RDF Semantics

W3C Recommendation 10 February 2004

New Version Available: "RDF 1.1 Semantics" (Document Status Update, 25 February 2014)

The RDF Working Group has produced a W3C Recommendation for a new version of RDF which adds features to this 2004 version, while remaining compatible. Please see "RDF 1.1 Semantics" for a new version of this document, and the "What's New in RDF 1.1" document for the differences between this version of RDF and RDF 1.1.

This Version:

http://www.w3.org/TR/2004/REC-rdf-mt-20040210/

Latest Version:

http://www.w3.org/TR/rdf-mt/

Previous Version:

http://www.w3.org/TR/2003/PR-rdf-mt-20031215/

Editor:

Patrick Hayes (IHMC)< phayes@ihmc.us>

Series Editor

Brian McBride (Hewlett Packard Labs)<bwm@hplb.hpl.hp.com>

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

See also translations.

Copyright © 2004 W3C^® (MIT, ERCIM, Keio), All Rights Reserved. W3C liability, trademark, document use and software licensing rules apply.

Abstract

This is a specification of a precise semantics, and corresponding complete systems of inference rules, for the Resource Description Framework (RDF) and RDF Schema (RDFS).

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 Note (Informative)
     1.2 URI references, Resources and Literals
     1.3 Interpretations
     1.4 Denotations of Ground Graphs
     1.5 Blank nodes as Existential variables
2. Simple Entailment between RDF graphs
     2.1 Vocabulary interpretations and vocabulary entailment
3. Interpreting the RDF vocabulary
     3.1 RDF Interpretations
     3.2 RDF Entailment
     3.3 Reification, Containers, Collections and rdf:value
       3.3.1 Reification
       3.3.2 RDF Containers
       3.3.3 RDF Collections
       3.3.4 rdf:value
4. Interpreting the RDFS Vocabulary
     4.1 RDFS Interpretations
     4.2 Extensional Semantic Conditions (Informative)
     4.3 A Note on rdfs:Literal
     4.4 RDFS Entailment
5. Interpreting Datatypes
     5.1 Datatyped Interpretations
     5.2 D-Entailment
6. Monotonicity of Semantic Extensions
7. Entailment Rules (Informative)
     7.1 Simple Entailment Rules
     7.2 RDF Entailment Rules
     7.3 RDFS Entailment Rules
       7.3.1 Extensional Entailment Rules
     7.4 Datatype Entailment Rules
Appendix A. Proofs of lemmas (Informative)
Appendix B. Glossary (Informative)
Appendix C. Acknowledgements
References
Appendix D. Change Log (Informative)

0. Introduction

0.1 Specifying a formal semantics: scope and limitations

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

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

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

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

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

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

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

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

0.2 Graph Syntax

Any semantic theory must be attached to a syntax. This semantics is defined as a mapping on the abstract syntax of RDF described in the RDF concepts and abstract syntax document [RDF-CONCEPTS]. This document uses the following terminology defined there: URI reference, literal, plain literal, typed literal, XML literal, XML value, node, blank node, triple and RDF graph. Throughout this document we use the term 'character string' or 'string' to refer to a sequence of Unicode characters, and 'language tag' in the sense of RFC 3066, c.f. section 6.5 in [RDF-CONCEPTS]. Note that strings in an RDF graph SHOULD be in Normal Form C.

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

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

Prefix rdf: namespace URI: http://www.w3.org/1999/02/22-rdf-syntax-ns#

Prefix rdfs: namespace URI: http://www.w3.org/2000/01/rdf-schema#

Prefix xsd: namespace URI: http://www.w3.org/2001/XMLSchema#

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

<ex:a> rdf:type rdfs:Class .

should be read as an abbreviation for the N-Triples syntax

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

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

0.3 Graph Definitions

An RDF graph, or simply a graph, is a set of RDF triples.

A subgraph of an RDF graph is a subset of the triples in the graph. A triple is identified with the singleton set containing it, so that each triple in a graph is considered to be a subgraph. A proper subgraph is a proper subset of the triples in the graph.

A ground RDF graph is one with no blank nodes.

A name is a URI reference or a literal. These are the expressions that need to be assigned a meaning by an interpretation. Note that a typed literal comprises two names: itself and its internal type URI reference.

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

Suppose that M is a mapping from a set of blank nodes to some set of literals, blank nodes and URI references; then any graph obtained from a graph G by replacing some or all of the blank nodes N in G by M(N) is an instance of G. Note that any graph is an instance of itself, an instance of an instance of G is an instance of G, and if H is an instance of G then every triple in H is an instance of some triple in G.

An instance with respect to a vocabulary V is an instance in which all the names in the instance that were substituted for blank nodes in the original are names from V.

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

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

An RDF graph is lean if it has no instance which is a proper subgraph of the graph. Non-lean graphs have internal redundancy and express the same content as their lean subgraphs. For example, the graph

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

is not lean, but

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

is lean.

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

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

1. Interpretations

1.1 Technical note (Informative)

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

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

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

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

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

1.2 URI references, Resources and Literals.

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

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

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

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

1.3 Interpretations

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

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

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

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

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

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

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

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

1.4 Denotations of Ground Graphs

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

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

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

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

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

IR = LV union{1, 2}

IP={1}

IEXT: 1=>{<1,2>,<2,1>}

IS: ex:a=>1, ex:b=>1, ex:c=>2

IL: "whatever"^^ex:b =>2

This interpretation makes these triples true:

  <ex:a> <ex:b> <ex:c> .

  <ex:c> <ex:a> <ex:a> .

  <ex:c> <ex:b> <ex:a> .

  <ex:a> <ex:b> "whatever"^^<ex:b> .

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

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

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

It makes these triples false:

  <ex:a> <ex:c> <ex:b> .

  <ex:a> <ex:b> <ex:b> .

  <ex:c> <ex:a> <ex:c> .

  <ex:a> <ex:b> "whatever" .

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

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

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

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

1.5. Blank Nodes as Existential Variables

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

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

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

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

  _:xxx <ex:a> <ex:b> .

  <ex:c> <ex:b> _:xxx .

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

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

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

2. Simple Entailment between RDF graphs

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

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

Any process which constructs a graph E from some other graph(s) S is said to be (simply) valid if S entails E in every case, otherwise invalid. Note that being an invalid process does not mean that the conclusion is false, and being valid does not guarantee truth. However, validity represents the best guarantee that any assertional language can offer: if given true inputs, it will never draw a false conclusion from them.

