- This version:
- http://www.w3.org/2002/06/lbase/20021106Snapshot
- Latest version:
- @@@UPDATE@@@http://www.w3.org/2002/06/lbase/
- Authors:
- R.V.Guha, IBM, < rguha@us.ibm.com>
- Patrick Hayes, IHMC, University of West Florida < phayes@ai.uwf.edu>

Copyright
© 2003 W3C^{®} (MIT, ERCIM, Keio), All
Rights Reserved. W3C liability,
trademark,
document use
and software
licensing rules apply.

This document presents a framework for specifying the semantics for the languages of the Semantic Web. Some of these languages (RDF [RDF], RDFS [RDFS] and OWL [OWL]) are currently in various stages of development and we expect others to be developed in the future.This framework is intended to provide a framework for specifying the semantics of all of these languages in a uniform and coherent way. specifying the semantics of all of these languages in a uniform and coherent way. The strategy is to translate the various languages into a common 'base' language thereby providing them with a single coherent model theory.

We describe a mechanism for providing a precise semantics for the Semantic Web Languages (referred to as SWELs from now on). The purpose of this is to define clearly the consequences and allowed inferences from constructs in these languages.

*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 Recommendations and other technical
reports is available at http://www.w3.org/TR/.*

This document is a working document for the use by W3C Members and other interested parties. It may be updated, replaced or made obsolete by other documents at any time.

This document results from discussions within the RDF Core Working Group concerning the formalization of RDF and RDF-based languages. The RDF Core Working Group is part of the W3C Semantic Web Activity. The group's goals and requirements are discussed in the RDF Core Working Group charter. These include requirements that...

*The RDF Core group must take into account the various formalizations of RDF that have been proposed since the publication of the RDF Model and Syntax Recommendation. The group is encouraged to make use both of formal techniques and implementation-led test cases throughout their work.**The RDF schema system must provide an extensibility mechanism to allow future work (for example on Web Ontology and logic-based Rule languages) to provide richer facilities.*

This document is motivated by these two requirements. It does not present an RDF Core WG design for Semantic Web layering. Rather, it documents a technique we are exploring to describe the semantics of the RDF Core specifications. The RDF Core WG solicit feedback from other Working Groups and from the RDF implementor community on the wider applicability of this technique.

Review comments on this document are invited and should be sent to the public mailing list www-rdf-comments@w3.org. An archive of comments is available at http://lists.w3.org/Archives/Public/www-rdf-comments/.

Discussion of this document is invited on the www-rdf-logic@w3.org list of the RDF Interest Group (public archives).

- 1. Model-theoretic semantics
- 2. Outline of Approach
- 2.1. Consistency
- 2.2. L
_{base}Syntax - 2.3. Interpretations
- 2.4. Axiom Schemas
- 2.5. Entailment
- 3.0. 3.0 Using L
_{base} - 3.1. Relation between the two...
- 4.0. 4.0 Inadequacies of
- 5.0 Acknowledgements
- 6.0 6.0 References

A model-theoretic
semantics for a language assumes that the language refers to a
'world', and describes the minimal conditions that a world must
satisfy in order to assign an appropriate meaning for every
expression in the language. A particular world is called an
*interpretation,* so that model theory might be better called
'interpretation theory'. The idea is to provide a mathematical
account of the properties that any such interpretation must have,
making as few assumptions as possible about its actual nature or
intrinsic structure. Model theory tries to be metaphysically and
ontologically neutral. It is typically couched in the language of
set theory simply because that is the normal language of
mathematics - for example, this semantics assumes that names denote
things in a *set* IR called the 'universe' - but the use of
set-theoretic language here is not supposed to imply that the
things in the universe are set-theoretic in nature.

The chief utility of such a semantic theory is not to suggest any particular processing model, or to provide any deep analysis of the nature of the things being described by the language, but rather to provide a technical tool to analyze the semantic properties of proposed operations on the language; in particular, to provide a way to determine when they preserve meaning. Any proposed inference rule, for example, can be checked to see if it is valid with respect to a model theory, i.e. if its conclusions are always true in any interpretation which makes its antecedents true.

We note that the word 'model' is often used in a rather different sense, eg as in 'data model', to refer to a computational system or data structures of some kind. To avoid misunderstanding, we emphasise that the interpretations referred to in a model theory are not, in general, intended to be thought of as things that can be computed or manipulated by computers.

