UnboundedSum SNorm -- UnboundedDifference TNorm about fuzzylite HOT 5 CLOSED

Hi Juan,

Well, I'm not an expert of fuzzy logic but I think here we have some theoretical issues.
For what I understand (e.g., see Scholarpedia, Wikipedia, and MATLAB),

• an operator T is a t-norm iff:

T: [0, 1] x [0, 1] ->  [0, 1]  (boundary)
T(x,y) = T(y,x)                (commutativity)
T(x,T(y,z)) = T(T(x,y),z)      (associativity)
y <= z -> T(x,y) <= T(x,z)     (monotonicity)
T(x,1) = x                     (neutral element 1) 

• an operator S is a t-conorm iff:

S: [0, 1] x [0, 1] ->  [0, 1]   (boundary)
S(x,y) = S(y,x)                 (commutativity)
S(x,S(y,z)) = S(S(x,y),z)       (associativity)
y <= z -> S(x,y) <= S(x,z)      (monotonicity)
S(x,0) = x                      (neutral element 0) 

Unfortunately, UnboundedSum cannot be a t-conorm operator since it violates some of the above properties, while UnboundedDifference (if defined as the difference between its operands) cannot be a t-norm operator since it violates all of such properties. For instance:

UnboundedSum(x,1) = x+1 > 1 for x >0 (violation of boundary)
UnboundedDifference(x,1) = x-1 != x for x != 2   (violation of neutrality)

Indeed, MATLAB allows sum to be only an aggregator operator.

Am I wrong?

Cheers,

Marco

References:

1. http://www.scholarpedia.org/article/Triangular_norms_and_conorms
2. https://en.wikipedia.org/wiki/T-norm
3. https://it.mathworks.com/help/fuzzy/foundations-of-fuzzy-logic.html#bp78l70-5

from fuzzylite.

Hi Marco,

thank you for your useful post.

For the sake of simplicity and design, the aggregation/accumulation operators in fuzzylite are modelled as SNorms, and hence the UnboundedSum needs to be modelled as such in order to be incorporated into fuzzylite.

As for the UnboundedDifference, it is more of a suggestion after observing that most TNorms have their SNorm counterparts, although I am not aware of its usage in anyway.

from fuzzylite.

OK, maybe, if you think it is a good idea, you could add a warning message in the API doc of the UnboundedSum class just to alert a fuzzylite user that this class should only be used for rule aggregation and not for other stuff (like for representing the OR operation in a fuzzy rule).

Regarding the UnboundedDifference, according to the aforementioned references, the t-norm counterpart of a t-conorm operator S is the operator T such that

T(x,y) = 1-S(1-x,1-y)

So, if we (wrongly) consider UnboundedSum a t-conorm operator, its t-norm counterpart would be:

T(x,y) = 1-(1-x+1-y) = 1-2+x+y = x+y-1

As expected, this is not a real t-norm operator since it violates some properties.

from fuzzylite.

Hi,

After such a long time, I have fixed this with your help. Thank you very much!

I understand the UnboundedSum is not a T-conorm, but it is the only way it can be added to fuzzylite for aggregation. I am not sure if i will add support for FCL yet.

Please, check 809675e

Cheers.

from fuzzylite.

Hi Juan,
Thank you for the fix. Unfortunately, I'm very busy now. I'll check it out ASAP.

Cheers,

Marco

from fuzzylite.