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

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

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

Subgraph Lemma. A graph entails all its subgraphs. [Proof]

Instance Lemma. A graph is entailed by any of its instances.[Proof]

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

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

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

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

The main result for simple RDF inference is:

Interpolation Lemma. S entails a graph E if and only if a subgraph of S is an instance of E.[Proof]

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

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

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

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

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

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

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

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

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

2.1 Vocabulary interpretations and vocabulary entailment

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

3. Interpreting the RDF Vocabulary

3.1 RDF Interpretations

The RDF vocabulary, rdfV, is a set of URI references in the rdf: namespace:

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

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

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

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

IR = LV union {1, 2, T , P}

IP = {1, T}

IEXT: 1=>{<1,2>,<2,1>}, T=>{<1,P>,<T,P>}

IS: ex:a=>1, ex:b=>1, ex:c=> 2, rdf:type=>T, rdf:Property=>P, rdf:nil=>1, rdf:List=>P,  rdf:Statement=>P, rdf:subject=>1, rdf:predicate=>1, rdf:object=>1, rdf:first=>1, rdf:rest=>1, rdf:Seq=>P, rdf:Bag=>P, rdf:Alt=>P, rdf:_1, rdf:_2, ... =>1

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

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

3.2. RDF entailment

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

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

rdf:type rdf:type rdf:Property .

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

3.3. Reification, Containers, Collections and rdf:value

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

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

3.3.1 Reification

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

aaa rdf:type rdf:Statement .

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.

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

and

_:xxx rdf:type rdf:Statement .
_:xxx rdf:subject <ex:a> .
_:xxx rdf:predicate <ex:b> .
_:xxx rdf:object <ex:c> .

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

aaa bbb ccc .

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

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

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

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

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

_:xxx rdf:type rdf:Statement .
_:xxx rdf:subject <ex:subject> .
_:xxx rdf:predicate <ex:predicate> .
_:xxx rdf:object <ex:object> .
_:yyy rdf:type rdf:Statement .
_:yyy rdf:subject <ex:subject> .
_:yyy rdf:predicate <ex:predicate> .
_:yyy rdf:object <ex:object> .
_:xxx <ex:property> <ex:foo> .

does not entail

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

3.3.2 RDF containers

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

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

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

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

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

_:xxx rdf:type rdf:Bag .
_:xxx rdf:_1 <ex:a> .
_:xxx rdf:_2 <ex:b> .

does not entail

_:xxx rdf:_1 <ex:b> .
_:xxx rdf:_2 <ex:a> .

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

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

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

_:xxx rdf:type rdf:Seq.
_:xxx rdf:_1 <ex:a> .
_:xxx rdf:_3 <ex:c> .

does not entail

_:xxx rdf:_2 _:yyy .

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

3.3.3 RDF collections

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

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

_:c1 rdf:first aaa .
_:c1 rdf:rest _:c2

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

_:c1 rdf:first <ex:aaa> .
_:c1 rdf:rest _:c2 .
_:c2 rdf:first <ex:bbb> .
_:c2 rdf:rest rdf:nil .

does not entail

_:c3 rdf:first <ex:bbb> .
_:c3 rdf:rest _:c4 .
_:c4 rdf:first <ex:aaa> .
_:c4 rdf:rest rdf:nil .

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

_:666 rdf:first <ex:aaa> .
_:666 rdf:first <ex:bbb> .
_:666 rdf:rest <ex:ccc> .
_:666 rdf:rest rdf:nil .

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

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

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

3.3.4 rdf:value

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

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

4. Interpreting the RDFS Vocabulary

4.1 RDFS Interpretations

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

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

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

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

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

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

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

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

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

cannot be true in any rdfs-interpretation; it is rdfs-inconsistent.

4.2 Extensional Semantic Conditions (Informative)

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

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

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

4.3 A Note on rdfs:Literal

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

For example, a triple of the form

<ex:a> rdf:type rdfs:Literal .

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

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

<ex:a> <ex:b> "10" .

entails

<ex:a> <ex:b> _:xxx .

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

4.4 RDFS Entailment

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

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

xxx rdf:type rdfs:Resource .

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

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

5. Interpreting Datatypes

5.1 Datatyped Interpretations

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

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

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

Formally, a datatype d is defined by three items:

1. a non-empty set of character strings called the lexical space of d;

2. a non-empty set called the value space of d;

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

The lexical-to-value mapping of a datatype d is written as L2V(d).

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

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

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

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

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

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

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

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

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

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

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

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

<ex:a> <ex:b> "25"^^xsd:decimal .
<ex:b> rdfs:range xsd:string .

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

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

_:x rdf:type xsd:string .
_:x rdf:type xsd:decimal .

or that a property with an 'impossible' range has a value:

<ex:p> rdfs:range xsd:string .
<ex:p> rdfs:range xsd:decimal .
_:x <ex:p> _:y .

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

<ex:a> <ex:p> "2.5"^^xsd:decimal .
<ex:p> rdfs:range xsd:integer .

or by the use of an ill-typed lexical form:

<ex:a> <ex:p> "abc"^^xsd:integer .
<ex:p> rdfs:range xsd:integer .

Datatype clashes are the only inconsistencies recognized by this model theory; note however that datatype clashes involving XML literals can arise in RDFS.

If D is a subset of D', then restricting interpretations of a graph to D'-interpretations amounts to a semantic extension compared to the same restriction with respect to D. In effect, the extension of the datatype map makes implicit assertions about typed literals, by requiring them to denote entities in the value space of a datatype. The extra semantic constraints associated with the larger datatype map will force interpretations to make more triples true, but they may also reveal datatype clashes and violations, so that a D-consistent graph could be D'-inconsistent.

Say that an RDF graph recognizes a datatype URI reference aaa when the graph rdfs-entails a datatyping triple of the form

aaa rdf:type rdfs:Datatype .

The semantic conditions for rdfs-interpretations require the built-in datatype URI reference 'rdf:XMLLiteral' to be recognized.

If every recognized URI reference in a graph is the name of a known datatype, then there is a natural datatype map DG which pairs each recognized URI reference to that known datatype (and 'rdf:XMLLiteral' to rdf:XMLLiteral). Any rdfs-interpretation I of that graph then has a corresponding 'natural' DG-interpretation which is like I except that I(aaa) is the appropriate datatype and the class extension of rdfs:Datatype is modified appropriately. Applications MAY require that RDF graphs be interpreted by D-interpretations where D contains a natural datatype map of the graph. This amounts to treating datatyping triples as 'declarations' of datatypes by the graph, and making the fourth semantic condition into an 'iff' condition. Note however that a datatyping triple does not in itself provide the information necessary to check that a graph satisfies the other datatype semantic conditions, and it does not formally rule out other interpretations, so that adopting this requirement as a formal entailment principle would violate the general monotonicity lemma described in section 6, below.

5.2 D-entailment

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

As noted above, it is possible that a graph which is consistent in one vocabulary becomes inconsistent in a semantic extension defined on a larger vocabulary, and D-interpretations allow for inconsistencies in an RDF graph. The definition of vocabulary entailment means that an inconsistent graph will entail any graph in the stronger vocabulary entailment. For example, 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 may not be valid entailments in a smaller vocabulary.