There will be many Semantic Web languages, most of which will be
built on top of more basic Semantic Web language(s). It is
important that this *layering* be clean and simple, not just
for human understandability, but also to enable the construction of
robust semantic web agents that use these languages. The emerging
current practice is for each of the SWELs to be defined in terms of
their own model theory, layering it on top of the model theories of
the languages they are layered upon. While having a model theory is
clearly desireable, and even essential, for a SWEL, this
direct-construction approach has several problems. It produces a
range of model theories, each with its own notion of consequence
and entailment. It requires expertise in logic to make sure that
model theories align properly, and model-theoretic alignment does
not always sit naturally with interoperability requirements.
Experience to date (particularly with the OWL standard under
development at the time of writing by the W3C Webont working
group) shows that quite difficult problems can arise when
layering model theories for extensions to the 'basic' RDF layer
[RDF] of the semantic web. Moreover, this strategy places a very
high burden on the 'basic' layer, since it is difficult to
anticipate the semantic demands which will be made by all future
higher layers, and the expectations of different development and
user communities may conflict. Further, we believe that a melange
of model theories will adversely impact developers building agents
that implement proof systems for these layers, since the proof
systems will likely be different for each layer, resulting in the
need to micro-manage small semantic variations for various dialects
and sub-languages (cf. the distinctions between various dialects of
OWL).

In this document, we use an alternative approach to for defining
the semantics for the different SWELs in a fashion which ensures
interoperability. We first define a basic language L_{base}
which is expressive enough to state the content of all currently
proposed web languages, and has a fixed, clear model-theoretic
semantics. Then, the semantics of each SWEL L_{i} is
defined by specifying how expressions in the L_{i} map into
equivalent expressions in L_{base}, and by providing axioms
written in L_{base} which constrain the intended meanings
of the SWEL special vocabulary. The L_{base} meaning of any
expression in *any* SWEL language can then be determined by
mapping it into L_{base} and adding the appropriate
language axioms, if there are any.

The intended result is that the model theory of L_{base}
is the model theory of *all* the Semantic Web Languages,
even though the languages themselves are different. This makes it
possible to use a single inference mechanism to work on these
different languages. Although it will possible to exploit
restrictions on the languages to provide better performance, the
existence of a reference proof system is likely to be of utility to
developers. This also allows the meanings of expressions in
different SWELs to be compared and combined, which is very
difficult when they all have distinct model theories.

The idea of providing a semantics for SWELs by
translating them into logic is not new [see for example
Marchiolri&Saarela, Fikes&McGuinness] but we plan to adopt
a somewhat different style than previous 'axiomatic semantics',
which have usually operated by mapping all RDF triples to instances
of a single three-place predicate. We propose rather to use the
logical form of the target language as an explication of the
intended meaning of the SWEL, rather than simply as an axiomatic
description of that meaning, so that RDF classes translate to unary
predicates, RDF properties to binary relations, the relation
rdf:type translates to application of a predicate to an argument,
and list-valued properties in OWL or DAML can be translated into
n-ary or variadic relations. The syntax and semantics of
L_{base} have been designed with this kind of translation
in mind. It is our intent that the model theory of L_{base}
be used in the spirit of its model theory and not as a programming
language, i.e., relations in L_{i} should correspond to
relations in L_{base}, variables should correspond to
variables and so on.

It is important to note that L_{base} is not
being proposed as a SWEL. It is a tool for specifying the semantics
of different SWELs. The syntax of L_{base} described here
is not intended to be accessible for machine processing; any such
proposal should be considered to be a proposal for a more
expressive SWEL.

By using a well understood logic (i.e., first order logic
[Enderton]) as the core of L_{base}, and providing for
mutually consistent mappings of different SWELs into
L_{base}, we ensure that the content expressed in several
SWELs can be combined consistently, avoiding paradoxes and other
problems. Mapping type/class language into predicate/application
language also ensures that set-theoretical paradoxes do not arise.
Although the use of this technique does not in itself guarantee
that mappings between the syntax of different SWELs will always be
consistent, it does provide a general framework for detecting and
identifying potential inconsistencies.

