The MEANING of fuzzy AND and OR.

Ellen Hisdal (ellen@ifi.uio.no)
Sun, 14 Jun 1998 19:52:29 +0200 (MET DST)

The proliferation of email letters concerning AND and OR
(one of these letters is reproduced at the end of this letter), shows that some
clarification is necessary concerning the precise meaning of a fuzzy
label containing one of these connectives. Such a meaning is
associated with the idea of a `composite experiment'.

Let A be the label of a fuzzy set
(e.g. `A=tall', more specifically `A=tall woman'),
`a' the grade of memberhip in the fuzzy set A of objects with
measured attribute calue `u' (e.g. u=175 cm).

Let B be the label of a fuzzy set (e.g. `medium'),
`b' the grade of memberhip in the fuzzy set B of objects with
measured attribute calue `u' (e.g. u=175 cm).

Let C be the label of the fuzzy set A AND B
(e.g. C = A AND B = tall AND medium),
`c' the grade of memberhip in in the fuzzy set C of objects with
measured attribute calue `u' (e.g. u=175 cm).

Let D be the label of the fuzzy set A ORA B ,
(e.g. D = A ORA B = tall ORA medium, ORA=inclusive OR),
`d' the grade of memberhip in the fuzzy set D of objects with
measured attribute calue `u' (e.g. u=175 cm).

We interpret the grade of membership `a' specified by some subject S
(who is acquainted with the value of `u' for the given object)
as S's estimate of the probability that objects with that value of `u'
will be labeled A (see references [1], [4], [5], [6]). For example,
the subject S may estimate that objects with height u=175 cm
will be labeled `A=tall' in 40% of all cases of objects with that height,
and `B=medium' in 60% of all cases of objects with that height.
S will than specify the membership value a=0.4 for objects with
height u=175 cm, and the membership value b=0.6 for objects with that height.
The reason why S believes in such variable labeling is due to her or
his ability to take the presence of different sources of fuzziness
into account. Three such different sources of fuzziness are described
in reference [2] below.
_______________________________________

COMPOSITE EXPERIMENTS FOR LABELS WITH CONNECTIVES

In a composite experiment the subject S imagines that an object of
height u is labeled twice by some other subject(s) in two different
labeling experiments. For example in connection with errors of observation
of the height value u (fuzziness #1), the height of the same object
may be estimated to lie in the interval (170, 180]cm in 40% of all
conditions of observation and in the interval (160, 170]cm
60% of all conditions of observation. In connection with different
threshold intervals for `tall' for different persons (fuzziness #3),
S may estimate that
for 40% of the human population the height u=175cm belongs to the threshold
interval for `tall', and
for 60% of the human population the height u=175cm belongs to the threshold
interval for `medium'.

S then interprets the membership value `c' in C = A AND B
for objects of measured height u=175cm
as the proportion of objects of that height which are labeled A in
one of the component experiments and B in the other..

Similarly S interprets the membership value `d' in D = A ORA B
for objects of measured height u=175cm
as the proportion of objects of that height which are labeled
A in one of the component experiments,
or B in one of the component experiments,
or A in one component experiment and B in the other component experiment.

It can then be shown that c = ab,
d = a+b-ab.
(see equations (17) (18) in reference [5] below).

These equations are valid when S refers to
RR (ReRandomizing) composite experiments
with distinguishable components. For reference to SIM (SIMultaneous)
composite experiments, and for reference to composite experiments
with indistinguishable components,
other formulas are valid. These differences are discussed in reference [5],
section 2.9 and in more detail in [6]. The last reference also derives
the formulas.

In the following papers the exact measured attribute value is denoted
by `u' with superscript `ex' (as opposed to `u' without superscript here).
`u' without superscript denotes the estimated attribute value in the
papers below.

The important subjects of the label set to which the subject S refers,
and of the different types of experiment (labeling (LB) or yes-no (YN)
versus grade of membership (MU) experiments) are discussed in [3].

REFERENCES
[1]
@article{inf1p1,
author = {Hisdal, E.},
title = {Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part~1.1. {D}ifficulties with Present-Day
Fuzzy Set Theory and their Resolution in the {TEE} Model},
journal = {Int. J. Man-Machine Studies},
year = {1986},
volume = {25},
pages = {89-111},
ignored = {page 94 for different words for grade of membership
page 95 for lack of difference between distr tall|u and u|tall},
}
[2]
@article{inf1p2,
author = {Hisdal, E.},
title = {Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part~1.2. {D}ifferent Sources of Fuzziness
and Uncertainty},
journal = {Int. J. Man-Machine Studies},
year = {1986},
volume = {25},
pages = {113-138} }

[3]
@techreport{inf1p3,
author = {Hisdal, E.},
title = {Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part~1.3. {R}eference Experiments and
Label Sets},
institution = {Institute of Informatics, University of Oslo, Box
1080 Blindern, 0316 Oslo 3, Norway},
year = {1988,1990},
type = {Research Report},
note = {ISBN~82-7368-053-3.
Can also be found on
http://www.ifi.uio.no/$\sim$ftp/publications/research-reports/Hisdal-3.ps},
number = {147} }

[4]
@techreport{inf1p4,
author = {Hisdal, E.},
title = {Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part~1.4. {T}he {TEE} Model},
institution = {Institute of Informatics, University of Oslo, Box
1080 Blindern, 0316 Oslo 3, Norway},
year = {1988,1990},
type = {Research Report},
note = {ISBN~82-7368-054-1.
Can also be found on
http://www.ifi.uio.no/$\sim$ftp/publications/research-reports/Hisdal-4.ps},
number = {148} }

[5]
@article{are,
author = {Hisdal, E.},
title = {Are Grades of Membership Probabilities?},
journal = {Fuzzy Sets and Systems},
year = {1988},
volume = {25},
pages = {325-348} }

[6]
@techreport{64,
author = {Hisdal, E.},
title = {A Theory of Logic Based on Probability},
institution = {Institute of Informatics, University of Oslo, Box
1080 Blindern, 0316 Oslo 3, Norway},
year = {1984},
type = {Research Report},
note = {ISBN~82-90230-60-5},
number = {64} }

------------------------------------------------------------------------------

Please send to me any possible email correspondence concerning this subject
after July 20-th, 1998.

