Copyright ©2002 W3C^{®} (MIT, INRIA, Keio), All Rights Reserved. W3C liability, trademark, document use and software licensing rules apply.
This is a specification of a modeltheoretic semantics for RDF and RDFS, and some basic results on entailment. This document was written with the intention of providing a precise semantic theory for RDF and RDFS, and to sharpen the notions of consequence and inference. It reflects the current understanding of the RDF Core working group at the time of writing. In some particulars this differs from the account given in Resource Description Framework (RDF) Model and Syntax Specification, and these exceptions are noted.
This work is part of the W3C Semantic Web Activity. It has been produced by the RDF Core Working Group which is chartered to address a list of issues raised since RDF 1.0 was issued. This document reflects the Working Group's recent deliberation of some of these issues, but the Working Group has not yet decided whether or how to integrate this document with the RDF 1.0 specification ultimately.
This document is a W3C Working Draft for review by W3C members and other interested parties. It is a draft document and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use W3C Working Drafts as reference material or to cite them as other than "work in progress". A list of current public W3C Working Drafts can be found as part of the W3C Technical Reports and Publications.
Please address feedback on this document to mailto:wwwrdfcomments@w3.org, a mailing list with public archive). The editors and the Working Group plan to address feedback in future revisons of this document.
0. Introduction
0.1 Modeltheoretic
semantics
0.2 Graph Syntax
0.3 Definitions
1. Interpretations
1.1 Technical notes
1.2 Urirefs, resources and
literals
1.3 Interpretations
1.4 Denotations of ground
graphs
1.5 Unlabeled nodes as
existential assertions
1.6 Comparison with formal
logic
2. Simple Entailment between RDF
graphs
2.1 Criteria for
nonentailment
3. Interpreting the RDF(S)
vocabulary
3.1 RDF
interpretations
3.2 Reification and
containers
3.2.1 Reification
3.2.2 RDF
Containers
3.3 RDFS
Interpretations
3.3.1 A note on
rdfs:Literal
4. Vocabulary entailment and
closure rules
4.1. Rdfentailment and rdf
closures
4.2. Rdfsentailment and
rdfs closures
4.3 A note on computing
and entailment
Appendix A. Summary of model theory
Appendix B. Proofs of lemmas
Appendix C. Acknowledgements
References
Change Log
A modeltheoretic semantics for a language assumes that the language refers to a 'world', and describes the minimal conditions that a world must satisfy in order to assign an appropriate meaning for every expression in the language. A particular world is called an interpretation, so that model theory might be better called 'interpretation theory'. The idea is to provide an abstract, mathematical account of the properties that any such interpretation must have, making as few assumptions as possible about its actual nature or intrinsic structure. Model theory tries to be metaphysically and ontologically neutral. It is typically couched in the language of set theory simply because that is the normal language of mathematics  for example, this semantics assumes that names denote things in a set IR called the 'universe'  but the use of settheoretic language here is not supposed to imply that the things in the universe are settheoretic in nature.
The chief utility of such a semantic theory is not to suggest any particular processing model, or to provide any deep analysis of the nature of the things being described by the language (in our case, the nature of resources), but rather to provide a technical tool to analyze the semantic properties of proposed operations on the language; in particular, to provide a way to determine when they preserve meaning.Model theory is usually most relevant to implementation via the notion of entailment, described later, and by making it possible to define valid inference rules.
This document describes a model theory for RDF(S) which treats the language as simple assertional language, in which each triple makes a distinct assertion and the meaning of any triple is not changed by adding other triples. This imposes a fairly strict monotonic discipline on the language, so that it cannot express closed world assumptions, local default preferences, and several other commonlyused nonmonotonic constructs. There are several aspects of meaning in RDF which are ignored by this semantics; in particular, it treats URIs as simple names, ignoring aspects of meaning encoded in particular URI forms [RFC 2396] and does not provide any analysis of timevarying data or of changes to URI denotations.
Any semantic theory must be attached to a syntax. Of the several syntactic forms for RDF, we have chosen the RDF graph as introduced in [RDFMS] as the primary syntax, largely for its simplicity. We understand linear RDF notations such as NTriples and rdf/xml [RDF/XML] as lexical notations for specifying RDF graphs. (There are wellformed graphs that cannot be described by these notations, however.) Two RDF documents, in whatever lexical form, are syntactically equivalent if and only if they map to the same RDF graph. The model theory assigns interpretations directly to the graph; we will refer to this as the 'graph syntax' to avoid ambiguity, since the bare term 'syntax' is often assumed to refer to a lexicalization.
To describe RDF graphs it is first necessary to define the things that can act as nodes and arcs of the graph. There are three kinds of node in any RDF graph: urirefs, literals, and blank nodes. A uriref is defined to be a URI reference in the sense of [RFC 2396]. An RDF literal has three parts ( a bit, a character string, and a language tag), but we will treat them simply as character strings, since the other parts of the literal play no role in the model theory. Blank (unlabeled) nodes are considered to be drawn from some set of 'anonymous' entities which have no label and are unique to the graph. Finally, every arc in an RDF graph is labelled with a uriref. The same uriref may label several arcs and also be a node in the graph. An RDF graph can then be defined as a set of triples of the form <S, P, O>, where P is a uriref, S is either a uriref or a blank node, and O is either a uriref, a blank node, or a literal. It is convenient to adopt a familiar abuse of terminology and identify a single triple with the graph consisting of the singleton set containing that triple.
We refer to urirefs and literals (but not blank nodes) as names; but note that there is no distinction between the name of a node and the node itself. A name may occur in more than one graph, but blank nodes are unique to each graph. This reflects the fact that literals always have the same meaning and urirefs are considered to have a 'global' meaning but blank nodes do not. (Note that this is a simplification from earlier versions, which distinguished between literals and literal nodes.)
The convention that relates such a set of triples to a picture of an RDF graph can then be stated as follows. Draw one oval for each blank node and uriref, and one rectangle for each literal, which occur in either the S or O position in any triple in the set, and write each uriref or literal as the label of its shape. Then for each triple <S,P,O>, draw an arrowed line from the shape produced from S to the shape produced from O, and label it with P.
In this document we will also use the Ntriples syntax described in [RDFTestCases] to describe RDF graphs. This notation uses a nodeID convention to indicate blank nodes in the triples of a graph. Note that while these node identifiers (formerly called bNodes) identify blank nodes in the surface syntax, these expressions are not considered to be the label of the graph node they identify; they are not names, and do not occur in the actual graph. In particular, two Ntriples documents which differ only by renaming their node identifiers will be understood to describe identical RDF graphs.
The Ntriples syntax requires that urirefs be given in full. In the interests of brevity, we use the imaginary URI prefix 'ex:' to provide "generic" examples. To obtain a more realistic view of the normal appearance of the Ntriples syntax, the reader should imagine this replaced with something like 'http://example.org/rdf/mt/artificialExample/'. We will also make extensive use of the Qname prefixes rdf: and rdfs: defined as follows:
Prefix rdf: namespace URI: http://www.w3.org/1999/02/22rdfsyntaxns#
Prefix rdfs: namespace URI: http://www.w3.org/2000/01/rdfschema#
Since Qname syntax is not legal in the Ntriples syntax, and in the interests of brevity and readability, we will sometimes use a convention whereby a Qname in square brackets is used to indicate the corresponding uriref enclosed in corner brackets, eg the triple
<ex:a> [rdf:type] [rdfs:Property] .
should be read as an abbreviation for the Ntriples syntax
<ex:a>
<http://www.w3.org/1999/02/22rdfsyntaxns#type>
<http://www.w3.org/2000/01/rdfschema#Property>
.
However, the reader is cautioned that this notation is not normative or correct Ntriples syntax, and is used in this document only for editorial and orthographic convenience.
Other RDF serializations may use other means of indicating the graph structure; for our purposes, the important syntactic property of RDF graphs is that each distinct item in an RDF graph is treated as a distinct referring entity in the graph syntax.
Several definitions will be important in what follows.
A subgraph of an RDF graph is simply a subset of the triples in the graph. Notice that each triple in a graph is considered to be a subgraph.
The result of taking the setunion of two or more RDF graphs (i.e. sets of triples) is another graph, which we will call the merge of the graphs. Each of the original graphs is a subgraph of the merged graph. Notice that when forming a merged graph, two occurrences of a given uriref or literal as nodes in two different graphs become a single node in the union graph (since by definition they are the same uriref or literal), but blank nodes are not 'merged' in this way; and arcs are of course never merged.In particular, this means that every blank node in a merged graph can be identified as coming from one particular graph in the original set of graphs.
Notice that one does not, in general, obtain the merge of a set of graphs by concatenating their corresponding Ntriples documents and constructing the graph described by the merged document, since if some of the documents use the same node identifiers, the merged document will describe a graph in which some of the blank nodes have been 'accidentally' merged. To merge Ntriples documents it is necessary to check if the same nodeID is used in two or more documents, and to replace it with a distinct nodeID in each of them, before merging the documents.