It is also important that the axioms defining the vocabulary
items introduced by a SWEL are internally consistent. Although
first-order logic (and hence L_{base}) is only
semi-decideable, we are confident that it will be routine to
construct L_{base} interpretations which establish the
relevant consistencies for all the SWELs currently contemplated. In
the general case, future efforts may have to rely on certifications
from particular automated theorem provers stating that they weren't
able to find an inconsistency with certain stated levels of effort.
The availablity of powerful inference engines for first-order logic
is of course relevant here.

In this document, we use a version of
first order logic with equality as L_{base}. This imposes a
fairly strict monotonic discipline on the language, so that it
cannot express local default preferences and several other
commonly-used non-monotonic constructs. We expect
that as the Semantic Web grows to encompass more and our
understanding of the Semantic Web improves, we will need to replace
this L_{base} with more expressive logics. However,
we expect that first order logic will be a proper subset of such
systems and hence we will be able to smoothly transition to more
expressive L_{base} languages in the future. We note that
the computational advantages claimed for various sublanguages of
first-order logic, such as description logics, logical programming
languages and frame languages, are irrelevant for the purposes of
using L_{base} as a semantic specification language.

We will use First Order Logic with suitable minor changes to account for the use of referring expressions (such as URIs) on the Web, and a few simple extensions to improve utility for the intended purposes.

Any first-order logic is based on a set of *atomic terms*,
which are used as the basic referring expressions in the syntax.
These include *names*, which refer to entities in the domain,
*special names*, and *variables*. L_{base} allows
character strings (not starting with the characters ')','(', '\',
'?','<' or ''' , and containing no whitespace characters) as
names, but distinguishes the special class of *urirefs*,
defined to be a URI reference in the sense of [URI]. Urirefs are
used to refer to both individuals and relations between the
individuals.

L_{base} allows for various collections of special names
with fixed meanings defined by other specifications (external to
the L_{base} specification). There is no assumption that
these could be defined by collections of L_{base} axioms,
so that imposing the intended meanings on these special names may
go beyond strict first-order expressiveness. (In mathematical
terms, we allow that some sets of names refer to elements of
certain fixed algebras, even when the algebra has no characteristic
first-order description.) Each such set of names
has an associated predicate which is true of the things denoted by
the names in the set. At present, we assume three categories
of such fixed names: numerals, quoted strings, and XML structures,
with associated predicate names 'NatNumber', 'String', and
'XmlThing' respectively.

Numerals are defined to be strings of the characters
'0123456789', and are interpreted as decimal numerals in the usual
way. Since arithmetic is not first-order definable, this is the
first and most obvious place that L_{base} goes beyond
first-order expressiveness.

Quoted strings are arbitrary character sequences enclosed in (single) quotation marks, and are interpreted as denoting the string inside the quotation marks. The backslash character is used as a self-escaping prefix escape to include a quote mark in a quoted string. Quoted strings denote the strings they contain, i.e. any such string denotes the string gotten by removing the two surrounding quote marks and unescaping any inner quote marks which are then at the beginning or end. For example,

'\'a\\\'b\''

is a quoted string which denotes the string:

'a\\\'b'

Double quote marks have no special interpretation.

An XML structure is a character string which is a well-formed piece of XML, possibly with an XML lang tag, and it is taken to denote an abstract structure representing the parsed XML with the lang tag attached to any enclosed literals.

The associated predicate names *NatNumber*,
*String*, *XmlThing* and *Relation* (see
below) are considered to be special names.

A variable is any non-white-space character string starting with the character '?'.

The characters '(', ',' and ')' are considered to be punctuation symbols.

The categories of punctuation, whitespace, names, special names and variables are exclusive and each such string can be classified by examining its first character. This is not strictly necessary but is a useful convention.

Any L_{base} language is defined with respect to a
*vocabulary*, which is a set of non-special names. We require
that every L_{base} vocabulary contain all urirefs, but
other expressions are allowed. (We will require that every
L_{base} interpretation provide a meaning for every special
name, but these interpretations are fixed, so special names are not
counted as part of the vocabulary.)

There are several aspects of meaning of expressions on the
semantic web which are not yet treated by this semantics; in
particular, it treats URIs as simple names, ignoring aspects of
meaning encoded in particular URI forms [RFC 2396] and does not
provide any analysis of time-varying data or of changes to URI
denotations. The model theory also has nothing to say about whether
an HTTP uri such as "http://www.w3.org/" denotes the World Wide Web
Consortium or the HTML page accessible at that URI or the web site
accessible via that URI. These complexities may be addressed in
future extensions of L_{base}; in general, we expect that
L_{base} will be extended both notationally and by adding
axioms in order to track future standardization efforts.

