SUO: Re: Re: Language And Module Processing (LAMP)
Jon,
This reply is associated with and to a certain extent relies upon the more
detailed discussion that I gave in this morning's message "Building the
Hierarchy by Language and Module Processing"
http://suo.ieee.org/email/msg09555.html.
The quotients formed in Information Flow use the notion of a *dual
invariant* as defined by Barwise and Seligman in their book on "Information
Flow". These correspond to the endorelations and quotients for IFF type
languages and theories -- see the IFF type language and IFF theory namespace
documents for further information and the axiomatizations:
http://suo.ieee.org/IFF/metalevel/lower/namespace/type-language/version20021
205.pdf
http://suo.ieee.org/IFF/metalevel/lower/namespace/theory/version20021205.pdf
These invariants, endorelations and quotients are exact and not soft notions
(neither rough nor fuzzy).
The "term logics" and the examples below correspond to FOL languages that
have only sorts (entity symbols), but no non-sort relation symbols and no
function symbols. The instances represented in the symmetric difference of
the extents of the vehicle_1 and vehicle_2 entity types are not permitted in
the notions of a dual invariant and quotient in Information Flow or in the
notions of an endorelation and quotient for IFF type languages and IFF
theories. In IF and the IFF any two equivalenced types must have the same
extent. I presume that that means that these follow the "denotative" or
"extensional" type of quotient operation. I am interested in extending the
IFF to fuzzy notions, and perhaps this is one place where that might occur.
Robert E. Kent
rekent@ontologos.org
----- Original Message -----
From: "Jon Awbrey" <jawbrey@oakland.edu>
To: "Robert E. Kent" <rekent@ontologos.org>; "SUO"
<standard-upper-ontology@ieee.org>
Sent: Sunday, June 01, 2003 8:08 PM
Subject: SUO: Re: Language And Module Processing (LAMP)
>
> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
>
> Robert, SUO WG, ...
>
> I would like to start thinking about this by beginning with simple term
logics.
> Let's say we have two formal languages Lex_1, Lex_2 over the same alphabet
A,
> so that Lex_1, Lex_2 c A*, and these understood as the basic lexicons of
terms
> that are available as building blocks for every higher level. For
example, we
> might be thinking of something like the following:
>
> Lex_1 = {..., "thing", "object", "vehicle", "truck", "semi_truck",
"trunk", ...}
>
> Lex_2 = {..., "thing", "entity", "vehicle", "lorry", "articulated_lorry",
"boot", ...}
>
> The quotient idea is very important. It is yet another one of the ways
that we have
> of talking about constraints, or controlled variety, or information, or
lawfulness.
> I can think of at least two different ways that quotients come up. Very
roughly,
> we might think of these as "denotative" and "connotative" information.
>
> For example, we might suspect that "thing"_1 and "object"_1 in Lex_1 mean
the same thing,
> or we might suspect that "object"_1 in Lex_1 and "entity"_2 in Lex_2 mean
the same thing.
>
> There are many way to test these sorts of hypotheses. Proceeding
"empirically",
> like the ever-popular "field linguist", we could go out into the "field",
find
> informants Inf_1 and Inf_2 for Lex_1 and Lex_2, respectively, and query
them
> about the instances of a suitable universe X to which they would apply the
> terms of Lex_1 and Lex_2.
>
> For example, suppose we collect data that falls
> into the pattern of the following scattergram:
>
> o-------------------------------------------------o
> | X . . . . . . . . . . |
> | . . . . . |
> | . . o-----------o . o-----------o . . |
> | / \ / . \ . |
> | . / . o \ . |
> | / /.\ . \ . |
> | . / / . \ . \ |
> | o . o . . o o . |
> | . | | . | | . |
> | . | vehicle_1 | . . | vehicle_2 | . |
> | | | . . | | . |
> | . o . o . o . o |
> | . \ \ . / / . |
> | . \ \ / / . |
> | . \ . o . / . |
> | . \ . / \ / . |
> | . . o-----------o . o-----------o . . |
> | . . . . . . . . . |
> | . . . . . . . . . . |
> o-------------------------------------------------o
>
> This would make it reasonable to say, approximately speaking,
> that "vehicle"_1 denotes the same things in X as "vehicle"_2,
> on the evidence of Inf_1 and Inf_2. Observe that the weight
> of the counter-evidence is measured by the relative frequency
> of the outlying instances, those in the "symmetric difference"
> of the extensions of the terms: Symdiff (Vehicle_1, Vehicle_2).
