[FOM] Fuzziness

[Rafee Kamouna]

Dear Charlie,
Confusing fuzzy logic with probability is a common misconception as some say:"Fuzzy set theory is wrong. I can design a controller using probability theory that will do everything a fuzzy controller does.".
However, this misconception is handled via possibility theory (studied by Didier Dubois & Henri Prade) which is different than probability theory. It is possible that one may eat 4 eggs in the morning but it is quite improbable. 
Despite many conceptions about fuzzy logic, the new discipline of:"Mathematical Fuzzy Logic" as a (legitimate) branch of mathematical logic emerged after Petr Hajek's mongraph in 1998.
In addition, fuzzy set theory generalized set theory which is seen as foundational resulting in fuzzy topology, fuzzy group theory, etc.
Apart from fuzzy databases, fuzzy logic programming and soft computing.
Kind regards,
Rafee Kamouna.

 

 Also, there some important books of Fuzzy Logic:
1. Mathematics Behind Fuzzy Logic, by Turnen.2. Mathematical Principles of Fuzzy Logic, by Novak & Perfilieva.
And Prof Petr Hajek acknowledged his colleagues at Barcelona: Frances Esteva and Lluis Godo who co-authored with him many papers.
Kind Regards,
Rafee Kamouna.
 



       On Thursday, July 21, 2016 1:55 AM, "Kreinovich, Vladik" <vladik at utep.edu> wrote:
 

 #yiv4332157811 -- filtered {font-family:Wingdings;panose-1:5 0 0 0 0 0 0 0 0 0;}#yiv4332157811 filtered {panose-1:2 4 5 3 5 4 6 3 2 4;}#yiv4332157811 filtered {font-family:Calibri;panose-1:2 15 5 2 2 2 4 3 2 4;}#yiv4332157811 p.yiv4332157811MsoNormal, #yiv4332157811 li.yiv4332157811MsoNormal, #yiv4332157811 div.yiv4332157811MsoNormal {margin:0in;margin-bottom:.0001pt;font-size:12.0pt;}#yiv4332157811 a:link, #yiv4332157811 span.yiv4332157811MsoHyperlink {color:blue;text-decoration:underline;}#yiv4332157811 a:visited, #yiv4332157811 span.yiv4332157811MsoHyperlinkFollowed {color:purple;text-decoration:underline;}#yiv4332157811 pre {margin:0in;margin-bottom:.0001pt;font-size:10.0pt;}#yiv4332157811 p.yiv4332157811MsoListParagraph, #yiv4332157811 li.yiv4332157811MsoListParagraph, #yiv4332157811 div.yiv4332157811MsoListParagraph {margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:.5in;margin-bottom:.0001pt;font-size:12.0pt;}#yiv4332157811 span.yiv4332157811hoenzb {}#yiv4332157811 span.yiv4332157811EmailStyle18 {color:#1F497D;}#yiv4332157811 span.yiv4332157811HTMLPreformattedChar {}#yiv4332157811 .yiv4332157811MsoChpDefault {}#yiv4332157811 filtered {margin:1.0in 1.0in 1.0in 1.0in;}#yiv4332157811 div.yiv4332157811WordSection1 {}#yiv4332157811 filtered {}#yiv4332157811 filtered {font-family:Symbol;}#yiv4332157811 filtered {}#yiv4332157811 filtered {font-family:Wingdings;}#yiv4332157811 filtered {font-family:Symbol;}#yiv4332157811 filtered {}#yiv4332157811 filtered {font-family:Wingdings;}#yiv4332157811 filtered {font-family:Symbol;}#yiv4332157811 filtered {}#yiv4332157811 filtered {font-family:Wingdings;}#yiv4332157811 ol {margin-bottom:0in;}#yiv4332157811 ul {margin-bottom:0in;}#yiv4332157811 While there does not seem to be a direct relation between fuzzy logic and neural networks, in both areas, there are some semi-empirical function that (somewhat mysteriously) turn out to lead to the most efficient results.     ·       In neural networks, this is the “activation function” f(x) that for each neuron, transform a linear combination x of inputs x1, …, xn into the output y=f(x); the most efficient choice is the sigmoid function f(x)=1/(1+\exp(-k*x)). ·       In fuzzy logic, it is the selection of a membership function – which assigns to each value of a quantity the degree from [0,1] to which this quantity satisfies the given informal property like “small”, and “and”- and “or”-operations  f_& and f_\/ that transform our degrees of belief a and b in statements A and B into estimates f_&(a,b) and f_\/(a,b) of degrees of belief in A & B and A \/ B. It turns out that in both cases (and also, for the similar selection of functions in evolutionary computations) the empirical selection of functions can be explained by the fact that all these functions are related to natural symmetries (re-scaling), like linear transformations that come from changing the measuring unit or starting point, or more general non-linear transformations. These results are described in detail e.g., in our book    Hung T. Nguyen and Vladik Kreinovich, "Applications of continuous mathematics to computer science", Kluwer, Dordrecht, 1997.    Vladik      On 22 June 2016 at 03:07, Harvey Friedman <hmflogic at gmail.com> wrote: 
"fuzzy logic" --  how does it relate to recent breakthroughs in machine learning,
deep leaning, etcetera?


 
   
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