6. Monotonicity of semantic extensions

Given a set of RDF graphs, there are various ways in which one can 'add' information to it. Any of the graphs may have some triples added to it; the set of graphs may be extended by extra graphs; or the vocabulary of the graph may be interpreted relative to a stronger notion of vocabulary entailment, i.e. with a larger set of semantic conditions understood to be imposed on the interpretations. All of these can be thought of as an addition of information, and may make more entailments hold than held before the change. All of these additions are monotonic, in the sense that entailments which hold before the addition of information, also hold after it. We can sum up this in a single lemma:

General monotonicity lemma. Suppose that S, S' are sets of RDF graphs with every member of S a subset of some member of S'. Suppose that Y indicates a semantic extension of  X, S X-entails E, and S and E satisfy any syntactic restrictions of Y. Then S' Y-entails E.

In particular, if D' is a datatype map, D a subset of D' and if S D-entails E then S also D'-entails E.

7. Entailment rules (Informative)

The following tables list some inference patterns which capture some of the various forms of vocabulary entailment, and could be used as a guide for the design of software to check RDF graphs for RDF and RDFS entailment. Implementations may be based on applying the rules forwards, or treating the conclusion as a pattern to be recognized in the consequent of a proposed entailment and searching backwards for an appropriate match with other rule conclusions or the proposed antecedent. Other strategies are also possible.

The rules are all stated with the form add a triple to a graph when it contains triples conforming to a pattern, and they are all valid in the following sense: a graph entails (in the appropriate sense listed) any larger graph that is obtained by applying the rules to the original graph. Notice that applying such a rule to a graph amounts to forming a simple union with the conclusion, rather than a merge, and hence preserves any blank nodes already in the graph.

These rules all use the following conventions: aaa, bbb, etc., stand for any URI reference, i.e. any possible predicate of a triple; uuu, vvv, etc. for any URI reference or blank node identifier, i.e any possible subject of a triple; xxx, yyy etc. for any URI reference, blank node identifier or literal, i.e. any possible subject or object of a triple; lll for any literal; and _:nnn, etc., for blank node identifiers.

7.1 Simple Entailment Rules

The interpolation lemma in Section 2 can be characterized by the following entailment rules which generate generalizations, i.e. graphs which have the original graph as an instance. Being a subgraph requires no explicit formulation as an inference rule on triples.

The terminology 'allocated to' means that the blank node must have been created by an earlier application of the specified rules on the same URI reference, blank node or literal, or if there is no such blank node then it must be a 'new' node which does not occur in the graph. This rather complicated condition ensures that the resulting graph, obtained by adding the new blank-node triples, has the original graph as a proper instance and that any such graph will have a subgraph which is the same as one which can be generated by these rules. For example, the graph

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

could be expanded as follows

_:x <ex:p> <ex:b> . by se1 using a new _:x which is allocated to ex:a
<ex:c> <ex:q> _:x . by se2 using the same _:x allocated to ex:a
_:x <ex:p> _:y . by se2 using a new _:y which is allocated to ex:b

but it would not be correct to infer

** _:x <ex:q> <ex:a> . ** by se2 (** since _:x is not allocated to ex:c )

These rules allow blank nodes to proliferate, producing highly non-lean graphs; they sanction entailments such as

<ex:a> <ex:p> <ex:b> .
<ex:a> <ex:p> _:x . by xse1 with _:x allocated to ex:b
<ex:a> <ex:p> _:y . by xse1 with _:y allocated to _:x
_:z <ex:p> _:y . by xse2 with _:z allocated to ex:a
_:u <ex:p> _:y . by xse2 with _:u allocated to _:z
_:u <ex:p> _:v . by xse1 with _:v allocated to _:y

It is easy to see that S is an instance of E if and only if one can derive E from S by applying these rules in a suitable sequence; the rule applications required can be discovered by examination of the instance mapping. So, by the interpolation lemma, S simply entails E iff one can derive (a graph containing) E from S by the application of these rules. However, it is also clear that applying these rules naively would not result in an efficient search process, since the rules will not terminate and can produce arbitrarily many redundant derivations of equivalent triples.

The general problem of determining simple entailment between arbitrary RDF graphs is decideable but NP-complete. This can be shown by encoding the problem of detecting a subgraph of an arbitrary directed graph as an RDF entailment, using only blank nodes to represent the graph (observation due to Jeremy Carroll.)

Subsequent rule sets for detecting RDF and RDFS entailment use a special case of rule se1 which applies only to literals:

Note that this rule is just sufficient to reproduce any subgraph of E consisting of triples all containing a given literal as an isomorphic subgraph in which the literal is replaced by a unique blank node. The significance of this lies in the fact the blank nodes can stand in a subject position in an RDF triple, allowing conclusions to be drawn by the other rules which express properties of the literal value denoted by that literal.

For RDFS entailment one additional rule is required, which inverts rule lg:

Clearly rule gl will simply produce a redundancy, except in cases where the allocated blank node has been introduced into the object position of a new triple by some other inference rule. The effect of these rules is to ensure that any triple containing a literal, and the similar triple containing the allocated blank node, are always derivable from one another, so that a literal can be identified with its allocated blank node for purposes of applying these rules.

7.2 RDF Entailment Rules

These rules are complete in the following sense.

RDF entailment lemma. S rdf-entails E if and only if there is a graph which can be derived from S plus the RDF axiomatic triples by the application of rule lg and the RDF entailment rules and which simply entails E. (Proof in Appendix A)

Note that this does not require the use of rule gl.

7.3 RDFS Entailment Rules

Stating completeness for these rules requires more care, since it is possible for a graph to be rdfs-inconsistent by virtue of containing unsatisfiable assertions about ill-typed XML literals, for example:

<ex:a> rdfs:subClassOf rdfs:Literal .
<ex:b> rdfs:range <ex:a> .
<ex:c> rdfs:subPropertyOf <ex:b>.
<ex:d> <ex:c> "<"^^rdf:XMLLiteral .

where the denotation of the ill-typed XML literal is required not to be a literal value but can be inferred to be on the basis of the other assertions. Since such an rdfs-inconsistent graph rdfs-entails all RDF graphs, even when they are syntactically unrelated to the antecedent, we have to exclude this case.

There is a characteristic signal of inconsistency which can be recognized by the entailment rules, by the derivation of a triple of the form

xxx rdf:type rdfs:Literal .

where xxx is allocated to an ill-typed XML literal by rule lg. Call such a triple an XML clash. The derivation of such a clash from the above example is straightforward:

<ex:d> <ex:c> _:1 .   (by rule lg, with _:1 allocated to "<"^^rdf:XMLLiteral)
<ex:d> <ex:b> _:1 .    (by rule rdfs7)
_:1 rdf:type <ex:a>.   (by rule rdfs3)
_:1 rdf:type rdfs:Literal .   (by rule rdfs9)

These rules are then complete in the following sense:

RDFS entailment lemma. S rdfs-entails E if and only if there is a graph which can be derived from S plus the RDF and RDFS axiomatic triples by the application of rule lg, rule gl and the RDF and RDFS entailment rules and which either simply entails E or contains an XML clash. (Proof in Appendix A)

The RDFS rules are somewhat redundant. All but one of the RDF axiomatic triples can be derived from the rules rdfs2 and rdfs3 and the RDFS axiomatic triples, for example.

