RDF Semantics

This is a specification of a precise semantics for RDF and RDFS, and of corresponding entailment and inference rules which are sanctioned by the semantics.

Status of this Document

This document is in the Last Call review period, which ends on 21 February 2003. This document has been endorsed by the RDF Core Working Group.

This document is being released for review by W3C Members and other interested parties to encourage feedback and comments, especially with regard to how the changes made affect existing implementations and content.

The Working Group notes that this Last Call Working Draft completes the group's design of the formal Semantics for RDF, however it may still need some editorial polishing and clarification following Last Call.

This is a public W3C Last Call Working Draft for review by W3C Members and other interested parties. This section describes the status of this document at the time of its publication. It is a draft document and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use W3C Working Drafts as reference material or to cite as other than "work in progress". A list of current W3C Recommendations and other technical documents can be found 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], and as a basic foundation for more advanced assertional languages. Exactly what is considered to be the 'meaning' of an assertion in RDF/S 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 here is not intended to provide an analysis of 'meaning' in this broad sense. 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. For @@check link@@further discussion of notions of meaning in RDF , see [RDF-CONCEPTS]

We use 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 C helpful; uses of terms in a technical sense are linked to their 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. 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.

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.

In this document we give two versions of the same semantic theory: directly, and also (in an informative appendix) an 'axiomatic semantics' in the form of a translation from RDF and RDFS into another formal language, L_base [LBASE] which has a pre-defined model-theoretic semantics. The translation technique offers some advantages for machine processing and may be more readable, so is described here as a convenience. We believe that both of these descriptions, and also the closure rules described in section 4, are all in exact correspondence, but only the directly described model theory in sections 1- 3 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 [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 language of [RFC 2119]. Semantic extensions of RDF MUST conform to the semantic conditions for simple and RDF entailment described in sections 1 and 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 errors.

An example of a semantic extension of RDF is RDF Schema, the semantics of which are defined in later parts of this document. RDF Schema imposes no extra syntactic restrictions.

0.2 Graph syntax

The convention that relates a set of triples to a picture of an RDF graph can be stated as follows. Draw one oval for each blank node and uriref which occur in either the S or O position in any triple in the set, and one rectangle for each literal, and write each uriref or literal as the label of its shape. Shapes corresponding to blank nodes have no label. Then for each triple <S,P,O>, draw an arrowed line from the shape produced from S to the shape produced from O, and label it with P. Technically, this is a picture of a mathematical structure which can be described as a partially labeled directed pseudograph with unique node labels, but we will simply refer to a set of triples as an RDF graph. Graphs with isomorphic pictures in this sense are considered to be identical; this means that we do not distinguish sets of triples which differ only in the identity of their blank nodes. This slight abuse of terminology allows us to simplify the presentation by ignoring questions of re-naming of bound variables.

In this document we will use the N-triples syntax described in [RDF-TESTS] to describe RDF graphs. This notation uses a nodeID convention to indicate blank nodes in the triples of a graph. Note that 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, two N-triples documents which differ only by re-naming their node identifiers will be understood to describe identical RDF graphs.

The N-triples syntax requires that urirefs be given in full, enclosed in angle brackets. In the interests of brevity, we use the imaginary URI scheme 'ex:' 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/'. We will also make extensive use of the Qname prefixes rdf:, rdfs: and xsd: defined as follows:

Since Qname syntax is not legal N-triples syntax, and in the interests of brevity and readability, we will use the convention whereby a Qname is used without surrounding angle brackets to indicate the corresponding uriref enclosed in angle brackets, eg the triple

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

In stating rules and giving general semantic conditions we will use single characters or character sequences without a colon to indicate an arbitrary name or blank node in a triple.

0.3 Graph Definitions

Several definitions will be important in what follows.They are stated together here for reference.

Consider a set S of graphs which share no blank nodes. The graph consisting of all the triples in all the graphs in S is another graph, which we will call the merge of S. Each of the original graphs is a subgraph of the merged graph. Notice that when forming a merged graph, two occurrences of a given uriref or literal as nodes in two different graphs become a single node in the union graph (since by definition they are the same uriref or literal), but blank nodes are not 'merged' in this way. If the members of the set S share some blank nodes, then we will define the merge of S to be the merge of a set obtained by replacing blank nodes in some members of S by distinct blank nodes to obtain another set S' of graphs which are isomorphic to those in S.

Notice that 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, since 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' merged. To merge Ntriples 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-SYNTAX].

This allows blank nodes in the second graph to be replaced by names in the instance (which might cause some nodes to be identified that were previously distinct) but it also allows them to be replaced by other blank nodes. For example, the graph:

In particular, any graph obtained by replacing all blank nodes by new blank nodes not in the original graph is an instance of the original and also, by inverting the mapping, has it as an instance. By our convention, such isomorphic graphs are considered to be identical.

1. Interpretations

1.1 Technical Notes

We do 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, we will allow them to contain themselves. This might seem to violate one of the axioms of standard (Zermelo-Fraenkel) set theory, the axiom of foundation, 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 system of 'layers', 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 such 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, and still retain much of the expressive power of RDFS for many practical purposes.

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, 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.

1.2 Urirefs, resources and literals.

RDF uses two kinds of referring expression, urirefs and literals. We make very simple and basic assumptions about these. Urirefs are treated as logical constants, i.e. as names which denote things (the things are called 'resources', following [RFC 2396], but no assumptions are made here about the nature of resources; we treat 'resource' here as synonymous with 'entity'.) The meaning of a literal is principally determined by its character string: it either refers to the value mapped from the string by the associated datatype, or if no datatype is provided then it refers to the literal itself, which is either a unicode character string or a pair of a string with a language tag.

We do not take any position here on the way that urirefs 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 uriref 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 urirefs 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 uriref and a document or network resource which can be obtained by using that uriref 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 urirefs and the uses of those urirefs in other protocols.

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 RDF interpretation. 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 uriref, 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 used to indicate a datatype, we assume 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.(We will show how to determine the truth-values of non-ground graphs in the following section.) Notice that if we left any of this information out, 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 urirefs, 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, so that we will speak of 'rdf-interpretations' , 'rdfs-interpretations', etc.. An interpretation with no particular extra conditions on a vocabulary will be called a simple interpretation, or simply an interpretation. A simple interpretation can be viewed as having an empty vocabulary.

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. 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; for this reason we postpone a full discussion of the meaning of typed literals until later sections, where we introduce a special notion of datatype interpretation. For now, we will assume that each interpretation defines a mapping IL from typed literals to their interpretations, and will impose stronger conditions on IL as the notion of 'interpretation' is extended in later sections. Simple literals, without embedded datatypes, are always interpreted as referring to themselves: either a character string or a pair consisting of two character strings, the second of which is a language tag.

The set of all possible values of all literals is assumed to be a set called LV. Since the set of datatypes is not restricted by RDF syntax, it is impossible to give a sharp definition of LV, but it is required to contain all literal strings, all pairs consisting of a literal string and a language tag, and all well-formed canonical XML documents.

1. A non-empty set IR of resources, called the domain or universe of I, which is a superset of LV.

3. A mapping IEXT from IP into the powerset of IR x IR i.e. the set of sets of pairs <x,y> with x and y in IR .

IEXT(x) is a set of pairs which identify the arguments for which the property is true, i.e. a binary relational extension, called the extension of x. 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 IR is a superset of LV amounts to saying that literal values are thought of as real entities that 'exist'. This assumption may seem controversial, since it amounts to saying that literal values are resources. We note however that this does not imply that literals should be identified with urirefs.

There is a technical reason why the range of IL is IR rather than being restricted to LV. When we consider interpretations which take account of datatype information, it is syntactically possible for a typed literal to be internally inconsistent, and we will require such badly typed literals to denote a non-literal value.

In the next sections we give the exact rules for 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. RDF has two kinds of denotation: names denote things in the universe, and sets of triples denote truth-values.These rules apply to all subsequent semantic extensions.

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 labels 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 constitutents of E, hence allowing the denotation of any piece of RDF to be determined by a kind of syntactic recursion.

If the vocabulary of an RDF graph contains urirefs 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.