An RDF graph will be said to be ground if it has no blank nodes.
We will refer to a set of urirefs as a vocabulary. The vocabulary of a graph is the set of urirefs that it contains. Intuitively, the vocabulary of a graph is the set of names that need to be given an interpretation. We do not include literals in the vocabulary because their interpretation is fixed.
An instance of an RDF graph is, intuitively, a similar graph in which some blank nodes may have been replaced by names. However, it is technically convenient to allow blank nodes to be 'replaced' by other blank nodes, so we need to state this rather more precisely. Say that one triple is an instance of another if it can be obtained by substituting zero or more urirefs, literals or blank nodes for blank nodes in the original; and that a graph is an instance of another just when every triple in the first graph is an instance of a triple in the second graph, and every triple in the second graph has an instance in the first graph. Note that any graph is an instance of itself.
This allows blank nodes in the second graph to be replaced by names in the instance (notice that this might cause some nodes to be identified that were previously distinct) but it also allows them to be replaced by other blank nodes. In particular, this means that the two graphs:
<ex:a> <ex:b> _:xxx
.
<ex:a> <ex:b> _:yyy
.
and
<ex:a> <ex:b> _:zzz
.
with, respectively, three nodes and two arcs, and two nodes and one arc, are instances of each other. Similarly,
_:xxx <ex:b> _:xxx .
is an instance of
_:xxx <ex:b> _:yyy .
A proper instance of a graph is an instance in which at least one blank node has been replaced by a name.The above examples are not proper instances.
Throughout this document, the fact that two sets are given different names should not be taken to imply that they are disjoint. We will explicitly state any disjointness or containment conditions as they arise. In the same spirit, the fact that one set is stated to be a subset of another should not be interpreted as saying that these sets cannot be identical, unless this is stated explicitly.(These conventions are normal in mathematical discourse, but may violate the conventions expected by some readers.)
We do not impose any restrictions on the domains and ranges of properties; in particular, a property may be applied to itself. When classes are introduced in RDFS, we will assume that they can contain themselves. This might seem to violate one of the axioms of standard (ZermeloFraenkel) set theory, the axiom of foundation, which forbids infinitely descending chains of membership. However, the semantic model given here distinguishes properties and classes as objects from their extensions  the sets of objectvalue 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], which gives a semantics for an infinitary extension of full firstorder logic.
In this respect, RDFS differs from many conventional ontology frameworks such as UML which assume a more structured system of 'layers', or draw a distinction between data and metadata. However, while RDFS does not assume the existence of such structure, it does not prohibit it. If this aspect of RDFS is found worrying, then it is possible to restrict oneself to a subset of RDF graphs which do not contain any such 'loops' of class membership or property application, and still retain almost all the expressive power of RDFS for almost all practical purposes. RDF allows such loops, but it does not mandate their use for all parts of a user vocabulary.
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.
RDF uses two kinds of referring expression; urirefs and literals. We make very simple and basic assumptions about these. Urirefs are treated as logical constants, i.e. as names which denote something (the things are called 'resources', but no assumptions are made about the nature of resources.) Literals are treated as denoting themselves, ie each literal is essentially a quoted character string. (A future extension of RDF is planned which will allow datatypes to associate values with literals.) We will define LV to be the set of all possible literals, ie all character strings that can occur as a literal label in an RDF graph.
An interpretation assigns meanings to symbols in a particular vocabulary of urirefs. Some interpretations may assign special meanings to the symbols in a particular namespace, which we will call a reserved vocabulary.We will use two reserved vocabularies in this document, defined using the Qname syntax with the prefixes rdf: and rdfs: as follows:
Interpretations which share the special meaning of a particular reserved vocabulary will be named for that vocabulary, so that we will speak of 'rdfinterpretations' and 'rdfsinterpretations', etc.. An interpretation with no reserved vocabulary will be called a simple interpretation, or simply an interpretation.
We do not take any position here on the way that urirefs may be composed from other expressions, e.g. from relative URIs or Qnames; the model theory simply assumes that such lexical issues have been resolved in some way that is globally coherent, so that a single uriref can be taken to have the same meaning wherever it occurs.
Similarly, the model theory given here has no special provision for tracking temporal changes. It assumes, implicitly, that urirefs have the same meaning whenever they occur.To provide an adequate semantics which would be sensitive to temporal changes is a research problem which is beyond the scope of this document.
The 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 truthvalue (true or false) of any ground RDF triple. It does this by specifying for each uriref, what it is supposed to be a name of; and also, if it is used to indicate a property, what values that property has for each thing in the universe. Together with our assumed fixed interpretation of literals, this is just enough to fix the truthvalue of any ground triple, and hence any ground RDF graph.(We will show how to determine the truthvalues of nonground graphs in the following section.) Notice that if we left any of this information out, it would be possible for some wellformed 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 truthvalues of any triple in the world being specified.
Asserting an RDF graph amounts to claiming that it is true, which is another way of saying that the world it describes is, in fact, so arranged as to be an interpretation which makes it true. In other words, asserting a piece of RDF amounts to asserting a constraint on the possible ways the world might be. Notice that there is no presumption here that any RDF graph contains enough information to specify a single unique interpretation. It is very difficult, and usually impossible, to assert enough in any language to completely constrain the interpretations to a single possible world, so there is no such thing as 'the' unique RDF interpretation. In general, increasing the size of a graph  saying more about the world  decreases the set of interpretations that an assertion of the graph allows to be true  the number of ways the world could be, while making the asserted graph true of it.
The use of 'public' URIs in an RDF graph is often taken to imply that an assertion of the graph implicitly assents to the truth of other RDF graphs that define the meaning of that URI. To apply the model theory to this kind of situation, one should think of the assertion of such a graph as amounting to an assertion of the merge of that graph together with whatever RDF graphs are assumed to define the public vocabulary, in order to fully convey the intended meaning of the 'public' assertion.
This only applies to uses of RDF that are intended to be the assertion of simple propositional content. To give a fully adequate account of what it means to make an assertion in a Web context is a research problem that is beyond the scope of this document.
All interpretations will be relative to a set of urirefs, called the vocabulary of the interpretation; so that one has to speak, strictly, of an interpretation of an RDF vocabulary, rather than of RDF itself.
A simple interpretation I of a vocabulary V is defined by:
1. A nonempty set IR of resources, called the domain or universe of I.
2. A mapping IEXT from IR into the powerset of IR x (IR union LV) i.e. the set of sets of pairs <x,y> with x in IR and y in IR or LV
3. A mapping IS from V into IR
IEXT(x) is a set of pairs which identify the arguments for which the property is true, i.e. a binary relational extension, called the extension of x. This trick of distinguishing a relation as an object from its relational extension allows a property to occur in its own extension, as noted earlier. Note that no particular relationship is assumed between IR and LV; an interpretation is allowed to treat them as disjoint or to include all or part of LV in IR.
It is convenient to define IP to be the subset of IR with a nonempty extension. Intuitively, IP is the set of properties.
In the next sections we give the exact rules for how an interpretation of a vocabulary determines the truthvalues of any RDF graph, by a recursive definition of the "denotation"  the semantic "value"  of any RDF expression in terms of those of its immediate subexpressions. RDF has two kinds of denotation: node labels denote things, and sets of triples denote truthvalues.
The denotation of a ground RDF graph in I is given recursively by the following rules, which extend the interpretation mapping I from labels to ground graphs. These rules (and extensions of them given later) work by defining the denotation of any piece of RDF syntax E in terms of the denotations of the immediate syntactic constitutents of E, hence allowing the denotation of any piece of RDF to be determined by a kind of syntactic recursion.
if E is a literal then I(E) = E 
if E is a uriref then I(E) = IS(E) 
if E is a triple s p o . then I(E) = true if <I(s),I(o)> is in IEXT(I(p)), otherwise I(E)= false. 
if E is a ground RDF graph then I(E) = false if I(E') = false for some triple E' in E, otherwise I(E) =true. 
Notice that if the vocabulary of an RDF graph contains urirefs that are not in the vocabulary of an interpretation I  that is, if I simply does not give a semantic value to some name that is used in the graph  then these truthconditions 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 denote something in the world. Similarly, the first rule implies that the universe must contain every literal in the graph.
