# updated draft of model theory

From: pat hayes <phayes@ai.uwf.edu>
Date: Fri, 27 Jul 2001 14:49:15 -0700
Message-Id: <v04210100b78784cc7ee9@[130.107.66.237]>

```RDF Model Theory StrawDog proposal (draft 7/27/01)(typos fixed in
sections 1 thru 5, sections 6,7,8 added.)

Pat Hayes, IHMC

In what follows, 'N-triples' refers to Dave Beckett's version 5,
relevant parts of which are:
ntripleDoc  ::=line*
line        ::=ws* (comment | blankline | triple) eoln
triple      ::=subject ws+ predicate ws+ object ws* '.' ws*
subject     ::=uriref | anonNode
predicate   ::=uriref
object      ::=uriref | anonNode | qLiteral

1. Introduction

The basic idea is to re-state the M&S notions of what a set of RDF
triples means in mathematical terms, making as few extra or
gratuitous assumptions as possible. By keeping things as general as
possible, any restrictions that someone might want to impose can be
captured by further restrictions on the model theory.

We assume that:
a. there is no restriction on what a resource can be
b. there is no restriction on the domains and ranges of properties;
in particular, a property may be applied to itself.
c. all nodes in an RDF graph play fundamentally the same role in the
language no matter how they are labelled (for comments on anonymous
nodes, see later.)

2. Interpretations

First, we assume a global set  LV of literal values and a fixed
mapping XL from the set of qLiterals to LV. All interpretations will
be required to conform to XL on qLiteral expressions.
<comment> This leaves open the question of the exact nature of
literals, their language-sensitivity and so on, while acknowledging
their special status as expressions with a 'fixed' interpretation.
</comment>

All interpretations will be relative to a set of RDF nodes, called
the vocabulary of the interpretation, so that one has to speak,
strictly, of an interpretation of an RDF 'nodespace' or of an RDF
graph, rather than of RDF itself.

This is stated in terms of RDF nodes and graphs to follow the M&S.
For N-triples, we can identify the vocabulary of I with a set of
expressions that would be mapped to nodes by a parser, so it would be
a subset of (uriref |anonNode)
<comment> Restricting an interpretation to a particular namespace is
normal practice in model theory, since an interpretation mapping is
expected to interpret only a given set of 'names'. It is possible to
adopt a 'global' approach where any interpretation assigns a meaning
to every possible expression, but this makes the subsequent
development more awkward. For example, there are no 'new names' in a
global model theory, since all names have already been given a fixed
meaning by every interpretation. </comment>

An interpretation I (of vocab(I)) is defined by:
1. A nonempty set R of resources, called the domain or universe of I.
<comment> Whatever resources are, this a set of those. </comment>
2. A non-empty subset P of R corresponding to properties
<comment> Note that  P could be R itself.</comment>
3. A mapping IEXT: P -> Rx(R union LV)   (ie the set of pairs <x,y>
with x in R and y in R or LV)
<comment> IEXT(x) is the extension of x. This allows a property to be
an object and so occur in its own extension, a neat trick I learned
from Chris Menzel. </comment>
4. A mapping IS:  vocab(I) -> R

The denotation of an RDF expression E in I is given by the following
rules (note, other rules follow later):

>> if E is a <uriref>  or <anonNode> then I(E) = IS(E)
>> if E is a <qLiteral> then I(E) = LV(E)
>> if E is an asserted triple with the form  s p o
then I(E) = true iff <I(s),I(o)> is in  IEXT(I(p)), otherwise I(E)= false.
>> if E is a set of triples then I(E) = false just in case I(E') =
>>false for some asserted triple E' in E, otherwise I(E) = true.

(Note that if E contains no asserted triples then I(E)= true. This is
the usual convention: an empty conjunction is considered to be true
since it contains nothing that would cause it to be false. This makes
empty assertions have vacuous content, as one would expect.)

Comments and blanklines have no semantics, of course.

