Fuzzy logic operators

What about..

(a AND b) = ab
(a OR b) = a + b - ab
(a XOR b) = a + b - 2ab
(NOT a) = 1 - a

xyzzy

[Edited by - xyzzy00 on March 17, 2005 8:36:20 PM]

fuzzy logic isn't about probabilities; it's about degrees of membership or the *possibility* of belonging to a set, not the probability. This is a common misunderstanding. See the Yen & Langari book pg 32.

xyzzy00, thanks, but these doesnt follow the requirements. For example,

0.5 AND 0.5 = 0.5 * 0.5 = 0.25 != 0.5 (third requirement)


fup: I read that in some papers too, but thats still debated in the AI research community, see the huge amount amount of papers solving fuzzy logic with bayesian methods. Anyway, the analogy is so close between *probability* and *possibility*. But thats out of the topic. Do you know other operators than min and max that would be better?

Ok, after doing some thinking, I decided to scrape the '0.5' requirement.
These are the rules Im considering now, if anyone is interested:

a AND b = min(a,b) - (min(a,b) * (1.0 - max(a,b)))

a OR b = max(a,b) + (1-max(a,b))*min(a,b)

I like thier behavior, and they keep everything nicely between 0 and 1.

Ok, sorry.. didn't realize you weren't looking for probabilites. I'm not really sure offhand what the difference between a *possibility* and a *probability* is.. sounds like I'll have to go read up on on Fuzzy Logic, thanks for the paper.

If you are trying to avoid probabilities, you might want to reconsider your formulas. They work out to exactly the same thing as the ones I wrote above:

(a AND b) = min(a,b) - (min(a,b) * (1.0 - max(a,b)))

if a > b: (a AND b) = b - b * (1 - a) = b - b + ab = ab
if a < b: (a AND b) = a - a * (1 - b) = a - a + ab = ab


(a OR b) = max(a,b) + (1-max(a,b))*min(a,b)

if a > b: (a OR b) = a + (1 - a) * b = a + b - ab;
if a < b: (a OR b) = b + (1 - b) * a = a + b - ab;

xyzzy

lol!

You are totally right, xyzzy00. I should have give it more tought. ++ for you!

Warning: heading off-topic... (sort of)

Quote:

Original post by Steadtler I read that in some papers too, but thats still debated in the AI research community, see the huge amount amount of papers solving fuzzy logic with bayesian methods.

Actually, that debate is pretty much one-sided, consisting of proponents of fuzzy logic attempting to justify their work (which doesn't require justification if you accept it for what it is). Most Bayesians discount FL has having anything to say about beliefs, since FL is a Truth Maintenance system, rather than a Belief Revision system. If you try to use FL to represent beliefs, you're just trying to knock in a nail with a screwdriver; you could probably do it, but it's going to hurt... and anyone holding a hammer is going to laugh at you.

Quote:

Anyway, the analogy is so close between *probability* and *possibility*.

The possibility of set membership has nothing to do with probability, particularly with regards to the use of each in AI. Probabilists, in particular Bayesians, almost exclusively use probabilities to measure the degree of belief held by an agent regarding a fact in the domain. Probabilities are then a measure of uncertainty of knowledge and the probability calculus provides a means of maintaining consistent beliefs.

FL on the other hand provides a means of describing systems that aren't adequately modelled by First Order (Aristotlean) Logic. FL permits the description of domains where items from a set belong to several distinct subsets. The universe of discourse in FL is specifically detailing the domain itself, rather than the beliefs held by an agent observing that domain. To use FL in this manner would be to use the proverbial screwdriver mentioned above.

As for operators... have you considered using the product operator for Fuzzy AND?

Regards,

Timkin