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[ontolog-forum] cyclic and acyclic definitions

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Bart Gajderowicz <bgajdero@xxxxxxxxxx>
Date: Tue, 21 Apr 2009 14:33:56 -0400
Message-id: <6b20199d0904211133uaf4ded7i406696ef63d8e7f9@xxxxxxxxxxxxxx>
*** Originally from [ontolog-forum] web-syllogism-and-worldview ***
DC> Well this thread is veering in directions I wouldn't
> have expected
It's an interesting analysis of NL and human development.
I decided to start a new thread as to not interrupt the conversion in
*web-syllogism-and-worldview*    (01)

Let me state that my original intention was to find a way to model
cyclic definitions in a complete system. My approach was to convert
cyclic to acyclic definitions, via context. My envisioned result is
purely hierarchical, for the purpose of comparing 2 hierarchies, not
their instances.    (02)

AA> There is much bigger problem here, the semantic one.
> Nobody can describe natural phenomena with logical syntax. Otherwise science
> would be redundant.
> Broadly, there are two types of conceptual systems:
> logical semantic systems (LOS), involving the so-called formal semantics,
> real semantic systems (RES), involving the real semantics, as in:    (03)

AA> 1. a denotation rule (reference), indicating the physical referent p;    (04)

I don't have to be concerned with indexicality of physical objects,
only classification through the fact that at some point, a particle
with similar attributes existed, and had a certain relationship to
something else, and was classified as type of X particle.    (05)

AA> 2. a representation rule (semantic assumption), measuring the position of
> the particle x(p, u,t) re. to some reference frame of units u at some moment
> of time t.
> With the RES you also find out that m(p) represents the inertia of particle
> (semantic axiom), and that it is subject to the conservation laws and
> Newton's law of motion (factual axioms).    (06)

I wouldn't use temporal information to identify a particular particle,
but I could use the recorded speed of some particle, to identify
others as belonging to a group that behave in a similar manner.    (07)

If I implement some uncertainty measures, something like fuzzy logic,
at some level in the system, most likely at the last steps of
inference, I could dynamically calculate class membership or inclusion
based on empirical data.  This may be used to either reach a new level
of granularity, or resolve conflicting definitions, that are not
available through reasoning alone. I can equate this to defining a
structure for 2 molecules, X and Z, but resorting to experiments to
see how they interact. If up to that point, X and a new molecule Y
were structurally equivalent, but reacted differently to Z, I could
say that structurally X and Y are the same, but they are different in
the context of "reaction with Z".    (08)

JSF> Formalization is not some kind of ultimate goal that we're
> striving to achieve.  It is a very mundane, everyday process
> that children learn in elementary arithmetic.    (09)

John, what I meant by formalizing, and I thought you meant as well,
was agreeing on some standard to create and interpret ontologies, the
topic of the Summit.  Could you please clarify what you meant.    (010)

BG>> Fields such as fuzzy logic and probability theory let us
>> make statements about the world based on empirical data.    (011)

JSF> Not quite.  Those two systems, fuzzy logic and probability
> theory, are defined formally by patterns.    (012)

If we're trying to model the world, wouldn't this be an appropriate
way of doing it, or at least approximating it? If we can't define it
precisely due to ambiguity, couldn't corpora in the form of
statistically built hierarchies, help us here? If a class is
structurally the same in two different systems, and we find, through
empirical means, 2 contradictory instances in the two systems, which
differ in some attribute we didn't consider, then hierarchical
information would not tell them apart.    (013)

JSF> Again, you have to recognize that the C4.5 procedure is purely
> formal....  The only things that they process are
> patterns of character strings that represent somebody's best
> guess about reality.
> The GIGO principle still holds:  Garbage In, Garbage Out.    (014)

I agree completely.    (015)

> BG> I'm wondering whether there is a point where statistical
>  > analysis can take over where we simply don't know enough
>  > about a topic to infer information using logic alone.
JSF> There is no difference in principle... Logic can be used to
> define what is an A, what is a B, and what is a C.
> Statistics counts how many As, how many Bs, and how many Cs.
> As for the C4.5 algorithm, it is more closely related to
> logic than it is to statistics.  It's a kind of learning
> algorithm, but what it learns is a *decision tree* that
> can be expressed as a very large nest of if-then-else
> statements.  That tree can be mapped to a program in any
> language that supports if-then-else statements, such as
> C or Java.  It can also be mapped to first-order logic.    (016)

Again, I completely agree.
Perhaps I envision it differently, because I'm not concerned with how
many A's, B's or C's, but with classifying where instances lie based
on their attributes, adding to the structural definition. C4.5 would
give me that. For example:    (017)

In addition to structural information:
(forall x,z (R(x,z) iff exists y, s.t. R(x,y) ^ R(y,z)))    (018)

... I add something along the lines of:
(forall x,z (R1(x,z) iff exists y, s.t. R(x,y) ^ R(y,z)) ^ R2(x,y) ^ R2(y,z) )    (019)

Where R2(a,b) could be:
R2(a,b) => a.id < b.id
R2(a,b) => a.country is part of b.continent    (020)

You can define ranges like this for error checking in XML,
specifically ranges in OWL to define variations in objects. So that's
nothing new.    (021)

Back to my proposed way of handling cyclic definitions, by changing
the attributes and properties to acyclic definitions:
Following Azamat's questions:
I could perhaps define relationships this way to determine if A and B
are transitive hierarchies with R10(A,B) (see below):
1. restrained by the class inclusion relationships
K-9 is a Species is-a Animal    (022)

2. restrained by the the class membership relationship
Fido is a K-9    (023)

In the context of **structural categories**:
R10(K-9, Animal) holds
R10(Fido, Animal) doesn't hold, because Fido is not a subcategory of
Animal, only an instance of it.    (024)

In the context of **instances**:
R10(K-9, Animal) does not hold because you need an instance attributes
to start with.    (025)

Both of these hold because Fido is an instance of K-9, which is a
subclassOf Animal.
R10(Fido, K-9) hold
R10(Fido, Animal) holds    (026)

I'm not sure how this distinction would work, but one possibility is types:
(forall x,y R10(x,y) iff (x.type == y.type))
... where R is a subclassOf relation, and R1 is a special R relation
with type check for transitivity using R10:
(forall x,z (R1(x,z) iff exists y, s.t. R(x,y) ^ R(y,z)) ^ R10(x,z))    (027)

Fido.type => 'instance'
K-9.type => 'category'
Species.type => 'category'
Animal.type => 'category'    (028)

Note that Fido could never be of type 'category' with Animal, only instance.
If we wanted to look at Fido's individual parts, they would be Fido's
paw, Fido's tail, etc, transitive to K-9's-paw, K-9's-tail. I'd have
to prevent Animals from having paws.  Perhaps through empirical means,
where I find an instance Giraffe named Mindy which is an animal but
does not have paws.    (029)

Bart Gajderowicz
MSc Candidate, '10
Dept. of Computer Science
Ryerson University
http://www.scs.ryerson.ca/~bgajdero    (030)

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