The use of  the phrase "asserted triple" is a deliberate
weasel-worded artifact, to allow an RDF graph/document to contain
triples which are being used for some non-assertional purpose (such
as expressing a query, implementing part of some expression of
another language, or forming part of a pattern intended to be matched
against some asserted RDF in another document, or just being
breezy:-). We may wish to say that strict conformity to the RDF M&S
assumes that all triples in a document are asserted triples, but
making the distinction allows RDF parsers and engines such as CWM and
Euler to conform to the RDF syntax and to respect the RDF model
theory without necessarily being fully committed to it. Also, I
expect that future extensions or versions of RDF will require such
non-asserted triples, and this allows the model theory to be extended
in those future directions.

(We will give examples of interpretations in an abbreviated format
simply by listing R, IEXT and IS in the form R/IEXT/IS. P is the
domain of IEXT. For example, for the vocabulary {a b c} the following
is a small interpretation:
{1 2}/1->{<1,1>,<2,2>}/a->1,b->1,c->2
which makes the set
a b a
c a c
true, and all the triples
a c b
a b c
c c c
false)

3. Anonymous nodes as existential assertions

This notion of interpretation treats anonymous nodes exactly like
URIs, semantically speaking. It amounts to adopting the view that an
anonymous node is equivalent to a node with an unknown URI. However,
we could adopt an extra clause which would treat an anonymous node
like an existentially quantified variable. To do so requires that we
decide on the 'scope' of the quantification. The easiest convention
to state would be that the scope was the triple, but that seems
unworkable since the same anonymous node may occur in several triples
in a given document. To make the quantification have the scope of the
entire document requires introducing the notion of 'document' into
the syntax, however. I am not sure how to do this in the RDF graph
model, but in N-triples I will take it that the appropriate syntax
class is <ntripleDoc>.

This will require some definitions. If I is an interpretation and A
is a mapping from some set of anonNodes to the domain of I, then
define I[A] to be the interpretation which is like I except that
I[A](x)=A(x)  wherever A is defined, ie I[A] uses the A-mapping to
assign values wherever it can, and otherwise is identical to I. If E
is an <ntripleDoc>, define anon(E) to be the set of anonymous nodes
in E, and set(E) to be the set of asserted triples in E. Then we have
the extra interpretation rule:

>> If E is an <ntripleDoc> then I(E) = true if I[A](set(E))=true for
>>some A defined on anon(E), otherwise I(E)= false .

This effectively treats all anonymous nodes as existentially
quantified at the boundary of the document containing them. This
requires us to make a conceptual distinction between a document
(containing anonNodes) and the set of triples in the document; the
anonymous nodes are free in the set, but bound by an implicit
existential quantifier in the document. (Notice that if the document
E contains no anonNodes, then I(E)=I(set(E)) ) The distinction
becomes important when we combine documents, since the result of
unifying two documents may not have the same meaning as combining the
two sets of triples into a single document. (The difference will be
significant only if the two documents share an anonymous node. If
this is guaranteed to be impossible, then we do not need to make the
distinction; still, however, it will not be true in general that any
document is equal to the union of the documents formed by dividing it
up into subsets.) We will use the neutral term 'RDF expression' to
refer to a triple, a set of triples, or a document.

It is convenient to define the vocabulary of an RDF expression to be
the set of 'names' in it which are 'free'. The vocabulary of a triple
is the set of <uriref>s and <anonNode>s in the triple; the vocabulary
of a set of triples is the union of the vocabularies of the triples
in the set; and the vocabulary of a document E is the set of
<urirefs> in the vocabulary of set(E). The anonNodes of a document
are bound and are not part of the vocabulary. (The correspondence to
the earlier usage of this word is that the vocabulary of an
interpretation of an expression must contain the vocabulary of the
expression in order for the interpretation to give a well-defined
value to the expression.)