We do not take any position here on the way that urirefs may be composed from other expressions, e.g. from relative URIs or Qnames; the model theory simply assumes that such lexical issues have been resolved in some way that is globally coherent, so that a single uriref can be taken to have the same meaning wherever it occurs.

Similarly, the model theory given here has no special provision
for tracking temporal changes. It assumes, implicitly, that urirefs
have the same meaning *whenever* they occur. To provide an
adequate semantics which would be sensitive to temporal changes is
a research problem which is beyond the scope of this document..

We will assume that there are three sets of names (not special
names) which together constitute the vocabulary: individual names,
relation names, and function names, and that each function name has
an associated *arity*, which is a non-negative integer. In a
particular vocabulary these sets may or may not be disjoint.
Expressions in L_{base} (speaking strictly,
L_{base} expressions in this particular vocabulary) are
then constructed recursively as follows:

A *term* is either a name or a special name or a variable,
or else it has the form f(t1,...,tn) where f is an n-ary function
name and t1,...,tn are terms.

A *formula* is either atomic or boolean or quantified,
where:

an atomic formula has the form (t1=t2) where t1 and t2 are terms, or else the form R(t1,...,tn) where R is a relation name or a variable and t1,...,tn are terms;

a boolean formula has one of the forms

(W1 and W2 and ....and Wn)

(W1 or W2 or ... or Wn)

(W1 implies W2)

(W1 iff W2)

(not W1)

where W1, ...,Wn are formulae; and

a quantified formula has one of the forms

(forall (?v1 ...?vn) W)

(exists (?v1 ... ?vn) W)

where ?v1,...,?vn are variables and W is a formula. (The
subexpression just after the quantifier is the *variable
list* of the quantifier. Any occurrence of a variable in W is
said to be *bound* in the quantified formula *by* the
nearest quantifer to the occurrence which includes that variable in
its variable list, if there is one; otherwise it is said to be
*free* in the formula.)

Finally, an L_{base} *knowledge base* is a set of
formulae.

Formulae are also called 'wellformed formulae' or 'wffs' or simply 'expressions'. In general, surplus brackets may be omitted from expressions when no syntactic ambiguity would arise.

Some comments may be in order. The only parts of this definition
which are in any way nonstandard are (1) allowing 'special names',
which was discussed earlier; (2) allowing variables to occur in
relation position, which might seem to be at odds with the claim
that L_{base} is first-order - we discuss this further
below - and (3) not assigning a fixed arity to relation names. This
last is a useful generalization which makes no substantial changes
to the usual semantic properties of first-order logic, but which
eases the translation process for some SWEL syntactic constructs.
(The computational properties of such 'variadic relations' are
quite complex, but L_{base} is not being proposed as a
language for computational use.)

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 L_{base} well formed
formula in that world. It does this by specifying for each uriref,
what it is supposed to be a name of; and also, if it is a function
symbol, what values the function has for each choice of arguments;
and further, if it is a relation symbol, which sequences of things
the relation holds between. This is just enough information to
determine the truth-values of all atomic formulas; and then this,
together with a set of recursive rules, is enough to assign a truth
value for any L_{base} formula.

In specifying the following it is convenient to define use some
standard definitions. A relation over a set S is a set of finite
sequences (tuples) of members of S. If R is a relation and all the
elements of R have the same length n, then R is said to have
*arity* n, or to be a *n-ary relation*. Not every
relation need have an arity. If R is an (n+1)-ary relation over S
which has the property that for any sequence <s1,...,sn> of
members of S, there is exactly one element of R of the form <s0,
s1, ..., sn>, then R is an n-ary *function*; and s0 is the
*value* of the function for the arguments s1, ...sn. (Note
that an n-ary function is an (n+1)-ary relation, and that, by
convention, the function value is the first argument of the
relation, so that for any n-ary function f, f(y,x1,...,xn) means
the same as y = f(x1,...,xn).)

