdf:rest property value.

Semantic extensions MAY place extra syntactic well-formedness restrictions on the use of this vocabulary in order to rule out such graphs. They MAY exclude interpretations of the collection vocabulary which violate the convention that the subject of a 'linked' collection of two-triple items of the form described above, ending with an item ending with rdf:nil, denotes a totally ordered sequence whose members are the denotations of the rdf:first values of the items, in the order got by tracing the rdf:rest properties from the subject to rdf:nil. This permits sequences which contain other sequences.

Note that the RDFS semantic conditions, described below, require that any subject of the rdf:first property, and any subject or object of the rdf:rest property, be of rdf:type rdf:List.

3.3.4 rdf:value

The intended use for rdf:value is explained intuitively in the RDF Primer document [RDF-PRIMER]. It is typically used to identify a 'primary' or 'main' value of a property which has several values, or has as its value a complex entity with several facets or properties of its own.

Since the range of possible uses for rdf:value is so wide, it is difficult to give a precise statement which covers all the intended meanings or use cases. Users are cautioned, therefore, that the meaning of rdf:value may vary from application to application. In practice, the intended meaning is often clear from the context, but may be lost when graphs are merged or when conclusions are inferred.

4. Interpreting the RDFS Vocabulary

4.1 RDFS Interpretations

RDF Schema [RDF-VOCABULARY] extends RDF to include a larger vocabulary rdfsV with more complex semantic constraints:

(rdfs:comment, rdfs:seeAlso, rdfs:isDefinedBy and rdfs:label are included here because some constraints which apply to their use can be stated using rdfs:domain, rdfs:range and rdfs:subPropertyOf. Other than this, the formal semantics does not assign them any particular meanings.)

Although not strictly necessary, it is convenient to state the RDFS semantics in terms of a new semantic construct, a 'class', i.e. a resource which represents a set of things in the universe which all have that class as the value of their rdf:type property. Classes are defined to be things of type rdfs:Class, and the set of all classes in an interpretation will be called IC. The semantic conditions are stated in terms of a mapping ICEXT (for the Class Extension in I) from IC to the set of subsets of IR. The meanings of ICEXT and IC in a rdf-interpretation of the RDFS vocabulary are completely defined by the first two conditions in the table of RDFS semantic conditions, below. Notice that a class may have an empty class extension; that (as noted earlier) two different class entities could have the same class extension; and that the class extension of rdfs:Class contains the class rdfs:Class.

An rdfs-interpretation of V is an rdf-interpretation I of (V union rdfV union rdfsV) which satisfies the following semantic conditions and all the triples in the subsequent table, called the RDFS axiomatic triples.

Since I is an rdf-interpretation, the first condition implies that IP = ICEXT(I(rdf:Property)).

These axioms and conditions have some redundancy: for example, all but one of the RDF axiomatic triples can be derived from the RDFS axiomatic triples and the semantic conditions on ICEXT, rdfs:domain and rdfs:range. Other triples which must be true in all rdfs-interpretations include the following:

Note that datatypes are allowed to have class extensions, i.e. are considered to be classes, in RDFS. As illustrated by the semantic condition on the class extension of rdf:XMLLiteral, the members of a datatype class are the values of the datatype. This is explained in more detail in section 5 below. The class rdfs:Literal contains all literal values; however, typed literals whose strings do not conform to the lexical requirements of their datatype are required to have meanings not in this class. The semantic conditions on rdf-interpretations imply that ICEXT(I(rdf:XMLLiteral)) contains all XML values of well-typed XML literals.

The conditions on rdf:XMLLiteral and rdfs:range taken together make it possible to write a contradictory statement in RDFS, by asserting that a property value must be in the class rdf:XMLLiteral but applying this property with a value which is an ill-formed XML literal, and therefore required to not be in that class: for example

<ex:a> <ex:p> "<notLegalXML"^^rdf:XMLLiteral 
  .

  <ex:p> rdfs:range rdf:XMLLiteral .

