From: Peter F. Patel-Schneider <pfps@research.bell-labs.com>

Date: Thu, 27 Sep 2001 10:37:29 -0400

To: phayes@ai.uwf.edu

Cc: www-rdf-logic@w3.org

Message-Id: <20010927103729F.pfps@research.bell-labs.com>

Date: Thu, 27 Sep 2001 10:37:29 -0400

To: phayes@ai.uwf.edu

Cc: www-rdf-logic@w3.org

Message-Id: <20010927103729F.pfps@research.bell-labs.com>

I thought I might give a stab at cleaning up some of the problems that I see with Pat's model theory. Here are the results for RDF (without reification or containers). Note that this should be considered a derivative work from Pat's work, and not a completely separate model theory. I have several themes here. First, I wanted to be very formal, so no misconceptions can arise. To this end, I have produced a very sparse document. (However, I have had to use some non-standard symbology for the set-theoretic relationships. Make the obvious fixes.) Second, I wanted to stick as close as possible to the ``Formal Model for RDF''. This drove some of the changes from Pat's model theory. Third, I wanted to annotate some of the issues and choices in the model theory. 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 [NB: This ignores non-string literals, but they wouldn't cause any more problems.] [NB: I'm sticking with graphs, even though they have some problems for RDF. Triples have their own problems with respect to RDF.] A basic untidy RDF graph, R, is a four-tuple (that can be considered to be a partially node labeled, totally edge labeled, directed graph) < N, E, LN, LE > where N is the set of nodes in the graph LN :(partial) N -> URI u L gives labels for nodes LE :(total) E -> URI gives labels for edges E <= N' x N is the set of edges in the graph where N' = { n : LN(n) in URI } [NB: This accounts for literals not being allowed as labels of edges, nor as the labels of nodes that are heads of edges, but does not account for edge labels being properties. It would be possible to make the latter distinction here.] [NB: I don't think that Pat's definition of RDF graphs has the requirement that heads of edges cannot be labeled with literals.] An basic untidy RDF graph is ground if LN is a total function on N. An basic untidy RDF graph is tidy (also called a basic RDF graph) if LN is injective (not necessarily total). Literal Values: LV is a non-empty set, currently the set of 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.] 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 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. 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. 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. 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. [NB: This claim is not true for more-powerful representation systems.]Received on Thursday, 27 September 2001 10:35:53 UTC

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