The conventional textbook treatment of first-order
interpretations assumes that relation symbols denote relations. We
will modify this slightly to require that relation symbols denote
entities with an associated relation, called the relational
extension, and will sometimes abuse terminology by referring to the
entities with relational extensions as relations. This device gives
L_{base} some of the freedom to quantify over relations
which would be familiar in a higher-order logic, while remaining
strictly a first-order language in its semantic and metatheoretic
properties. We will use the special name
*Relation* to denote the property of having a relational
extension.

Let VV be the set of all variables, and NN be the set of all special names.

We will assume that there is a globally *fixed* mapping
SN from elements of NN to a domain ISN (i.e, consisting of
character strings, integers and XML structures). The exact
specification of SN is given for numerals by the usual reading of a
decimal numeral to denote a natural number; for quoted strings by
the dequotation rules described earlier; and for XML structures by
the XML 1.0 specification [XML].

An *interpretation* I of a vocabulary V is then a
structure defined by:

- a set ID, called the domain or universe of I;
- a mapping IS from (V union VV) into ID;
- a mapping IEXT from IR, a subset of ID, into a relation over ID+ISN (ie a set of tuples of elements of ID+ISN).

which satisfies the following conditions:

- for any n-ary function symbol f in V, IEXT(I(f)) is an n-ary function over ID+ISN.
- IEXT(I(
*NatNum*)) = {<n>, n a natural number} - IEXT(I(
*String*)) = {<s>, s a character string} - IEXT(I(
*XmlThing*)) = {<x>, x an XML structure} - IEXT(I(
*Relation*)) = IR

An interpretation then specifies the value of any other
L_{base} expression E according to the following rules:

if E is: |
then I(E) is: |

a name or a variable | IS(E) |

a special name | SN(E) |

a term f(t1,...,tn) | the value of IEXT(I(f)) for the arguments I(t1),...,I(tn) |

an equation (A=B) | true if I(A)=I(B), otherwise false |

a formula of the form R(t1,...,t2) | true if IEXT(I(R)) contains the sequence <I(t1),...,I(tn)>, otherwise false |

(W1 and ...and Wn) | true if I(Wi)=true for i=1 through n, otherwise false |

(W1 or ...or Wn) | false if I(Wi)=false for i=1 through n, otherwise true |

(W1 <=> W2) | true if I(W1)=I(W2), otherwise false |

(W1 => W2) | false if I(W1)=true and I(W2)=false, otherwise true |

not W | true if I(W)=false, otherwise false |

If B is a mapping from a set W of variables into ID, then define [I+B] to be the interpretation which is like I except that [I+B](?v)=B(?v) for any variable ?v in W.

if E is: |
then I(E) is: |

(forall (?v1,...,?vn) W) | false if [I+B](W)=false for some mapping B from {?v1,...,?vn} into ID, otherwise true |

(exist (?v1,...,?vn) W) | true if [I+B](W)=true for some mapping B from {?v1,...,?vn} into ID, otherwise false |

Finally, a knowledge base is considered to be true if and only if all its elements are true, .i.e. to be a conjunction of its elements.

Intuitively, the meaning of an expression containing free variables is not well specified (it is formally specified, but the interpretation of the free variables is arbitrary.) To resolve any confusion, we impose a familiar convention by which any free variables in a sentence of a knowledge base are considered to be universally quantified at the top level of the expression in which they occur. (Equivalently, one could insist that all variables in any knowledge-base expression be bound by a quantifier in that expression; this would force the implicit quantification to be made explicit.)

These definitions are quite conventional. The only unusual features are the incorporation of special-name values into the domain, the use of an explicit extension mapping, the fact that relations are not required to have a fixed arity, and the description of functions as a class of relations.

The explicit extension mapping is a technical device to allow
relations to be applied to other relations without going outside
first-order expressivity. We note that while this allows the same
name to be used in both an individual and a relation position, and
in a sense gives relations (and hence functions) a 'first-class'
status, it does not incorporate any comprehension principles or
make any logical assumptions about what relations are in the
domain. Notice that no special semantic conditions were invoked to
treat variables in relation position differently from other
variables. In particular, the language makes no comprehension
assumptions whatever. The resulting language is first-order in all
the usual senses: it is compact and satisfies the
downward Skolem-Lowenheim property, for example, and the
usual machine-oriented inference processes still apply, in
particular the unification algorithm. (One can obtain a translation
into a more conventional syntax by re-writing every atomic sentence
using a rule of the form R(t1,...,tn) => Holds(R, t1,...,tn),
where 'Holds' is a 'dummy' relation indicating that the relation R
is true of the remaining arguments. The presentation given here
eliminates the need for this artificial translation, but its
existence establishes the first-order properties of the language.
To translate a conventional first-order syntax
into the L_{base} form, simply qualify all quantifiers to
range only over non-*Relation*s. The issue is further
discussed in (Hayes & Menzel ref). )