As an illustrative example, the following
is a small interpretation for the artificial vocabulary
{ex:a, ex:b, ex:c
}. We use integers to
indicate the 'things' in the universe. This is not meant to
imply that RDF interpretations should be interpreted as being
about arithmetic, but more to emphasize that the exact nature
of the things in the universe is irrelevant.
IR= {1, 2}; IP = {1}
IEXT: 1>{<1,2>,<2,1>}
IS: ex:a
>1, ex:b
>1,
ex:c
>2
Figure 1: An example of an interpretation. Note, this
is not a picture of an RDF graph.
This interpretation makes these triples true:
<ex:a> <ex:b> <ex:c>
.
<ex:c> <ex:a> <ex:a>
.
<ex:c> <ex:b> <ex:a>
.
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.
It makes these triples false:
<ex:a> <ex:c> <ex:b>
.
<ex:a> <ex:b> <ex:b>
.
<ex:c> <ex:a> <ex:c>
.
For example, I(<ex:a ex:c ex:b>
) = true
if <I(ex:a
),I(ex:b
)>,
i.e.<1,2>, is in IEXT(I(ex:c
)); but
I(ex:c
)=2 and IEXT is not defined on 2, so the
condition fails and I(<ex:a ex:c ex:b>
) =
false.
Since the universe of this interpretation contains no character strings as objects, any triple with a literal object would be false.
To emphasize; this is only one possible interpretation of this vocabulary; there are (infinitely) many others. For example, if we modified this interpretation by attaching the property extension to 2 instead of 1, none of the above six triples would be true.
Blank nodes are treated as simply indicating the existence of a thing, without using, or saying anything about, the name of that thing. (Note that this is not the same as assuming that the blank node indicates an 'unknown' uriref; for example, it does not assume that there is any uriref which refers to the thing.See http://www.w3.org/2000/03/rdftracking/#rdfmsidentityanonresources for a summary and pointers to the extended discussions on this issue. The discussion of skolemization in section 2.1 is also relevant.)
We now show how an interpretation can specify the truthvalue of a graph containing blank nodes. This will require some definitions, as the theory so far provides no meaning for unlabeled nodes. Suppose I is an interpretation and A is a mapping from some set of unlabeled 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 unlabeled nodes. Define anon(E) to be the set of unlabeled nodes in E. Then we can extend the above rules to include the two new cases that are introduced when unlabeled nodes occur in the graph:
If E is an unlabeled node then [I+A](E) = A(E) 
If E is an RDF graph then I(E) = true if [I+A'](E) = true for some mapping A' from anon(E) to IR, otherwise I(E)= false. 
Notice that we have not changed the definition of an interpretation; it still consists of the same values IR, IEXT and IS. We have simply extended the rules for defining denotations under an interpretation, so that the same interpretation that provides a truthvalue for ground graphs also assigns truthvalues to graphs with unlabeled nodes, even though it provides no denotation for the unlabeled nodes themselves. Notice also that the unlabeled nodes themselves are perfectly welldefined entities with a robust notion of identity; they differ from other nodes only in not being assigned a denotation by an interpretation, reflecting the intuition that they have no 'global' meaning (i.e. outside the graph in which they occur).
This effectively treats all blank nodes as having the same meaning as existentially quantified variables in the RDF graph in which they occur. Notice however that that since two unlabeled nodes cannot have the same label, there is no need to specify the scope of the quantifier within a graph, and no need to use any explicit quantifier syntax.( If we were to apply the semantics directly to Ntriples syntax, we would need to indicate the quantifier scope, since in this lexicalization syntax the same node identifier may occur several times corresponding to a single blank node in the graph. The above rule amounts to the convention that would place the quantifiers just outside, or at the outer edge of, the Ntriple document corresponding to the graph.)
For example, with this convention, the graph defined by the following triples is false in the interpretation shown in figure 1:
_:xxx <ex:a> <ex:b>
.
<ex:c> <ex:b> _:xxx
.
since if A' maps the unlabeled 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, when taken as a single graph, is true in I, but their conjunction is not; and that if a different node ID 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.
With this semantics, it is simple to translate an RDF graph into a logical expression with essentially the same meaning, as several authors have noted previously [Marchiori&Saarela], [Fikes&McGuinness].To handle literals properly the logic should allow quoted character string constants.
Each uriref translates to a logical constant, each literal to a quoted string, and each unlabeled node to a distinct variable. An arc labeled with p from a node n1 to a node n2 maps to an atomic assertion that the relation p holds true between the expressions s and o gotten by translating n1 and n2 respectively (written as (p s o) in KIF syntax); and finally, an RDF graph translates to the existential closure of the conjunction of the translations of all the arcs in the graph.This requires us to introduce bound variables to correspond to the blank nodes of the graph, similarly to the use of node identifiers in the NTriples syntax.
For example, the graph defined in the above example translates to the logical expression (written in the extended KIF syntax defined in [Hayes&Menzel])
(exists (?y)(and (ex:a ?y ex:b)(ex:b ex:c
?y)))
This translation maps the model theory exactly. Notice
however that the resulting expression may contain the same
symbol in both relation and object positions (e.g.
'ex:b
' in this example), which is considered
syntactically illegal in many versions of logic.To map to a
more conventional logical syntax one can use a 'dummy' ternary
relation symbol to assert that a binary relation holds between
two arguments. For example,[Fikes&McGuinness]
translate the RDF triple
s p o
into the KIF expression
(PropertyValue p s o)
The above example would then map to
(exists (?y)(and (PropertyValue ex:a ?y
ex:b)(PropertyValue ex:b ex:c ?y)))
Under this translation, to obtain the appropriate KIF interpretation one has to interpret (PropertyValue x y z) to mean ((IEXT x) y z).
Following conventional terminology, we say that I satisfies E if I(E)=true, and that a set S of expressions (simply) entails E if every interpretation which satisfies every member of S also satisfies E. If the singleton set {E} entails E' then we will simply say that E entails E'. In later sections these notions will be adapted to classes of interpretations with particular reserved vocabularies, but throughout this section entailment should be interpreted as simple RDF entailment.
Entailment is the key idea which connects modeltheoretic semantics to realworld applications. As noted earlier, making an assertion amounts to claiming that the world is an interpretation which assigns the value true to the assertion. If A entails B, then any interpretation that makes A true also makes B true, so that an assertion of A already contains the same "meaning" as an assertion of B; we could say that the meaning of B is somehow contained in, or subsumed by, that of A. If A and B entail each other, then they both "mean" the same thing, in the sense that asserting either of them makes the same claim about the world. The interest of this observation arises most vividly when A and B are different expressions, since then the relation of entailment is exactly the appropriate semantic licence to justify an application inferring or generating one of them from the other. Through the notions of satisfaction, entailment and validity, formal semantics gives a rigorous definition to a notion of "meaning" that can be related directly to computable methods of determining whether or not meaning is preserved by some transformation on a representation of knowledge.
Any process or technique which constructs a graph E from some other graphs S is said to be (simply) valid if S entails E, 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.
In this section we give a few basic results about simple entailment and valid inference. Simple entailment can be recognized by relatively simple syntactic comparisons. The two basic forms of simply valid proof step 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)).
Note, these results apply only to simple entailment, not to the more subtle notions of entailment introduced in later sections. Proofs, all of which are straightforward, are given in appendix B, which also describes some other properties of entailment which may be of interest.
Subgraph Lemma. A graph entails all its subgraphs .
Instance Lemma.A graph is entailed by any of its instances.
The relationship between merging and entailment is simple, and obvious from the definitions:
Merging lemma. The merge of a set S of RDF graphs is entailed by S, and entails every member of S.
This means that a set of graphs can be treated as equivalent to a single graph as far as the model theory is concerned.
The main result for simple RDF inference is:
The interpolation lemma completely characterizes simple RDF entailment in syntactic terms. To tell whether a set of RDF graphs entails another, find a subgraph of their merge and replace names by unlabeled nodes to get the second. Of course, there is no need to actually construct the merge. If working backwards from the consequent E (the graph that may be entailed by the others), the most efficient technique would be to treat unlabeled nodes as variables in a process of subgraphmatching, 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 subgraphmatching algorithm is. The existence of complete subgraphchecking algorithms also shows that RDF is decidable, i.e. there is a terminating algorithm which will determine for any finite set S and any graph E, whether or not S entails E.
Notice however that such a variablebinding 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 unlabeled nodes, since (by the anonymity lemmas) the resulting graph would not be entailed by the assertion.
It might be thought that the operation of changing a bound variable would be an example of an inference which was valid but not covered by the interpolation lemma, e.g. the inference of
_:x <ex:a> <ex:b> .
from
_:y <ex:a> <ex:b> .