The outputs of these rules will often trigger others. These rules will propagate all rdf:type assertions in the graph up the subproperty and subclass hierarchies, re-asserting them for all super-properties and superclasses. rdf1 will generate type assertions for all the property names used in the graph, and rdfs3 together with the last RDFS axiomatic triple will add all type assertions for all the class names used. Any subproperty or subclass assertion will generate appropriate type assertions for its subject and object via rdfs2 and rdfs3 and the domain and range assertions in the RDFS axiomatic triple set. The rules will generate all assertions of the form

uuu rdf:type rdfs:Resource .

for every uuu in V, and of the form

uuu rdfs:subClassOf rdfs:Resource .

for every class name uuu; and several more 'universal' facts, such as

rdf:Property rdf:type rdfs:Class .

7.3.1 Extensional Entailment Rules

The stronger extensional semantic conditions described in Section 4.1 sanction further entailments which are not covered by the RDFS rules. The following table lists some entailment patterns which are valid in this stronger semantics. This is not a complete set of rules for the extensional semantic conditions. Note that none of these rules are rdfs-valid; they apply only to semantic extensions which apply the strengthened extensional semantic conditions described in Section 4.1. These rules have other consequences, eg that rdfs:Resource is a domain and range of every property.

Rules ext5-ext9 follow a common pattern; they reflect the fact that the strengthened extensional conditions require domains (and ranges for transitive properties) of the properties in the rdfV and rdfsV vocabularies to be as large as possible, so any attempt to restrict them will be subverted by the semantic conditions. Similar rules apply to superproperties of rdfs:range and rdfs:domain. None of these cases are likely to arise in practice.

7.4 Datatype Entailment Rules

In order to capture datatype entailment in terms of assertions and entailment 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.

For each kind of information which is available about a datatype, inference rules for information of that kind can be stated, which can be thought of as extending the table of RDFS entailment rules. These should be understood as applying to datatypes other than the built-in datatype, the rules for which are part of the RDFS entailment rules. The rules stated below assume that information is available about the datatype denoted by a recognized URI reference, and they use that URI reference to refer to the datatype.

The basic information specifies, for each literal string, whether or not it is a legal lexical form for the datatype, i.e. one which maps to some value under the lexical-to-value mapping of the datatype. This corresponds to the following rule, for each string sss that is a legal lexical form for the datatype denoted by ddd:

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

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

In addition, if it is known that the value space of the datatype denoted by ddd is a subset of that of the datatype denoted by eee, then it would be appropriate to assert that

ddd rdfs:subClassOf eee .

but this needs to be asserted explicitly; it does not follow from the subset relationship alone.

Assuming that the information encoded in these rules is correct, applying these and the earlier rules will produce a graph which is D-entailed by the original.

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 possible to add any number of leading zeros to any lexical form for xsd:integer without it ceasing to 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 not to omit some valid entailments, however, such canonicalization rules should 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 also define further such datatype-specific meanings.

These rules allow one to conclude that any well-formed typed literal of a recognized datatype will denote something in the class rdfs:Literal.

<ex:a> <ex:p> "sss"^^<ex:dtype> .
<ex:dtype> rdf:type rdfs:Datatype .

<ex:a> <ex:p> _:nnn . (by rule lg)
_:nnn rdf:type <ex:dtype> . (by rule rdfD1)
<ex:dtype> rdfs:subClassOf rdfs:Literal . (by rule rdfs11)
_:nnn rdf:type rdfs:Literal . (by rule rdfs9)

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

<ex:p> rdfs:range <ex:dtype> .
<ex:a> <ex:p> "sss"^^<ex:otherdtype> .

<ex:a> <ex:p> _:nnn .
_:nnn rdf:type <ex:otherdtype> . (by rule rdfD1)
_:nnn rdf:type <ex:dtype> . (by rule rdfs3)

These rules may not provide a complete set of inference principles for D-entailment, since there may be valid D-entailments for particular datatypes which depend on idiosyncratic properties of the particular datatypes, such as the size of the value space (e.g. xsd:boolean has only two elements, so anything established for those two values must be true for all literals with this datatype.) In particular, the value space and lexical-to-value mapping of the XSD datatype xsd:string sanctions the identification of typed literals with plain literals without language tags for all character strings which are in the lexical space of the datatype, since both of them denote the Unicode character string which is displayed in the literal; so the following inference rule is valid in all XSD-interpretations. Here, 'sss' indicates any RDF string in the lexical space of xsd:string.

Again, as with the rules rdfD2 and rdfD3, applications may use a systematic replacement of one of these equivalent forms for the other rather than apply these rules directly.

Appendix A: Proofs of Lemmas (Informative)

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

Proof. Let N be the empty set of triples. The semantic conditions on graphs require that N is true in I, for any I; so the first part follows from the definition of entailment. Suppose G is any nonempty graph and s p o . is a triple in G, then an interpretation I with IEXT(I(p)) = { } does not satisfy G but does satisfy N; so N does not entail G. QED.

This means that most of the subsequent results are trivial for empty graphs, which is what one would expect.

Subgraph Lemma. A graph entails all its subgraphs.

Proof. Obvious, from definitions of subgraph and entailment. If the graph is true in I then for some A, all its triples are true in I+A, so every subset of triples is true in I. QED

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

Proof. Obvious, from definitions of entailment and merge. All members of S are true if and only if all triples in the merge of S are true. QED.

This means that, as noted in the text, a set of graphs can be treated as a single graph when discussing satisfaction and entailment. This convention will be adopted in the rest of the appendix, where a reference to an interpretation of a set of graphs, a set of graphs entailing a graph, and so on, should be understood 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.

Instance Lemma. A graph is entailed by all its instances.

Proof. Suppose I satisfies E' and E' is an instance of E. Then for some mapping A on the blank nodes of E', I+A satisfies every triple in E'. For each blank node b in E, define B(b)=I+A(c), where c is the blank node or name that is substituted for b in E', or c=b if nothing was substituted for it. Then I+B(E)=I+A(E')=true, so I satisfies E. But I was arbitrary; so E' entails E. QED.

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. In RDF, Skolemization amounts to replacing every blank node in a graph by a 'new' name, i.e. a URI reference which is guaranteed to not occur anywhere else. In effect, it gives 'arbitrary' names to the anonymous entities whose existence was asserted by the use of blank nodes: the arbitrariness of the names ensures that nothing can be inferred that would not follow from the bare assertion of existence represented by the blank node. (Using a literal would not do. Literals are never 'new' in the required sense.)

