Re: entailment on sets of RDF graphs (union vs merge semantics)

Here are the changes (plus a fix to finite interpretations).   I didn't find 
any substantive changes, and the changes were not substantial after all.  By 
far the biggest differences are in the first two modifications, everything 
else amounts to eliminating sets of graphs on the LHS of entailment.

If everyone is happy with these, I'll update the document.

peter

REMOVED
If S is a set of graphs then S simply entails E when any every
interpretation which satisfies every member of the <a>unionizing</a> of S
also satisfies E.
/REMOVED

CHANGED
<p class="changenote"> Before determining whether a set of graphs entails
another graph, which was directly defined in the 2004 RDF 1.0 Semantics, the
graphs have to first be unionized or merged (if there is some reason to
treat shared blank nodes as if they were different in different graphs).
If there are no shared blank nodes unioning and merging produce the same
result, which is the also the same as the direct 2004 definition. </p>

     <p><a id="defvalid">Any process which constructs a graph E from
     some other graph S is (simply) <dfn>valid</dfn> if S
     simply entails E in every case, otherwise <dfn>invalid.</dfn></a></p>
/CHANGED


CHANGED
<p> completely characterizes simple entailment in syntactic
   terms. To detect whether an RDF graph simply entails another, check that
   there is some instance of the entailed graph which is a subset of the
entailing graph. </p>
/CHANGED

CHANGED
<p>A graph is (simply) <dfn>D-satisfiable</dfn> or <dfn>satisfiable
recognizing D</dfn> when it has the value true in some D-interpretation, and
a graph S (simply) <dfn>D-entails</dfn> or <dfn>entails recognizing D</dfn>
a graph G when every D-interpretation which makes S true also D-satisfies
G.</p>
/CHANGED

CHANGED
In all these cases, a pre-interpretation of the vocabulary of a graph may be
extended to a full interpretation of the appropriate type without changing
the truth-values of any triples in the graph.</p>
/CHANGED

CHANGED for other reasons
<p>Basically, it is only necessary for an interpretation structure to
interpret the <a>name</a>s actually used in the graphs whose entailment is
being considered, and to consider interpretations whose universes are at
most as big as the number of names and blank nodes in the graphs. More
formally, we can define a <dfn>pre-interpretation</dfn> over a
<a>vocabulary</a> V to be a structure I similar to a <a>simple
interpretation</a> but with a mapping only from V to its universe IR.  Then
when determining whether G entails E, consider only pre-interpretations over
the finite vocabulary of <a>name</a>s actually used in G union E. The
universe of such a pre-interpretation can be restricted to the cardinality
N+B***+1***, where N is the size of the vocabulary and B is the number of
blank nodes in the graphs. Any such pre-interpretation may be extended to
<a>simple interpretation</a>s, all of which will give the same truth values
for any triples in G or E. Satisfiability, entailment and so on can then be
defined with respect to these finite pre-interpretations, and shown to be
identical to the ideas defined in the body of the specification.</p>
/CHANGED

Received on Thursday, 20 June 2013 06:00:24 UTC