4.2 Extensional Semantic Conditions (Informative)

The semantics given above is deliberately chosen to be the weakest 'reasonable' interpretation of the RDFS vocabulary. Semantic extensions MAY strengthen the range, domain, subclass and subproperty semantic conditions to the following 'extensional' versions:

which would guarantee that the subproperty and subclass properties were transitive and reflexive, but would also have further consequences.

These stronger conditions would be trivially satisfied when properties are identified with property extensions, classes with class extensions, and rdfs:SubClassOf understood to mean subset, and hence would be satisfied by an extensional semantics for RDFS. In some ways the extensional versions provide a simpler semantics, but they require more complex inference rules. The 'intensional' semantics described in the main text provides for most common uses of subclass and subproperty assertions, and allows for simpler implementations of a complete set of RDFS entailment rules, described in section 7.3.

4.3 A Note on rdfs:Literal

Although the semantic conditions on rdfs-interpretations include the intuitively sensible condition that ICEXT(I(rdfs:Literal)) must be the set LV, there is no way to impose this condition by any RDF assertion or inference rule. This limitation is due to the fact that RDF does not allow literals to occur in the subject position of a triple, so there are severe restrictions on what can be said about literals in RDF. Similarly, while properties may be asserted of the class rdfs:Literal, none of these can be validly transferred to literals themselves.

is consistent even though 'ex:a' is a URI reference rather than a literal. What it says is that I(ex:a) is a literal value, ie that the URI reference 'ex:a' denotes a literal value. It does not specify exactly which literal value it denotes.

The semantic conditions guarantee that any triple containing a plain literal object entails a similar triple with a blank node as object:

This means that the literal denotes an entity, which could therefore also be named, at least in principle, by a URI reference.

4.4 RDFS Entailment

Since every rdfs-interpretation is an rdf-interpretation, if S rdfs-entails E then it rdf-entails E; but rdfs-entailment is stronger than rdf-entailment. Even the empty graph has a large number of rdfs-entailments which are not rdf-entailments, for example all triples of the form

are true in all rdfs-interpretations of any vocabulary containing the URI reference xxx.

An rdfs-inconsistent graph rdfs-entails any graph, by the definition of entailment; such 'trivial entailments' by an inconsistent set are not usually considered useful inferences to draw in practice, however.

5. Interpreting Datatypes

5.1 Datatyped Interpretations

RDF provides for the use of externally defined datatypes identified by a particular URI reference. In the interests of generality, RDF imposes minimal conditions on a datatype. It also includes a single built-in datatype rdf:XMLLiteral.

This semantics for datatypes is minimal. It makes no provision for associating a datatype with a property so that it applies to all values of the property, and does not provide any way of explicitly asserting that a blank node denotes a particular datatype value. Semantic extensions and future versions of RDF may impose more elaborate datatyping conditions. Semantic extensions may also refer to other kinds of information about a datatype, such as orderings of the value space.

A datatype is an entity characterized by a set of character strings called lexical forms and a mapping from that set to a set of values. Exactly how these sets and mappings are defined is a matter external to RDF.

3. a mapping from the lexical space of d to the value space of d, called the lexical-to-value mapping of d.

In stating the semantics we assume that interpretations are relativized to a particular set of datatypes each of which is identified by a URI reference.

Formally, let D be a set of pairs consisting of a URI reference and a datatype such that no URI reference appears twice in the set, so that D can be regarded as a function from a set of URI references to a set of datatypes: call this a datatype map. (The particular URI references must be mentioned explicitly in order to ensure that interpretations conform to any naming conventions imposed by the external authority responsible for defining the datatypes.) Every datatype map is understood to contain <rdf:XMLLiteral, x> where x is the built-in XML Literal datatype whose lexical and value spaces and lexical-to-value mapping are defined in the RDF Concepts and Abstract Syntax document [RDF-CONCEPTS].

