ANDs and ORs

ANDs and ORs

Post by William Sile » Sat, 21 Sep 1996 04:00:00



DIGEST OF BUCKLEY, JJ AND SILER, W: A NEW T-NORM. SUBMITTED TO
FUZZY SETS AND SYSTEMS, 1996.

Fuzzy systems theory has been criticized for not obeying all
the laws of classical set theory and classical logic. The
t-norm and t-conorm here presented obey all the laws of the
corresponding classical theory. A somewhat similar theory has
been proposed by Thomas (1994), except that he does not claim
that the distributive property is maintained.

We first propose a source of fuzziness. We suppose that the
truth value > 0 and < 1 of a fuzzy logical statement A is drawn
from a number of underlying (probably implicit) correlated
random variables from a Bernoulli process whose values alpha[i]
are binary, i.e. 0 or 1 with a Bernoulli distribution, and that
the truth value of A is a simple average of these binary
values. (George Klir (1994) proposed a similar process where
the random values are binary opinions of experts as to truth or
falsehood of a statement.) If this is so, then

a = truth(A) = sum(alpha[i] / n
b = truth(B) = sum(beta[i] / n)
r = correlation coefficient(alpha[i], beta[i])
aANDb = truth value(A AND B)
aORb = truth value(A OR B)
sa = standard deviation(alpha) = sqrt(p(alpha)*(1 - p(alpha))
sb = standard deviation(beta) = sqrt(p(beta)*(1 - p(beta))
aANDb = a*b + r*sa*sb
aORb = a + b - a*b - r*sa*sb
rmax = (min(a,b) - a*b) / (sa*sb)
for r = rmax, min(a,b) = a*b + r*sa*sb
rmin =  (a*b - max(a+b-1, 0)) / (sa*sb)
for r = rmin, max(a+b-1, 0) = a*b + rmin*sa*sb

Proofs of the following theorems are in the appendices of our
paper.

Theorem 1:
  1. maxr = ru, ru <= 1
  2. minr = rl, rl >= -1
  3. rl <= r <= ru

Theorem 2:
  1. aANDb = a*b + r*sa*sb = a*b + cov(a,b)
  2. aORb = a + b - a*b - r*sa*sb = a + b - a*b - cov(a,b)

Theorem 3:
  1. If r = ru, aANDb = min(a,b) and aORb = max(a,b)
  2. If r = 0, aANDb = a*b and aORb = a+b-ab
  3. If r = rl, aANDb = max(a+b-1, 0) and aORb = max(a+b, 1)

We now suppose that this basic process is inaccessible to us,
but that we do have a history of a number of instances of the
truths of statement A and statement B. Now, given a value of r,
the correlation coefficient between a, the truth values of A,
and b, the truth values of B, the t-norm and t-conorm
appropriate to this history, T (t-norm) and C (t-conorm) are
defined for [a, b] on S, a restricted subset of [0,1]x[0,1].

Theorem 4.
  (The 5 parts of this theorem define the subset S of
[0,1]x[0,1]
  possible for r = 1, 0 < r < 1, r = 0, -1 < r < 0 and r = -1.)
  Given a value of r, it may be that not all (a,b) combinations
  are possible; e.g. a = .25, b = .75 is not possible for r = 1
  in the binary process described above.)

Theorem 5.
  1. (Shows that for 0 < r < 1 and (a,b) in S,
    ab < T(a,b) <= min(a,b).)

  2. (Shows that for -1 < r < 0 and (a,b) in S,
    max(a+b-1) <= T(A,B) < ab)

  3. (Shows that for -1 <= r <= 1 and (a,b) in S,
    max(a+b-1, 0) <= T(a,b) <= min(a,b) and
    max(a,b) <= C(a,b) <= min(a+b, 1).

Theorem 6. Shows that T is a t-norm and C is a t-conorm on S.

Theorem 7.
  1. A AND A = A, r is 1.
  2. A OR 0 = 0, any r.
  3. A OR X = A, any r.
  4. A AND NOT-A = 0, r is -1.
  5. A OR A = A, r is 1.
  6. A OR X = A, any r.
  7. A OR 0 = A, any r.
  8. A OR NOT-A = X, r is -1.
  9. NOT-(A AND B) = NOT-A OR NOT-B, any r.
  10. NOT-(A AND B) = NOT-A AND NOT-B, any r.
  11. A OR (A AND B) = A, any appropriate r.
  12. A AND (A OR B) = A, any appropriate r.
  13. A AND (B OR C) = (A AND B) OR (A AND C), any appropriate
r.
  14. A OR (B AND C) = (A OR B) AND (A OR C), any appropriate
r.

References:

Klir, GJ (1994). Multivalued logics vesus model logics:
alternate frameworks for uncertainty modelling. In: Advances in
Fuzzy Theory and Technology, Vol II: 3-47. Duke University
Press, Durham, NC.

Thomas, SF (1994). Fuzzy Logic and Probability. ACG Press,
Wichita, KS.

 
 
 

ANDs and ORs

Post by William Sile » Sun, 22 Sep 1996 04:00:00



>  2. A OR 0 = 0, any r.
>  3. A OR X = A, any r.

Sorry - typos. In Theorem 7 parts 2 and 3, OR should be AND.

Bill Siler

 
 
 

ANDs and ORs

Post by William Sile » Sun, 22 Sep 1996 04:00:00



>Theorem 7.
>  1. A AND A = A, r is 1.
>  2. A OR 0 = 0, any r.
>  3. A OR X = A, any r.

Sorry for the typos. Theorem 7 parts 2 and 3 should read AND rather than OR.

Bill Siler