# Re: the beginning

cargan (cargan@delrio.com)
Sat, 21 Mar 1998 05:24:29 +0100 (MET)

Fuzzification: Fuzzy Logic: In Fuzzy Logic, truth values are real values in
the closed interval
[0..1]. The definitions of the boolean operators are extended to fit this
continuous domain. By
avoiding discrete truth-values, Fuzzy Logic avoids some of the problems
inherent in either-or
judgments and yields natural interpretations of utterances like "very hot".
Fuzzy Logic has
applications in control theory.

Fuzzy logic extends conventional (Boolean) logic to handle the concept of
partial truth --
truth values between "completely true" and "completely false". Fuzzy Logic
was introduced by
Dr. Lotfi Zadeh of UC/Berkeley in the early 1960's as a means to model the
uncertainty of natural
language. Zadeh, Lotfi, "Fuzzy Sets," Information and Control 8:338-353,
"Outline of a New Approach to the Analysis of Complex Systems", IEEE Trans.
on Sys., Man
and Cyb. 3, 1973. Zadeh, Lotfi, "The Calculus of Fuzzy Restrictions", in
Fuzzy Sets and
Applications to Cognitive and Decision Making Processes, edited by L. A.
Academic Press, New York, 1975, pages 1-39.
What is the difference between the fuzzy logic (of FuziCalc) versus the
Monte Carlo
simulation (used in @Risk and Crystal Ball), and how might the results
differ? You may get
results which are very similar, depending on how you set each up. They
both work with numeric
distributions which can be shaped pretty much however one likes. J Quelch &
T T Cameron
"Uncertainty representation and propagation in quantified risk assessments
using fuzzy sets"
Journal of Loss Prevention in the Process Industries, 1994, Vol 7, No 6, PP
463-473
Zadeh says that rather than regarding fuzzy theory as a single theory, we
should regard
the process of ``fuzzification'' as a methodology to generalize ANY
specific theory from a crisp
(discrete) to a continuous (fuzzy) form. Thus recently researchers have
also introduced "fuzzy
calculus", "fuzzy differential equations", and so on. This is essentially
the same as the
Dempster-Shafer model which distributes belief-function probabilities over
intervals in the range
[1,0]. See A. P. DEMPSTER, Upper and lower probabilities induced by a
multivalued mapping,
Annals of Mathematical Statistics, AMS-38 (1967), pp. 325-339. G. SHAFER, A
mathematical
theory of evidence, Princeton University Press, Princeton, N.J., 1976.
Fuzzy Subsets: Just as there is a strong relationship between Boolean
logic and the
concept of a subset, there is a similar strong relationship between fuzzy
logic and fuzzy subset
theory. In classical set theory, a subset U of a set S can be defined as a
mapping from the
elements of S to the elements of the set {0, 1}, U: S --> {0, 1} This
mapping may be represented
as a set of ordered pairs, with exactly one ordered pair present for each
element of S. The first
element of the ordered pair is an element of the set S, and the second
element is an element of
the set {0, 1}. The value zero is used to represent non-membership, and
the value one is used
to represent membership. The truth or falsity of the statement x is in U
is determined by finding
the ordered pair whose first element is x. The statement is true if the
second element of the ordered pair is 1, and the statement is false if it
is 0.
Similarly, a fuzzy subset F of a set S can be defined as a set of ordered
pairs, each with
the first element from S, and the second element from the interval [0,1],
with exactly one ordered
pair present for each element of S. This defines a mapping between elements
of the set S and
values in the interval [0,1]. The value zero is used to represent complete
non-membership, the
value one is used to represent complete membership, and values in between
are used to
represent intermediate degrees of membership. The set S is referred to as
the UNIVERSE OF
DISCOURSE for the fuzzy subset F. Frequently, the mapping is described as
a function, the
MEMBERSHIP FUNCTION of F. The degree to which the statement x is in F is
true is
determined by finding the ordered pair whose first element is x. The
DEGREE OF TRUTH of the