Recall however that by our conventions, these two expressions describe identical RDF graphs.
Notice that unlabeled nodes are not identified with other nodes in a merge, and indeed this reflects a basic principle of RDF graph inference: in contrast to names, which have a global identity which carries across all graphs, blank nodes should not be identified with other nodes or relabeled with urirefs, in order to ensure that the resulting graph is entailed by what one starts with. To state this condition precisely, we need to first exclude a counterexample. It is possible for a graph to contain two triples one of which is an instance of the other, for example:
<ex:a> <ex:b> _:xxx
.
<ex:a> <ex:b> <ex:c>
.
Such an internally redundant graph is equivalent
to one of its own instances, since replacing the blank node by
<ex:c>
would result in a singletriple graph
which is a subgraph of the original. To rule out such cases of
internal redundancy, we will say
that an RDF graph is lean if none of its triples is a
proper instance of any other. Then the above principle is
made precise in the following two lemmas concerning criteria
for nonentailment:
This means that there is no valid RDF inference process which can produce an RDF graph in which a single unlabeled node occurs in triples originating from several different graphs. (Of course, such a graph can be constructed, but it will not be entailed by the original documents. An assertion of such a graph would reflect the addition of new information about the identity of two unlabeled nodes.)
We emphasise again that these results apply only to simple entailment, not to the namespace entailment relationships defined in rest of the document.
So far we have considered only the model theory of what might be called the logical form of the RDF graph itself, without imposing any special interpretations on any reserved vocabulary. In the rest of the document we will extend the model theory to describe the semantic conditions reflecting the intended meanings of the rdf: and rdfs: namespaces. Some of these interpretations diverge in some ways from the meanings described in [RDFMS] and [RDFSchema] , and we will note these differences as they arise.
Although we will do this in stages, the same general technique is used throughout. First we describe a reserved vocabulary, i.e. a set of urirefs which will be given a special meaning; then we give the extra conditions on an interpretation which capture those meanings; then we restrict the notions of satisfiability and entailment to apply to these interpretations only. This essentially imposes an a priori restriction on the world being described that it satisfy the extra conditions. The new semantic conditions are automatically assumed to be true; an interpretation which would violate them is simply not allowed to count as a possible world.
Since there are now many distinct notions of interpretation, entailment and satisfiability, we use the Qname namespace prefixes to identify the various distinctions, eg an rdfinterpretation is an interpretation satisfying the rdf semantic conditions, rdfentailment means entailment relative to such interpretations, and so on.We call this general idea vocabulary entailment , i.e. entailment relative to a set of interpretations which satisfy extra semantic conditions on a reserved vocabulary. Vocabulary entailment is more powerful than simple entailment, in the sense that a given set of premises entails more consequences. In general, as the reserved vocabulary is increased and extra semantic conditions imposed, the class of satisfying interpretations is restricted, and hence the corresponding notion of entailment becomes more powerful. For example, if S simply entails E then it also rdfentails E, since every rdfinterpretation is also a simple interpretation; but S may rdfentail E even though it does not simply entail it. Intuitively, a conclusion may follow from some of the extra assumptions incorporated in the semantic conditions imposed on the reserved vocabulary.
Another way of expressing this is that any restriction on interpretations decreases the number of possible ways that an interpretation might be a counterexample to E's following from S.
Simple entailment is the vocabulary entailment of the empty vocabulary. It is therefore the weakest form of RDF entailment, which holds for any reserved vocabulary; it is the entailment which depends only on the basic logical form of RDF graphs, without making any further assumptions about the meaning of any urirefs.
We will consider the syntactic criteria for recognizing vocabulary entailment in the next section. This and subsequent sections of the document make extensive use of the notational convention described in section 0.2.
Consider the following (rather small) reserved vocabulary, which we will call rdfV:
RDF reserved vocabulary 
rdf:type
rdf:Property 
(The parts of the RDF vocabulary concerned with reification and containers are considered in the next subsection.)
IP contains
I(rdf:type ) 
if x is in IP then
IEXT(I(rdf:type )) contains <x,
I(rdf:Property )> 
This forces every rdf interpretation to contain a thing
which can be interpreted as the 'type' of properties. (The
second condition could be regarded as defining IP to be
the set of resources in the universe of the interpretation
which have the value I(rdf:Property
) of the
property I(rdf:type
). This way of construing
subsets of the universe will be central in interpretations of
RDFS.)
For example, the following rdfinterpretation extends the simple interpretation in figure 1:
IR = {1, 2, T }; 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
Figure 2: An example of an rdfinterpretation.
This is not the smallest rdfinterpretation which extends
the earlier example, since we could have made
I(rdf:Property
) be 2 and IEXT(T) be
{<1,2>,<T,2>}, and managed without having P in the
universe. In general, a given entity in an interpretation may
play several 'roles' at the same time, as long as this can be
done without violating any of the required semantic
conditions.
It is important to note that every rdfinterpretation is also a simple interpretation.The 'extra' structure does not prevent it acting in the simpler role.
RDF provides vocabularies which are intended for use in describing containers, and a reification vocabulary to enable an RDF graph to describe, as well as exhibit, triples. These vocabularies do not involve any extra formal semantic conditions, however, so the notions of rdfentailment and rdfinterpretation apply to them without further change. When the RDFS vocabulary is added, this part of the RDF vocabulary will have nontrivial consequences; but until then, it can be added to the RDF vocabulary without needing any further semantic conditions.
The RDF reification vocabulary consists of a class name and three property names.
RDF reification vocabulary 
rdf:Statement rdf:subject rdf:predicate
rdf:object 
The intended interpretation of these are that a triple of the form
aaa [rdf:type] [rdf:Statement] .
is true in I just when I(aaa) is an RDF triple in some RDF document. This requires triples to exist as firstclass objects in the universe, so that the semantic domain of the language contains its own syntax. 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. Let us call the node which is
intended to refer to the first triple  the blank node in the
second graph  the center of the reification. (This can
be a blank node or a uriref.) The intended interpretation of
the reification vocabulary is that the reification graph would
be made true in I by interpreting the center to refer to the
triple in the first graph, and using I to interpret that triple
(so that the subject, predicate and object of that triple would
be interpreted in the same way in the reification as in the
reified triple.) Formally, <x,y> is in
IEXT(I(rdf:subject
)) just when x is an occurence
of an RDF triple with the form
aaa bbb ccc .
and y is I(aaa). Notice that this condition involves a twostage interpretation process: we have to interpret the center  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 the object.
Note that the intended interpretation here is that the triple that the center refers to I(aaa) in the above example is a particular token or instance of a triple in a particular 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 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. The decision to use this particular interpretation of reification was made on the basis of use cases where properties such as dates of composition or provenance information have been applied to the center, which are meaningful only when thought of as applying to a particular instance or token of a triple.
While this intended interpretation is intuitively clear, RDF syntax provides no means to 'connect' an RDF triple to its reification. Since an assertion of a reification of a triple does not implicitly assert the triple itself, this means that there are no entailment relationships which hold between a triple and a reification of it. Thus the reification vocabulary has no effective semantic constraints on it, other than those that apply to an RDF interpretation. The chief facts that are worthy of note about RDF reification, in fact, are examples of nonentailments.
A reification of a triple does not entail the triple, and is not entailed by it. (The reason for first is clear, since the reification only asserts that the triple exists, not that it is true. The second nonentailment arises because asserting a triple does not automatically assert that any triples exist in the universe. For example, the triple might be part of an ontology describing animals, which could be satisfied by an interpretation in which the universe did not contain any triples.)
Since the relation between triples and reifications of triples in any RDF graph or graphs need not be onetoone, 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:
_: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> .
RDF provides for three classes of container, defined by three class names, and an infinite set of property names each of which relates a particular 'position' in a container to the entity, if there is one, which is 'at' that position. (The rdfs vocabulary described below adds a generic membership property which holds regardless of position.)
RDF Container Vocabulary 
rdf:Seq rdf:Bag rdf:Alt rdf:_1 rdf:_2
... 
One should understand this RDF container vocabulary is as providing a very weak language for describing containers, rather than as a vocabulary for constructing containers, 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 container 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 automatically entail that they must be true.
