Hi all, esp. Lluis Godo and John Lemmer,
Lluis: Thanks for the reminder about Enrique Ruspini's interpretation of
fuzzy membership as similarity to a prototype.
wrt clarity, and truth functionality versus Boolean tautologies:
Once one takes the Bayesian interpretation that fuzzy membership of "Height
is 180cm" in Tall means prob(Oracle says x is Tall | Height of x is
180cm), then formally the propositions involved no longer lack clarity.
The problematic nature of fuzzy logic combination rules then applies, cf.
my earlier posting. The work I know of on possibility theory does not
solve this (it could be viewed as formalizing the problem and restating its
relationships to Bayesian conditioning etc.).
wrt fuzzy logic for applications in control:
The view of fuzzy logic, in control applications, as an interpolation
scheme is a familiar one. It's fine as far as it goes -- indeed, I agree
with it and find it useful. But it's very shallow semantically. I would
like a deeper account, connected to Bayesian and/or classical-logic
knowledge representation.
Benjamin
Lluis Godo <godo@iiia.csic.es>@CS.ORST.EDU on 02/28/2000 07:49:03 PM
Sent by: owner-uai@CS.ORST.EDU
To: uai@CS.ORST.EDU
cc:
Subject: Re: [UAI] Fuzzy sets vs. Bayesian Network
Hi all,
I would like to contribute to this interesting discussion with some
comments on Kathy's and Benjamin's opinions.
Kathy:
>I have heard exactly one proposed ontology for fuzzy membership functions
>(proposed by Judea Pearl, among others) that makes sense to me. Under
this
>proposed ontology, the fuzzy membership of Robbie's adult height in the
set
>"tall," in a given context, should be taken as proportional to the
>probability that a generic person in that context would use the label
>"tall" to describe Robbie. Thus, fuzzy memberships are likelihood
>functions. We can think of them as soft evidence applied to numerical
>crisp set height measurements.
This "soft evidence" is actually what possibility theory tries to capture
and handle and I think there is some work done in relating likelihood
functions and possibility distributions.
There is another ontology, pushed by Enrique Ruspini (SRI), for fuzzy
membership functions which I personally like, it is to interpret
membership degrees in terms of similarity in the following terms. Using
your example, the fuzzy membership of Robbie's adult height in the set
"tall," in a given context, should be taken as proportional to how similar
(in terms of height) Robbie is (or will be) to some pre-determined
prototypical "tall" people in the given context. However, this ontology
does not give rise to a fully truth-functional framework.
>I might go beyond this and suggest an alternate criterion, that the fuzzy
>membership be proportional to the *utility* for an appropriately defined
>decision maker in that context of using the term "tall" to describe
Robbie
>(this, for example, would allow us to weigh costs of inappropriate usage
of
>the term).
>
>This proposed ontology makes a lot of sense at a surface level. However,
I
>know it is not what most fuzzy set researchers think they are talking
about
>when they use fuzzy memberships. I've never seen its mathematical
>implications worked out, or seen any discussions about whether or under
>what circumstances it gives rise to combination rules that look anything
>like what the fuzzy people now use.
I agree that one thing is to provide meaningful ontologies and another is
to try to fit them with a bunch of combination rules proposed for fuzzy
sets, some of them clearly not justifiable. This is a harder problem,
although there is a growing interest in the fuzzy community to constrain
arbitariness and at the same time to justify reasoning models on ground
logical bases. This may lead hopefully to a convergent process.
Benjamin:
>Wrt fuzzy logic overall:
> I always have found problematic the justification of the min or product
>rules for the conjunction-combination operation in fuzzy logic, for a
>couple reasons:
>- each makes a very strong and rigid assumption about dependency
(complete
>dependence for min, complete independence for product), when viewed in
>Bayesian terms.
>- there is no sensitivity to what propositions are being conjunctively
>combined, e.g., it could be P has membership of 0.4, not-P has membership
>0.6, and then the logical contradiction (P and not-P) is assigned
non-zero
>membership: 0.4 (min rule) or 0.24 (product rule).
>Boolean tautologies are thus not respected.
I think here there is clear misunderstanding. Using Kathy's terminology,
fuzzy logic is only appropriate for sets NOT satisfying the "clarity
test."
Therefore a "crisp" proposition (i.e. satisfying the "clarity test") even
in a fuzzy logic setting has to be constrained to only take the Boolean
truth-values 0 or 1. So, for crisp propositions, Boolean tautologies are
preserved, not for fuzzy propositions. This can be readily seen by looking
at the main systems of propositional fuzzy (many-valued) logic: Goedel
logic, Lukasiewicz logic and Product logic. If you add to any of these
systems the axiom
A v not-A
where v is max-disjunction, then all those systems collapse to Boolean
classical logic. I might recommend the book "Meta-mathematics of fuzzy
logic" by Peter Hajek (Kluwer, 98), to my mind the best manuscript about
fuzzy logic from the strict and formal logical point of view.
>
>Wrt applications of fuzzy logic to control (where they have had a lot of
>practical success and popularity in the last decade or so):
>Here fuzziness is applied to beliefs about what actions to take.
>I have yet to see a terribly satisfactory or deep intuitive
>*Bayesian*-flavored account of fuzziness in this context, esp. of fuzzy
>logic in the ways it is usually applied to control applications. One way
>to think about this at a high-level would be to view fuzzy beliefs about
>the control variables in terms of an oracle/person who would say what is
>the probability, or expected utility, of the right thing to do, given
>various evidences. But needed is a detailed Bayesian interpretation of
>the kind of fuzzy logic rules that are typically found in successful
>practice, in relation to this high-level interpretation.
To my opinion the best way of interpreting what fuzzy logic controllers do
is that fuzzy control rules are just providing a simple and efficient
interpolation method. A set of fuzzy rules is just describing a set of
(fuzzy) points (x,y) in a multi-dimensional space belonging to an
"unknown" graph. Then, given an input, the output is computed by applying
a simple analogical reasoning principle: the closer the input is to x, the
closer the output will be to y. That's all. But whether this has to do
with logic or not it is matter of current discussions.
Lluis.
--------------------------------------
Lluis Godo
IIIA - CSIC
Campus Univ. Autonoma de Barcelona s/n
08193 Bellaterra, Spain
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