From: "Kreinovich, Vladik"
Date: December 22, 2009 7:19:41 PM PST
Subject: Ladislav Kohout: sad news
Reply-To: North American Fuzzy Information Processing Society list
I have just learned that Ladislav Kohout died on November 20, 2009 in
Ladislav was one of the pioneers of interval-valued fuzzy techniques.
His idea of using intervals started with the observation that for each
statement A, one of the known ways to estimate its degree of truth d is
to ask several (n) people and take d = m/n, where m is the number of
people who believe that A is true.
If we have two statements A and B, we can thus estimate the degrees of
truth d(A), d(B), and d(A & B) for A, for B, and for A & B. In practice,
it is not feasible to ask the experts about the truth values of all
possible Boolean combinations of the original statements. Thus, we must
be able, e.g., given d(A) and d(B), to estimate d(A & B).
If we only know the degrees d(A) and d(B), then we cannot uniquely
determine p(A & B). For example, if d(A) < d(B) and all people who
believe in A also believe in B, then d(A & B) = d(A) = min(d(A), d(B)).
On the other hand, if people who believe in A tend not to believe in B,
we can have d(A & B) = max(d(A)+d(B)-1, 0). In general, the degree d(A &
B) can take any value from the interval [max(d(A)+d(B)-1, 0),
min(d(A),d(B))]. It is therefore sometimes reasonable, instead of
assigning a single value to d(A & B), to conclude that this whole
interval represents our degree of believe in A & B. This is important in
critical applications when we want to make only conclusions that follow
from the original expert information -- Dr. Kohout himself successfully
applied this idea to medical expert systems.
Of course, when we switch from numbers to intervals, we increase the
computational complexity of the corresponding problem. However, in many
cases, Dr. Kohout was able to decrease this complexity by using
techniques that are well known to simplify problems in physics (and in
science and engineering in general) -- the technique of symmetries.
In the traditional logic, & and \/ are "symmetric" in the sense that due
to de Morgan rules, negation transforms & into \/ and vice versa: ~(~A &
~B) = A \/ B and ~(~A \/ ~B) = A & B. In the interval-valued case, each
logical operation leads, in effect, to two function corresponding to the
bottom and the top of the corresponding interval. It turned out that the
resulting larger class of functions has its own symmetries, and these
symmetries help process these interval-valued degrees.
Ladislav published many papers and had many interesting ideas.
We will remember him.
From: "Kreinovich, Vladik"
στις 12:48 AM