Greetings,

Ellen Hisdal

-----------------------------------------------------------------------------
Address, etc.:
Ellen Hisdal | Email: ellen@ifi.uio.no
(Professor em.) |
Mail: Department of Informatics | Fax: +47 22 85 24 01
University of Oslo | Tel: (office): 47 22 85 24 39
Box 1080 Blindern |
0316 Oslo, Norway | Tel: (secr.): 47 22 85 24 10
Location: Gaustadalleen 23, |
Oslo | Tel: (home): 47 22 49 56 53
---------------------------------------------------------------------

----------------------------------------------------------------------------
> Date: Sun, 7 Jun 1998 02:24:54 +0200 (MET DST)
> Errors-To: fuzzy-owner@dbai.tuwien.ac.at
> Reply-To: wsiler@aol.com
> Originator: fuzzy-mail@dbai.tuwien.ac.at
> Sender: fuzzy-mail@dbai.tuwien.ac.at
> Precedence: bulk
> From: wsiler@aol.com (WSiler)
> X-Listprocessor-Version: 6.0c -- ListProcessor by Anastasios Kotsikonas
> X-Comment: Fuzzy Distribution List
>
> CA13159 wrote:
>
> > If A = percentage of students failing question A and,
> > B = percentage of students failing question B,
> > then if
> > C = percentage of students failing both questions A and B,
> > C will lie in these ranges, and C < = A ZAND B,
> > what is the rational for A ZAND B = min(A(x),B(x)) then ?
> >
>
> It is probably improper to use the min-max operator here. If we are using
> probabilities, then (if we want to preserve the laws of excluded middle and
> contradiction)the proper operator to use depends on the prior association (or
> correlation) between A and B. In your case, the probability of failing question
> A and of failing question B are almost certainly somewhat imperfectly
> correlated.
>
> Let me review three of the inifinity of possible operators.
>
> The Zadeh operators, valid for max possible association are:
> A ZAND B = min(A, B)
> A ZOR B = max(A, B)
>
> The probabilistic operators, valid for zero correlation, are:
> A PAND B = A * B
> A POR B = A + B - A * B
>
> The Lukasiewicz operators, valid for max negative correlation, are:
> A LAND B = max(0, A + B - 1)
> A LOR B = min(1, A + B)
>
> If the probability of failing question A and of failing question B are
> positively correlated to the maximum extent possible, then the Zadeh operators
> are the correct ones to use. For example, if all students failing question A
> (say 45%) also fail question B, and some students fail question B (say 55%) but
> don't fail question A, then the probability of failing both questions is the
> mininum of .45 and .55, or point 45. The probabililty of failing at least one
> question is the maximum of .45 and .55, or .55.
>
> If the probability of failing question A and of failing question B are
> negatively correlated to the maximum extent possible, then the Lukasiewz
> operators are the correct ones to use. For example, if all students failing
> question A (say 45%) do not fail question B, and some students fail question B
> (say 55%) but only 10% fail question A, then the probability of failing both
> questions is the maximum of 0 and (.45 + .55 -1), or 0.1. The probabililty of
> failing at least one question is the minimum of 1 and (.45 + .55), or 1.
>
>
> If the probability of failing question A and of failing question B are
> uncorrelated then the probilistic operators are the correct ones to use. For
> example, if students have a random probility of failing question A (say .45 and
> also a random probiltiy of failing question B (say .55), then the probability
> of failing both questions is the (.45 * .55), or 0.2475. The probabililty of
> failing at least one question is (.45 + .55 - .45 * .55), or .7525.
>
> This example is absurdly simple. However, we can extend it to more interesting
> applications to fuzzy expert systems.
>
> First, consider the probability of failing question A (.45) and of NOT failing
> question A (.55). These are maximally negatively associated, so we should use
> the Lukasiewicz operators, so the A LAND NOT A = max(0, (.45 + .55 - 1) or
> zero, and intuitively correct answer.
>
> Since the probability of failing question A can be used as a definition of the
> truth of the statement "Student fails question A", and similarly for B to
> "Student fails question B", then we can use the above ideas in evaluating the
> truth of the rule:
>
> IF (Student fails question A) and (student fails question B) THEN....
>
> and on to more complex rules.
>
> We can extend the idea of maximally negative association to the notion of
> semantic inconsistency. For example, when combining Slow = .45 OR Medium = .25,
> since Slow and Medium are semantically inconsistent we should use the Lukasiewz
> OR in combining their weighted membership functions. This would probably play
> merry hell with fuzzy control engineers, but for more general fuzzy expert
> systems it gives intuitively reasonable results.
>
> William Siler
>
>
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(5) WWW archive: http://www.dbai.tuwien.ac.at/marchives/fuzzy-mail/index.html