As an illustrative example, the following is a small interpretation for the artificial vocabulary {

ex:a, ex:b,
    ex:c

}. We use integers to indicate the non-literal 'things' in the universe. This is not meant to imply that RDF interpretations should be interpreted as being about arithmetic, but more to emphasize that the exact nature of the things in the universe is irrelevant.(In this and subsequent examples we use the greater-than and less-than symbols in several ways: following mathematical usage to indicate abstract pairs and n-tuples; following Ntriples syntax to enclose urirefs, and also as arrowheads when indicating mappings. We apologize for any confusion.)

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 literal is a consequence of the rather idiosyncratic interpretation chosen here for typed literals; this kind of oddity will be ruled out when we consider datatyped intepretations.

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 and IEXT is not defined on 2, so the condition fails and I(

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

) = false.

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

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

1.5. Blank nodes as existential assertions

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' uriref; for example, it does not assume that there is any uriref which refers to the thing. The discussion of skolemization in the proof appendix is relevant to this point.)

We now show how 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 we can extend the above rules to include the two new cases that are introduced when blank nodes occur in the graph:

Notice that we have not changed the definition of an interpretation; it still consists of the same values IR, IP, IEXT, IS and IL. We have simply extended 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).

This effectively treats all blank nodes as having the same meaning as existentially quantified variables in the RDF graph in which they occur. However, there is no need to specify the scope of the quantifier within a graph, and no need to use any explicit quantifier syntax.( If we were to apply the semantics directly to N-triples syntax, we would need to indicate the quantifier scope, since in this lexicalization syntax the same node identifier may occur several times corresponding to a single blank node in the graph. The above rule 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.)

For example, with this convention, 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.

2. Simple entailment between RDF graphs

Following conventional terminology, we say that I satisfies E if I(E)=true, and that a set S of expressions (simply) entails E if every interpretation which satisfies every member of S also satisfies E. In later sections these notions will be adapted to classes of interpretations with particular vocabularies, 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; we 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 licence 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.

In this section we give 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 the appendix, which also describes some other properties of entailment which may be of interest.

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.

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. In what follows, therefore, we will often not bother to distinguish between a set of graphs and a single graph. This can be used to simplify the terminology somewhat: for example, we can paraphrase the definition of S entails E, above, by saying that S entails E when every interpretation which satisfies S also satisfies E.

The interpolation lemma completely characterizes simple RDF entailment in syntactic terms. To tell whether a set of RDF graphs entails another, find a subgraph of their merge and replace names by blank nodes to get the second. Of course, there is no need to actually construct the merge. If working backwards from the consequent E, the most 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.

Notice however that 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 would not be entailed by the assertion, as explained in the next section.

Finally, the following is a trival but important consequence of the definition of entailment:

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

2.1 Criteria for non-entailment

In contrast to names, which have a global identity which carries across all graphs, blank nodes should not be identified with other nodes or replaced with urirefs, in order to ensure that the resulting graph is entailed by what one starts with. To state this condition precisely, we need to first exclude a counterexample. It is possible for a graph to contain two triples one of which is an instance of the other, for example:

Such an internally redundant graph is equivalent to one of its own instances, since replacing the blank node by <ex:c> would result in a single-triple graph which is a subgraph of the original. To rule out such cases of internal redundancy, we will say that an RDF graph is lean if none of its triples is an instance of any other. Then the above principle is made precise in the following two lemmas concerning criteria for non-entailment:

Anonymity lemma 1. Suppose E is a lean graph and E' is a proper instance of E. Then E does not entail E'.

Anonymity lemma 2. Suppose that E is a lean graph and that E' is like E except that two distinct blank nodes in E have been identified in E'. Then E does not entail E'.

This means that there is no valid RDF inference process which can produce an RDF graph in which a single blank node occurs in triples originating from several different graphs. (Of course, such a graph can be constructed, but it will not be entailed by the original documents. An assertion of such a graph would reflect the addition of new information about identity.)

We emphasise again that these results apply only to simple entailment, not to the vocabulary entailment relationships defined in rest of the document.

3. Interpreting the RDF(S) vocabulary

So far we have considered only the model theory of what might be called the logical form of the RDF graph itself, without imposing any special interpretations on any vocabulary. In the rest of the document we will extend the model theory to describe the semantic conditions reflecting the intended meanings of the rdf: and rdfs: namespaces.

Although we will do this in stages, the same general technique is used throughout. First we describe a special vocabulary, i.e. a set of urirefs which will be given a special meaning; then we give the extra conditions on an interpretation which capture those meanings; then we restrict the notions of satisfiability and entailment to apply to these interpretations only. This essentially imposes an a priori restriction on the world being described that it satisfy the extra conditions. The new semantic conditions are automatically assumed to be true; an interpretation which would violate them is simply not allowed to count as a possible world.

Since there are now several distinct notions of interpretation, entailment and satisfiability, we use the Qname namespace prefix conventions to identify the various distinctions, eg an rdf-interpretation is an interpretation satisfying the rdf semantic conditions, rdf-entailment means entailment relative to such interpretations, and so on. We call this general idea vocabulary entailment, i.e. entailment relative to a set of interpretations which satisfy extra semantic conditions on a vocabulary. Vocabulary entailment is more powerful than simple entailment, in the sense that a given set of premises entails more consequences. In general, as the vocabulary is increased and extra semantic conditions imposed, the class of satisfying interpretations is restricted, and hence the corresponding notion of entailment becomes more powerful. For example, if S simply entails E then it also rdf-entails E, since every rdf-interpretation is also a simple interpretation; but S may rdf-entail E even though it does not simply entail it. Intuitively, a conclusion may depend on some of the extra assumptions incorporated in the semantic conditions imposed on the vocabulary.

Another way of expressing this is that any restriction on interpretations decreases the number of possible ways that an interpretation might be a counterexample to E's following from S.

Simple entailment is the vocabulary entailment of the empty vocabulary. It is therefore the weakest form of RDF entailment, which holds for any vocabulary; it is the entailment which depends only on the basic logical form of RDF graphs, without making any further assumptions about the meaning of any urirefs.

We will consider syntactic criteria for recognizing vocabulary entailment in the next section.

3.1. RDF Interpretations

RDF imposes some extra semantic conditions on the following (rather small) vocabulary, which we will call rdfRV:

The first condition forces every rdf interpretation to contain a thing denoted by rdf:type; which will be used as a property to associate 'type' values with resources. The second 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. The third condition says that the empty list object is classified as being a list: this is the only formal condition the RDF semantics places on the collection vocabulary, described later. The fourth condition defines the built-in RDF datatype; here 'canonical form' is understood in the sense of [XML-C14N]. For an exact statement of the precise conditions for attaching language tags to XML documents see [RDF-CONCEPTS].

For example, the following rdf-interpretation extends the simple interpretation in figure 1:

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

This is not the smallest rdf-interpretation which extends the earlier example, since we could have made I(rdf:Property) be 2 and 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.

It is important to note that every rdf-interpretation is also a simple interpretation.The 'extra' structure does not prevent it acting in the simpler role.

The RDF vocabulary contains several other items. Some of these are omitted because they have no formal semantic meaning, or have a meaning which can only be described using the RDFS vocabulary.

3.2. Reification, containers, collections and rdf:value

RDF provides vocabularies which are 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. Although these vocabularies have reasonably clear informally intended conventional meanings, we do not impose any further formal semantic conditions on them, so the notions of rdf-entailment and rdf-interpretation apply to them without further change. They are discussed here in order to explain both the intuitive meanings intended, and also to note the intuitive consequences which are not supported by the formal model theory. Constraints are imposed on the meanings of these vocabularies in semantic extensions. The RDFS assigns range and domain conditions for some of the properties used in this vocabulary.We will refer to the complete set of all rdf urirefs, consisting of the RDF vocabulary and all of the reification, container and collection vocabularies and the uriref rdf:value , as the RDF vocabulary, rdfV.

The lack of a formal semantics for these vocabularies does not reflect any technical semantic problems, but rather is a design decision to make it easier to implement processes to check formal RDF entailment. Since no extra formal semantic conditions are imposed on them, they are not considered to be restricted vocabularies in RDF. In RDFS, however, the entire RDF vocabulary is considered to be a restricted vocabulary.

3.2.1 Reification

The RDF reification vocabulary consists of a class name and three property names.

