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viewOf / complementOf discussion in 201-12-15 telecon

From: Graham Klyne <graham.klyne@zoo.ox.ac.uk>
Date: Thu, 15 Dec 2011 18:19:15 +0000
Message-ID: <4EEA3A23.8040304@zoo.ox.ac.uk>
To: Paolo Missier <Paolo.Missier@ncl.ac.uk>, W3C provenance WG <public-prov-wg@w3.org>
Paolo, and all,

Re: viewOf / complementOf discussion in 201-12-15 telecon

Prompted by discussion in today's teleconferences, and in particular by Paolo's 
articulation of the intuition behind "complementOf" (as was), here are some 
thoughts...

It seems we have two competing intuitions, yet much of the contention is about 
naming and how to formalize or otherwise define them.  So I'd like to take the 
following approach:

1. describe the intuitions, with examples

2. assign names to the intuitions

3. discuss the extent to which they can be formalized, and how


== Two intutiions ==

1. two entities that are constrained forms of the same real-world object; e.g.
       (a) Bob as Twitter account holder
       (b) Bob as Facebook account holder
     or
       (a) Luc in Boston
       (b) Luc in Southampton

I think this intuition is clearly symmetric and not, in general, transitive.

The intuition has been further constrained in some discussions as requiring some 
overlap between the two entities, so the first example might apply, but the 
second would not.

2. an entity that is a constrained form of some other entity; e.g.
       (a) Luc in his office
       (b) Luc in Southampton
   or
       (a) Luc in Southampton
       (b) Luc through his lifetime
   or
       (a) Luc in Boston
       (b) Luc through his lifetime
   or
       (a) Bob as a Twitter account holder
       (b) Bob as a computer user

This intuition is transitive non-symmetric.


== Naming ==

For me, the name "complementOf" applies reasonably to the first intuition about 
two entities that are some facet of the same real-world entity.

The term "viewOf" applies to the second intuition.

I'll use these terms for the discussion that follows.


== Formalization ==

What can we say about these?

Notation used below:
   ':' such that
   '==' defined as
   '=>' implies (logical implication)
   '|=' entails

=== complementOf ===

We might capture the intuition thus:

complementOf(a, b0 == exists r : isRealWorldThing(r) and
                                  isAbout(a, r) and isAbout(b, r)

but this begs a formalization of isRealWorldThing and isAbout

Previously, there was an appeal to attributes, but that seems somewhat 
arbitrary, and for me not directly reflecting the original intuition.

(I'm not sure where to go from here.)


=== viewOf ===

I start by suggesting that an entity denotes a set of instances. Thus, when we 
talk about "Luc in Boston", we mean the set of all (instantaneous) instances of 
Luc for which Luc is in Boston.  This is presumed to be a primitive assertion 
(rather like a primitive class in a Description Logic).

For some entity a, let us call this set instances(a) (somewhat as RDF formal 
semantics introduces a class extension ICext(c) to denote the members of a class c)

Then we can formalize
   viewOf(a, b) == forall(x) : x in instances(a) => x in instances(b)

A corollory of this would be that if a provenance assertion A[p](a) is an 
assertion about a using some predicate p such that:

   A[p](a) == forall(x) : x in instances(a) => p(a)
        (i.e. A[p] asserts that p is true for all instances of a, which
         captures the original notion we discussed months ago that
         provenance assertions are invariant with respect to an entity)

then

   viewOf(a, b) |= A[p](a) => A[p](b)

I think the transitivity of viewOf follows from the above.


=== viewOf and complementOf ===

Given this formalism of viewOf, I think it is now possible to propose a more 
complete formalism of complementOf:

   complementOf(a, b) == exists(x) : viewOf(x, a) and viewOf(x, b)

<aside>
Note that the existential x here replaces the need for the predicate 
isRealWorldThing, but is not necessarily itself a real world thing, whatever 
that may be.  We might try and define isRealWorldThing thus:

   isRealWorldThing(x) == not exists(y) : isView(x, y)

so one might say that real world things are anything that sit at the top of the 
isView hierarchy.

Similarly, one might also define:
    isAbout(a, b) == viewOf(a, b)
</aside>

This definition of complementOf does not capture the notion of overlap between 
complements.  But we could do that too, if needed, e.g.

   strictComplementOf(a, b) == complementOf(a, b) and
                               exists(x) : viewOf(x,a) and viewOf(x,b)


== Conclusion ==

I believe this substantiates my previous claim that viewOf is somehow more 
fundamental.  Based on just a simple set-theoretic definition of viewOf, I have 
been able to construct a formal definition of complementOf.  But I don't believe 
it would be as easy to construct a primitive definition of complementOf and use 
just that to define viewOf.


#g
Received on Thursday, 15 December 2011 18:23:30 UTC

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