From: pat hayes <phayes@ai.uwf.edu>

Date: Thu, 19 Sep 2002 17:39:42 -0500

Message-Id: <p05111b15b9b001c3fb2e@[65.217.30.172]>

To: w3c-rdfcore-wg@w3.org

Date: Thu, 19 Sep 2002 17:39:42 -0500

Message-Id: <p05111b15b9b001c3fb2e@[65.217.30.172]>

To: w3c-rdfcore-wg@w3.org

Since RDF now has DAML_style lists (right?), I thought it would be good to get feedback on the proposed MT for them. So here it is: toss back any comments, please. ----------- Given a set S, we define a set of lists over S to be a set L containing the empty sequence <> and all structures of the form <s, l> where s is in S and l is in L. As with any recursive 'definition', this in fact is an equation with many possible solutions. The usual way to interpret this kind of definition is in terms of a minimal solution of the equation. That means that one would understand the set of lists to be the smallest collection of things that would satisfy the recursion, which would be all finitely deep lists which have no loops, i.e. structures of the form <s1 <s2 <...<sn <>>...>>. On this view, every list defines a finite sequence of elements of S. Other lists are possible, however, which would also satisfy the recursive definition: infinitely deep lists, for example, or looping lists of the form l = <s, l>. Although it is possible to axiomatize a recursive 'definition' as a logical assertion, there is no way to finitely axiomatize the least fixed-point solution. We could impose it as a semantic condition; but this condition, although intuitively sensible and in correspondence to the usual semantics for computational languages, may have some regrettable consequences when used, as here, in the context of a descriptive language. In particular, there would be no way to establish the completeness of any finitary inference process relative to such a semantics. We therefore avoid making this stipulation, and simply require that the set of lists in any interpretation be *some* set which satisfies the recursive definition. Note that this means that any set of lists will at least contain all the finite non-looping lists. The semantics of the rdf list vocabulary is then straightforward. In any RDF interpretation I, we assume that ICEXT(I(rdf:List)) is a set of lists over IR @@Note the use of 'a' rather than 'the'.@@ I(rdf:nil) = <> <x, y> in IEXT(I(rdf:first)) iff x = <y, l> for some l in ICEXT(I(rdf:List)) <x, y> in IEXT(I(rdf:rest)) iff x = <s, y> for some s in IR We note in passing that this semantics requires that the universe IR is closed under the operation of constructing lists. Any interpretation I of any RDF graph of the form A1 rdf:type rdf:List . A1 rdf:first B1 . A1 rdf:rest A2 . A2 rdf:first B2 . A2 rdf:rest A3 . ... An rdf:first Bn . An rdf:rest rdf:nil . has I(A1) = <I(B1), <I(B2), <... <I(Bn), <> >...>>. We will describe this as a sequence and write it as [I(B1), ... , I(Bn)]. Sequences are the ordered multisets of the elements of finite lists. -------- -- --------------------------------------------------------------------- IHMC (850)434 8903 home 40 South Alcaniz St. (850)202 4416 office Pensacola, FL 32501 (850)202 4440 fax phayes@ai.uwf.edu http://www.coginst.uwf.edu/~phayesReceived on Thursday, 19 September 2002 18:39:42 UTC

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