The only 'structure' which RDF presumes its containers to
have is what can be inferred from the use of this vocabulary
and the semantic conditions on the rest of the RDF vocabulary.
Since the membership properties rdf:_1, rdf:_2
,
... are implicitly ordered by their very names, that order can
be thought of as an ordering of the elements of the container.
This implicit ordering of members of a container applies to all
three kinds of container, even though bags are normally thought
of as unordered. RDF does not support any entailments which
could arise from reordering the elements of an rdf:Bag. 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 an consequence of the fact that RDF is a purely assertional language.)
There is no assumption that a property of a container applies to any of the elements of the container, or that if a property applies to a container then the property applies to any of the members of the container, or vice versa. There is no requirement that the three container classes are disjoint, so that for example something can be consistently asserted to be both an rdf:Bag and an rdf:Seq. There is no assumption that containers are gapfree, 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 builtin assumption that an RDF container has only finitely many members.
The informal purpose of the three container types is to
allow applications to encode the various intentions or
expectations about different kinds of containers. Sequences are
totally ordered, bags are unordered (that is, equivalent under
reorderings) and rdf:Alt containers are intended to convey a
series of alternative values of a property, which an
application can choose from. However, these informal
interpretations are not reflected in any RDF entailments. In
particular, a triple with a rdf:Alt
as a subject
or object should not be thought of as an encoding of a
logical disjunction.
RDF Schema [RDFSchema] extends RDF to include a larger reserved vocabulary rdfsV with more complex semantic constraints:
RDFS reserved vocabulary 
rdf:type rdf:Property rdfs:domain rdfs:range
rdfs:Resource rdfs:Literal rdfs:Class rdfs:subClassOf
rdfs:subPropertyOf rdfs:member 
(rdfs:seeAlso
, rdfs:isDefinedBy
,
rdfs:comment
and rdfs:label
are
omitted, as the model theory places no constraints on their
meanings. The account given here of
rdfs:domain
and rdfs:range
reflect
our
current understanding of multiple domain and range
restrictions under which such assertions are understood
conjunctively, and they allow
cyclic assertions of rdfs:subClassOf
and
rdfs:subPropertyOf
. These differ from earlier
specifications of RDF and RDFS.)
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 have the same value of the
rdf:type
property. We will define a mapping ICEXT
(for the Class Extension in I) from classes to their
extensions, in terms of the relational extension of
rdf:type
, as follows:
ICEXT(x) = {y  <y,x> is in
IEXT(I(rdf:type
)) }
An rdfsinterpretation of V is a simple interpretation of (V union rdfsV) which satisfies the following semantic conditions. The first two of these are simply definitions of IC and ICEXT, which could be used to eliminate these concepts from the rest of the conditions; as noted earlier, the conditions on IR and IP could also be regarded as definitions.
x is in ICEXT(y) iff <x,y> is in
IEXT(I( IC = ICEXT(I( 
ICEXT(I( IP is a subset of
ICEXT(I( 
IC contains: I( 
IP contains: I( 
IEXT(I( <I( 
IEXT(I( <I( 
If <x,y> is in
IEXT(I( 
If <x,y> is in
IEXT(I( 
If <x,y> is in
IEXT(I( 
If <x,y> is in
IEXT(I( 
The IEXT condition on
rdf:Property
in an rdfinterpretation is
equivalent to the ICEXT condition on rdf:Property
above; so these clearly imply all the conditions on an
rdfinterpretation. It follows that any rdfsinterpretation is
also an rdfinterpretation of the same vocabulary.
We will not attempt to give a pictorial diagram of an rdfsinterpretation.
The semantic conditions on
rdfsinterpretations do not include the condition that
ICEXT(I(rdfs:Literal
)) must be a subset of LV. While this would seem to be required for
conformance with [RDFMS], there is no
way to impose this condition by any RDF assertion or syntactic
closure rule. This limitation is due to the fact that
RDF does not allow literals to occur in the subject position of
a triple, so there are severe restrictions on what can be said
about literals in RDF. Similarly, while properties may
be asserted of the 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 legal even though 'ex:a
' is a uriref rather
than a literal. What it says is that I(ex:a
) is a
literal, ie that the uriref 'ex:a
' denotes
a literal. There is however no way in current RDF to specify
exactly what literal it denotes.
We will say that S rdfentails E (S rdfsentails E) when every rdfinterpretation (every rdfsinterpretation) which satisfies every member of S also satisfies E. This follows the wording of the definition of simple entailment in section 2, but refer only to rdf or rdfsinterpretations instead of all simple interpretations. These are examples of vocabulary entailment, i.e. entailment relative to a set of interpretations which satisfy extra semantic conditions on a reserved vocabulary.
It is easy to see that the lemmas in section 2 do not hold for vocabulary entailment. For example, the triple
[rdf:type] [rdf:type] [rdf:Property] .
is true in every rdfinterpretation, and hence rdfentailed by the empty set, which immediately contradicts the interpolation lemma for rdfentailment.
Rather than develop a separate theory of the syntactic conditions for recognising entailment for each reserved vocabulary, we will use a general technique for reducing these broader notions of entailment to simple entailment, by defining the closure of an RDF graph relative to a set of semantic conditions. The basic idea is to rewrite the semantic conditions as a set of syntactic inference rules, and define the closure to be the result of applying those rules to exhaustion. The resulting graphs will contain RDF triples which explicitly state all the special meanings embodied in the extra semantic conditions, in effect axiomatizing them in RDF itself. A graph rdfentails (rdfsentails) another just when its rdfclosure (rdfsclosure) simply entails it.
The notion of closure used here is purely a formal device to relate two notions of entailment. We do not mean to suggest that closure rules should be used as a computational technique, or that actually generating the full closure would be the best process to use in order to determine vocabulary entailment. Implementors who wish to check any kind of entailment should use a process which is optimised for the combinatorics of the particular set of use cases that are most likely to arise in a given application area. In many cases it may be more efficient to use a process of backchaining on the closure rules, for example.
1. Add the following triple (which is true in any rdfinterpretation):
[rdf:type] [rdf:type] [rdf:Property] .
2. Apply the following rule recursively to generate all legal RDF triples (i.e. until none of the rules apply or the graph is unchanged.) Here xxx and yyy stand for any uriref, bNode or literal, aaa for any uriref.
if E contains  then add  
rdf1  xxx aaa yyy .  aaa [rdf:type] [rdf:Property]
. 
(This rule will generate the triple mentioned in 1 in two steps from any RDF triple; nevertheless, we mention the triple explicitly since it is required to be in the closure of even an empty set of graphs.)
The following lemma is the basic result on rdfentailment, and illustrates a general pattern of how to characterize vocabulary entailment syntactically.
For proof, see appendix B.
RDFS closures require more complex rules to reflect the
consequences of the more elaborate semantic constraints on the
rdfs reserved vocabulary. For example, the fact that the subset
relationship is transitive means that two
subClassOf
assertions may entail a third,
different, triple. Again, we emphasize that these closure rules
are not being recommended as an efficient computational
process.
1. Add the following triples, which are true in any rdfsinterpretation. These assign classes, domains and ranges to the properties in the rdfs vocabulary. (There are several other triples which are true in every rdfsinterpretation, but they will be generated from these by other rules.)
[rdfs:Resource] [rdf:type] [rdfs:Class] .
[rdfs:Literal] [rdf:type] [rdfs:Class] .
[rdfs:Class] [rdf:type] [rdfs:Class] .
[rdf:Property] [rdf:type] [rdfs:Class] .
[rdf:Seq] [rdf:type] [rdfs:Class] .
[rdf:Bag] [rdf:type] [rdfs:Class] .
[rdf:Alt] [rdf:type] [rdfs:Class] .
[rdf:Statement] [rdf:type] [rdfs:Class] .
[rdf:type] [rdf:type] [rdf:Property] .
[rdf:type] [rdfs:domain] [rdfs:Resource] .
[rdf:type] [rdfs:range] [rdfs:Class] .
[rdfs:domain] [rdf:type] [rdf:Property] .
[rdfs:domain] [rdfs:domain] [rdf:Property]
.
[rdfs:domain] [rdfs:range] [rdfs:Class] .
[rdfs:range] [rdf:type] [rdf:Property] .
[rdfs:range] [rdfs:domain] [rdf:Property] .
[rdfs:range] [rdfs:range] [rdfs:Class] .
[rdfs:subPropertyOf] [rdf:type] [rdf:Property]
.
[rdfs:subPropertyOf] [rdfs:domain] [rdf:Property]
.
[rdfs:subPropertyOf] [rdfs:range] [rdf:Property]
.
[rdfs:subClassOf] [rdf:type] [rdf:Property]
.
[rdfs:subClassOf] [rdfs:domain] [rdfs:Class]
.
[rdfs:subClassOf] [rdfs:range] [rdfs:Class] .
[rdf:subject] [rdf:type] [rdf:Property] .
[rdf:subject] [rdfs:domain] [rdf:Statement] .
[rdf:predicate] [rdf:type] [rdf:Property] .
[rdf:predicate] [rdfs:domain] [rdf:Statement] .
[rdf:object] [rdf:type] [rdf:Property] .
[rdf:object] [rdfs:domain] [rdf:Statement] .
2. Add all triples of the following forms, which are true in any rdfsinterpretation. This is an infinite set because the RDF container vocabulary is infinite. However, since none of these triples entail any of the others, it is only necessary, in practice, to add the triples which use those container properties which actually occur in any particular graph or set of graphs in order to check the rdfsentailment relation between those graphs.
[rdf:_1] [rdf:type] [rdfs:Property] .
[rdf:_2] [rdf:type] [rdfs:Property] .
...
[rdf:_1] [rdfs:subPropertyOf] [rdfs:member] .
[rdf:_2][rdfs:subPropertyOf] [rdfs:member]
.
...
3. Apply the following rules recursively to generate all legal RDF triples (i.e. until none of the rules apply or the graph is unchanged.) Here, xxx, yyy and zzz stand for any uriref, bNode or literal, aaa for any uriref, and uuu for any uriref or bNode (but not a literal).
If E contains:  then add:  