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 uriref.) 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 uriref itself but its interpretation, and so this condition involves a two-stage interpretation process: we have 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.

We emphasize that 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 RDF. RDF syntax provides no means to 'connect' an RDF triple to its reification. 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. The chief facts that are worthy of note about RDF reification, in fact, are examples of non-entailments.

A reification of a triple does not entail the triple, and is not entailed by it. (The reason for first is clear, since the reification only asserts that the triple token exists, 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.)

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> .

3.2.2 RDF containers

RDF provides vocabularies for describing three classes of containers. Containers have a type, and their members can be listed 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.

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 semantic conditions on the rest of the RDF vocabulary. 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. However, these informal interpretations are not reflected in any formal RDF entailments.

RDF does not support any entailments which could arise from re-ordering the elements of an rdf:Bag. 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 that if a property applies to a container then the property applies to any of the members 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.

3.2.3 RDF collections

RDF provides a vocabulary for describing collections, ie.'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 nil being 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 three triples of the form

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. Clearly, any such graph amounts to an assertion that the collection, and all its sub-collections, exist, 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:type rdf:List .

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

     _:c1 rdf:rest _:c2 .

     _:c2 rdf:type rdf:List .

    <ex:bbb> .

     _:c2 rdf:rest rdf:nil .

_:c3 rdf:type rdf:List .

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

     _:c3 rdf:rest _:c4 .

     _:c4 rdf:type rdf:List .

    <ex:aaa> .

     _:c4 rdf:rest rdf:nil .

Also, RDF imposes no 'wellformedness' 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:type rdf:List .

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

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

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

     _:666 rdf:rest _:777 .

     _:666 rdf:rest rdf:nil .

As this example shows, 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, and MAY exclude interpretations of the collection vocabulary which violate the convention that the subject of a 'linked' collection of three-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.

3.2.4 rdf:value

The intended use for rdf:value is explained intuitively in [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 impossible to give a precise model-theoretic statement which covers all the intended meanings or use cases. Users are cautioned, therefore, that the use of rdf:value is somewhat risky, and that it should be treated as a 'blank' piece of RDF syntax whose meaning in any particular case should be defined by the user, and 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.

3.3 RDFS interpretations

RDFSchema 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. We will assume that there is a mapping ICEXT (for the Class Extension in I) from classes to their extensions; the first semantic condition in the table below amounts to the following definition of this mapping in terms of the relational extension of rdf:type:

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 given the above definition, 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, which we will call axiomatic triples. The first condition can be understood as a definition of ICEXT and hence of IC, the set of classes. Since I is an rdf-interpretation, this means that IP = ICEXT(I(rdf:Property))

x is in ICEXT(y) iff <x,y> is in IEXT(I(rdf:type))

IC = ICEXT(I(rdfs:Class))

IR = ICEXT(I(rdfs:Resource))

ICEXT(I(rdfs:Literal)) is a subset of LV

If <x,y> is in IEXT(I(rdfs:range)) and <u,v> is in IEXT(x) then v is in ICEXT(y)

If <x,y> is in IEXT(I(rdfs:domain)) and <u,v> is in IEXT(x) then u is in ICEXT(y)

<x,y> is in IEXT(I(rdfs:subClassOf)) if and only if x and y are in IC and ICEXT(x) is a subset of ICEXT(y)

<x,y> is in IEXT(I(rdfs:subPropertyOf)) if and only if x and y are in IP and IEXT(x) is a subset of IEXT(y)

If x is in ICEXT(I(rdfs:ContainerMembershipProperty)) then <x,I(rdfs:member)> is in IEXT(I(rdfs:subPropertyOf))

ICEXT(I(rdf:XMLLiteral)) is the set of all canonical XML documents.

IC contains:

I(rdfs:Resource), I(rdf:Property), I(rdfs:Class), I(rdfs:Literal), I(rdfs:Datatype), I(rdf:Seq), I(rdf:Bag), I(rdf:Alt), I(rdfs:Container), I(rdf:List), I(rdfs:ContainerMembershipProperty), I(rdf:Statement)

IP contains:

I(rdf:type), I(rdfs:domain), I(rdfs:range), I(rdfs:subPropertyOf), I(rdfs:subClassOf), I(rdfs:comment), I(rdfs:label), I(rdf:subject), I(rdf:predicate), I(rdf:object), I(rdfs:member), I(rdf:_1), I(rdf:_2), ... , I(rdf:first), I(rdf:rest), I(rdfs:seeAlso), I(rdfs:isDefinedBy), I(rdfs:comment), I(rdfs:label), I(rdf:value)

rdfs:subPropertyOf rdfs:domain rdf:Property .
rdfs:subClassOf rdfs:domain rdfs:Class .
rdfs:domain rdfs:domain rdf:Property .
rdfs:range rdfs:domain rdf:Property .
rdf:subject rdfs:domain rdf:Statement .
rdf:predicate rdfs:domain rdf:Statement .
rdf:object rdfs:domain rdf:Statement .
rdf:first rdfs:domain rdf:List .
rdf:rest rdfs:domain rdf:List . rdfs:subPropertyOf rdfs:range rdf:Property .
rdfs:subClassOf rdfs:range rdfs:Class .
rdfs:domain rdfs:range rdfs:Class .
rdfs:range rdfs:range rdfs:Class .
rdf:type rdfs:range rdfs:Class .
rdfs:comment rdfs:range rdfs:Literal .
rdfs:label rdfs:range rdfs:Literal .
rdf:rest rdfs:range rdf:List .

rdf:Alt rdfs:subClassOf rdfs:Container .
rdf:Bag rdfs:subClassOf rdfs:Container .
rdf:Seq rdfs:subClassOf rdfs:Container .
rdfs:ContainerMembershipProperty rdfs:subClassOf rdfs:Property .

rdfs:isDefinedBy rdfs:subPropertyOf rdfs:seeAlso . rdf:XMLLiteral rdf:type rdfs:Datatype .
rdfs:Datatype rdfs:subClassOf rdfs:Class .
rdf:_1 rdf:type rdfs:ContainerMembershipProperty .
rdf:_2 rdf:type rdfs:ContainerMembershipProperty .
...

The truth of the axiomatic triples could be stated as conditions on IEXT and ICEXT, but it is convenient to use the truth-of-triples formulation. Similarly, the conditions on IC and IP in the first table could be stated as axiomatic triples with property rdf:type and objects rdfs:Class and rdfs:Property respectively.

Since ICEXT(I(rdfs:Resource)) is the universe, everything has rdfs:Resource as an rdf:type value, and every class is a subclass of rdfs:Resource.Such assertions would be redundant, therefore.

Similarly, some domain and range assertions are omitted from the above table; in those cases, the domain or range of the property may be taken to be rdfs:Resource, i.e. the universe; such range and domain assertions are essentially vacuous.

The semantics given here for rdfs:range and rdfs:domain do not entail that superclasses of domains or ranges of a property must also be domains and ranges of that property. Semantic extensions MAY strengthen the domain and range semantic conditions to the following:

This stronger condition will not effect any class-membership entailments on the elements of the domains and ranges of the property. The semantics given here was chosen because it is sufficient for all normal uses of these terms and allows some subtleties in class reasoning.

Note that datatypes are considered to be classes. 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 3.4 below.

3.3.1 A note on rdfs:Literal

Although the semantic conditions on rdfs-interpretations include the intuitively sensible condition that ICEXT(I(rdfs:Literal)) must be a subset of LV, there is no way to impose this condition by any RDF assertion or syntactic closure 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 uriref rather than a literal. What it says is that I(ex:a) is a literal value, ie that the uriref 'ex:a' denotes a literal value. It does not specify exactly which literal value it denotes.

Note that the interpolation lemma guarantees that any triple containing a simple literal object entails a similar triple with a bnode as object:

This means that literal denotes 'something', which could therefore also be named, at least in principle, by a uriref.

3.4 Datatyped interpretations

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. (The built-in datatype rdf:XMLLiteral, exceptionally, allows pairs in its lexical space.) Exactly how these sets are defined is a matter external to RDF. Since the set of possible datatypes is open-ended, we will assume that datatype interpretations are defined relative to a particular set of datatypes, and refer to D-interpretations where D is a set of datatypes, which we will call recognized datatypes. Urirefs which denote recognized datatypes are required to have the same denotation in all D-interpretations, so recognizing a datatype amounts to fixing the meaning of a uriref.

