Re: Semantic Predication: 5 - model theory

Dear Enrico,

Before joining the discussion and also explaining our rationale why we chose for referential opacity in the first place, I would like to better understand your semantics here. So, I start with a rather short question:

Am 14.02.2023 um 17:27 schrieb Franconi Enrico <franconi@inf.unibz.it<mailto:franconi@inf.unibz.it>>:

Here I introduce the exact model theory for semantic embedded triple, as a simple extension of the RDF 1.1. model theoretical semantics.

A RDF-star "sem" interpretation is a structure consisting of (IR, IP, IS, IL, IT, IEXT) such that:
- (IR, IP, IS, IL, IEXT) is a RDF 1.1 simple interpretation.
- IT is a mapping from IRxIPxIR to IR.

Add a section "RDF-star semantically quoted triples interpretation"
   between "5.1 Blank Nodes" and "5.2 Simple Entailment"

Extend the mapping [I+A] as follows:
[I+A](<<<s,p,o>>>) = IT( ([I+A](s),[I+A](p),[I+A](o)) ).


What exactly is that b? Should it be a blank node or an IRI? I ask because I am bothered by the last line,


Extend the semantic conditions for an RDF-star graph as follows:
[I+A](b,unstar-sem:subject,s)=true   iff ∃p,o. [I+A](<<<s,p,o>>>)=[I+A](b)
[I+A](b,unstar-sem:predicate,p)=true iff ∃s,o. [I+A](<<<s,p,o>>>)=[I+A](b)
[I+A](b,unstar-sem:object,o)=true    iff ∃s,p. [I+A](<<<s,p,o>>>)=[I+A](b)
[I+A](s,p,o)=true                    iff ∃b.   [I+A](<<<s,p,o>>>)=[I+A](b)

This is always true since I know that [I+A](<<<s,p,o>>>)=[I+A](<<<s,p,o>>>), so there exists alway a b, namely <<<s,p,o>>>, or was that the idea? That the quoted triple is true if we have an interpretation for it, that is, if it is not malformed? Mmm, but what if

([I+A](s),[I+A](o)) not in IEXT([I+A](p))?

Then, the triple is true and false at the same time? That can’t be it.

So, could you please explain here? (And sorry for the „writing while-thinking-style“ you see some representation of my thinking process here ;) )


Kind regards,
Dörthe


Add to the "5.3 Properties of simple entailment (Informative)" section:

The following holds: given the mapping LS from RDF-star graphs to RDF-1.1 graphs defined below, a RDF-star graph G entails a RDF-star graph E under the "sem" interpretation if and only if LS(G) entails LS(E) under RDF-1.1 simple interpretation.

LS(G) is defined as follows.
While G contains sem-quoted triples:
1) Pick an RDF-star sem-quoted triple (s, p, o) in the constituents of G such that neither s nor o is a sem-quoted triple.
2) Mint a fresh blank node b (i.e., such that b is not in the constituents of G).
3) Replace with b all occurrences of (s, p, o) in the subject or object position of an asserted or sem-quoted triple of G.
4) Add the following asserted triples to G:
   (b, unstar-sem:subject, s)
   (b, unstar-sem:predicate, p)
   (b, unstar-sem:object, o)
   (s, p, o)

This semantics has the following properties:

  *   Semantic embedded triples are fully transparent.
  *   Semantic embedded triples are always asserted.
  *   Semantic embedded triples entail their reification *and* vice-versa (logical equivalence).



  *   Monotonically extends RDF 1.1 and full backward compatible with RDF 1.1: if a RDF-star graph does not contain semantically quoted triples, then then the RDF-star "sem" semantics coincides with the RDF 1.1 semantics.
  *   The semantics is compositional.