rdf1 
xxx aaa yyy 
aaa [rdf:type]
[rdf:Property] 
rdfs2 
xxx aaa yyy aaa 
xxx [rdf:type] zzz 
rdfs3 
xxx aaa uuu aaa 
uuu [rdf:type] zzz 
rdfs4a  xxx aaa yyy  xxx [rdf:type]
[rdfs:Resource] 
rdfs4b  xxx aaa uuu  uuu [rdf:type]
[rdfs:Resource] 
rdfs5 
aaa bbb 
aaa [rdfs:subPropertyOf]
ccc 
rdfs6 
xxx aaa yyy aaa 
xxx bbb yyy 
rdfs7 
xxx 
xxx
[rdfs:subClassOf]
[rdfs:Resource] 
rdfs8 
xxx yyy 
xxx [rdfs:subClassOf]
zzz 
rdfs9 
xxx aaa 
aaa [rdf:type] yyy 
Unlike the simpler rdf closure rules, the outputs of some of these rules may trigger others. For example, these rules will generate the complete transitive closures of all subclass and subproperty heirarchies, together with all of the resulting type information about everything which can be inferred to be a member of any of the classes, and will propagate all assertions in the graph up the subproperty heirarchy, reasserting them for all superproperties.They will generate all assertions of the form
xxx [rdf:type] [rdfs:Resource] .
for every xxx in V, and of the form
xxx [rdfs:subClassOf] [rdfs:Resource] .
for every class name; and several more 'universal' facts, such as
[rdf:Property]
[rdf:type]
[rdfs:Class] .
[rdf:Property]
[rdfs:subClassOf]
[rdfs:Resource] .
However, it is easy to see that (with the restriction noted of the infinite sets to those membership properties which occur in the graph) the rules will indeed terminate on any finite RDF graph, since there are only finitely many triples that can be formed from a given finite vocabulary.
A similar result applies here as in the case of rdfentailment, though it takes considerably longer to prove:
As noted earlier, we do not intend to suggest that these closure rules be used directly in this form as a mechanism to compute rdfs entailment between graphs. They are intended only as a formal specification of a syntactic criterion corresponding to rdfs entailment. Taken together with the interpolation lemma for simple entailment, however, they do define a search space within which a computational system could find proofs of rdfsentailment between graphs.
The exact computational significance of finding such a proof
is not determined by the proof itself. For example, one view of
domain and range assertions is that they should be viewed as
constraints on the legality of assertions involving a
property. On this view, the rules rdfs2 and rdfs3 would be most
naturally seen as applying 'backwards', and rdfs2 would be
better phrased as 'If aaa [rdfs:domain]
zzz , then
if xxx is NOT [rdf:type]
zzz, then xxx aaa yyy is
illegal', and similarly for rdfs3. However, this is the same
inference, differently phrased, as that expressed by the rules
in the table above. In other words, these rules should not be
read as defining only a lefttoright inference pattern. The
entailments they define can be 'used' in any direction by an
inference engine, depending on what its task or purpose is. All
that the model theory defines is the fact of entailment or
nonentailment between RDF(S) expressions. What use is made of
that fact of entailment is up to the particular
application.
RDF/RDFS model theory summary 


0. Domains and mappings of interpretation I 

vocab(I): set of urirefs ; LV: (global) set of literals ; IR: set of resources (universe); IP: subset of IR (properties) ; IC: subset of IR (classes). 

IS: vocab(I) > IR IEXT: IP > subsets of (IR x (IR union LV)) ICEXT: IC > subsets of IR 

1. Basic equations 

E is: 
I(E) is: 
a literal 
E 
a uriref 
IS(E) 
a triple <s p o> 
true if <I(s), I(o)> is in IEXT(I(p)), otherwise false 
a ground RDF graph 
false if I(E') =false for any asserted triple E' in E, otherwise true 
an unlabeled node (blank node) 
not defined ; but [I+A](E) = A(E) 
an RDF graph 
true if [I+A'](E) = true for some A': anon(E) > IR, otherwise false. 
2. Class extensions 

E is: 
I(E) is in IC; ICEXT(I(E)) includes: 

IR (The universe of the interpretation) 

IP (Properties; subset of IR, domain of IEXT) 

IC (Classes; subset of IR, domain of ICEXT) 

LV (Literals) 
3. Property extensions 

E is: 
I(E) is in IP; <x,y> is in IEXT(I(E)) iff: 

x is in ICEXT(y) 
E is: 
I(E) is in IP; if <x,y> is in IEXT(I(E)) then: 

if <u,v> is in IEXT(x) then u is in ICEXT(y) 

if <u,v> is in IEXT(x) then v is in ICEXT(y) 