To be precise, a Skolemization of E (with respect to V) is a ground instance of E with respect to V with a 1:1 instance mapping that maps each blank node in G to a URI reference that does not appear in G (so the Skolem vocabulary V must be disjoint from the vocabulary of E)

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 names from the Skolem vocabulary:

Skolemization Lemma. Suppose sk(E) is a skolemization of E with respect to V. Then sk(E) entails E; and if sk(E) entails F and the vocabulary of F is disjoint from V, then E entails F .

Proof. sk(E) entails E by the instance lemma.

Now, suppose that sk(E) entails F where F shares no vocabulary with V; and suppose I is some interpretation satisfying E. Then for some mapping A from the blank nodes of E, I+A satisfies E. Define an interpretation I' of the vocabulary of sk(E) by: IR'=IR, IEXT'=IEXT, I'(x)=I(x) for x in the vocabulary of E, and I'(x)=[I+A](y) for x in V, where y is the blank node in E that is replaced by x in sk(E). Clearly I' satisfies sk(E), so I' satisfies F. But I'(F)=[I+A](F) since the vocabulary of F is disjoint from that of V; so I satisfies F. But I was arbitrary; so E entails F.

QED.

Intuitively, this lemma shows that asserting a Skolemization expresses a similar content to asserting the original graph, in many respects. However, a graph should not be thought of as being equivalent to its Skolemization, since these 'arbitrary' names would have the same status as any other URI references 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. Nevertheless, for many purposes when proving results about entailment, we need only consider ground graphs: for provided E does not contain any Skolem vocabulary, S entails E iff S' entails E.

The proof of the subsequent lemmas uses a way of constructing an interpretation of a graph by using the lexical items in the graph itself. (This was Herbrand's idea; we here modify it slightly to handle literals appropriately.) Given a nonempty graph G, the (simple) Herbrand interpretation of G, written Herb(G), is the interpretation defined as follows.

Clearly Herb(G)+B, where B is the identity map on blank nodes in G, satisfies every triple in G, by construction, so Herb(G) satisfies G.

Herbrand interpretations treat URI references and typed literals (and blank nodes) in the same way as plain 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 construction shows that any graph can be interpreted in this way. This therefore establishes that any RDF graph has a satisfying simple interpretation, i.e. there cannot be a simple inconsistency in RDF.

Notice that the universe of the Herbrand interpretation of G contains the blank nodes of G; they are 'standing for' the entities that they assert the existence of, in effect. Since blank nodes must be interpreted to denote themselves in order to satisfy the graph, the Herbrand interpretation of a Skolemization of a graph is isomorphic with the Herbrand interpretation of the graph together with the blank node mapping: Herb(sk(G)) = Herb(G)+B (using a familiar abuse of notation where a blank node in a Herbrand interpretation is treated as a Skolem name.)

Interpolation Lemma. S entails E if and only if a subgraph of S is an instance of E.

Proof. 'if' follows from the subgraph and instance lemmas.

'only if' uses the Herbrand construction. Suppose S simply entails E. Herb(S) satisfies S, so Herb(S) satisfies E, i.e. for some mapping A from the blank nodes of E to IR_Herb(S), [Herb(S)+A] satisfies every triple
s p o .
in E, so S must contain the triple
[Herb(E)+A](s) p [Herb(E)+A](o) .
which is the instance of the previous triple under the instance mapping A. So the set of all such triples is a subgraph of S which is an instance of E.

QED

The following are direct consequences of the interpolation lemma:

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

Proof. Suppose E entails E', then a subgraph of E is an instance of E' and therefore a proper instance of E; so E is not lean , contrary to hypothesis. So E does not entail E'.
QED

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

Proof. By the interpolation lemma, a subgraph S' of S is an instance of E; so S' is finite, and S' entails E.
QED

Although compactness is trivial for simple entailment, it becomes progressively less trivial in more elaborate semantic extensions.

Monotonicity Lemma. Suppose S is a subgraph of S' and S entails E. Then S' entails E. (Special case of general monotonicity lemma) QED

General monotonicity lemma. Suppose that S, S' are sets of RDF graphs with every member of S a subset of some member of S'. Suppose that Y indicates a semantic extension of  X, S X-entails E, and S and E satisfy any syntactic restrictions of Y. Then S' Y-entails E.

Proof. This follows simply by tracing the definitions. Suppose that I is a Y-interpretation of S'; then since Y is a semantic extension of X, I is an X-interpretation; and by the subgraph and merge lemmas, I satisfies S; so I satisfies E.
QED

Both of the following proofs follow a common pattern which generalizes that used in the proof of the interpolation lemma, by using a modification of the Herbrand construction applied to a 'closure' obtained by applying rules to exhaustion. The proofs operate by showing that the resulting interpretation is both appropriate to the vocabulary and also acts similarly to the Herbrand interpretation. Much of the complexity of the proofs arises from the need to adapt the Herbrand construction in order to take proper account of literal values. Herbrand interpretations ignore literal typing and treat all typed literals as non-literal values; this is irrelevant when considering simple entailment, which treats typed literals simply as denoting names; but more care will be needed when considering rdf- and rdfs-interpretations.

Both proofs use a single basic idea which requires rather an awkward notation but is basically straightforward to understand. The simple Herbrand interpretation treats all vocabulary items as denoting themselves, and builds the interpretation out of these syntactic items. The semantic conditions on rdf- and rdfs-interpretations do not permit this in all cases: XML literals in particular are required to denote other kinds of entity. We therefore distinguish between the 'real' semantic values and their syntactic 'surrogates' in these cases, by defining a mapping sur from the universe of the intepretation to the vocabulary of the graph (plus blank nodes) which is as close as possible to being an identity mapping, but which when applied to these special literal values, identifies the particular blank node which acts as a witness for that value in the graph. In the case of RDFS, the surrogate mapping is extended to all literal values, since the blank node allocated to the literal can occur in a subject position, and hence record information about the literal value which must be applied back to that value in the interpretation.

RDF entailment lemma. S rdf-entails E if and only if there is a graph which can be derived from S plus the RDF axiomatic triples by the application of rule lg and the RDF entailment rules and which simply entails E.

Proof. To show 'if' one has only to check that the rules are rdf-valid, which is left as an exercise for the reader; and if S or E is empty then the result follows trivially; so suppose they are both non-empty.

To establish 'only if', the proof proceeds by constructing an rdf Herbrand interpretation RH of S which serves the same role for rdf-interpretations that the simple Herbrand interpretation does for simple interpretations. The construction follows the Herbrand construction as far as possible, but interprets well-formed XML literals so as to satisfy the RDF semantic conditions, guided by the triples in the RDF closure, C, defined to be the graph resulting from the following process:

add to S all the RDF axiomatic triples;
apply rule lg to any triple containing a well-typed XML literal until the graph is unchanged by the rule;
apply rule rdf2 until the graph is unchanged;
apply rule rdf1 until the graph is unchanged.