4. Terminology

We say that I satisfies E if I(E)=true, and that E entails E' iff
every interpretation which satisfies E also satisfies E'.

More subtle relationships require some definitions. (These are
standard mathematical definitions for arbitrary mappings, given here
for completeness.) If M is a subset of the vocabulary of I, define
I\M to be the restriction of I to M, ie I\M(x)=I(x) on M. We will
also say that I is an extension of I\M. If I and I' have the same
restriction to the intersection of their vocabularies, then say they
are compatible, and define the union I+I' of I and I' to be the
interpretation whose vocabulary is the union of the vocabularies of I
and of I', and which agrees with them on their vocabularies, ie
(I+I')(x)=I(x) or =I'(x) wherever the latter are defined. Notice that
if vocab(I) contains no anonNodes, then I[A] = I+A. Obviously, I
satisfies E iff I\vocabulary(E) satisfies E.

5. Skolemisation

If we replace an anonNode by a uriref, the result is not entailed by
the original document. In fact, in general, if E' contains any
urirefs not contained in E then E cannot entail E'.  For example, the
document containing the single triple
<foo> <baz> _:xxx
obviously bears some relation to the triple
<foo> <baz> <bar>
but it does not entail it, since the interpretation
{1,2}/1->{<1,1>}/{foo->1,baz->1,bar->2}
satisfies the first document but not the second. (Notice this
interpretation assigns a value to something - <bar> - outside the
vocabulary of the first document, which is how the entailment fails.)
However, the second triple entails the first document, since any
interpretation which satisfies <foo> <bar> <baz> can be used to make
the first document true by restricting it to the smaller vocabulary
and assigning the interpretation of <bar> to the anonNode to provide
a suitable I[A] mapping. More significantly, any RDF expression which
is entailed by the second triple and does not contain <bar> is also
entailed by the first document.

To see why, suppose E'' is such an expression and I is an
interpretation on {<foo>,<bar>} which satisfies the first document.
Then there must be a mapping A from _:xxx to the universe of I such
that I+A satisfies the first triple. Define I' on the second triple
by I'(<bar>)=A(_:xxx), then I' satisfies the second triple, so it
must satisfy E''; but E'' does not contain <bar>, so it must be
satisfied by I'\{<foo>,<bar>}, which is I; so, I satisfies E''; but I
was arbitrary, so, E'' is entailed by the first document. QED.

This is in fact a general result. We can characterise skolemisation
as follows. Consider a document E and another sk(E) got from E by
replacing all its anonNodes by urirefs drawn from a 'new' vocabulary
V which does not intersect vocab(E). Then sk(E) entails E, and if E'
is any expression with no vocabulary in V and sk(E) entails E', then
E entails E'. (The proof follows exactly the special case given in
the previous paragraph, but takes longer to state because it has to
refer to anon(E).) Notice that since sk(E) contains no anonNodes,
I(sk(E))=I(set(sk(E))) for any I, so there is no need to distinguish
a skolemised document from its set of triples.
The key to proper skolemisation is the use of skolem names which are
guaranteed to never occur in any other document, thereby ensuring
that any other expression in any other document which is entailed by
the skolem form is also entailed by the original existential form.
(The reverse follows trivially.) This is the sense in which anonNodes
used in asserted triples can be replaced by urirefs without any
significant 'loss of content', if the document is considered to be an
assertion.

6. Entailment

In this section we give a few basic results about entailment in RDF.
(Note, these results do not take account of extra semantic
restrictions imposed by rdfs, and will be reconsidered when those are
introduced.) Since the language is so simple, entailment can be
recognized by relatively simple syntactic comparisons. The two basic
forms of valid proof step in RDF are, in logical terms, the inference
of P from P and Q, and the inference of (exists (?x) (foo ?x)) from
(foo baz).  Proofs are given in an appendix.

Lemma 1. (subset lemma) Suppose E and E' are sets of triples (not
documents!). Then E entails E' iff E  contains E'.

The basic insight here is that the triples in a set (ie with no
existential quantification) can be assigned interpretations
essentially independently of one another simply by adjusting the
extensions of the properties; there is no 'connection' between the
truth of one triple and the truth of another. This in turn is a
reflection of the fact that one cannot express the material
conditional in RDF, because RDF has no disjunction.

To describe implication relationships between documents we need to be
precise about instantiating 'variables'. If E is an expression and v
is a (syntactic) mapping from a set of anonNodes to urirefs, then let
v(E) be the expression obtained by replacing every anonNode x in E by
v(x) when v(x) is defined. v(E) will be called an instance of E. An
expression containing no anonNodes will be called a ground RDF
expression.

Lemma 2. (instance lemma) Suppose E is a document and v(E) an
instance of E. Then v(E) entails E .

Lemma 3 (RDF interpolation lemma) Suppose E and E' are documents.
Then E entails E' iff there is a set F of triples such that set(E)
contains F and F is an instance of set(E').

What this means, in effect, is that pure RDF proofs need only be one
step 'deep'.

7. Publishing content: assertion versus query.

The model theory characterizes truth-preserving relations between
expressions (documents) but it does not specify what exactly is being
'said' when a document is published. <comment> This is what one would
expect, since publications are tantamount to speech acts rather than
expressions. The same expression can, notoriously, be used to do
various different things: it can be asserted, questioned, doubted,
assented to, etc. </comment>

The most obvious assumption is that to publish an RDF expression is
to assert that it is true, thereby in effect offering a warranty for
anyone else to draw valid conclusions from it. Let us call this a
descriptive or asserting publication. It can be characterized as
making a public claim about the appropriate uses of the expression;
in effect, it says: this can be correctly used to make inferences
from. However, the model theory would apply just as well to a
different kind of publication, where the intended meaning is not that
it is appropriate to draw conclusions from the expression, but that
inference is intended to go in the other direction, so that the
publication says, in effect: this should be used to make inferences
to; or, in other words, can anyone prove this? If we represent E
entails E' by writing (E |=> E'), then assertional publication of E
might be represented as (E |=> ??) , putting E at the blunt end of
the arrow, while the other case - which we can call a querying
publication -  might be written as putting E at the sharp end of the
entailment, (?? |=> E);  where in each case the world in general (or
the community to which the publication is addressed) is being invited
to fill in the ?? blanks.

This picture is almost certainly too simplistic as it stands, since
one would presume that the intention of publishing a query is not
merely to advertise a potential conclusion, but would include an
implicit request to be told about any assumptions out there that
would entail it. This could be handled by a general assumption about
various kinds of publication acts and their associated protocols; but
in any case, RDF has no way to make this distinction at present. I
mention it only to try to clarify the distinctions that have arisen
in discussions.

We could propose a related language to RDF which might be called
Resource Querying Format, which is identical to RDF except that RQF
expressions are understood to be queries rather than assertions.
<joke> The syntax of RQF could be identical to RDF except that an RQF
document must end with the string ",eh?". </joke>
The processing appropriate to RQF would be somewhat different from
that for RDF. In particular, anonNodes in RQF documents would have to
be treated as genuine variables which can be bound to values at run
time. A unification process which binds an RQF variable to an RDF
uriref or anonNode would be a central part of the machinery for
linking RDF assertions to the RQF queries which they entail.

8. Shared meanings and relative entailment.

Some rdf and rdfs triples are considered to have a fixed meaning, and
these meanings can be captured by requiring that all interpretations
satisfy some additional equations or rules, which might include some
constraints on the universe. For example, the RDF machinery for
reification requires that the universe contain enough things to be
the reifications of the reified expressions, and that the denotations
of the urirefs rdf:Statement, rdf:subject, rdf:predicate and
rdf:object are all given fixed values which produce a 'mirror' of the
syntax in the interpretation.  These extra constraints wil be
introduced as the various rdf and rdfs constructs are mentioned in
later sections.