ICEXT(x) is a subset of ICEXT(y) 

IEXT(x) is a subset of IEXT(y) 
4. Domain and Range  
IEXT(I(rdfs:domain ))
contains: 
<I( <I( <I( <I( <I( <I( 
IEXT(I(rdfs:range ))
contains: 
<I( <I( <I( 
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
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.
If an instance of a graph E' is a subgraph of another graph E then E entails E'; this follows from the subgraph and instance lemmas. As we show below, this is in fact a necessary as well as sufficient condition for entailment, so it is useful to give a name to the syntactic condition that captures nonentailment. Say that a graph E' is separable from a graph E if no instance of E' is a subgraph of E. In particular, a ground graph is separable from E just when it is not a subgraph of E, and a ground triple is separable just in case it isn't in the graph. Graphs which are not separable from E are entailed by E; but for all others, there is a way to arrange the world so that they are false and E true.
For ground graphs, the subgraph lemma can be strengthened to provide simple necessary and sufficient conditions for entailment.
Conjunction Lemma.If E is ground, then I satisfies E if and only if it satisfies every triple in E.
Proof. Obvious, from definition of denotation for ground graphs. QED
Plain Subgraph Lemma. If E and E' are ground, then E entails E' if and only if E' is a subgraph of E.
Proof. 'If' follows directly from subgraph lemma; 'only if' follows from previous lemma and definition of entailment. QED
Herbrand Lemma. Any RDF graph has a satisfying interpretation.
Proof. We will construct the interpretation from the graph, by providing 'just enough' entities and extensions to make the graph true. Since the exact nature of the things in the universe is irrelevant, it is convenient to use the nodes of the graph themselves as their own denotations. (That was Herbrand's idea.)
The universe of I is defined to be the set of names and blank nodes in the graph.
Define IS to be the identity mapping on the vocabulary of the graph, and IEXT as follows: <x,y> is in IEXT(z) just when there is a triple in the graph of the form x z y . Define the mapping A to be the identity mapping on blank nodes of the graph.
Clearly I satisfies all ground triples in the graph, and [I+A] satisfies the entire graph; so I satisfies the graph. QED
An interpretation constructed in this way, so that the IS mapping is the identity mapping, is called a Herbrand interpretation. A Herbrand interpretation treats urirefs in the same way as literals, i.e. as denoting their own syntactic form. Of course this may not be what was intended by the writer of the RDF, but the lemma shows that any graph can be intepreted in this way.
If I satisfies E, then I may contain more information than is necessary to specify the truth of E; an interpretation  a world  can be larger than strictly needed to establish the truthvalues of a particular set of triples. It is therefore useful to define a notion of the minimal part of an interpretation which is just enough to make a given graph true.
Say that I' is a subinterpretation of I when vocab(I') is a subset of vocab(I), IR'is a subset of IR, I'(x)=I(x) wherever I'(x) is defined, and IEXT'(x) is a subset of IEXT(x) wherever IEXT'(x) is defined. Intuitively, I' defines a 'part' of the world defined by I. If a subinterpretation of I satisfies E, then I must also satisfy E.
Herbrand interpretations are minimal, and every minimal interpretation has a corresponding Herbrand interpretation which assigns the same truthvalues to every triple, and hence to every graph.
It is clear that if I satisfies E, then a minimal satisfying interpretation exists with a vocabulary precisely the vocabulary of E. The minimal interpretations can be characterized by the following lemma.
Minimality lemma. If I is a minimal satisfying interpretation of E, then I fails to satisfy every triple which has no instance in E.
Proof. We will argue by reductio. Suppose I satisfies some such triple S P O, i.e.. IEXT(I(P)) contains <I(S),I(O)>, and consider the subinterpretation I' which is like I except that IEXT(I'(P)) does not contain that pair. Since S P O has no instances in E, [I'+A](x)=[I+A](x) for any mapping A from blank nodes and any triple x in E, and I satisfies E, so I' satisfies E; so I is not minimal. QED
The property extensions in a minimal interpretation are 'shrinkwrapped' onto the assertions in the graph. Notice that every thing in the universe of a minimal interpretation of E must be the denotation of at least one node in E, and that every pair in any property extension must have at least one corresponding triple in E that it makes true; for if not, one could delete some of the interpretation and still satisfy E. We will make use of this property in later proofs.
Strong Herbrand Lemma. Any RDF graph E has a satisfying interpretation which does not satisfy any graph which is separable from E.
Proof.The construction in the proof of the Herbrand Lemma in fact establishes this result for arbitrary separable graphs. Consider the Herbrand interpretation I constructed in the proof of the Herbrand lemma, and let <S P O> be a triple which has no instances in E. Then either S is a name and there are no triples of the form S P O' in E, or O is a name and there are no triples of the form S' P O in E. Consider the first case (the other case is similar); then by the construction in the earlier proof, IEXT(I(P)) contains no pairs of the form <I(S), x>; so there is no mapping A from blank nodes to IR that could make the triple true in [I+A]; so the triple is false in I. Similarly for the other case. 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 iff all triples in the merge of S are true. QED.
Anonymity lemma 1. Suppose E is a lean graph and E' is a proper instance of E. Then E does not entail E'.
Proof. Since E' is a proper instance and E is lean, E' contains a triple which has no instances in E; otherwise the triple in E which it is a proper instance of would have had a proper instance in E. By the strong Herbrand lemma, there exists an interpretation which satisfies E but not E'. So E does not entail E'. QED
Proof. First we assume that the blank nodes occur in two distinct triples in E. Suppose that E contains the triples
S1 P1 _:x1 .
S2 P2 _:x2 .
where E' contains the triples
S1 P1 _:x .
S2 P2 _:x .
(The arguments for the cases where the blank nodes occur in other positions in the triples are similar.) Since E is lean, it contains no other triples of the form S1 P1 O' or S2 P2 O'. Let I be a Herbrand interpretation of E; then I(S1) is distinct from I(S2) and IEXT(I(P1)) ={<I(S1), _:x1>}and IEXT(I(P2))={<I(S2), _:x2>}. Let A be any mapping from the blank nodes of E' to IR, then in order for both triples to be true in [I+A], [I+A](_:x) would have to equal both _:x1 and _:x2; but these are distinct; so I does not satisfy E'.
The only remaining case is where E contains a single triple with two blank nodes which are identified in E':
_:x1 P _:x2 .
where E' contains
_:x P _:x .
The argument here is similar; the Herbrand interpretation I now has IEXT(I(P)) = {<_:x1,_:x2>} and there is no mapping from the second triple that could satisfy this, so again I satisfies E but not E'. QED.
Note that the 'minimal' nature of the Herbrand construction provides an interpretation that is sufficient to make a graph true, but only just sufficient. This is a basic technique for showing that one graph does not entail another and for establishing a precise correspondence between syntactic relationships and entailment.
Interpolation Lemma. S entails E if and only if a subgraph of the merge of S is an instance of E.
Proof. 'if' is a direct consequence of the merging and instance lemmas.
To prove 'only if' we will show the converse. This is just a restatement of the strong Herbrand lemma. Assume that no subgraph of the merge of S is an instance of E, i.e. that all subgraphs of the merge of S fail to be instances of E; i.e., that E is separable from the merge of S. Then by the strong Herbrand lemma the merge of S does not entail E. So, by the merging lemma, S does not entail 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. While not itself strictly a valid operation, skolemization adds no new content to an expression, in the sense that a skolemized expression has the same entailments as the original expression provided they do not contain the new skolem functions.
In RDF, skolemization simplifies to the special case where an existential variable is replaced by a 'new' name, i.e. a uriref which is guaranteed to not occur anywhere else.(Using a literal would not do. Literals are never 'new' in the required sense, since their meaning is fixed.) To be precise, a skolemization of E (with respect to V) is a ground instance of E with respect to a vocabulary V which is disjoint from the vocabulary of E.
The following lemma shows that skolemization has the same properties in RDF as it has in conventional logics. Intuitively, this lemma shows that asserting a skolemization expresses a similar content to asserting the original graph, in many respects. In effect, it simply gives 'arbitrary' names to the anonymous entities whose existence was asserted by the use of blank nodes. However, care is needed, since these 'arbitrary' names have the same status as any other urirefs once published. Also, skolemization would not be an appropriate operation when applied to anything other than the antecendent of an entailment. A skolemization of a query would represent a completely different query.
Proof. sk(E) entails E by the interpolation 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. So E entails F. QED.
RDF closure lemma. Any satisfying rdfinterpretation of E satisfies the rdfclosure of E; and any minimal simple satisfying interpretation of the rdfclosure of E is a satisfying rdfinterpretation of E.
Proof. This follows from a comparison of the rdf closure rules with the semantic conditions on an rdfinterpretation. Although the argument is very simple in this case, we give it here in full to illustrate the general technique.
The first part follows from the fact that the closure rules are all rdfvalid. To show this, suppose I is an rdfinterpretation; then for any aaa in the vocabulary of I, if a triple of the form xxx aaa yyy is true in I, then IEXT(I(aaa)) is nonempty then I(aaa) is in IP, so IEXT(I(rdf:type)) contains <I(aaa),I(rdf:Property)>, so the triple aaa rdf:type rdf:Property is true in I. Since I is an rdfinterpretation, its vocabulary contains rdf:type and IP contains I(rdf:type), so in particular the triple
[rdf:type] [rdf:type] [rdf:Property] .
is true in I. That establishes that the closure rules are rdfvalid.
To prove the other part of the lemma we must show that the closure rules are together sufficient to force any minimal interpretation to be an rdfinterpretation of E. The simplest way to argue this is to show the converse, viz. that any minimal simple interpretation of the rdfclosure that violates one of the semantic conditions for an rdfinterpretation of E would thereby fail to satisfy the closure. Suppose therefore that I is a minimal simple interpretation of the rdfclosure of E.
If I violates the first constraint then IP does not contain I(rdf:type); in that case, the added triple in the first closure rule is false in I. So assume that I violates the second constraint. Then there is some x in IP for which IEXT(I(rdf:type)) does not contain <x,I(rdf:Property)>. Since I is minimal, there is some node aaa in E with I(aaa)=x; and since I(aaa) is in IP, there is a pair <y,z> in IEXT(I(aaa)) and a triple
bbb aaa ccc .
in E with I(bbb)=y and I(ccc)=z. Then the closure of E contains the triple
aaa
[rdf:type] [rdf:Property] .
which is false in I. So I fails to satisfy the rdfclosure. QED.
Notice the need for the minimality assumption, which 'forces' the semantic violation to be made explicit in the syntax of the graph itself. The second part of the lemma could be false for an arbitrary simple interpretation of the closure, which might fail to meet the required semantic conditions on some part of the universe that was not referred to in the graph itself. In general, one cannot infer, from the lack of an assertion in a graph, that what that assertion would say if it were in the graph must be false in a satisfying interpretation of the graph. Minimal interpretations, however, embody a ' closed world assumption' which would sanction such an inference. To prove an entailment we need to prove something about all interpretations; but to prove the converse, it is enough to show that a single interpretation exists with the right properties, and this is where the special properties of minimal interpretations are useful.
RDF entailment lemma. S rdfentails E if and only if the rdfclosure of the merge of S simply entails E.
Proof. Follows from the merging lemma, the RDF closure lemma and the definition of entailment. By the merging lemma, we can identify S with the merge of S, i.e. we can treat a set of graphs as a single graph MS.
So suppose that MS rdfentails E, and let I be a simple interpretation of the rdfclosure c(MS) of of MS. Then there is a minimal simple subinterpretation I' of I which satisfies c(MS); so, by the previous lemma, I' is a satisfying rdfsinterpretation of E. Therefore I satisfies E (since I' is a subinterpretation of I).
Conversely, suppose that c(MS) simply entails E, and let I be an rdfinterpretation of MS; then by the previous lemma, I satisfies c(MS), so I satisfies E (since every rdfinterpretation is a simple interpretation). QED.
RDFS Closure Lemma. Any satisfying rdfsinterpretation of E satisfies the rdfsclosure of E; and any minimal simple satisfying interpretation of the rdfclosure of E is a satisfying rdfsinterpretation of E.
Proof.(Sketch) As in the proof of the RDF closure lemma, this follows from a pointbypoint comparison of the rdfs closure rules with the semantic conditions on an rdfsinterpretation. A full proof would be long but tedious.We will illustrate the form of the argument by considering some typical cases in detail.
The first part follows from the fact that the rdfs closure rules are all rdfsvalid, which can be checked case by case. For example, consider the closure rule rdfs5, and suppose I is an rdfsinterpretation which satisfies
aaa
[rdfs:subPropertyOf]
bbb .bbb
[rdfs:subPropertyOf]
ccc .Then by the semantic conditions on an rdfsinterpretation, IEXT(I(aaa)) is a subset of IEXT(I(bbb)) and IEXT(I(bbb)) is a subset of IEXT(I(ccc)); so IEXT(I(aaa)) is a subset of IEXT(I(ccc)); so I satisfies
aaa
[rdfs:subPropertyOf]
ccc .The other cases are similarly straightforward.
To demonstrate the other part of the lemma we must show that the closure rules are together sufficient to restrict any minimal interpretation to be an rdfsinterpretation. The simplest way to argue this is to show the converse, by demonstrating that any minimal simple interpretation of the rdfsclosure c(E) of E that violates one of the semantic conditions for an rdfsinterpretation of E would thereby make some triple in c(E) false. Again, this can be checked by a detailed examination of the cases that arise.
Suppose that I is a minimal simple interpretation of E. If I violates any of the conditions involving the rdfsvocabulary then it is easy to check that one of the 'added' triples would be false, eg if IEXT(I(rdfs:domain) does not contain <I(rdfs:domain),I(rdf:Property)> then the triple
[rdfs:domain] [rdfs:domain] [rdf:Property] .
is false in I.
If I violates the condition on IEXT(I(
rdfs:range
)), then there exist x, y, u and v in IR with <x,y> in IEXT(I(rdfs:range
)), <u,v> in IEXT(x) but v not in ICEXT(y). Since I is a minimal interpretation and satisfies c(E), the closure must contain two triplesaaa
[rdfs:range]
bbb .ccc aaa ddd .
where I(aaa)=x, I(bbb)=y, I(ccc)=u and I(ddd)=v; but I makes the triple
ddd
[rdf:type]
bbb .false, since I(ddd) is not in ICEXT(I(bbb)); but by the closure rule rdfs3, this triple is in c(E); so I fails to satisfy c(E). The IEXT(I(
rdfs:domain
)) case is similar.Finally, suppose that I violates the condition on
rdfs:subClassOf
. Then for some x and y in IR, <x,y> is in IEXT(I(rdfs:subClassOf
)) but there is some z in ICEXT(x) but not in ICEXT(y). Again, since I is minimal, these entities must occur in triples in c(E) of the form respectivelyaaa
[rdfs:subClassOf]
bbb .ccc
[rdf:type]
aaa .where I(aaa)=x, I(bbb)=y and I(ccc)=z, and where the triple
ccc
[rdf:type]
bbb .is false in I; so I does not satisfy c(E) by a similar argument, using the closure rule rdfs9. Again, the case for a violation of the the condition on
rdfs:subPropertyOf
is similar. QED.
RDFS Entailment Lemma. S rdfsentails E iff the rdfsclosure of the merge of S simply entails E.
Proof. Exactly similar to proof of RDF entailment lemma. QED.
This document reflects the joint effort of the members of the RDF Core Working Group. Dan Connolly clarified the relationship between RDF and RDFS and suggested the 'set of triples' definition of RDF graph. The idea of graph syntax is from Ora Lassila, who also helped clarify the distinction between rules and computation. Sergey Melnick suggested using a translation into logic. Jeremy Carroll noticed the need for the lean graph condition. Jos deRoo, Graham Klyne, Jeremy Carroll and Patrick Stickler found errors in earlier drafts and suggested many stylistic improvements.
The use of an explicit extension mapping to allow selfapplication without violating the axiom of foundation was suggested by Chris Menzel. Peter PatelSchneider found several major errors in an earlier draft, and suggested several important technical improvements.
Changes from Working Draft March 2002