Note that C contains precisely one new blank node _:nnn allocated to each literal in S by the rule rule lg, and that the subgraph of triples in S containing any well-typed XML literal is reproduced exactly in C with this blank node replacing the literal and with the extra triple
_:nnn rdf:type rdf:XMLLiteral .
introduced by rule rdf2. Note also that the proof only requires that rule lg is used on well-typed XML literals, so that it actually establishes a slightly tighter result.

Blank nodes introduced by rule lg stand as surrogates for well-formed XML literals in the subject position of a triple. (In the proof of the next lemma, this will be extend to all literals.) In order to construct an RDF interpretation, XML literals and their surrogates must be replaced by the appropriate literal values in the domain of the interpretation, but the proof requires that each XML literal value be uniquely associated with a lexical item that denotes it. This requires some delicacy in the following construction.

If lll is a well-formed XML literal, let xml(lll) be the XML value of lll; and for each XML value x of any well-formed XML literal in C, let sur(x) be the blank node allocated to that XML literal by rule lg; and extend sur to be the identity mapping on URI references, blank nodes and all other literals in C.

RH is then defined by:

LVRH = all plain literals in C plus {xml(x): x a well-typed XML literal in S} IRRH = LVRH plus the set of URI references, blank nodes and other typed literals occurring in C IPRH = {x: C contains the triple: x rdf:type rdf:Property .} If x is in IPRH then IEXTRH(x) = {<s,o>: C contains the triple sur(s) x sur(o) . } ISRH is the identity map on URIrefs in S If x is a well-formed XML literal in S then ILRH(x) = xml(x), otherwise ILRH(x) = x

Define a mapping B on blank nodes in C as follows: B(x)=xml(lll) if x is allocated to a well-formed XML literal lll, otherwise B(x)=x, then clearly [RH+B] satisfies C and therefore S, so RH satisfies S.

Since C contains all the required RDF axiomatic triples, RH satisfies them.

It is easy to see that J satisfies the first two RDF semantic conditions, by construction; for the triples introduced by rule rdf2 require that IEXT_RH(rdf:type) contains <xml(lll),rdf:XMLLiteral> for every well-typed XML literal lll.

The third RDF semantic condition is the only negative semantic condition which cannot be satisfied simply by construction, but it is satisfied trivially. Ill-typed XML literals denote themselves in RH, and so are excluded from LV_RH by construction. The pair <lll, rdf:XMLLiteral> cannot occur in IEXT_RH(rdf:type) because a literal cannot occur in subject position; so the condition is satisfied, so RH is an rdf-interpretation.

Since S rdf-entails E, RH satisfies E: so for some mapping A from the blank nodes of E to IR_RH, [RH+A] satisfies every triple
s p o .
in E, i.e. IEXT_RH(p) contains <[RH+A](s),[RH+A](o)>, i.e.C contains a triple
sur([RH+A](s)) p sur([RH+A](o)).
but this is an instance of the first triple under the instantiation mapping x -> sur(A(x)); so a subgraph of C is an instance of E; so C simply entails E.

QED

This lemma also shows that any graph has a satisfying rdf-interpretation, and the proof illustrates how to construct it from a Herbrand interpretation of the closure, by interpreting well-formed XML literals appropriately and allowing the possible existence of properties which have no extensions. Note that if E is finite then the derived subgraph of C is also finite.

The proof of the RDFS entailment lemma is similar in structure and uses closely similar definitions, but is of course longer and requires a more elaborate construction to ensure that the class extensions of rdfs:Literal and rdfs:Resource contain all literal values.

RDFS entailment lemma. S rdfs-entails E if and only if there is a graph which can be derived from S plus the RDF and RDFS axiomatic triples by the application of rule lg, rule gl and the RDF and RDFS entailment rules and which either simply entails E or is an XML clash.

Proof. Again, to show 'if' it is sufficient to show that the RDFS entailment rules are rdfs-valid, which is again left as an exercise; and again, the empty cases are trivial.

The proof of 'only if' is similar to that used in the previous lemma, and similar constructions and terminology will be used, except that the RDFS closure, D, is defined to be the graph resulting from the following process:

add to S all the RDF and RDFS axiomatic triples;
apply rule lg to any triple containing a literal until the graph is unchanged by the rule;
apply rules rdf2 and rdfs1 until the graph is unchanged;
apply rule rdf1, rule gl and the remaining RDFS entailment rules until the graph is unchanged.

Unlike the previous lemma, this proof requires that rule lg is applied to all literals, even ill-typed XML literals, and it requires the inverse rule gl. Rule gl needs to be used only after an application of rules rdfs6 or rdfs10, since those are the only rules which can move a blank node from subject to object position. Note that D contains precisely one new blank node _:nnn allocated to each literal in S by the rule rule lg, and that the subgraph of triples in S containing any literal is reproduced exactly in D with this blank node replacing the literal and with the extra triple
_:nnn rdf:type rdfs:Literal .
introduced by rule rdfs1, and possibly also
_:nnn rdf:type rdf:XMLLiteral .
introduced by rule rdf2 when appropriate. This means that after this point in the construction, literals can effectively be ignored, since any of the subsequent rules which applies to triples containing a literal will also apply equally well to the similar triples where the literal is replaced by its allocated blank node. The rest of the proof uses this by requiring literal values in the interpretation to satisfy just the semantic conditions imposed on the blank nodes allocated to the corresponding literal, and ignoring triples in the graph which contain literals. The use of rule gl ensures that D contains any triple containing a literal if and only if it contains the similar triple with the literal replaced by its allocated blank node.

As in the previous proof, if lll is a well-formed XML literal, let xml(lll) be the XML value of lll; the surrogate mapping sur is then extended as follows. First, the domain of sur is the set containing just the URI references, literals and blank nodes occurring in D and all XML values of well-formed XML literals in D. (This is the universe of the rdfs-Herbrand interpretation, defined below.) Now, if lll is a well-formed XML literal in D, let sur(xml(lll)) be the blank node allocated to lll by rule lg; for any other literal lll in D, let sur(lll) be the blank node allocated to lll by rule lg, and for all URI references and blank nodes in D, let sur(x) = x . Note that the range of sur contains only URI references and blank nodes which occur in D.

The rdfs-Herbrand interpretation SH of S is then constructed similarly to the previous lemma.

LVSH = {x: D contains the triple: sur(x) rdf:type rdfs:Literal .} IRSH = LVSH plus the set of URI references, blank nodes and literals other than well-formed XML literals occurring in D IPSH = {x: D contains the triple: sur(x) rdf:type rdf:Property .} If x is in IPRH then IEXTRH(x) = {<s,o>: D contains the triple sur(s) x sur(o) . } ISSH is the identity map on URI references in S If x is a well-formed XML literal in S then ILSH(x) = xml(x), otherwise ILSH(x) = x

Define B(x) as follows: if x is a blank node allocated to a well-formed XML literal lll in D then B(x) = xml(lll); if it is allocated to any other literal lll in D then B(x)=lll; and otherwise B(x)=x; then clearly [SH+B] satisfies D and therefore S, so SH satisfies S.