>
> The "denotative" or "extensional" type of quotient operation
> amounts to ignoring the outliers and the regions of the universe
> that they occupy, collapsing the topology into the following form:
>
> o-------------------------------------------------o
> | X |
> | |
> | o-------------o |
> | / \ |
> | / \ |
> | / \ |
> | / \ |
> | o o |
> | | vehicle_1 | |
> | | = | |
> | | vehicle_2 | |
> | o o |
> | \ / |
> | \ / |
> | \ / |
> | \ / |
> | o-------------o |
> | |
> | |
> o-------------------------------------------------o
>
> Jon Awbrey
>
> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
>
>
> > Robert E. Kent wrote:
> > >
> > > All,
> > >
> > > I have posted a file called "Language and Module Processing" at the
address:
> > >
> > > http://suo.ieee.org/IFF/language-module-processing.html
> > >
> > > and linked off the main SUO IFF page.
> > >
> > > In this file I have offered a possible diagrammatic scenario for
> > > the creation of the John Sowa's library of modules diagram:
> > >
> > > http://www.jfsowa.com/figs/suohier2.gif
> > >
> > > which (as John mentioned in a previous message) shows how
> > > the modules derived from SUMO and OpenCyc could fit together.
> > >
> > > There are six diagrams linked by five processing steps.
> > > The first two processing steps occur in the *context of theories*,
> > > whereas the last three processing steps occur in the *lattice of
theories*.
> > >
> > > The first diagram assumes two distinct ontologies SUMO and OpenCyc
have been
> > > resolved into several submodules resulting in two libraries of modules
within
> > > two lattices of theories for two distinct first order logic (FOL)
languages
> > > L_SUMO and L_CpenCyc, respectively. Recall that an FOL language is
the
> > > terminological content of an ontology consisting of relation symbols,
> > > function symbols and constants with appropriate arities specified.
> > >
> > > The first step forms the sum of the two FOL languages using the
disjoint union
> > > of the sets of relation symbols, function symbols and constants. This
results
> > > in the second diagram which has only one summed FOL language, but
consists of
> > > the two ontologies arranged as two distinct libraries of modules in a
single
> > > lattice of theories (recall that each FOL language determines its own
> > > lattice of theories.
> > >
> > > The second step involves two substeps: (1) specifying equivalent
relation symbols,
> > > equivalent function symbols, and equivalent constant symbols; and (2)
forming the
> > > quotient of the sum language modulo these equivalence relations. This
results in
> > > the third diagram which has only one quotiented FOL language, but
still consists
> > > of the two ontologies arranged as two distinct libraries of modules in
a single
> > > lattice of theories.
> > >
> > > The third step works inside a lattice of theories. It forms a single
library
> > > of modules by summing the two previous libraries of modules. This
results in
> > > the fourth diagram which has the two ontologies arranged within one
library
> > > of modules in a single lattice of theories.
> > >
> > > The fourth step accomplishes two objectives: (1) it extracts various
> > > sub-sub-modules from the various sub-modules of the two ontologies;
> > > and (2) it may create from scratch several generic modules that may
> > > be needed later. This results in the fifth diagram which has these
> > > generic modules situated at the highest level below the top, with
> > > the two ontologies arranged below these.
> > >
> > > The fifth and final step creates a customized theory by forming the
meet
> > > of some of the original submodules from the two ontologies plus some
of the
> > > generic modules. The meet is formed by taking the union of the axioms
in the
> > > appropriate modules.
> > >
> > > Please look this over and send me your comments.
> > >
> > > Robert E. Kent
> > > rekent@ontologos.org
> >
> > o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
>
>