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Re: model theory for RDF/S

From: Dan Connolly <connolly@w3.org>
Date: Thu, 27 Sep 2001 10:40:57 -0500
Message-ID: <3BB34889.C4A0063E@w3.org>
To: "Peter F. Patel-Schneider" <pfps@research.bell-labs.com>
CC: phayes@ai.uwf.edu, www-rdf-logic@w3.org
"Peter F. Patel-Schneider" wrote:
[...]
> Syntax:
> 
> URI is the collection of URI names, i.e., some collection of strings
> 
>   [NB:  This ignores all aspects of the structure of URIs.]
> 
> L is the collection of literals, i.e., some collection of strings,
> disjoint from URI

I think the mention of "string" here is a misnomer/red-herring, no?

i.e. take the string "http://example/". It's a URI, and
you say URIs are disjoint from literals... hence it's not a literal.

But it's a perfectly legal property value in RDF:

	<rdf:Description rdf:about="#something">
	  <dc:identifier>http://example/</dc:identifier>
	</rdf:Description>

I think if you just say URI and L are disjoint, denumerable
sets, you've said what you need to say.

>   [NB:  This ignores non-string literals, but they wouldn't cause
>         any more problems.]


> Literal Values:
> 
> LV is a non-empty set, currently the set of strings.

again, why bother mentioning "strings"?

> XL : L -> LV, giving meaning to literals
> 
>   [NB: This ignores many aspects of the intended meaning of strings,
>        such as whether the mapping is surjective or injective.]


> 
> Interpretations:
> 
> Let V be a subset of URI.
> 
>   [NB: Does V have to be non-empty, does it have to be finite?  So far
>        there are no such requirements, but we have to be careful that this
>        is OK, and that we don't make unnecessary requirements on V.]
> 
> An interpretation, I, on a vocabulary V, is a four-tuple < IR, IP, IEXT, IS>
>         where IR is a non-empty set, called resources
>               IP <= IR, called properties
>               IEXT : IP -> powerset ( IR x (IR u LV) )
>               IS : V -> IR
> 
>   [NB: There is no requirement that IR and LV be disjoint nor is there any
>        requirement that they be subsets of one another.   We have to be
>        careful that this is OK, and that we don't accidentally disturb this
>        (non-)relationship.]
> 
> Interpretations of ground basic untidy RDF graphs:
> 
> Let R = < N, E, LN, LE > be a ground basic untidy RDF graph.
> 
>   [NB: There appears to be no harm in allowing untidy graphs here, but this
>        could easily be restricted to tidy graphs.]

restricting to tidy graphs would be my preference. I'm not
sure exactly why just now...

> An interpretation I on the vocabulary V with
>         { LN(n) | n in N } ^ URI <= V
>         { LN(e) | e in E } <= V
> is extended to R as follows
> 
>   [NB: I am being a bit vague here in just what is the domain of I.]
> 
>   [NB: It is explicit here that the vocabulary of the interpretation can
>        have ``names'' that do not appear in the graph.  Pat's theory is
>        vague on this point.]
> 
> 1. if LN(n) is defined and in L then I(n) = XL(LN(n))
> 
> 2. if LN(n) is defined and in V then I(n) = IS(LN(n))
> 
>    [NB: These two give an interpretation for all nodes in R.]
>    [NB: The ``defined'' part is used later.]
> 
> 3. if <f,g> is in E
>    then I(E) = true

you mean I(<f,g>) rather than I(E), no?


> if IS(LE(<f,g>)) in IP and <I(f),I(g)> in IEXT(IS(LE(<f,g>))),
>         I(E) = false otherwise
> 
>    [NB: This cleans up the meaning of edges whose labels are not mapped to
>         properties.

Ah! now I understand your earlier comment about that.

