From: Dan Connolly <connolly@w3.org>

Date: Thu, 17 May 2001 21:41:28 -0500

Message-ID: <3B048BD8.3FE779A6@w3.org>

To: Graham Klyne <GK@ninebynine.org>

CC: www-rdf-logic@w3.org

Date: Thu, 17 May 2001 21:41:28 -0500

Message-ID: <3B048BD8.3FE779A6@w3.org>

To: Graham Klyne <GK@ninebynine.org>

CC: www-rdf-logic@w3.org

Graham Klyne wrote: > > At 10:15 PM 5/16/01 -0500, Dan Connolly wrote: [...] > >II. Semantics. > > > >An interpretation I is a directed, > >labelled graph; the vertices > >and the edge labels come from the > >same set; let's call it N[I], for nodes. > >Let's call the labelled > >vertices S[I]; so S[I] \subset N[I] x N[I] x N[I]. > > Did you really mean labelled vertices here? Labelled edges makes more > sense to me. Labelled edges. Right. > >An interpretation I satisfies a formula F > >if there exists an assignment A of > >the existentially quantified variables > >in F to N[I] such that > >for each atom (p, s, o) in F, > >the triple of what I and A say those terms > >denote, (p', s', o'), is in S[I]. > > This seems to miss something. I thought the definition of satisfaction > required some notion of truth value. Also, my reference defines > satisfaction in terms of a formula F, an interpretation I and a > substitution A, as "A satisfies F (under interpretation I)..." I lumped it all together: RDF formulas can only have existentially quantified variables; they're true iff there exists an assignment of the variables that works. > It might > be easier to consider F by cases. E.g. for a given formula F and > interpretation I, and some assignment A of members of N to variables in F: > > Case 1: F contains a single atomic formula with no variables; i.e. the > interpretation of all terms in F is a specific member of N. > A satisfies F iff I assigns the truth value TRUE to the member of > F. (i.e. irrespective of A) er... I assigning TRUE to F's interpretation, (p', s', o'), is the same as (p', s', o') \elt S[I]. [...] > Case 4: [by convention?] if F is the empty set, A satisfies F (for any A > or I). It's not by convention; it's by definition: I and A satisfy F if all the (p, s, o) are in S[I]; if there are no (p, s, o), it falls out in the wash. > >An interpretation is a model for a > >set of formulas if it satisfies all the formulas. > > The definition of model that I have is more strict here: > > defn: A formula F is true for I iff every assignment A of N to variables > satisfies F Er... that's for universally quantified variables, no? No such thing in RDF 1.0. > defn: An interpretation I is a model for a set of formulae {Fj} iff every > Fj is true for I. That's what I said. > The key difference here is that, to be a model, the interpretation must > provide satisfaction for *every* set of variable assignments. Not for existentially quantified variables, I don't think. > >That's it for the basics. > > > >Note that the prinicple of erasure holds: > >any interpretation that satisfies {A1, A2, A3} > >also satsifes {A1, A2}. > > I think this follows pretty directly from my case 2 above. I think this > bit captures the "conjunctive semantics" of RDF. Yup. That's the point. > >Does that answer the sorts of questions > >that a model-theoretic semantics is supposed > >to answer? > > It seems to tell us that for every valid interpretation of RDF, for each > statement there exists a unique resource with certain properties. It don't > think it says anything about how statements about this resource relate to > the corresponding statement. Er... right; would you expect it to? > Question: If we apply model operators to this > resource, do we need something outside the semantics of the model operator > to tell us that it applies to the corresponding statement? I don't know what a model operator is. > I'm not sure that having a unique resource for each statement is quite what > RDF anticipates. It seems to me that RDF allows multiple resources (i.e. > interpretations of different URIs) that have a similar relationship to the > original statement. Maybe Rf would be more usefully defines as a relation? I don't think I follow you. But I'm quite sure that the spec says that if subject(x)=subject(y) and predicate(x)=predicate(y) and object(x)=object(y), then x=y. That's what "triple" means, no? > (e.g. Rf( p, s, o, r ) being TRUE if r is a reification of (p,s,o) in S.) No, that doesn't seem right. > >Pat asks, in his message of Wed, 16 May 2001 20:54:38 -0400 (EDT) > >| it isnt entirely clear whether or not relations can > >| be objects > > > >Yes and no; existential variables can go in the 'predicate' > >slot in formulas, but the (p, s, o) structure in > >an interpretation is just like one first-order relation; > >so there aren't any actual logical relations > >in N. > > I don't understand this bit. Look at it this way: an RDF formula <rdfs:Class about="http://www.w3.org/2000/01/rdf-schema#Class"/> might look like (rdfs:Class rdfs:Class) but this semantics makes it out more like (PropertyValue rdf:type rdfs:Class rdfs:Class) so the terms in the RDF syntax all go in argument slots, not the predicate slot. Otherwise, we'd have rdfs:Class denoting an element of itself. Bzzt. > >| (if so, the model theory > >| required is going to be a bit more complicated) > > > >I have been trying, without success, to get my head > >around semantics of lambda calculus and all that. > >By this account of things, it seems that I don't need to. > > Hmmm... echoes of Scott&Strachey denotational semantics? Dunno. > [...] > >| And > >| when reification appears, it seems that we have to put the triples > >| themselves into the domain. > > > >By Sergey's account, only indirectly, thru the Rf thingy. > > I'm getting out by depth, but... I think this is a difference between an > interpretation to (semantics over?) a reflexively defined domain compared > with semantics of relexion. I think that using a denumerable, reflexively > defined domain doesn't introduce any hard problems. I dunno about reflexion. It looks to me like RDF syntax has one function symbol: reify(p, s, o). > >| What *is* a triple, then? What kinds of > >| properties does it have? Are the triples in the domain of > >| quantification the same triples used in RDF syntax, > > > >No; the syntax has URIs and anonymous terms (i.e. existentially > >quantified variables); the domain of quantification > >is N. > > "Domain of quantification" is a new one on me. Is this the same as the > "domain of dioscourse"? That's the way I understood the question. > >(of course, N may contain some strings and URIs and such, > >but that's beside the point, right?) > > Ummm... that might be the edge of a precipice. It seems to be OK for the > model theory you've outlined, but also seems to fall outside the expected > interpretation of URIs discussed elsewhere. I don't see that. > I guess the real problem comes if you then want to introduce (and define > semantics for) some function that operates in the domain of discourse to > map a string to some proposition. Why does that look like a problem? > >| or are they > >| abstract things denoted by the RDF syntax? > > > >Yes. > > > >Of course, this is that funny use the word 'abstract'; > >in fact, an interpretation's domain might be > >very concrete things: people, places, rocks, houses, no? > > Hmmm... see above. I don't follow. -- Dan Connolly, W3C http://www.w3.org/People/Connolly/Received on Thursday, 17 May 2001 22:41:34 UTC

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