The datatype map which also contains the set of all pairs of the form <http://www.w3.org/2001/XMLSchema#sss, sss>, where sss is a built-in datatype named sss in XML Schema Part 2: Datatypes [XML-SCHEMA2] and listed in the following table, is referred to here as XSD.

The other built-in XML Schema datatypes are unsuitable for various reasons, and SHOULD NOT be used: xsd:duration does not have a well-defined value space (this may be corrected in later revisions of XML Schema datatypes, in which case the revised datatype would be suitable for use in RDF datatyping); xsd:QName and xsd:ENTITY require an enclosing XML document context; xsd:ID and xsd:IDREF are for cross references within an XML document; xsd:NOTATION is not intended for direct use; xsd:IDREFS, xsd:ENTITIES and xsd:NMTOKENS are sequence-valued datatypes which do not fit the RDF datatype model.

If D is a datatype map, a D-interpretation of a vocabulary V is any rdfs-interpretation I of V union {aaa: < aaa, x > in D for some x } which satisfies the following extra conditions for every pair < aaa, x > in D:

The first condition ensures that I interprets the URI reference according to the datatype map provided. Note that this does not prevent other URI references from also denoting the datatype.

The second condition ensures that the datatype URI reference, when used as a class name, refers to the value space of the datatype, and that all elements of a value space must be literal values.

The third condition ensures that typed literals in the vocabulary respect the datatype lexical-to-value mapping. For example, if I is an XSD-interpretation then I("15"^^xsd:decimal) must be the number fifteen. The condition also requires that an ill-typed literal, where the literal string is not in the lexical space of the datatype, not denote any literal value. Intuitively, such a name does not denote any value, but in order to avoid the semantic complexities which arise from empty names, the semantics requires such a typed literal to denote an 'arbitrary' non-literal value. Thus for example, if I is an XSD-interpretation, then all that can be concluded about I("arthur"^^xsd:decimal) is that it is not in LV, i.e. not in ICEXT(I(rdfs:Literal)). An ill-typed literal does not in itself constitute an inconsistency, but a graph which entails that an ill-typed literal has rdf:type rdfs:Literal, or that an ill-typed XML literal has rdf:type rdf:XMLLiteral, would be inconsistent.

Note that this third condition applies only to datatypes in the range of D. Typed literals whose type is not in the datatype map of the interpretation are treated as before, i.e. as denoting some unknown thing. The condition does not require that the URI reference in the typed literal be the same as the associated URI reference of the datatype; this allows semantic extensions which can express identity conditions on URI references to draw appropriate conclusions.

The fourth condition ensures that the class rdfs:Datatype contains the datatypes used in any satisfying D-interpretation. Notice that this is a necessary, but not a sufficient, condition; it allows the class I(rdfs:Datatype) to contain other datatypes.

The semantic conditions for rdf-interpretations impose the correct interpretation on literals typed by 'rdf:XMLLiteral'. However, a D-interpretation recognizes the datatype to exist as an entity, rather than simply being a semantic condition imposed on the RDF typed literal syntax. Semantic extensions which can express identity conditions on resources could therefore draw stronger conclusions from D-interpretations than from rdfs-interpretations.

If the datatypes in the datatype map D impose disjointness conditions on their value spaces, it is possible for an RDF graph to have no D-interpretation which satisfies it. For example, XML Schema defines the value spaces of xsd:string and xsd:decimal to be disjoint, so it is impossible to construct a XSD-interpretation satisfying the graph

This situation could be characterized by saying that the graph is XSD-inconsistent, or more generally as a datatype clash. Note that it is possible to construct a satisfying rdfs-interpretation for this graph, but it would violate the XSD conditions, since the class extensions of I(xsd:decimal) and I(xsd:string) would have a nonempty intersection.

Datatype clashes can arise in several other ways. For example, any assertion that something is in both of two disjoint dataype classes:

<ex:p> rdfs:range xsd:string .

  <ex:p> rdfs:range xsd:decimal .

  _:x <ex:p> _:y .

would constitute a datatype clash. A datatype clash may also arise from the use of a particular lexical form, for example: