Copyright © 2003 W3C ® ( MIT , ERCIM , Keio ), All Rights Reserved. W3C liability , trademark , document use and software licensing rules apply.
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).
This section describes the status of this document at the time of its publication. Other documents may supersede this document. A list of current W3C publications and the latest revision of this technical report can be found in the W3C technical reports index at http://www.w3.org/TR/.
Publication as a Proposed Recommendation does not imply endorsement by the W3C Membership. This is a draft document and may be updated, replaced or obsoleted by other documents at any time. It is inappropriate to cite this document as other than work in progress.
This is one document in a set of six ( Primer , Concepts , Syntax , Semantics , Vocabulary , and Test Cases ) intended to jointly replace the original Resource Description Framework specifications, RDF Model and Syntax (1999 Recommendation) and RDF Schema (1999 Proposed Recommendation) . It has been developed by the RDF Core Working Group as part of the W3C Semantic Web Activity ( Activity Statement , Group Charter ) for publication on 15 December 2003.
The design expressed in earlier versions of these documents has been widely reviewed and satisfies the Working Group's chartered technical requirements . The Working Group has addressed all comments received , making changes as necessary. Many implementations have been reported, covering among them all features of the specification. Changes to this document since the Second Last Call Working Draft are detailed in the change log .
W3C Advisory Committee Representatives are now invited to submit their formal review via Web form, as described in the Call for Review. Additional comments may be sent to a Team-only list, w3t-semweb-review@w3.org . The public is invited to send comments to www-rdf-comments@w3.org ( archive ) and to participate in general discussion of related technology on www-rdf-interest@w3.org ( archive ). The review period extends until 19 January 2004 .
The W3C maintains a list of any patent disclosures related to this work .
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)
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.
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.
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 name s: itself and its internal type URI reference.
A set of name s 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 name s in the instance that were substituted for blank nodes in the original are name s 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 ].
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 ].
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 name s 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
name
s
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
.
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 name s, 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-interpretation s', ' rdfs-interpretation s', 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 .
A simple interpretation I of a vocabulary V is defined by: 1. A non-empty set IR of resources, called the domain or universe of I. 2. A set IP, called the set of properties of I. 3. A mapping IEXT from IP into the powerset of IR x IR i.e. the set of sets of pairs <x,y> with x and y in IR . 4. A mapping IS from URI references in V into (IR union IP) 5. A mapping IL from typed literals in V into IR. 6. A distinguished subset LV of IR, called the set of literal values, which contains all the plain literals in V |
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: name s denote things in the universe, and sets of triples denote truth-values.
The denotation of a ground RDF graph in I is given recursively by the following rules, which extend the interpretation mapping I from name s 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 E is a plain literal "aaa" in V then I(E) = aaa |
if
E
is
a
plain
literal
"aaa"
@
ttt
in
V
then
I(E)
=
<aaa,
ttt>
|
if E is a typed literal in V then I(E) = IL(E) |
if E is a URI reference in V then I(E) = IS(E) |
if
E
is
a
ground
triple
s
p
o
s, p and o are in V, I(ppp) I(p) is in IP and <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. |
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 name s 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
Figure
1
:
An
example
of
an
interpretation.
Note,
this
is
not
a
picture
of
an
RDF
graph.
The
figure
does
not
show
the
infinite
number
of
members
of
LV.
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.
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:
If E is a blank node and A(E) is defined 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 blank(E) to IR, otherwise I(E)= false. |
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.
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' name s 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.
Simple interpretations and simple entailment capture the semantics of RDF graphs when no attention is paid to the particular meaning of any of deleted text: 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.
The
RDF
vocabulary,
rdfV,
is
a
set
of
URI
references
in
the
rdf:
namespace:
RDF vocabulary |
rdf:type
rdf:Property
rdf:XMLLiteral
rdf:nil
rdf:List
rdf:Statement
rdf:subject
rdf:predicate
rdf:object
rdf:first
rdf:rest
rdf:Seq
rdf:Bag
rdf:Alt
rdf:_1
rdf:_2
...
rdf:value
|
rdf-interpretation
s
impose
extra
semantic
conditions
on
rdfV
and
on
typed
literals
with
the
type
rdf:XMLLiteral
,
which
is
referred
to
as
the
RDF
built-in
datatype.
This
datatype
is
fully
described
in
the
RDF
Concepts
and
Abstract
Syntax
document
[
RDF-CONCEPTS
].
Any
character
string
sss
which
satisfies
the
conditions
for
being
in
the
lexical
space
of
rdf:XMLLiteral
will
be
called
a
well-typed
XML
literal
string
.
The
corresponding
value
will
be
called
the
XML
value
of
the
literal.
Note
that
the
XML
values
of
well-typed
XML
literals
are
in
precise
1:1
correspondence
with
the
XML
literal
strings
of
such
literals,
but
are
not
themselves
character
strings
.
An
XML
literal
whose
literal
string
is
well-typed
will
be
called
a
well-typed
XML
literal
;
other
XML
literals
will
be
called
ill-typed
.
An rdf-interpretation of a vocabulary V is a simple interpretation I of (V union rdfV ) which satisfies the extra conditions described in the following table and all the triples in the subsequent table. These triples are called the rdf axiomatic triples .
The
first
condition
could
be
regarded
as
defining
IP
to
be
the
set
of
resources
in
the
universe
of
the
interpretation
which
have
the
value
I(
rdf:Property
)
of
the
property
I(
rdf:type
).
Such
subsets
of
the
universe
will
be
central
in
interpretations
of
RDFS.
Note
that
this
condition
requires
IP
to
be
a
subset
of
IR.
The
third
condition
requires
that
ill-typed
XML
literals
denote
something
other
than
a
literal
value:
this
will
be
the
standard
way
of
handling
ill-formed
typed
literals.
The rdfs-interpretation s 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
Figure
2
:
An
rdf-interpretation.
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.
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-interpretation s 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.
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.
RDF reification vocabulary |
rdf:Statement
rdf:subject
rdf:predicate
rdf:object
|
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>
.
RDF Container Vocabulary |
rdf:Seq
rdf:Bag
rdf:Alt
rdf:_1
rdf:_2
...
|
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.
RDF Collection Vocabulary |
rdf:List
rdf:first
rdf:rest
rdf:nil
|
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
.
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.
RDF Schema [ RDF-VOCABULARY ] extends RDF to include a larger vocabulary rdfsV with more complex semantic constraints:
RDFS vocabulary |
rdfs:domain
rdfs:range
rdfs:Resource
rdfs:Literal
rdfs:Datatype
rdfs:Class
rdfs:subClassOf
rdfs:subPropertyOf
rdfs:member
rdfs:Container
rdfs:ContainerMembershipProperty
rdfs:comment
rdfs:seeAlso
rdfs:isDefinedBy
rdfs:label
|
(
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
C
lass
Ext
ension
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
condiions,
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-interpretation
s
include
the
following:
rdfs:Resource
rdf:type
rdfs:Class
.
|
Note
that
datatype
s
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-interpretation
s
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 .
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:
<x,y>
is
in
IEXT(I(
|
<x,y>
is
in
IEXT(I(
|
<x,y>
is
in
IEXT(I(
|
<x,y>
is
in
IEXT(I(
|
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
.
Although
the
semantic
conditions
on
rdfs-interpretation
s
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.
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-interpretation s 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-interpretation s 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.
RDF
provides
for
the
use
of
externally
defined
datatype
s
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.
XSD datatypes |
xsd:string
,
xsd:boolean
,
xsd:decimal
,
xsd:float
,
xsd:double
,
xsd:dateTime
,
xsd:time
,
xsd:date
,
xsd:gYearMonth
,
xsd:gYear
,
xsd:gMonthDay
,
xsd:gDay
,
xsd:gMonth
,
xsd:hexBinary
,
xsd:base64Binary
,
xsd:anyURI
,
xsd:normalizedString
,
xsd:token
,
xsd:language
,
xsd:NMTOKEN
,
xsd:Name
,
xsd:NCName
,
xsd:integer
,
xsd:nonPositiveInteger
,
xsd:negativeInteger
,
xsd:long
,
xsd:int
,
xsd:short
,
xsd:byte
,
xsd:nonNegativeInteger
,
xsd:unsignedLong
,
xsd:unsignedInt
,
xsd:unsignedShort
,
xsd:unsignedByte
,
xsd:positiveInteger
|
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> < aaa, x > in D:
if <aaa,x> is in D then I(aaa) = x |
if <aaa,x> is in D then ICEXT(x) is the value space of x and is a subset of LV |
if
<aaa,x>
is
in
D
then
for
any
typed
literal
"sss"^^ddd
in
V
with
I(ddd)
=
x
,
if sss is in the lexical space of x then IL("sss"^^ddd) = L2V(x)(sss), otherwise IL("sss"^^ddd) is not in LV |
if
<aaa,x>
is
in
D
then
I(aaa)
is
in
ICEXT(I(
rdfs:Datatype
))
|
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 datatype s 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
datatype
s
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
datatype
s.
The
semantic
conditions
for
rdf-interpretation
s
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-interpretation
s
than
from
rdfs-interpretation
s.
If
the
datatype
s
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-interpretation
s
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-interpretation
s
where
D
contains
a
natural
datatype
map
of
the
graph.
This
amounts
to
treating
datatyping
triples
as
'declarations'
of
datatype
s
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.
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-interpretation s 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-interpretation s 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-entail s 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.
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-entail s E then S also D'-entails E.
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.
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.
Rule name | If E contains | then add |
se1 |
uuu
aaa
xxx
.
|
uuu
aaa
_:
nnn
identifies
a
blank
node
allocated
to
xxx
by
rule
se1
or
se2.
|
se2 |
uuu
aaa
xxx
.
|
_:
nnn
identifies
a
blank
node
allocated
to
uuu
by
rule
se1
or
se2.
|
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
by
se2
using
the
same
<ex:c>
<ex:q>
_:x
.
_:x
allocated
to
ex:a
by
se2
using
a
new
_:x
<ex:p>
_:y
.
_: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:
Rule Name | If E contains | then add |
lg |
uuu
aaa
lll
|
uuu
aaa
_:
nnn
identifies
a
blank
node
allocated
to
the
literal
lll
by
this
rule.
|
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:
Rule Name | If E contains | then add |
gl |
uuu
aaa
_:nnn
where
|
uuu
aaa
lll
|
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.
Rule Name | if E contains | then add |
rdf1 |
uuu
aaa
yyy
.
|
aaa
rdf:type
rdf:Property
.
|
rdf2 |
uuu
aaa
lll
|
where _:nnn identifies a blank node allocated to lll by rule lg . |
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 .
Rule Name | If E contains: | then add: |
---|---|---|
rdfs1 |
uuu
aaa
lll
where lll is a plain literal (with or without a language tag). |
where
|
rdfs2 |
aaa
|
uuu
rdf:type
xxx
.
|
rdfs3 |
aaa
|
vvv
rdf:type
xxx
.
|
rdfs4a |
uuu
aaa
xxx
.
|
uuu
rdf:type
rdfs:Resource
.
|
rdfs4b |
uuu
aaa
vvv
.
|
vvv
rdf:type
rdfs:Resource
.
|
rdfs5 |
uuu
|
uuu
rdfs:subPropertyOf
xxx
.
|
rdfs6 |
uuu
rdf:type
rdf:Property
.
|
uuu
rdfs:subPropertyOf
uuu
.
|
rdfs7 |
aaa
|
uuu
bbb
yyy
.
|
rdfs8 |
uuu
|
uuu
rdfs:subClassOf
rdfs:Resource
.
|
rdfs9 |
uuu
|
vvv
rdf:type
xxx
.
|
rdfs10 |
uuu
rdf:type
rdfs:Class
.
|
uuu
rdfs:subClassOf
uuu
.
|
rdfs11 |
uuu
|
uuu
rdfs:subClassOf
xxx
.
|
rdfs12 |
uuu
rdf:type
rdfs:ContainerMembershipProperty
.
|
uuu
rdfs:subPropertyOf
rdfs:member
.
|
rdfs13 |
uuu
rdf:type
rdfs:Datatype
.
|
uuu
rdfs:subClassOf
rdfs:Literal
.
|
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
.
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.
ext1 |
uuu
|
uuu
rdfs:domain
zzz
.
|
ext2 |
uuu
|
uuu
rdfs:range
zzz
.
|
ext3 |
uuu
rdfs:domain
vvv
.
www
rdfs:subPropertyOf
uuu
.
|
www
rdfs:domain
vvv
.
|
ext4 |
uuu
rdfs:range
vvv
.
www
rdfs:subPropertyOf
uuu
.
|
www
rdfs:range
vvv
.
|
ext5 |
rdf:type
rdfs:subPropertyOf
www
.
www
rdfs:domain
vvv
.
|
rdfs:Resource
rdfs:subClassOf
vvv
.
|
ext6 |
rdfs:
subClassOf
rdfs:subPropertyOf
www
.
www
rdfs:domain
vvv
.
|
rdfs:Class
rdfs:subClassOf
vvv
.
|
ext7 |
rdfs:subPropertyOf
rdfs:subPropertyOf
www
.
www
rdfs:domain
vvv
.
|
rdf:Property
rdfs:subClassOf
vvv
.
|
ext8 |
rdfs:
subClassOf
rdfs:subPropertyOf
www
.
www
rdfs:range
vvv
.
|
rdfs:Class
rdfs:subClassOf
vvv
.
|
ext9 |
rdfs:subPropertyOf
rdfs:subPropertyOf
www
.
www
rdfs:range
vvv
.
|
rdf:Property
rdfs:subClassOf
vvv
.
|
In order to capture datatype entailment in terms of assertions and entailment rules, the rules need to refer to information supplied by the datatype s 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 datatype s 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:
rdfD1 |
ddd
|
_:nnn
where _:nnn identifies a blank node allocated to "sss"^^ddd by rule rule lg . |
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:
rdfD2 |
ddd
|
uuu
aaa
"ttt"^^ddd
.
|
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:
rdfD3 |
ddd
|
uuu
aaa
"ttt"^^eee
.
|
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-entail ed 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
)
(by
rule
rdfD1
)
_:nnn
rdf:type
<ex:dtype>
.
(by
rule
rdfs11
)
<ex:dtype>
rdfs:subClassOf
rdfs:Literal
.
(by
rule
rdfs9
)
_:nnn
rdf:type
rdfs:Literal
.
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
.
(by
rule
rdfD1
)
_:nnn
rdf:type
<ex:otherdtype>
.
_: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
.
xsd 1a |
uuu
aaa
"sss"
.
|
uuu
aaa
"sss"^^
xsd:string
.
|
xsd 1b |
uuu
aaa
"sss"^^
xsd:string
.
|
uuu
aaa
"sss"
.
|
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.
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:
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.
LV Herb(G) = the set of all plain literals in G; IR Herb(G) = the set of all name s and blank nodes which occur in a subject or object position in a triple in G; IP Herb(G) = the set of URI references which occur in the property position of a triple in G;
IEXT
Herb(G)
=
{<s,o>:
G
contains
a
triple
s
p
o
IS Herb(G) and IL Herb(G) are both identity mappings on the appropriate parts of the vocabulary of G. |
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 interpretation s 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
_:nnnrdf: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:
LV RH = all plain literals in C plus { xml (x): x a well-typed XML literal in S}
IR RH = LV RH plus the set of URI references, blank nodes and other typed literals occurring in C
IP RH = {x: C contains the triple: x
rdf:type rdf:Property .
}If x is in IP RH then IEXT RH (x) = {<s,o>: C contains the triple sur (s) x sur (o)
.
}IS RH is the identity map on URIrefs in S
If x is a well-formed XML literal in S then IL RH (x) = xml (x), otherwise IL RH (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.
deleted text: <strong> deleted text: </strong> 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
_:nnnrdf:type
rdfs:Literal .
introduced by rule rdfs1 , and possibly also
_:nnnrdf: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.
LV SH = {x: D contains the triple: sur (x)
rdf:type rdfs:Literal .
}IR SH = LV SH plus the set of URI references, blank nodes and literals other than well-formed XML literals occurring in D
IP SH = {x: D contains the triple: sur (x)
rdf:type rdf:Property .
}If x is in IP RH then IEXT RH (x) = {<s,o>: D contains the triple sur (s) x sur (o)
.
}IS SH is the identity map on URI references in S
If x is a well-formed XML literal in S then IL SH (x) = xml (x), otherwise IL SH (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 triplessur (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
rdfs4aLiteral:
_: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
rdfs9URIref or bnode in object:
a b x
x rdf:type rdfs:Resource
rdfs4bURIref in predicate:
a x b
x rdf:type rdf:Property
rdf:type rdfs:domain rdfs:Resource
x rdf:type rdfs:Resource
rdf1
axiomatic
rdfs2x 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
rdfs9other literal lll :
a b lll
_:x rdf:type rdfs:Literal
lg , rdfs1 , sur (lll)=_:xif
<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 .
rdfs2if
<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 .
rdfs3if
<x,rdf:Property> in IEXT(rdf:type)
then
<x,x> in IEXT(rdfs:subPropertyOf)x rdf:type rdf:Property
x rdfs:subPropertyOf x
rdfs6if
<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
rdfs5if
<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
rdfs7if
<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
rdfs8if
<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
rdfs9if
<x,rdfs:Class> in IEXT(rdf:type)
then
<x,x> in IEXT(rdfs:subClassOf)x rdf:type rdfs:Class
x rdfs:subClassOf x
rdfs10if
<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
rdfs11if
<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
rdfs12if
<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
rdfs13so 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 triplesur ([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.
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 entailment s 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 an 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'.
Infer ence (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.)
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.
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 were corrected.
Following comments by ter Horst , the definition of D-interpretation has been re-worded.
Changes since the 10 October last call working draft/ .
The definition of "name" now includes simple literals (formerly this was restricted to typed literals and URIrefs), and the plain-literal semantic conditions on simple interpretations only apply to the plain literals occurring in a given vocabulary. This makes the development more conventional and provides a more uniform treatment of names and instantiation: it also corrects a previously unnoticed error in the anonymity lemma, and simplifies the proof constructions used in the proofs of the entailment lemmas, which were previously faulty (noted by ter Horst.) This change does not affect any entailments or test cases.
An error in the statement of the RDFS semantic condition concerning rdfs:Datatype was corrected. An error in the statement of the RDFS semantic conditions was corrected.
The entailment rules in section 7 have been re-stated. The rules for simple entailment are now complete for simple entailment,and a special case (rule lg) stated which applies only to literals, for use in defining the RDF and RDFS entailment lemmas. The effect of this is that the rule sets for rdf and rdfs entailment are now more tightly described and generate fewer redundant and irrelevant inferences. The presentation of the rules and examples in this section has been rendered more coherent and two errors in rules rdfs5 and rdfs9 corrected.
A rule gl was added to cover the case where a literal can be inferred to be a class or a property, which can arise in RDFS; the need for this was noted by ter Horst.
The proof appendix A has been completely rewritten. Proofs of the lemmas in the text are now given in full detail.
Following email comments by ter Horst , the definitions of entailment have been simplified so as to make no reference to vocabularies. This makes the definition(s) more conventional and ensures that entailment is transitive. The statement of the rdf and rdfs entailment lemmas has been strengthened slightly to accomodate this change. The content of the lemmas is the same as it was before this change. No test cases are changed by this alteration, but one entailment pattern is now valid that formerly was not, viz.
{ } entails { _:x rdf:type rdfs:ContainerMembershipProperty . }
The definition of 'vocabulary of a graph' in section 0.3 has been simplified to exclude URIreferences inside typed literals. This condition was not needed. Note that test caes and entailments involving datatype entailment all invoke the name of the datatype in a subject position, outside any typed literals.
Several pieces of explanatory prose have been added, in response to requests for clarification.
Changes since the 5 September 2003 working draft.
The appendix describing the Lbase translation (formerly Appendix A) has been removed, with corresponding minor editoral changes to the text to refer to the Lbase Note. Editorial (wording and notation) changes made to the proof appendix for clarity (following comments by ter Horst ) and also to correct errors found in earlier versions.
Some items (Complete, Decideable) and comments added to the glossary.
The rules in section 7 are no longer considered normative, and a comment has been added to the text concerning implementation strategies. The wording of the text in 7.1 concerning the simple entailment rules has been modified for clarity, in response to email comments by Patel-Schneider , and a pointer added to new text in the Appendix describing and discussing a complete rule set for simple entailment.
Text has been added, with examples, to section 5 to better explain the concept of a datatype clash, following email suggestions by Horrocks and Pan , with corresponding minor wording changes made to section 7. Text has been added to 7.3 after the statement of the RDFS entailment lemma pointing out that it does not apply to inconsistent graphs, with a brief discussion; the proof of the lemma has been rewritten.
Character strings in RDF graphs SHOULD (not MUST) conform to normal form C; (WG decision Oct 3 2003) and the text in section 0.2 has been modified to reflect this.
Several minor errors (typos, broken links, redundant references) have been corrected.
Changes since the 23 January 2003 last call working draft.
The following changes do not affect the normative technical content
Many
small
changes
to
wording
and
document
organization
to
improve
clarity,
remove
ambiguities,
make
definitions
clearer,
etc.
and
for
conformity
with
other
RDF
documents
and
W3C
house
style,
following
review
comments
by
Lech.
The
definition
of
'graph
merge'
has
been
rewritten,
following
review
comments
by
Patel-Schneider
and
Beckett.
The
background
colors
have
been
changed
to
avoid
red/green
confusion
and
internal
links
highlighted
with
background
color
in
the
text.
Some
of
the
lemmas
have
been
re-stated
more
economically
and
the
proofs
rephrased
for
clarity.
The
semantic
conditions
are
now
aligned
exactly
with
the
vocabularies,
so
that
RDF
interpretations
exactly
constrain
the
rdf:
vocabulary,
etc..
Some
of
the
section
numbers
and
titles
have
been
changed
to
better
reflect
this
realignment.
The
term
'namespace'
has
been
replaced
by
'vocabulary'
,
cf
pfps-21
.
The
description
of
reification
refers
to
rdf:ID
.
The informative appendices on Lbase and the proof appendix have been extensively rewritten and errors corrected, cf. pfps-02 . Some parallel changes have been made to the Lbase note. The text of the datatypes section been extensively rewritten following technical changes noted below; the older version was ambiguous and contained errors . The XSD datatypes suitable for RDF use are listed explicitly in the text, cf. pfps-01 .
Following email by Patel-Schneider, the possibility of an rdfs-inconsistency has been noted in the text (section 4, end), with an example, a comment added on trivial entailments in section 4.3, and the wording of the rdfs entailment lemma has been restricted to consistent antecedents (section 7.3). This corrects an omission and a consequent error in the text.
The following changes do not affect any entailments or test cases.
rdf:type
rdf:List
triples
for
all
sublists.
(WG
decision
recorded
http://lists.w3.org/Archives/Public/w3c-rdfcore-wg/2003May/0199.html
)
The
wording
of
the
text
and
examples
in
section
3.2.3
have
been
modified
to
suit.
All the following changes cause changes to some entailments.
1. The treatment of language tags in literals has been simplified. Typed literals, including XML literals, no longer have associated language tags. (WG decision recorded http://lists.w3.org/Archives/Public/w3c-rdfcore-wg/2003May/0138.html) There are therefore three types of literal: plain without a language tag, plain with a language tag, and typed literals.
XML literals are required to be in canonical form, and to denote entities which are distinct from any character string.(WG decision recorded http://lists.w3.org/Archives/Public/w3c-rdfcore-wg/2003Jun/0156.html )
The
chief
effect
of
these
decisions
in
this
document
is
that
XML
literals
can
be
treated
uniformly
with
other
typed
literals.
However,
the
rdf
semantic
conditions
on
rdfs:XMLLiteral
are
stated
without
explicit
reference
to
datatypes
in
order
to
make
it
possible
for
a
conforming
RDF
inference
system
to
avoid
full
datatyping.
The
accounts
of
rdfs:XMLLiteral
given
for
RDF,
and
for
RDF
with
datatyping,
are
in
exact
correspondence.
This
change
is
relevant
to
the
last
call
comments:
pfps-02
,
pfps-06
,
pfps-07
.
2.
The
treatment
of
rdfs:subClassOf
and
rdfs:subPropertyOf
has
been
changed.
They
are
no
longer
required
to
satisfy
'iff'
semantic
conditions,
but
only
to
be
transitive
and
reflexive.
This
decision
was
taken
as
a
result
of
the
felt
need
to
support
simple
complete
sets
of
inference
rules
for
RDFS.
We
are
grateful
to
Herman
ter
Horst
and
Peter
Patel-Schneider
for
noticing
the
complexities
which
arose
from
the
older
conditions,
c.f.
horst-01
et.
seq..
(WG
decision
recorded
http://lists.w3.org/Archives/Public/w3c-rdfcore-wg/2003Jul/0025.html
)
This has required changes to the RDFS entailment rules table. The older conditions are now explained in informative sections as 'extensional' conditions, and corresponding entailment rules discussed.
3.
Plain
literals,
and
literals
typed
with
xsd:string
,
both
denote
character
strings.
(WG
decision
recorded
http://lists.w3.org/Archives/Public/w3c-rdfcore-wg/2003Jul/att-0393/rdfcore-2003-07-25-minutes.txt
)
The
semantics
now
states
explicitly
that
these
are
the
coextensive
in
XSD
interpretations
and
describes
a
corresponding
inference
rule.
4. The account of datatyped interpretations has been stated relative to a 'datatype map' from URI references to datatypes. This change was introduced after email discussions with Patel-Schneider, cf pfps-08 and pfps-09 et. seq.. The previous treatment was not adequate to support the intended entailments . (WG decision recorded http://lists.w3.org/Archives/Public/w3c-rdfcore-wg/2003May/0199.html )
5.
LV
is
required
to
be
the
class
extension
of
rdfs:Literal
in
all
RDFS
interpretations,
cf.
pfps-06
.
This
clarifies
and
rationalizes
the
treatment
of
rdfs:Literal
and
supports
the
entailments
noted
in
pfps-10
.
6. The rule rdfD4 has been removed: given the change mentioned in 2. above, it was no longer valid. The text notes that it is consistent to assert a subClass relation between datatypes in this case.