statement is the second element of the ordered pair. In practice, the terms
"membership
function" and fuzzy subset get used interchangeably.
Dempster-Shafer model is a more general approach to representing
uncertainty than the
Bayesian approach. This is a currently popular amalgam of two earlier
theories. The
Dempster-Shafer theory of evidence comes from the concepts of upper and
lower probability
introduced by Dempster in the 1960's.
See A. P. DEMPSTER, Upper and lower probabilities induced by a
multivalued mapping, Annals of Mathematical Statistics, AMS-38 (1967),
pp. 325-339.
In his 1978 possibility paper (L.A. Zadeh. fuzzy Sets as a Basis for a
Theory of
Possibility. Fuzzy Sets and Systems, 1, 1978: 3-28.) Zadeh defines the
consistency of a
probability distribution Prob(ui) with a possibility distribution
Poss(ui) as the sum of the
products Prob(ui)Poss(ui) over all attribute values ui (e.g. height values
u quantized in cm
intervals). This quantity lies always between 0 and 1 because Prob(ui) sums
up to 1.
Let `a' be the label of a fuzzy set, e.g. `tall' or `short'. Both Prob(ui)
and Poss(ui) must
refer to this label. The TEE model for grades of membership interprets
these as follows:
Prob(ui)=P(ui|a)=probability that an object which has been labeled yes-a
by a subject has the
(exactly measured) height ui. Poss(ui)=Prob(a|ui)=probability that a
subject will assign to an
object of (exactly measured) height ui the label yes-a.
The reason why Poss(ui)=Prob(a|ui) is not always 1 or 0 (which it would be
if the subject
had used exact, nonfuzzy threshold for `a' (tall)) (in the universe U of
exactly measured height
values) is the presence of one or more sources of fuzziness).(Ellen Hisdal.
Infinite-Valued Logic
Based on Two-Valued Logic and Probability, Part 1.2 Different Sources of
Fuzziness and
Uncertainty. Int. J. Man-Machine Studies 25, 1986: 113-38)
One of these sources can be that a subject (who assigns the grade of
membership or
possibility value in the fuzzy set `a=tall' to an object whose exact height
ui is specified to him)
takes into account that other subjects only estimate the height value of an
object in everyday life.
An object of exact height ui may thus be estimated by another subject to
have a slightly different
height.
Another source of fuzziness can be that the subject (who assigns the
membership or
possibility value in a=tall to the object of exact height ui) takes into
account that other subjects
have slightly variable thresholds in U (universe of exactly measured height
values) for the exact
height value of an object to which they would assign the label yes-a.
Zadehs `possibility-probability consistency' is equal to P(a|a), the
probability that an
object labeled `a' in one experiment will be labeled `a' again in a second
experiment (assuming
that the conditions of observation of the object are chosen at random from
a set of
conditions of observation in each of the two experiments. Or that the
assumed subject who
assigns the label `a' is chosen at random from a set of subjects with
somewhat different
thresholds for, e.g. a=tall). (Ellen Hisdal. Infinite-Valued Logic Based on
Two-Valued Logic and
Probability, Part 1.2 Different Sources of Fuzziness and Uncertainty. Int.
J. Man-Machine
Studies 25, 1986: 113-38
Ellen Hisdal. A Theory of Logic Based on Probability. Institute of
Informatics Research
Report 64 1984.
Ellen Hisdal. Are Grades of Membership Probabilities? Fuzzy Sets and
Systems 25,
1988: 325-348.
E. Hisdal. "Open-Mindedness and Probabilities versus Possibilities" in
Fuzzy Logic
Foundations and Industrial Applications. Kluwer Academic Publishers,
Ellen Hisdal. Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part 1.3.
Reference Experiments and Label Sets. Institute of Informatics, University
of Oslo. 1988, 1990.
Ellen Hisdal. Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part 1.4.
The TEE Model. Institute of Informatics, University of Oslo, 1988, 1990.
Ellen Hisdal. Infinite-Valued Logic Based on Two-Valued Logic and
Probability, Part 1.1.
Difficulties with Present-Day Fuzzy Set Theory and their Resolution in the
{TEE} Model},
Int. J. Man-Machine Studies 25, 1986: 89-111.