The set of recognized datatypes always includes rdf:XMLLiteral and may include the XML Schema, part 2 built-in datatypes defined in [XML-SCHEMA2], which we will refer to as XSD and use the Qname prefix xsd:. In any XSD-interpretation, any uriref of the form http://www.w3.org/2001/XMLSchema#sss will be understood to denote the datatype named sss in [XML-SCHEMA2]

We will describe the semantic conditions in terms of a mapping L2V from datatypes to their lexical-to-value mappings; the valid lexical forms of a datatype d constitute the domain of L2V(d), and the range of L2V(d) is the set of elements of the value space of d. Recall that the set LV is required to include all members of all datatype value spaces, so that the range of L2V(d) must be a subset of LV.

A D-interpretation of a graph G is an rdfs-interpretation I of V, where V contains the vocabulary of G, which satisfies the following extra conditions on all datatypes other than the built-in datatype:

The first condition says that membership in the class rdfs:Datatype means the same as being a recognized datatype. Thus, the inclusion of a triple of the form

in an RDF graph can be understood as a claim that ex:somedatatype identifies a recognized datatype. The semantic conditions on rdfs-interpretations have the consequence that the built-in datatype rdf:XMLLiteral is a recognized datatype in all RDFS interpretations, and therefore in all D-interpretations for any D.

The second condition says that the meaning of any typed literal which uses a recognized datatype is the value of the literal character string under that datatype, and that the language tag, if present, is ignored. For example, if I is an XSD-interpretation then I("15"^^xsd:decimal) must be the number fifteen. Notice that this applies only to datatypes in D; typed literals whose type is not a recognized datatype are treated as before, i.e. as denoting some unknown thing. This means that their meanings can be further restricted by adding a suitable extra datatype to the set of recognized datatypes.

The third condition requires that an 'ill-formed' typed literal, i.e. one 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, we require such a typed literal to denote an 'arbitrary' value. Thus for example, if D contains the XML schema datatypes, then all that can be concluded about I("arthur"^^xsd:decimal) is that it is not in ICEXT(I(rdfs:Literal)). Any graph in which a typed literal denotes a non-literal value is a datatype violation.

The semantic conditions for the built-in datatype rdf:XMLLiteral have been described in previous sections; but in a datatyped interpretation, in addition, a graph which contains a literal with a non-well-formed XML string or an illegal language tag, and which is typed with rdf:XMLLiteral is always considered a datatype violation. These semantic conditions are exactly similar to the above if one defines the lexical space of rdf:XMLLiteral as the set of all XML documents and all pairs of XML documents and language tags, and L2V(I(rdf:XMLLiteral)) as XML canonicalization. The possible inclusion of language tags makes this a special case, however: in all other cases, RDF ignores any language tags which occur in typed literals.

The final condition indicates that RDF treats datatypes as classes, i.e. they are assumed to have a class extension, which is required to be the set of items in the value space of the datatype. All the members of all such datatype classes are required to be literal values. ICEXT(I(rdf:XMLLiteral)) is the set of all well-formed canonical XML documents in any D-interpretation I.

RDF does not itself impose any identity conditions on elements in value spaces, nor assume any subclass relationships between datatype value classes. Information about such relationships should be obtained from the specifications of the datatypes themselves.

Datatype monotonicity lemma. If D is a subset of D' and S D-entails E, then S D'-entails E.

It is possible for an RDF graph to have no D-interpretation which satisfies it. For example, XML Schema requires that 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 any such interpretation, when extended to an XSD-interpretation, would violate the XSD conditions, since the class extensions of I(xsd:decimal) and I(xsd:string) would have a nonempty intersection.

Datatype clashes are the only inconsistencies recognized by this model theory. The definition of entailment means that a D-inconsistent graph D-entails any RDF graph; however, it will usually not be appropriate to consider such 'trivial' entailments as useful consequences, since they are not valid rdf- or rdfs- entailments.

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, for example, and does not provide any way of explicitly asserting that a blank node denotes a particular value under a datatype mapping. We expect that semantic extensions and future versions of RDF will 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.

4. Vocabulary entailment and closure rules

We will say that S rdf-entails E (S rdfs-entails E, S D-entails E) when every rdf-interpretation (every rdfs-interpretation, every interpretation datatyped with respect to D) 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- , rdfs- or D-interpretations instead of all simple interpretations. These are examples of vocabulary entailment, i.e. entailment relative to a set of interpretations which satisfy extra semantic conditions on a vocabulary.

It is easy to see that the lemmas in section 2 do not hold for vocabulary entailment. For example, the triple

is true in every rdf-interpretation, and hence rdf-entailed by the empty graph, which immediately contradicts the interpolation lemma for rdf-entailment.

Rather than develop a separate theory of the syntactic conditions for recognizing entailment for each special vocabulary, we will use a general technique for reducing these broader notions of entailment to simple entailment, by defining the closure of an RDF graph relative to a set of semantic conditions. The basic idea is to rewrite the semantic conditions as a set of syntactic inference rules, and define the closure to be the result of applying those rules to exhaustion. The resulting graphs will contain RDF triples which explicitly state all the special meanings embodied in the extra semantic conditions, in effect axiomatizing them in RDF itself. A graph rdf-entails (rdfs-entails) another just when its rdf-closure (rdfs-closure) simply entails it. It is not possible to provide such a tight result for D-entailment closures since the relevant semantic conditions require identities which cannot be stated in RDF.

The notion of closure used here is purely a formal device to relate two notions of entailment. We do not mean to suggest that closure rules should be used as a computational technique, or that actually generating the full closure would be the best process to use in order to determine vocabulary entailment.

Closure rules correspond to implication axioms in the L_base translation given in the appendix.

4.1 Rdf-entailment and rdf closures (Informative)

2. Apply the following rules recursively to generate all legal RDF triples (i.e. until none of the rules apply or the graph is unchanged.) Here xxx and yyy stand for any uriref, bNode or literal, aaa for any uriref.

The following lemma is the basic result on rdf-entailment, and illustrates a general pattern of how to characterize vocabulary entailment syntactically.

The result is rather obvious, but a complete proof is given in the appendix to illustrate the proof method.

The rather awkward statement of rules rdf2a and b is necessary to ensure that the entailment lemma holds when the conclusion E contains a non-canonicalized XML literal. In practice, it would be sufficient to apply a canonicalization operation to both S and E, taking appropriate account of language tags, and then ignore these rules.

4.2 RDFS-entailment and RDFS closures (informative)

RDFS closures require more complex rules to reflect the consequences of the more elaborate semantic constraints on the rdfs vocabulary.

1. Add the RDFS axiomatic triples from the table in section 3.3, and all the following triples. There are many other triples which are true in every rdfs-interpretation, but they will be generated from these by the closure rules.

rdfs:Resource rdf:type rdfs:Class .

rdfs:Literal rdf:type rdfs:Class .

     rdf:Statement rdf:type rdfs:Class .

     rdf:nil rdf:type rdf:List .

     rdf:XMLLiteral rdf:type rdfs:Datatype .

rdf:subject rdf:type rdf:Property .

     rdf:predicate rdf:type rdf:Property .

     rdf:object rdf:type rdf:Property .

     rdf:first rdf:type rdf:Property .

rdf:rest rdf:type rdf:Property .

2. Add all triples of the following forms. This is an infinite set because the RDF container vocabulary is infinite. However, since none of these triples entail any of the others, it is only necessary, in practice, to add the triples which use those container properties which actually occur in any particular graph or set of graphs in order to check the rdfs-entailment relation between those graphs.

rdf:_1 rdf:type rdfs:ContainerMembershipProperty .

     rdf:_2 rdf:type rdfs:ContainerMembershipProperty .

...

3. Apply the following rules recursively to generate all legal RDF triples (i.e. until none of the rules apply or the graph is unchanged.) Here, xxx, yyy and zzz stand for any uriref, bNode or literal, aaa for any uriref, and uuu for any uriref or bNode (but not a literal).