Allowing relations with no fixed arity is a technical
convenience which allows L_{base} to accept more natural
translations from some SWELs. It makes no significant difference to
the metatheory of the formalism compared to a fixed-arity sytnax
where each relation has a given arity. Treating functions as a
particular kind of relation allows us to use a function symbol in a
relation position (albeit with a fixed arity, which is one more
than its arity as a function); this enables some of the
translations to be specified more efficiently.

As noted earlier, incorporating special name interpretations (in
particular, integers) into the domain takes L_{base}
outside strict first-order compliance, but these domains have
natural recursive definitions and are in common use throughout
computer science. Mechanical inference systems typically have
special-purpose reasoners which can effectively test for
satisfiability in these domains. Notice that the incorporation of
these special domains into an interpretation does not automatically
incorporate all truths of a full theory of such structures into
L_{base}; for example, the presence of the integers in the
semantic domain does not in itself require all truths of arithmetic
to be valid or provable in L_{base}.

An axiom scheme stands for an infinite set of L_{base}
sentences all having a similar 'form'. We will allow schemes which
are like L_{base} formulae except that expressions of the
form "<exp1>**...**<expn>", ie two
expressions of the same syntactic category separated by three dots,
can be used, and such a schema is intended to stand for the
infinite knowledge base containing all the L_{base}
formulae gotten by substituting some actual sequence of appropriate
expressions (terms or variables or formulae) for the expression
shown, which we call the *L _{base} instances* of the
scheme. (We have in fact been using this convention already, but
informally; now we are making it formal.) For example, the
following is an L

(forall
(?v1**...**?vn)(R(?v1**...**?vn) implies
Q(a, ?v2**...**?vn)))

- where the expression after the first quantifier is an actual scheme expression, not a conventional
abbreviation - which has the following L_{base}
instances, among others:

(forall (?x)(R(?x) implies Q(a, ?x)))

(forall (?y,?yy,?z)(R(?y, ?yy, ?z) implies Q(a,?y,?yy,?z)))

Axiom schemes do not take the language beyond first-order, since
all the instances are first-order sentences and the language is
compact, so if any L_{base} sentence follows from (the
infinite set of instances of) an axiom scheme, then it must in fact
be entailed by some finite set of instances of that scheme.

We note that L_{base} schemes should be understood only
as syntactic abbreviations for (infinite) sets of L_{base}
sentences when stating translation rules and specifying axiom sets.
Since all L_{base} expressions are required to be finite,
one should not think of L_{base} schemes as themselves
being sentences; for example as making assertions, as being
instances or subexpressions of L_{base} sentences, or as
being posed as theorems to be proved. Such usages would go beyond
the first-order L_{base} framework. (They amount to a
convention for using infinitary logic: see [Hayes& Menzel] for
details.) This kind of restricted use of 'axiom schemes' is
familiar in many textbook presentations of logic.

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

Any process or technique which
constructs a well formed formula F_{output} from some other
F_{input} is said to be *valid* if F_{input}
entails F_{output}, 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.

a procedure for translating expressions in L_{i} to
expressions in L_{base}. This process will also
consequently define the subset of L_{base} that is used by
L_{i}.

a set of vocabulary items introduced by L_{i}

a set of axioms and/or axiom schemas (expressed in
L_{base} or L_{base} schema) that define the
semantics of the terms in (2).

Given a set of expressions G in L_{i}, we apply the
procedure above to obtain a set of equivalent well formed formulae
in L_{base}. We then conjoin these with the axioms
associated with the vocabulary introduced by L_{i} (and any
other language upon which L_{i} is layered). If there are
associated axiom schemata, we appropriately instantiate these and
conjoin them to these axioms. The resulting set, referred to as
A(G), is the *axiomatic equivalent* of G.