More generally, the successful operation of the web seems to require
that some meanings are 'shared' between those who publish content on
the web and those who read it. For example, the file-locating
protocols that utilize URI syntax to enable web browsing software to
locate web pages consitutute a commonly accepted core of 'shared
meaning' which is assumed to be available to the entire community
without needing to be stated explicitly in assertional form. Exactly
what counts as 'shared meaning' seems to be a matter for further
discussion, but we can represent the idea here in a simple way by
characterizing such sharing as a set of interpretations (for part of
the shared vocabulary) which are 'accepted as reasonable' by all
members of a community, so that they consider semantic notions such
as entailment to be restricted only to interpretations which extend
one or more of those shared interpretations.

To make this precise we need to define a notion of 'relative
entailment'.  Suppose COM is a set of interpretations. Then E entails
E' relative to COM just in case every interpretation which is
compatible with an interpretation in COM and satisfies E, also
satisfies E'. What this does, in effect, is to rule out
interpretations that are inconsistent with everything in COM as
possible counterexamples to a claim of valid entailment, thereby
making it 'easier' for one expression to entail another. Intuitively,
E does not have to explain so much in order to make E' follow from
it, since the content expressed by the restriction to compatibility
with a member of COM can be taken for granted in the communication.

Clearly, if C is a subset of D then validity relative to C implies
validity relative to D. Ordinary entailment is then the special case
where COM is empty, i.e. the strongest kind of entailment. (Note that
COM is a set of interpretations, not an expression, so the question
of how the shared content is expressed is here left open.) As an
example, consider the set of URIs that a browser could utilize to
locate a web page, consider the interpretation ADDR which maps each
such uriref to the web page it locates according to the hypertext
transfer protocol (and to the string "404error" if there is no such
web page, say) and define HTTP-COM to be {ADDR}. Then entailment
relative to HTTP-COM does not permit arbitrary re-interpretations of
uri's starting with 'http://'.

We will call such a 'shared' set of interpretations an interpretation
core, or simply a core. The exact analysis of what constitutes the
interpretation core for a community is beyond the scope of this
document, but we will use the idea when extending the model theory to
rdfs.

This idea can be extended to describe assumptions shared between
communities. If one community's core is S and the other is using
entailment relative to T, then entailment relative to the
intersection of S and T represents their 'common core' of agreed
content. In the worst case this will be empty, and they will only
have logical validity in common. (Actually, the worst case in
practice is likely to be represented by the interpretation core of
rdfs, see later sections.)

The lemmas in section 6 above need to be stated more carefully when
applied to relative entailment, since the conclusions of relative
entailments can be derived in part from the core, in ways that are
not easy to characterize.  For example, it may be that the core
interpretations impose a condition that requires two urirefs to
denote the same resource, so that anything said using one would also
be true if said using the other. Such an entailment would be
relatively valid but might have no syntactic trace in the expressions
themselves. Right now (7/27/01) I am not quite sure how to approach
studying this aspect of relative validity.

9. rdfs interpretations
(to be written)

10. reification and containers
(to be written)

11. relative and absolute URIs.
(to be written)

-------------------------------
Appendix: proofs
Lemma 1. Suppose E and E' are sets of triples. Then E entails E' iff
E  contains E'.
Proof.
If is trivial. For only if, suppose that  (a b c) is a triple in E'
which is not in E, and let I be an interpretation satisfying E. We
will construct an interpretation I' which satisfies E but not E'. I'
is like I except that I'EXT(I(b)) does not contain <I(a), I(c)>.
Clearly I' does not satisfy E'. However, I(e)=I'(e) for all triples
except (a b c), and I satisfies E, so I' satisfies E.
QED.

Lemma 2. Suppose E is a document. Then v(E) entails E.
Proof (trivial).

Lemma 3. (Proof to be written.)

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Received on Friday, 27 July 2001 17:49:21 UTC

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