[FOM] Fuzziness

Rafee Kamouna rafee102000 at yahoo.com
Thu Jul 21 01:31:49 EDT 2016

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:

 #yiv3382986892 #yiv3382986892 -- _filtered #yiv3382986892 {font-family:Wingdings;panose-1:5 0 0 0 0 0 0 0 0 0;} _filtered #yiv3382986892 {panose-1:2 4 5 3 5 4 6 3 2 4;} _filtered #yiv3382986892 {font-family:Calibri;panose-1:2 15 5 2 2 2 4 3 2 4;}#yiv3382986892 #yiv3382986892 p.yiv3382986892MsoNormal, #yiv3382986892 li.yiv3382986892MsoNormal, #yiv3382986892 div.yiv3382986892MsoNormal {margin:0in;margin-bottom:.0001pt;font-size:12.0pt;}#yiv3382986892 a:link, #yiv3382986892 span.yiv3382986892MsoHyperlink {color:blue;text-decoration:underline;}#yiv3382986892 a:visited, #yiv3382986892 span.yiv3382986892MsoHyperlinkFollowed {color:purple;text-decoration:underline;}#yiv3382986892 pre {margin:0in;margin-bottom:.0001pt;font-size:10.0pt;}#yiv3382986892 p.yiv3382986892MsoListParagraph, #yiv3382986892 li.yiv3382986892MsoListParagraph, #yiv3382986892 div.yiv3382986892MsoListParagraph {margin-top:0in;margin-right:0in;margin-bottom:0in;margin-left:.5in;margin-bottom:.0001pt;font-size:12.0pt;}#yiv3382986892 span.yiv3382986892hoenzb {}#yiv3382986892 span.yiv3382986892EmailStyle18 {color:#1F497D;}#yiv3382986892 span.yiv3382986892HTMLPreformattedChar {}#yiv3382986892 .yiv3382986892MsoChpDefault {} _filtered #yiv3382986892 {margin:1.0in 1.0in 1.0in 1.0in;}#yiv3382986892 div.yiv3382986892WordSection1 {}#yiv3382986892 _filtered #yiv3382986892 {} _filtered #yiv3382986892 {font-family:Symbol;} _filtered #yiv3382986892 {} _filtered #yiv3382986892 {font-family:Wingdings;} _filtered #yiv3382986892 {font-family:Symbol;} _filtered #yiv3382986892 {} _filtered #yiv3382986892 {font-family:Wingdings;} _filtered #yiv3382986892 {font-family:Symbol;} _filtered #yiv3382986892 {} _filtered #yiv3382986892 {font-family:Wingdings;}#yiv3382986892 ol {margin-bottom:0in;}#yiv3382986892 ul {margin-bottom:0in;}#yiv3382986892 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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