Fuzzy Logic

[steve_bank]

The book I read back in 90s used a seesaw as an analogy.

Any number of weights at different positions on a seesaw can result in a balance or the same position of the seesaw.

Decision points are not defined by deterministc logic.

Fuzzy control systems go back to the 80s. Fuzzy processors learn or are taught from data not coded with explicit logic.

You could look at fuzzy logic as intuition.

AI is not fuzzy logic rather, fuzzy logic is a technique used within AI to help systems make decisions based on vague or imprecise information, similar to how humans do. While traditional AI might rely on strict, binary logic (true or false), fuzzy logic allows for degrees of truth, which makes it useful for complex, real-world applications where data is not perfectly clear.

https://dl.ebooksworld.ir/motoman/CRC.A.First.Course.in.Fuzzy.Logic.www.EBooksWorld.ir.pdf

https://www.studiecd.dk/cahiers_techniques/Fuzzy_logic.pdf

 

[Jarhyn]

Well, in a larger or broader sphere of things, fuzzy logic is really analog referenced logic: instead of the switches yielding a transform to exactly 1 or exactly 0 based on inputs of exactly 1 or 0, the the input and outputs are "analog numbers".

This IS AI in the sense that the neuron is a special case of fuzzy logic switch.

Edit: or rather, neural AI is all fuzzy logic. It's just the case that while all AI is fuzzy logic, not all fuzzy logic is AI unless it has an error function and conformance response to the error function.

 

[lpetrich]

I have a thread on the subject of fuzzy logic: Fuzzy Logic | Internet Infidels Discussion Board

Fuzzy logic is the logic of partial truth values, truth values as numbers from 0 (false) to 1 (true).

One can take most of the axioms of 2-valued or crisp logic and extrapolate them to fuzzy logic. To have well-behaved functions, one introduces an additional axiom, monotonicity: increasing (or decreasing) parameter values will not make the function values reverse direction of change, though no change is still possible.

Axioms for "not", "and" and "or" that one carries over into fuzzy logic:

• end values: not 0 = 1 ... not 1 = 0
• involution (self-inversion, double negation): not (not a) = a
• commutativity: a and b = b and a ... a or b = b or a
• associativity: (a and b) and c = a and (b and c) ... (a or b) or c = a or (b or c)
• Identity: a and 1 = a ... a or 0 = a
• Zero: a and 0 = 0 ... a or 1 = 1
• DeMorgan Inversion: not (a and b) = (not a) or (not b) ... not (a or b) = (not a) and (not b)

In fuzzy logic, (not a) decreases as a increases. One can also express (not a) as fni(-fn(a)) where fni(fn(a)) = fn(fni(a)) = a. There will be a crossover point r where not r = r.

(and) and (or) can be expressed as transforms into addition or multiplication:
function of a, b: fai(fa(a) + fa(b)) = fmi(fm(a)*fm(b)) where fa(fai(a)) = fai(fa(a)) = fmi(fm(a) = fm(fmi(a)) = a

(and) and (or) either increase or stay fixed as their parameters increase.

Here are some axioms of crisp logic that hold for only some fuzzy-logic functions:

• Distributivity: a and (b or c) = (a and b) or (a and c) ... a or (b and c) = (a or b) and (a or c)
• Absorption: (a and b) or a = (a or b) and a = a
• Idempotence: a or a = a and a = a

Distributivity with c = 1 and 0 gives absorption. Absorption with b = 1 or 0 gives idempotence.

From monotonicity and idempotence, one finds Zadeh or minmax norm-conorm or and-or: a and b = min(a,b) ... a or b = max(a,b)

Minmax is not only idempotent, but also absorptive and distributive: the only and-or combination to have these properties.

Another axiom of crisp logic is complementation: non-contradiction or excluded middle.

a and (not a) = 0 ... a or (not a) = 1

But minmax violates this axiom for every value of a: 0 < a < 1.

There is an and-or combination that does satisfy this axiom, the Lukasiewicz one:

a and b = max(a+b-1, 0) ... a or b = min(a+b, 1)