The outputs of these rules will often trigger others. For example, these rules will generate the complete transitive closures of all subclass and subproperty heirarchies, together with all of the resulting type information about everything which can be inferred to be a member of any of the classes, and will propagate all assertions in the graph up the subproperty heirarchy, re-asserting them for all super-properties. rdfs1 will generate type assertions for all the property names used in the graph, and rdfs3 together with the first triple in the above list will add all the types for all the class names used. Any subproperty or subclass assertion will generate appropriate type assertions for its subject and object via rdfs 2 and 3 and the domain and range assertions in the RDFS axiomatic triple set. The rules will generate all assertions of the form

However, it is easy to see that (with the restriction noted of the infinite sets to those membership properties which occur in the graph) the rules will indeed terminate on any finite RDF graph, since there are only finitely many triples that can be formed from a given finite vocabulary.

A similar result applies here as in the case of rdf-entailment, though it takes considerably more work to prove:

	If E contains:	then add:
rdf1	xxx aaa yyy `.`	aaa `rdf:type rdf:Property .`
rdfs2	xxx aaa yyy `.` aaa `rdfs:domain` zzz `.`	xxx `rdf:type` zzz `.`
rdfs3	xxx aaa uuu `.` aaa `rdfs:range` zzz `.`	uuu `rdf:type` zzz `.`
rdfs4a	xxx aaa yyy `.`	xxx `rdf:type rdfs:Resource .`
rdfs4b	xxx aaa uuu `.`	uuu `rdf:type rdfs:Resource .`
rdfs5a	aaa `rdfs:subPropertyOf` bbb `.` bbb `rdfs:subPropertyOf` ccc `.`	aaa `rdfs:subPropertyOf` ccc `.`
rdfs5b	xxx `rdf:type rdf:Property .`	xxx `rdfs:subPropertyOf` xxx `.`
rdfs6	xxx aaa yyy `.` aaa `rdfs:subPropertyOf` bbb `.`	xxx bbb yyy `.`
rdfs7a	xxx `rdf:type rdfs:Class .`	xxx `rdfs:subClassOf rdfs:Resource .`
rdfs7b	xxx `rdf:type rdfs:Class .`	xxx `rdfs:subClassOf` xxx `.`
rdfs8	xxx `rdfs:subClassOf` yyy `.` yyy `rdfs:subClassOf` zzz `.`	xxx `rdfs:subClassOf` zzz `.`
rdfs9	xxx `rdfs:subClassOf` yyy `.` aaa `rdf:type` xxx `.`	aaa `rdf:type` yyy `.`
rdfs10	xxx `rdf:type rdfs:ContainerMembershipProperty .`	xxx `rdfs:subPropertyOf rdfs:member .`

We note in passing that the stronger 'iff' semantic conditions on rdfs:domain and rdfs:range mentioned in section 3.3 would be captured by removing rules rdfs4a and rdfs4b, and adding the additional rules

and that these would provide a redundant inference path to the conclusions of rdfs 2 and 3.

4.3 Datatype entailments (Informative)

In order to capture datatype entailment in terms of assertions and closure rules, the rules need to refer to information supplied by the datatypes themselves; and to state the rules it is necessary to assume syntactic conditions which can only be checked by consulting the datatype sources. Since such questions are beyond the scope of RDF, it is impossible to prove an entailment lemma for datatype closures analogous to those for RDF and RDFS.

Since language tags play no role in the meaning of a typed literal, they can in practice be ignored, and any literal of the form "sss"@ttt^^ddd, where ddd is not rdf:XMLLiteral, treated as identical to the same literal without the language tag, "sss"@ddd. We can capture this convention by special rules which allow language tags to be inserted or removed:

Here, ttt and uuu are any legal language tags and ddd is anything other than rdf:XMLLiteral. Clearly, these rules together can replace any language tag by any other.We also have a rule applying to all recognized datatypes, which requires their elements to be recognized as elements of rdfs:Literal:

For each kind of information which is available about a datatype, we can give inference rules for information of that kind, which can be thought of as extending the table of RDFS closure rules. These should be understood as applying to datatypes other than the built-in datatype, the rules for which are part of the RDF closure rules.

The basic information specifies, for each literal string, whether or not it is a legal lexical form for the dataype. This corresponds to the following rule, for each string sss that is a legal lexical form for the datatype ddd. Here _:xxx is a new blank node, i.e. one that does not appear elsewhere in the graph:

Suppose it is known that two lexical forms sss and ttt map to the same value under the datatype ddd; then the following rule applies:

Suppose it is known that the lexical form sss of the datatype ddd and the lexical form ttt of the datatype eee map to the same value. Then the following rule applies:

Suppose that it is known that the value space of the datatype ddd is a subset of that of the datatype eee. Then the following rule applies:

The rules rdfD2 and 3 are essentially substitutions by virtue of equations between lexical forms. Such equations may be capable of generating infinitely many conclusions, e.g. it is valid to add any number of leading zeros to any numeral and still be a correct lexical form for xsd:integer. To avoid such correct but unhelpful inferences, it is sufficient to restrict rdfD2 to cases which replace a lexical form with the canonical form for the datatype in question, when such a canonical form is defined. In order for the rules to be complete, however, such canonicalization rules would need to be applied to the conclusions as well as the antecedents of any proposed entailments, and the corresponding rules of type rdfD3 would need to reflect knowledge of identities between canonical forms of the distinct datatype.

In particular cases other information might be available, which could be expressed using a particular RDFS vocabulary. Semantic extensions MAY define further such datatype-specific meanings.

Note that the rule rdfD 0c allows us to conclude that any well-formed typed literal of a recognized datatype must denote something in the class rdfs:Literal.

aaa ppp _:xxx .

     _:xxx rdf:type ddd .

(by rule rdfD 1)



     ddd rdfs:subClassOf rdfs:Literal .

(by rule rdfD 0c)



     _:xxx rdf:type rdfs:Literal .

(by rule rdfs9)

The rule rdfD1 is sufficient to expose a datatype clash, by a chain of reasoning of the following form:

aaa ppp _:xxx .

     _:xxx rdf:type eee .

(by rule rdfD 1)
_:xxx rdf:type ddd . (by rule rdfs 3)

It may be useful to incorporate the assumption that any uriref appearing in a typed literal is presumed to be a datatype, which would be captured by the following rule. Note however that this is not strictly valid, so represents a (small) semantic extension.

Datatype clashes and violations may be considered to be error conditions. However, we note that such graphs are not strictly ill-formed, and can be used to support valid RDFS entailments which might be meaningful in certain contexts.

Appendix A: Translation into L_base(Informative)

As noted in the introduction, an alternative way to specify RDF interpretations 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]. Here we use a version of first-order logic which was designed to provide a semantic reference for such translations from web-based languages, called L_base [LBASE], which uses a particularly efficient syntax.

To translate an RDF graph into the semantic reference language L_base, apply the following rules to each expression noted. Each rule gives a translation TR[E] for the expression E, to be applied recursively. To achieve a translation which reflects a vocabulary entailment, add the axioms specified. Each vocabulary includes all axioms and rules for preceding vocabularies, so that the RDFS translation of a graph should include the RDF translation as well as the RDFS axioms, and so on.

This translation uses the L_base logical expressions Lbase:String and Lbase:XMLthing, which are true respectively of unicode character strings, and anything that is denoted by a piece of well-formed XML syntax; and it introduces some terminology in order to give a logical account of the meanings implicit in the various literal constructions. The axioms given are sufficient to define the intended meanings of the vocabulary used.

L_base translation rules
RDF expression E	L_base expression TR[E]
a plain literal "sss"	'sss', with all occurrences of the symbols ',\,(,),<,> prefixed with \
a plain literal "sss"@tag	the term `pair(`TR["sss"], 'tag'`)`
a typed literal "sss"(@tag)^^ddd	the term `L2V(`TR["sss"],TR[ddd]`)`
the uriref `rdfs:Resource`	T
a uriref of the form `rdf:_nnn`	`rdf-member(`nnn`)`
any other uriref aaa	aaa
a blank node	a variable (one distinct variable per blank node)
a triple aaa rdf:type bbb .	TR[bbb]`(`TR[aaa]`)` and `rdfs:Class(`TR[bbb]`)`
any other triple aaa bbb ccc .	TR[bbb]`(`TR[aaa], TR[ccc]`) and rdf:Property(`TR[bbb]`)`
an RDF graph	The existential closure of the conjunction of the translations of all the triples in the graph.
a set of RDF graphs	The conjunction of the translations of all the graphs.

