**From:** Peter F. Patel-Schneider (*pfps@research.bell-labs.com*)

**Date:** 10/10/01

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From: Pat Hayes <phayes@ai.uwf.edu> Subject: Re: new model theory for DAML+OIL Date: Wed, 10 Oct 2001 15:08:25 -0500 > >I didn't account for blank nodes in the model theory in a way that is easy > >to define a decent notion of entailment, so I put together a revision of > >the model theory. The development of interpretations and models is now > >similar to yours. > > > >I believe that entailment has > > > > G |= G' > > > >for G' isomorphic to G, which would have been somewhat difficult to achieve > >in the previous model theory. > > ? Not sure I follow you. If you mean (labelled)graph-isomorphism, > that is trivial in the rdf MT, whereas I think it actually does not > follow in yours, since the same literal may denote something > different in G and in G' (?) I think that it does now, with the new changes. Think of literal-labeled nodes as restricted existentials. [...] > >An untidy RDF graph, R, is a three-tuple (that can be considered to be a > >partially node labeled, directed triple-graph) > > < N, E, LN > > >where N is the set of nodes in the graph > > LN :(partial) N -> URI u L gives labels for nodes > > E <= N' x N'' x N is the set of edges in the graph > > where N' = { n : LN(n) is undefined or LN(n) in URI } > > where N'' = { n : LN(n) is defined and LN(n) in URI } > > ? Why is an edge a triple of *nodes*? I would prefer to have edges as > a distinct set with (total) mappings to their endpoints, and LN > defined on N u E. Just stylistic, not a significant change. [...] > >2/ Literal Values > > > >LV is some collection of literal values. > > > > Literal values form the value space, in XML Schema datatype terms, not > > the lexical space! > > > >XLS : L -> powerset ( LV ), maps literals into the set of literal values > > that they might have. > > > > Here is the first substantive difference from Pat Hayes's model theory. > > The XLS mapping does not provide a definitive answer for the meaning of a > > literal. The reason for not pinning down the mapping for literals is to > > allow different mappings for different datatypes. For example, a node > > with literal label 05 might be mapped into the integer 5 or the string "05". > > See my recent message; could you change this to XLS: LL -> LV, where > LL is the set of all *occurrences* of literal labels (or, maybe, of > nodes which have literals as labels, given that we do not require > tidiness on such nodes) ? This would allow a node labelled with 05 > to be mapped into either 5 or "5", but not both. (I think this might > also simplify the later treatment of DTs, since the extra conditions > they introduce would indeed be semantic conditions in the usual > sense, ie restrictions on the class of interpretations.) I think that the next step is for me to try to come up with some (larger) set of examples to see if that approach will work. [...] > >6/ Datatypes (general version) > > > >Datatypes add extra structure to literals and literal values. > > > >A datatype theory is a four-tuple <LV,DT,DTC,DTS> > >where LV is a collection of literal values > > DT is a collection URIs that are also datatypes > > DTC : DT -> powerset ( LV ) > > DTS : DT -> ( L -> LV ), with DTS(d) potentially partial > > and DTS(d)(L) <= DTC(d) for all d > > > >DTC maps a datatype to its extension (or value space). > >DTS maps a datatype to a partial map from literals (or lexical space) to > >literal values (or value space). > > This seems rather complicated. Couldn't you define DTC as a pullback from DTS? Perhaps. I'm trying to stay close to the XML Schema Datatype terminology. > > Each datatype provides at most one literal value for each literal via > > the DTS mapping. > > Quite. Now, Peter, what is wrong with my original MT in which > 'global' is understood as relative to a datatyping scheme, and XL is > defined to be > (lambda x. DTS(datatype(x))(x)), where datatype is whatever function > from literals to their type determines that unique literal value? > Could we resolve this just by my altering the text to make this > possibility clear? > > >Given a datatype theory <LV,DT,DTC,DTS> > >define XLS(l) = { lv in LV : for some d in DT with DT(d) defined on l > > lv = DT(d)(l) } > > Wait a minute. Surely once we have a datatyping scheme fixed, the > denotation of any literal should be uniquely defined? When is this > actually going to home in on a referent? It seems to me that this is > getting out of hand. The semantics of '5' is more complicated than > the entire semantic theory for all of RDF. Not necessarily. Consider ``literals'' that have no datatype information. > >Given a datatype theory <LV,DT,DTC,DTS> > >a datatype RDFS model for a core RDFS graph R is a core RDFS model I for R, > >with the following extra conditions: > > > > if <s,p,l> is in E with LN(l) in L > > and <M(p),c> in IEXT(IS(rdfs:range)) > > then for any node nc with LN(nc) in DT and M(nc) = c > > M(l) = DTS(LN(nc)) > > so M(l) is a mapping, not a literal value? Or should that be > DTS(c)(LN(nc)) (??) Sorry, it should have been M(l) = DTS(LN(nc))(l) so a datatype determines the denotation of a literal that it is the type of. [...] > > These conditions are rather complicated for semantic conditions, > > Worse, seems to me: they aren't really semantic conditions. The first > one should be phrased as a(n extra) truth-condition on E in terms of > <s, p, l>. Just having that triple in the graph shouldn't be a > semantic constraint in itself; what if the graph is false in I? Agreed, they ain't pretty. > Also, what set (of nodes) does nc range over? Nodes of the graph that is being ``modelled''. > >so some > > explanation is in orer. The first condition says that literals (n) that > > are objects of statements must denote according to any datatype range for > > the predicate (M(p)) of the statement. > > Even if the assertion that is made about them is false? To be a model the assertions must be true, right? > > The second condition says that > > literals values (y) that are in relationships must belong to the value > > space (DTC(LN(c))) of any range of the relationship. > > > > > >A core RDFS graph R datatype-entails another core RDFS graph R' iff > >R u R' is defined and > >every datatype RDFS model for R is also a datatype RDFS model for R u R'. > > > > With the same datatype [theory], right(?) Right, that needs to be added. > Pat

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