As an illustrative example, we give in the following table a
**sketch** of the axiomatic equivalent for a fragment
of RDF(S), in the form of a translation from N-triples.
**This is incomplete and should not be referred to as an
accurate semantic description**. It is given here only to
illustrate the axiomatic technique which L_{base} is
designed to support.

rdf:Resource | T |

any other uriref aaa |
a name aaa |

a blank node _:xxx |
a variable ?xxx |

an untyped literal "xxx" |
a character string 'xxx' |

a typed literal "xxx"^^rdf:XMLLiteral |
the XML special name xxx |

any other typed literal |
a term aaa('xxx') |

a triple aaa bbb ccc . |
bbb(aaa, ccc) |

an RDF graph | the existential closure of the conjunction of the translations of all the triples in the graph |

T(?x) |

The following diagram illustrates the relation between Li, L

The important point to note about the above diagram is that if
the Li to L_{base} mapping and model theory for Li are done
consistently, then the two 'routes' from G to a satisfying
interpretation will be equivalent. This is because the
L_{i} axioms included in the L_{base} equivalent of
G should be sufficient to guarantee that any satisfying
interpretation in the L_{base} model theory of the
L_{base} equivalent of G will contain a substructure which
is a satisfying interpretation of G according to the Li model
theory, and vice versa.

The utility of this framework for combining assertions in
several different SWELs is illustrated by the following diagram,
which is an 'overlay' of two copies of the previous diagram.

Note that the G1+G2 equivalent in this case contains axioms for
both languages, ensuring (if all is done properly) that any
L_{base} interpretation will contain appropriate
substructures for both sentences.

If the translations into L_{base} are appropriately
defined at a sufficient level of detail, then even tighter semantic
integration could be achieved, where expressions which 'mix'
vocabulary from several SWELs could be given a coherent
interpretation which satisfies the semantic conditions of both
languages. This will be possible only when the SWELS have a
particularly close relationship, however. In the particular case
where one SWEL (the one used by G2) is layered on top of another
(the one used by G1), the interpretations of G2 will be a subset of
those of G1

- It does not capture the social meaning of URIs. It merely treats them as opaque symbols. A future web logic should go further towards capturing this intention.
- At the moment, L
_{base}does not provide any facilities related to the representation of time and change. However, many existing techniques for temporal representation use languages similar to L_{base}in expressive power, and we are optimistic that L_{base}can provide a useful framework in which to experiment with temporal ontologies for Web use.

- It might turn out that some aspects of what we want to
represent on the the semantic web requires more than can be
expressed using the L
_{base}described in this document. In particular, L_{base}does not provide a mechanism for expressing propositional attitudes or true second order constructs.

- [Enderton] A Mathematical Introduction to Logic, H.B.Enderton,
2
^{nd}edition, 2001, Harcourt/Academic Press. - [Fikes & McGuinness] R. Fikes, D. L. McGuinness, An Axiomatic Semantics for RDF, RDF Schema, and DAML+OIL, KSL Technical Report KSL-01-01, 2001
- [Hayes & Menzel] P. Hayes, C. Menzel, A Semantics for the Knowledge Interchange Format , 6 August 2001 (Proceedings of 2001 Workshop on the IEEE Standard Upper Ontology)
- [Marchiori & Saarela]M. Marchioi, J. Saarela, Query + Metadata + Logic = Metalog, 1998
- [OWL] F. van Harmelen, P. F. Patel-Schneider, Ian Horrocks (editors), Reference Description of the DAML+OIL (March 2001) ontology markup language
- [RDF] O. Lassila, Ralph R. Swick (editors), Resource Description Framework (RDF) Model and Syntax Specification, 22 February 1999.
- [RDFS] D. Brickley, R.V. Guha (editors), @@@TODO: update reference.@@@ Resource Description Framework (RDF) Schema Specification 1.0, 27 March 2000 (W3C Candidate Recommendation).
- [URI] T. Berners-Lee, Fielding and Masinter, RFC 2396 - Uniform Resource Identifiers (URI): Generic Syntax, August 1998.
- [WebOnt] The Web Ontology Working Group
- [XML] T. Bray, J. Paoli, C.M. Sperberg.McQueen, E. Maler. Extensible Markup Language (XML) 1.0 (Second Edition), W3C Recommendation 6 October 2000