L2V(?y,?x) is a term denoting the value of the lexical form ?y under the L2V mapping of the datatype ?x. Datatype specifications in L_base must include sufficient axioms to define the appropriate values of L2V. This can be simply a countably infinite set of equations stating the value for each legal lexical string, eg all equations of the form

RDF axioms
`rdf:type(?x,?y) implies ?y(?x) rdf:Property(rdf:type) rdf:List(rdf:nil) pair(?x,?y)=pair(?u,?v) iff (?x=?u and ?y=?v)` ;; uniqueness for pairs, required by graph syntax rules. `Lbase:XMLThing(?x) implies L2V(?x,rdf:XMLLiteral)=XMLCanonical(?x) (Lbase:XMLThing(?x) and LanguageTag(?y)) implies L2V(pair(?x,?y),rdf:XMLLiteral)=XMLCanonical(withLang(?x,?y))`

The RDF treatment of XML special forms does not lend itself to axiomatic description. The special function XMLCanonical is intended to map well-formed XML strings to their canonical forms. The meaning of this can be expressed by a countable set of L_baseequations, but is probably more usefully encoded in practice by an XML canonicalization algorithm. Similarly, the function withLang maps an XML document and a language tag to the similar document with that language tag attached; again, this would be more usefully specified by an algorithm than by the infinite collection of corresponding axioms, in practice. The predicate LanguageTag is intended to be true of syntactically correct XML language tags. Such axioms could be given as a list of atomic assertions, one for each of the valid language tags, and an assertion that all language tags were in the list.

The third and fourth axioms for domain and range can be changed to 'if and only if' as suggested in section 3; these would then have the consequence

RDFS axioms
T(?x) rdfs:Class(?y) implies (?y(?x) iff rdf:type(?x,?y)) (rdfs:range(?x,?y) and ?x(?u,?v)) implies ?y(?v) (rdfs:domain(?x,?y) and ?x(?u,?v)) implies ?y(?u) (rdfs:subClassOf(?x ?y) iff (rdfs:Class(?x) and rdfs:Class(?y) and (forall (?u)(?x(?u) implies ?y(?u))) ) (rdfs:subPropertyOf(?x,?y) iff (rdf:Property(?x) and rdf:Property(?y) and (forall (?u ?v)(?x(?u,?v) implies ?y(?u,?v))) ) rdfs:ContainerMembershipProperty(?x) implies rdfs:subPropertyOf(?x,rdfs:member) rdf:XMLLiteral(?x) implies rdfs:Literal(?x) Lbase:String(?y) implies rdfs:Literal(?y) `LanguageTag(?x) implies` `Lbase:String(?x)` `(Lbase:String(?x) and LanguageTag(?y)) implies rdfs:Literal(pair(?x,?y))` `rdfs:Datatype(rdf:XMLLiteral) Lbase:NatNumber(?x) implies rdfs:ContainerMembershipProperty(rdf-member(?x))` `rdfs:Class(T)` `rdfs:Class(rdf:Property)` `rdfs:Class(rdfs:Class)` `rdfs:Class(rdfs:Datatype)` `rdfs:Class(rdf:Seq)` `rdfs:Class(rdf:Bag)` `rdfs:Class(rdf:Alt)` `rdfs:Class(rdfs:Container)` `rdfs:Class(rdf:List)` `rdfs:Class(rdfs:ContainerMembershipProperty)` `rdfs:Class(rdf:Statement)` `rdf:Property(rdfs:domain)` `rdf:Property(rdfs:range)` `rdf:Property(rdfs:subClassOf)` `rdf:Property(rdfs:subPropertyOf)` `rdf:Property(rdfs:comment)` `rdf:Property(rdf:predicate)` `rdf:Property(rdf:subject)` `rdf:Property(rdf:object)` `rdf:Property(rdf:first)` `rdf:Property(rdf:rest)` `rdf:Property(rdfs:seeAlso)` `rdf:Property(rdfs:isDefinedBy)` `rdf:Property(rdfs:label)` `rdf:Property(rdf:value)` ;; the rest of the axioms are direct transcriptions of the RDFS axiomatic triples listed earlier, using the RDF to L_base transcription rules. `rdfs:domain(rdfs:subPropertyOf,rdf:Property) rdfs:domain(rdfs:subClassOf,rdfs:Class) rdfs:domain(rdfs:domain,rdf:Property)` `rdfs:domain(rdfs:range,rdf:Property)` `rdfs:domain(rdf:subject,rdf:Statement)` `rdfs:domain(rdf:predicate,rdf:Statement)` `rdfs:domain(rdf:object,rdf:Statement)` `rdfs:domain(rdf:first,rdf:List)` rdfs:domain(rdf:rest,rdf:List) rdfs:range(rdfs:subPropertyOf,rdf:Property) rdfs:range(rdfs:subClassOf,rdfs:Class) rdfs:range(rdfs:domain,rdfs:Class) rdfs:range(rdfs:range,rdfs:Class) rdfs:range(rdf:type,rdfs:Class) rdfs:range(rdfs:comment,rdfs:Literal) rdfs:range(rdfs:label,rdfs:Literal) rdfs:range(rdf:rest,rdf:List) rdfs:subClassOf(rdfs:Datatype,rdfs:Literal) rdfs:subClassOf(rdf:Alt,rdfs:Container) rdfs:subClassOf(rdf:Bag,rdfs:Container) rdfs:subClassOf(rdf:Seq, rdfs:Container) rdfs:subClassOf(rdfs:ContainerMembershipProperty,rdf:Property) rdfs:subPropertyOf(rdfs:isDefinedBy,rdfs:seeAlso)

RDF-D axioms
`rdfs:Datatype(?x) implies ( LegalLexicalForm(?y,?x)) iff ?x(L2V(?y,?x)) ) ( rdfs:Datatype(?x) and LegalLexicalForm(?y,?x) and ?x(?y)) implies rdfs:Literal(?y) Lbase:XMLThing(?x) iff LegalLexicalForm(?x,rdf:XMLLiteral)`

Here, LegalLexicalForm(?y,?x) is true when ?y is a character string which is a legal lexical for for the datatype ?x. As with L2V, the corresponding axiomatic theory depends on the particular datatype, but in the limit can be two countably infinite sets of atomic sentences asserting for each string whether or not it is a legal form for that datatype. Usually these will conform to some regular pattern; for example, the lexical forms for the datatype xsd:decimal can be defined by infinitely many axioms of the form

Appendix B: Proofs of lemmas (informative).

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

This means that, as noted in the text, we can treat a set of graphs as a single graph when discussing satisfaction and entailment. We will do this from now on, and refer to an interpretation of a set of graphs, a set of graphs entailing a graph, and so on, meaning in each case to refer to the merge of the set of graphs, and references to 'graph' in the following can be taken to refer to graphs or to sets of graphs.

For ground graphs, the subgraph lemma can be strengthened to provide simple necessary and sufficient conditions for entailment.

Plain Subgraph Lemma. If E and E' are ground, then E entails E' if and only if E' is a subgraph of E.

To prove the subsequent lemmas we introduce a way of constructing an interpretation of a graph by using the lexical items in the graph itself. (This was Herbrand's idea.) Given a graph G, the Herbrand interpretation of G , Herb(G), is the interpretation I defined as follows. The universe IR of I is the set of names and blank nodes in G; IS is the identity mapping on the vocabulary of G, IL maps XML documents to their canonical forms, and is otherwise is the identity mapping on typed literals, IEXT is defined by: <s,o> is in IEXT(p) just when there is a triple in the graph of the form s p o ., and IP is defined to be the set of urirefs which occur either as arc labels in the graph, or as subjects of triples of the form s rdf:type rdf:Property .

(This follows the standard definition of Herbrand interpretation but has some special features for RDF. The first part of the IP condition ensures that Herb(G) is an interpretation, and the second is a technical requirement for some of the later proofs. The XML condition is needed to make sure that Herbrand interpretations satisfy the built-in datatype rules.)