As in the previous lemma, SH satisfies all the required RDF and RDFS axiomatic triples and the first two RDF semantic conditions by construction.

SH satisfies the third RDF semantic condition just in case D does not contain an XML clash. Note that the presence of surrogates for ill-typed XML literals invalidates the argument used in the previous lemma to the effect that this condition is trivally satisfied. So assume that D does not contain an XML clash.

As noted in the text, we can regard the first RDFS semantic condition as defining ICEXT and IC: we will do so without further comment and describe all the conditions in terms of IEXT. To show that SH satisfies the remaining RDFS semantic conditions we argue case by case, using the minimality of the Herbrand interpretation and the completeness of the closure.

All of these conditions can be mirrored by a corresponding sequence of rule applications. The general form of the argument can be illustrated with the case of the second RDFS semantic condition. Suppose <x,y> is in IEXT_SH(rdfs:domain) and <u,v> is in IEXT_SH(x); then D must contain the triples

sur(x) rdfs:domain sur(y) .
sur(u) x sur(v).

so x must be a URIref, so sur(x)=x; and then by rule rdfs2, it must also contain the triple

sur(u) rdf:type sur(y).

so IEXT_SH(rdf:type) contains <u,v> ; so the condition is satisfied.

The other cases proceed similarly, by translating the semantic condition into a derivation using the rules and axiomatic triples. The argument forms are summarized in the following table. Some of the semantic conditions split into several subconditions, and some also have special subcases.

RDFS semantic condition	Derivation
if x in IR then <x,rdfs:Resource> in IEXT(rdf:type)	URIref or bnode in subject: x a b x rdf:type rdfs:Resource	rdfs4a
	Literal: _:x rdf:type rdfs:Literal rdfs:type rdfs:range rdfs:Class rdfs:Literal rdf:type rdfs:Class rdfs:Literal rdfs:subClassOf rdfs:Resource _:x rdf:type rdfs:Resource	see below axiomatic rdfs3 rdfs8 rdfs9
	URIref or bnode in object: a b x x rdf:type rdfs:Resource	rdfs4b
	URIref in predicate: a x b x rdf:type rdf:Property rdf:type rdfs:domain rdfs:Resource x rdf:type rdfs:Resource	rdf1 axiomatic rdfs2
x in LV iff <x,rdfs:Literal> in IEXT(rdf:type)	well-typed XML literal lll: a b lll _:x rdf:type rdf:XMLLiteral rdf:XMLLiteral rdfs:subClassOf rdfs:Literal _:x rdf:type rdfs:Literal	lg, rdf2, sur(xml(lll))=_:x axiomatic rdfs9
	other literal lll : a b lll _:x rdf:type rdfs:Literal	lg, rdfs1, sur(lll)=_:x
if <x,y> in IEXT(rdfs:domain) and <u,v> in IEXT(x) then <u,y> in IEXT(rdf:type)	x rdfs:domain y . u x v . u rdf:type y .	rdfs2
if <x,y> in IEXT(rdfs:range) and <u,v> in IEXT(x) then <v,y> in IEXT(rdf:type)	x rdfs:range y . u x v . v rdf:type y .	rdfs3
if <x,rdf:Property> in IEXT(rdf:type) then <x,x> in IEXT(rdfs:subPropertyOf)	x rdf:type rdf:Property x rdfs:subPropertyOf x	rdfs6
if <x,rdf:Property> in IEXT(rdf:type) <y,rdf:Property> in IEXT(rdf:type) <z,rdf:Property> in IEXT(rdf:type) <x,y> in IEXT(rdfs:subPropertyOf) <y,z> in IEXT(rdfs:subPropertyOf) then <x,z> in IEXT(rdfs:subPropertyOf)	x rdfs:subPropertyOf y y rdfs:subPropertyOf z x subPropertyOf z	rdfs5
if <x,y> in IEXT(rdfs:subPropertyOf) <u,v> in IEXT(x) then <x,rdf:Property> in IEXT(rdf:type) <y,rdf:Property> in IEXT(rdf:type) <u,v> in IEXT(y)	x rdfs:subPropertyOf y u x v rdfs:subPropertyOf rdfs:domain rdf:Property x type rdf:Property rdfs:subPropertyOf rdfs:domain rdf:Property y rdf:type rdf:Property u y v	axiomatic triple rdfs2 axiomatic triple rdfs3 rdfs7
if <x,rdfs:Class> in IEXT(rdf:type) then <x,rdfs:Resource> in IEXT(rdfs:subClassOf)	x rdf:type rdfs:Class x rdfs:subClassOf rdfs:Resource	rdfs8
if <x,y> in IEXT(rdfs:subClassOf) <u,x> in IEXT(rdf:type) then <x,rdfs:Class> in IEXT(rdf:type) <y,rdfs:Class> in IEXT(rdf:type) <u,y> in IEXT(rdf:type)	x rdfs:subClassOf y u rdf:type x rdfs:subClassOf rdfs:domain rdfs:Class x rdf:type rdfs:Class rdfs:subClassOf rdfs:range rdfs:Class y rdf:type rdfs:Class u rdf:type y	axiomatic triple rdfs2 axiomatic triple rdfs3 rdfs9
if <x,rdfs:Class> in IEXT(rdf:type) then <x,x> in IEXT(rdfs:subClassOf)	x rdf:type rdfs:Class x rdfs:subClassOf x	rdfs10
if <x,rdfs:Class> in IEXT(rdf:type) <y,rdfs:Class> in IEXT(rdf:type) <z,rdfs:Class> in IEXT(rdf:type) <x,y> in IEXT(rdfs:subClassOf) <y,z> in IEXT(rdfs:subClassOf) then <x,z> in IEXT(rdfs:subClassOf)	x rdfs:subClassOf y y rdfs:subClassOf z x rdfs:subClassOf z	rdfs11
if <x,rdfs:ContainerMembershipProperty> in IEXT(rdf:type) then <x,rdfs:member> in IEXT(rdfs:subPropertyOf)	x rdf:type rdfs:ContainerMembershipProperty x rdfs:subPropertyOf rdfs:member	rdfs12
if <x,rdfs:Datatype> in IEXT(rdf:type) then <x,rdfs:Literal> in IEXT(rdfs:subClassOf)	x rdf:type rdfs:Datatype x rdfs:subClassOf rdfs:Literal	rdfs13

so SH is an rdfs-interpretation.