>  It might also be possible to make graphs with such
>         edges ill-formed in some way.]
> 
> 4. I(R) = false if I(e) = false for some e in E,
>    I(R) = true otherwise
> 
> Interpretations of basic untidy RDF graphs:
> 
>   [NB: I'm writing out extended interpretations completely for purposes of
>        being completely pedantic.
> 
> An extended interpretation on a vocabularly V is a two-tuple
>         <<IR,IP,IEXT,IS>,A'>
> where <IR,IP,IEXT,IS> is an interpretation on V
>   and A' is a mapping to IR.
> 
>   [NB: This means that unnamed nodes have to be resources, not literals
>        that are not resources.  This may have consequences!]
> 
> Let R = < N, E, LN, LE > be a basic untidy RDF graph and
> let U = { n in N | LN(n) is not defined }
> 
> An extended interpretation I = <<IR,IP,IEXT,IS>,A'> on the vocabulary V with
>         { LN(n) | n in N } ^ URI <= V
>         { LN(e) | e in E } <= V
>         and the domain of A' = U
> is extended to R as follows
> 
> 0. if LN(n) is not defined I(n) = A'(n)
> 
> 1. if LN(n) is defined and in L then I(n) = XL(LN(n))
> 
> 2. if LN(n) is defined and in V then I(n) = IS(LN(n))
> 
> 3. if <f,g> is in E
>    then I(E) = true if IS(LE(<f,g>)) in IP and <I(f),I(g)> in IEXT(IS(LE(<f,g>))),
>         I(E) = false otherwise
> 
> 4. I(R) = false if I(e) = false for some e in E,
>    I(R) = true otherwise
> 
> Let R = < N, E, LN, LE > be a basic untidy RDF graph and
> let U = { n in N | LN(n) is not defined }
> 
> An interpretation I = <IR,IP,IEXT,IS> on the vocabulary V with
>         { LN(n) | n in N } ^ URI <= V
>         { LN(e) | e in E } <= V
> is extended to R as follows:
> 
>   I(R) = true if there is some mapping A' from U to IR such that <I,A'>(R) = true,
>   I(R) = false otherwise
> 
> An interpretation I with vocabulary V is called a model of a
> basic untidy RDF graph R = <N,E,LN,LE> iff
> 1.      { LN(n) | n in N } ^ URI <= V
>         { LN(e) | e in E } <= V, and
> 2. I(R) = true
> 
> Taking care of rdf:type:
> 
> A core RDF interpretation, i.e., RDF without reification or containers, is
> an interpretation over a vocabularly that includes rdf:type with the
> following extra conditions
> 
>      1. IS(rdf:type) is in IP
>      2. IEXT(IS(rdf:type)) <= IR x IR
> 
> Now for the claims:
> 
> All these sorts of claims have to be backed up with theorems and lemmas
> like the ones that Pat has.  I would go even further than Pat has,
> extending to RDF query languages, at least whenever such become
> well-defined.
> 
> Claim 1:
> 
> A core RDF interpretation that is a model of a basic untidy RDF graph R
> contains everything (and more) that is in the intended core RDF (i.e., RDF
> without reification or containers) meaning of R.  In other words, a model
> contains all the information implicitly (or explicity) in the graph, and
> maybe more, and contradicts nothing that is implicitly (or explicitly) in
> the graph.

Hmm... at this point, the formality fails me. I would need
an example to provide intiutions about whether I believe this
or not.

> Claim 2:
> 
> A core RDF interpretation that is not a model of a basic untidy RDF graph R
> has something that is not compatible the intended core RDF meaning of R.
> In other words, a non-model is missing or incorrect on something that is
> implicitly (or explicitly) in the graph.

That makes sense relative to claim 1 (though I don't have
a good feel for claim 1 itself).

> Claim 3:
> 
> For every basic untidy RDF graph R there is a core RDF interpretation that
> captures exactly the closure of the intended core RDF meaning of R and that
> is a model for R.  That is, roughly, that there is a model that makes
> everything implicitly (or explicitly) in the graph true, and everything
> else false.

I have no idea what the significance of this claim is, let
alone any sense of whether I agree or disagree with it.

>   [NB:  This claim is not true for more-powerful representation systems.]

-- 
Dan Connolly, W3C http://www.w3.org/People/Connolly/
Received on Thursday, 27 September 2001 11:41:08 UTC

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