It is easy to see that Herb(G) is an interpretation which satisfies G. Clearly it satisfies all the ground triples in G; and if A is the identity mapping on blank nodes of G, then Herb(G)+A satisfies the entire graph; so Herb(G) satisfies G.

Herbrand interpretations treat urirefs and non-XML typed literals in the same way as simple literals, i.e. as denoting their own syntactic forms. Of course this may not be what was intended by the writer of the RDF, but the lemma shows that any graph can be interpreted in this way. This therefore establishes the

Herbrand interpretations have some very useful properties. The Herbrand interpretation of a graph is a 'minimal' interpretation, which is 'just enough' to make the graph true; and so any interpretation which satisfies the graph must in a sense agree with the Herbrand interpretation; and of course any interpretation which does agree with the Herbrand interpretation will satisfy the graph. Taken together and made precise, these observations provide a way to characterize entailment between graphs in terms of Herbrand interpretations.

Herb(G) satisfies G, but only just. Anything that goes beyond what is said by the graph itself is not satisfied by the Herbrand interpretation. This sense of 'going beyond' can be characterized both semantically and syntactically.

The semantic version refers to subinterpretations and isomorphisms between interpretations. Given two interpretations I and J, say that I is a subinterpretation of J , and write I << J, if the vocabulary of I is a subset of the vocabulary of J and there is a projection mapping from IR into JR, IS into JS, IL into JL and IEXT into JEXT such that any triple is true in J if it is true in I; and that I and J are isomorphic if each is a subinterpretation of the other. Obviously if I << J and I satisfies E then J satisfies E, so if I and J are isomorphic then they satisfy the same graphs. The key property of Herbrand interpretations, proved below, is that I satisfies E just when Herb(E) << I.

The syntactic version can be described in terms of instances and subgraphs. Say that a graph E' is separable from a graph E if no instance of E' is a subgraph of E. In particular, a ground graph is separable from E just when it is not a subgraph of E, and a ground triple is separable just in case it isn't in the graph. Graphs which are not separable from E are entailed by E, by the subgraph and instance lemmas; but for all others, there is a way to arrange the world so that they are false and E true.

In particular, if E' is separable from E then Herb(E) does not satisfy E'; for suppose that it did, then for some mapping B from the blank nodes of E' to the blank nodes and vocabulary of E, Herb(E)+B satisfies E', which means that for every triple

occurs in E, by definition of Herb(E). But the set of these triples is an instance of E', by construction; so E' is not separable from E.

This means that we can state an exact correspondence between separability and Herbrand interpretations:

Herbrand separation lemma. Herb(E) satisfies E' if and only if E' is not separable from E. QED

Probably the most useful property of Herbrand interpretations is the following. The version of this lemma for first-order logic, called Herbrand's theorem, is the basis of all the logical completeness results.

Putting the separation and entailment results together, it is obvious that S entails E if and only if E is not separable from S. This is simply a restatement of the:

Skolemization is a syntactic transformation routinely used in automatic inference systems in which existential variables are replaced by 'new' functions - function names not used elsewhere - applied to any enclosing universal variables. While not itself strictly a valid operation, skolemization adds no new content to an expression, in the sense that a skolemized expression has the same entailments as the original expression provided they do not contain the new skolem functions.

In RDF, skolemization simplifies to the special case where an existential variable is replaced by a 'new' name, i.e. a uriref which is guaranteed to not occur anywhere else.(Using a literal would not do. Literals are never 'new' in the required sense, since their meaning is fixed.) To be precise, a skolemization of E (with respect to V) is a ground instance of E with respect to a vocabulary V which is disjoint from the vocabulary of E.

The following lemma shows that skolemization has the same properties in RDF as it has in conventional logics. Intuitively, this lemma shows that asserting a skolemization expresses a similar content to asserting the original graph, in many respects. In effect, it simply gives 'arbitrary' names to the anonymous entities whose existence was asserted by the use of blank nodes. However, care is needed, since these 'arbitrary' names have the same status as any other urirefs once published. Also, skolemization would not be an appropriate operation when applied to anything other than the antecendent of an entailment. A skolemization of a query would represent a completely different query.

RDF closure lemma. The Herbrand interpretation of the rdf-closure of E is an rdf-interpretation of E.

RDF entailment lemma. S rdf-entails E if and only if the rdf-closure of S simply entails E.

RDFS Closure Lemma. The Herbrand interpretation of the rdfs-closure of E is an rdfs-interpretation of E.

Appendix C: Glossary of terms (informative).

Antecedent (n.) In an inference, the expression(s) from which the conclusion is derived. In an entailment relation, the entailer. Also assumption.

Assertion (n.) (i) Any expression which is claimed to be true. (ii) The act of claiming something to be true.

Class (n.) A general concept, category or classification. Something used primarily to classify or categorize other things. Formally, a resource of type rdfs:Class with an associated set of resources all of which have the class as a value of the rdf:type property. Classes are often called 'predicates' in the formal logical literature.

(RDF distinguishes class from set, although the two are often identified. Distinguishing classes from sets allows RDF more freedom in constructing class heirachies, as explained earlier.)

Complete (adj., of an inference system). Able to draw all valid inferences. See Inference. Also used with a qualifier: able to draw all valid inferences in a certain limited form or kind (e.g.between expressions in a certain normal form, or meeting certain syntactic conditions.)

Consequent (n.) In an inference, the expression constructed from the antecedent. In an entailment relation, the entailee. Also conclusion.

Correct (adj., of an inference system). Unable to draw any invalid inferences. See Inference.

Entail (v.), entailment(n.). A semantic relationship between expressions which holds whenever the truth of the first guarantees the truth of the second. Equivalently, whenever it is logically impossible for the first expression to be true and the second one false. Equivalently, when any interpretation which satisfies the first also satisfies the second. (Also used between a set of expressions and an expression.)

Equivalent (prep., with to) True under exactly the same conditions; making identical claims about the world, when asserted. Entails and is entailed by.

Extensional (adj., of a logic) A set-based theory or logic of classes, in which classes are considered to be sets, properties considered to be sets of <object, value> pairs, and so on. A theory which admits no distinction between entities with the same extension. See Intensional.

Formal (adj.) Couched in language sufficiently precise as to enable results to be established using conventional mathematical techniques.

Iff (conj.) Conventional abbreviation for 'if and only if'. Used to express necessary and sufficient conditions.

Inconsistent (adj.), false under all interpretations; impossible to satisfy. Inconsistency (n.), any inconsistent expression or graph.

Indexical (adj., of a logic expression) having a meaning which implicitly refers to the context of use. Examples from English include words like 'here', 'now', 'this'.

Inference (n.) An act or process of constructing new expressions from existing expressions, or the result of such an act or process. Inferences corresponding to entailments are described as correct or valid. Inference rule, formal description of a type of inference; inference system, set of inference rules; also, software which generates inferences or checks inferences for validity.

Intensional (adj., of a logic) Not extensional. A logic which allows distinct entities with the same extension.

(The merits and demerits of intensionality have been extensively debated in the philosophical logic literature. Extensional semantic theories are simpler, and conventional semantics for formal logics usually assume an extensional view, but conceptual analysis of ordinary language often suggests that intensional thinking is more natural. Examples often cited are that an extensional logic is obliged to treat all 'empty' extensions as identical, so must identify 'round square' with 'santa clause', and is unable to distinguish concepts that 'accidentally' have the same instances, such as human beings and bipedal hominids without body hair. RDFS model theory is basically intensional but has some extensional aspects, most notably in the 'if and only if' conditions in the definitions of rdfs:subClassOf and rdfs:subPropertyOf, which force these properties to take account only of the class and property extensions.)

Interpretation (of) (n.) A minimal formal description of those aspects of a world which is just sufficient to establish the truth or falsity of any expression of a logic.

(Some logic texts distinguish between a interpretation structure, which is a 'possible world' considered as something independent of any particular vocabulary, and an interpretation mapping from a vocabulary into the structure. The RDF semantics takes the simpler route of merging these into a single concept.)

Metaphysical (adj.). Concerned with the true nature of things in some absolute or fundamental sense.

Model Theory (n.) A formal semantic theory which relates expressions to interpretations.