Since S rdfs-entails E, SH satisfies E: so for some mapping A from the blank nodes of E to IR_SH, [SH+A] satisfies every triple
s p o .
in E, i.e. IEXT_SH(p) contains <[SH+A](s),[SH+A](o)>, i.e.D contains a triple

sur([SH+A](s)) p sur([SH+A](o)).

which is an instance of the first triple under the instantiation mapping x -> sur(A(x)), unless o is a literal. If o is a literal, then sur([SH+A](o) is the blank node allocated to o, and so D also contains the triple

sur([SH+A](s)) p o .

which is an instance of the first triple under the instantiation mapping x -> sur(A(x)). So a subgraph of D is an instance of E under the instantiation mapping x -> sur(A(x)); so D simply entails E.

So either D contains an XML clash or D simply entails E, so D satisfies the conditions of the lemma.

QED

Note that if E is finite, or if D contains an XML clash, then a finite subgraph of D also satisfies the conditions of the lemma.

Appendix B: 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, in RDF, 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 hierarchies, as explained earlier.)

Complete (adj., of an inference system). (1) Able to detect all entailments between any two expressions. (2) Able to draw all valid inferences. See Inference. Also used with a qualifier: able to detect entailments or 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.)

(These definitions are not exactly equivalent, since the first requires that the system has access to the consequent of the entailment, and may be unable to draw 'trivial' inferences such as (p and p) from p. Typically, efficient mechanical inference systems may be complete in the first sense but not necessarily in the second.)

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

Consistent (adj., of an expression) Having a satisfying interpretation; not internally contradictory. (Also used of an inference system as synonym for Correct.)

Correct (adj., of an inference system). Unable to draw any invalid inferences, or unable to make false claims of entailment. See Inference.

Decideable (adj., of an inference system). Able to determine for any pair of expressions, in a finite time with finite resources, whether or not the first entails the second. (Also: adj., of a logic:) Having a decideable inference system which is complete and correct for the semantics of the logic.

(Not all logics can have inference systems which are both complete and decideable, and decideable inference systems may have arbitrarily high computational complexity. The relationships between logical syntax, semantics and complexity of an inference system continue to be the subject of considerable research.)

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.

(Entailment and inconsistency are closely related, since A entails B just when (A and not-B) is inconsistent, c.f. the second definition for entailment. This is the basis of many mechanical inference systems.

Although the definitions of consistency and inconsistency are exact duals, they are computationally dissimilar. It is often harder to detect consistency in all cases than to detect inconsistency in all cases.)

Indexical (adj., of an 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, organized system 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. The semantics described in this document is basically intensional.)

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.)

Logic (n.) A formal language which expresses propositions.

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 has been avoided 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 (birds normally 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 he or she 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 emphasize 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.

Well-formed (adj., of an expression). Syntactically legal.

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 C: Acknowledgements

This document reflects the joint effort of the members of the RDF Core Working Group. Particular contributions were made by Jeremy Carroll, Dan Connolly, Jan Grant, R. V. Guha, Graham Klyne, Ora Lassilla, Brian McBride, Sergey Melnick, Jos deRoo and Patrick 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.

Peter Patel-Schneider and Herman ter Horst found several major problems in earlier drafts, and suggested several important technical improvements.

Patrick Hayes' work on this document was supported in part by DARPA under contract #2507-225-22.

References

Normative

[RDF-CONCEPTS]

Resource Description Framework (RDF): Concepts and Abstract Syntax, Graham Klyne and Jeremy J. Carroll, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/ . Latest version available at http://www.w3.org/TR/rdf-concepts/ .

[RDF-SYNTAX]

RDF/XML Syntax Specification (Revised), Dave Beckett, Editor, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-syntax-grammar-20040210/ . Latest version available at http://www.w3.org/TR/rdf-syntax-grammar/ .

[RDF-TESTS]

RDF Test Cases, Jan Grant and Dave Beckett, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-testcases-20040210/ . Latest version available at http://www.w3.org/TR/rdf-testcases/ .

[RDFMS]

Resource Description Framework (RDF) Model and Syntax Specification , O. Lassila and R. Swick, Editors. World Wide Web Consortium. 22 February 1999. This version is http://www.w3.org/TR/1999/REC-rdf-syntax-19990222/. The latest version of RDF M&S is available at http://www.w3.org/TR/REC-rdf-syntax/.

[RFC 2119]

RFC 2119 - Key words for use in RFCs to Indicate Requirement Levels , S. Bradner, IETF. March 1997. This document is http://www.ietf.org/rfc/rfc2119.txt.

[RFC 2396]

RFC 2396 - Uniform Resource Identifiers (URI): Generic Syntax Berners-Lee,T., Fielding and Masinter, L., August 1998

[XSD]

XML Schema Part 2: Datatypes, Biron, P. V., Malhotra, A. (Editors) World Wide Web Consortium Recommendation, 2 May 2001

Non-Normative

[OWL]

OWL Web Ontology Language Reference, Mike Dean and Guus Schreiber, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-owl-ref-20040210/ . Latest version available at http://www.w3.org/TR/owl-ref/ .

[Conen&Klapsing]

A Logical Interpretation of RDF, Conen, W., Klapsing, R..Circulated to RDF Interest Group, August 2000.

[DAML]

Frank van Harmelen, Peter F. Patel-Schneider, Ian Horrocks (editors), Reference Description of the DAML+OIL (March 2001) ontology markup language

[Hayes&Menzel]

A Semantics for the Knowledge Interchange Format, Hayes, P., Menzel, C., Proceedings of 2001 Workshop on the IEEE Standard Upper Ontology, August 2001.

[KIF]

Michael R. Genesereth et. al., Knowledge Interchange Format, 1998 (draft American National Standard).

[Marchiori&Saarela]

Query + Metadata + Logic = Metalog, Marchiori, M., Saarela, J. 1998.

[LBASE]

Lbase: Semantics for Languages of the Semantic Web, Guha, R. V., Hayes, P., W3C Note, 10 October 2003.

[McGuinness&al]

DAML+OIL:An Ontology Language for the Semantic Web, McGuinness, D. L., Fikes, R., Hendler J. and Stein, L.A., IEEE Intelligent Systems, Vol. 17, No. 5, September/October 2002.

[RDF-PRIMER]

RDF Primer, Frank Manola and Eric Miller, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-primer-20040210/ . Latest version available at http://www.w3.org/TR/rdf-primer/ .

[RDF-VOCABULARY]

RDF Vocabulary Description Language 1.0: RDF Schema, Dan Brickley and R. V. Guha, Editors, W3C Recommendation, 10 February 2004, http://www.w3.org/TR/2004/REC-rdf-schema-20040210/ . Latest version available at http://www.w3.org/TR/rdf-schema/ .

Appendix D: Change Log. (Informative)

Changes since the 15 December 2003 Proposed Recommendation.

An error in the wording of the RDFS entailment lemma in appendix A was corrected. Some typos in the glossary and main text were corrected.

After considering comments by ter Horst, the definition of D-interpretation has been modified so as to apply to an extended vocabulary including the datatype names.

Older entries in the change log were removed. They can be found in the previous version.