(The name 'model theory' arises from the usage, traditional in logical semantics, in which a satisfying interpretation is called a "model". This usage is often found confusing, however, as it is almost exactly the inverse of the meaning implied by terms like "computational modelling", so we have avoided it in this document.)

Monotonic (adj., of a logic or inference system) Satisfying the condition that if S entails E then (S + T) entails E, i.e. adding information to some antecedents cannot invalidate a valid entailment.

(All logics based on a conventional model theory and a standard notion of entailment are monotonic. Monotonic logics have the property that entailments remain valid outside of the context in which they were generated. This is why RDF is designed to be monotonic.)

Nonmonotonic (adj.,of a logic or inference system) Not monotonic. Non-monotonic formalisms have been proposed and used in AI and various applications. Examples of nonmonotonic inferences include default reasoning, where one assumes a 'normal' general truth unless it is contradicted by more particular information (eg birds usually fly, but penguins don't fly); negation-by-failure, commonly assumed in logic programming systems, where one concludes, from a failure to prove a proposition, that the proposition is false; and implicit closed-world assumptions, often assumed in database applications, where one concludes from a lack of information about an entity in some corpus that the information is false (e.g. that if someone is not listed in an employee database, that hse is not an employee.)

(The relationship between monotonic and nonmonotonic inferences is often subtle. For example, if a closed-world assumption is made explicit, e.g. by asserting explicitly that the corpus is complete and providing explicit provenance information in the conclusion, then closed-world reasoning is monotonic; it is the implicitness that makes the reasoning nonmonotonic. Nonmonotonic conclusions can be said to be valid only in some kind of 'context', and are liable to be incorrect or misleading when used outside that context. Making the context explicit in the reasoning and visible in the conclusion is a way to map them into a monotonic framework.)

Ontological (adj.) (Philosophy) Concerned with what kinds of things really exist. (Applied) Concerned with the details of a formal description of some topic or domain.

Proposition (n.) Something that has a truth-value; a statement or expression that is true or false.

(Philosophical analyses of language traditionally distinguish propositions from the expressions which are used to state them, but model theory does not require this distinction.)

Reify (v.), reification (n.) To categorize as an object; to describe as an entity. Often used to describe a convention whereby a syntactic expression is treated as a semantic object and itself described using another syntax. In RDF, a reified triple is a description of a triple-token using other RDF triples.

Resource (n.)(as used in RDF)(i) An entity; anything in the universe. (ii) As a class name: the class of everything; the most inclusive category possible.

Satisfy (v.t.), satisfaction,(n.) satisfying (adj., of an interpretation). To make true. The basic semantic relationship between an interpretation and an expression. X satisfies Y means that if the world conforms to the conditions described by X, then Y must be true.

Semantic (adj.) , semantics (n.). Concerned with the specification of meanings. Often contrasted with syntactic to emphasise the distinction between expressions and what they denote.

Skolemization (n.) A syntactic transformation in which blank nodes are replaced by 'new' names.

(Although not strictly valid, Skolemization retains the essential meaning of an expression and is often used in mechanical inference systems. The full logical form is more complex. It is named after the logician A. T. Skolem)

Token (n.) A particular physical inscription of a symbol or expression in a document. Usually contrasted with type, the abstract grammatical form of an expression.

Universe (n., also Universe of discourse) The universal classification, or the set of all things that an interpretation considers to exist. In RDF/S, this is identical to the set of resources.

Use (v.) contrasted with mention; to use a piece of syntax to denote or refer to something else. The normal way that language is used.

("Whenever, in a sentence, we wish to say something about a certain thing, we have to use, in this sentence, not the thing itself but its name or designation." - Alfred Tarski)

Valid (adj., of an inference or inference process) Corresponding to an entailment, i.e. the conclusion of the inference is entailed by the antecedent of the inference. Also correct.

World (n.) (with the:) (i) The actual world. (with a:)(ii) A way that the actual world might be arranged. (iii) An interpretation (iv) A possible world.

(The metaphysical status of ' possible worlds' is highly controversial. Fortunately, one does not need to commit oneself to a belief in parallel universes in order to use the concept in its second and third senses, which are sufficient for semantic purposes.)

Appendix D: Acknowledgements

This document reflects the joint effort of the members of the RDF Core Working Group. Particular contributions were made by D. Connolly, J. Carroll, J. Grant, R. V. Guha, G. Klyne, O. Lassilla, B. McBride, S. Melnick, J. deRoo and P. Stickler.

The basic idea of using an explicit extension mapping to allow self-application without violating the axiom of foundation was suggested by Christopher Menzel.

Several people found errors in earlier drafts. Herman ter Horst suggested improvements to the proofs. Peter Patel-Schneider found several major problems in earlier drafts, and suggested several important technical improvements.

	if E contains	then add
rdf1	xxx aaa yyy `.`	aaa `rdf:type rdf:Property .`
rdf2a	xxx aaa `"`yyy`"^^rdf:XMLLiteral .` where yyy is a well-formed XML document	xxx aaa `"`zzz`"^^rdf:XMLLiteral .` where zzz is a well-formed XML document with the same XML canonical form as yyy.
rdf2b	xxx aaa `"`yyy`"@`ttt`^^rdf:XMLLiteral .` where yyy is a well-formed XML document and ttt is a language tag	xxx aaa `"`zzz`"@`ttt`^^rdf:XMLLiteral .` where zzz is any well-formed XML document such that the canonical form of the XML document yyy with the language tag ttt is the same as the canonical form of zzz with language tag ttt.

rdfs 2a	xxx `rdfs:domain` yyy `.` yyy `rdfs:subClassOf` zzz `.`	xxx `rdfs:domain` zzz `.`
rdfs 3a	xxx `rdfs:range` yyy `.` yyy `rdfs:subClassOf` zzz `.`	xxx `rdfs:range` zzz `.`
rdfs 4a'	xxx aaa yyy .	aaa `rdfs:domain rdfs:Resource .`
rdfs 4b'	xxx aaa yyy .	aaa `rdfs:range rdfs:Resource .`

rdfD 0a	aaa ppp "sss"@ttt^^ddd `.`	aaa ppp "sss"^^ddd `.`
rdfD 0b	aaa ppp "sss"^^ddd `.`	aaa ppp "sss"@uuu^^ddd `.`

RDF Semantics

W3C Working Draft 23 January 2003

Abstract

Status of this Document

Table of Contents

0. Introduction

0.1 Specifying a formal semantics: scope and limitations

0.2 Graph syntax

0.3 Graph Definitions

1. Interpretations

1.1 Technical Notes

1.2 Urirefs, resources and literals.

1.3 Interpretations

1.4 Denotations of ground graphs

1.5. Blank nodes as existential assertions

2. Simple entailment between RDF graphs

2.1 Criteria for non-entailment

3. Interpreting the RDF(S) vocabulary

3.1. RDF Interpretations

3.2. Reification, containers, collections and rdf:value

3.2.1 Reification

3.2.2 RDF containers

3.2.3 RDF collections

3.2.4 rdf:value

3.3 RDFS interpretations

3.3.1 A note on rdfs:Literal

3.4 Datatyped interpretations

4. Vocabulary entailment and closure rules

4.1 Rdf-entailment and rdf closures (Informative)

4.2 RDFS-entailment and RDFS closures (informative)

4.3 Datatype entailments (Informative)

Appendix A: Translation into L_base(Informative)

Appendix B: Proofs of lemmas (informative).

Appendix C: Glossary of terms (informative).

Appendix D: Acknowledgements

References

Normative

Non-Normative

RDF Semantics

W3C Working Draft 23 January 2003

Abstract

Status of this Document

Table of Contents

1.2 Urirefs, resources and literals.

3.2. Reification, containers, collections and rdf:value

3.2.3 RDF collections

3.2.4 rdf:value

3.4 Datatyped interpretations

4.1 Rdf-entailment and rdf closures (Informative)

4.2 RDFS-entailment and RDFS closures (informative)

4.3 Datatype entailments (Informative)

Appendix A: Translation into Lbase(Informative)

Appendix B: Proofs of lemmas (informative).

Appendix C: Glossary of terms (informative).

Appendix D: Acknowledgements

Appendix A: Translation into L_base(Informative)