Editing Fuzzy logic (section)

===Decidability Issues===
The notions of a "decidable subset" and "[[recursively enumerable]] subset" are basic ones for classical mathematics and [[classical logic]]. Thus the question of a suitable extension of them to [[fuzzy set theory]] is a crucial one. The first proposal in such a direction was made by E. S. Santos by the notions of ''fuzzy [[Turing machine]]'', ''Markov normal fuzzy algorithm'' and ''fuzzy program'' (see Santos 1970). Successively, L. Biacino and G. Gerla argued that the proposed definitions are rather questionable. For example, in <ref>{{Cite journal|last=Gerla|first=G.|year=2016|title=Comments on some theories of fuzzy computation|journal= International Journal of General Systems|volume=45|issue=4|pages=372–392|doi=10.1080/03081079.2015.1076403|bibcode=2016IJGS...45..372G|s2cid=22577357}}</ref> one shows that the fuzzy Turing machines are not adequate for fuzzy language theory since there are natural fuzzy languages intuitively computable that cannot be recognized by a fuzzy Turing Machine. Then they proposed the following definitions. Denote by ''Ü'' the set of rational numbers in [0,1]. Then a fuzzy subset ''s'' : ''S'' <math>\rightarrow</math>&nbsp;[0,1] of a set ''S'' is recursively enumerable if a recursive map ''h'' : ''S''×'''N''' <math>\rightarrow</math>''Ü'' exists such that, for every ''x'' in ''S'', the function ''h''(''x'',''n'') is increasing with respect to ''n'' and ''s''(''x'') = lim ''h''(''x'',''n'').
We say that ''s'' is ''decidable'' if both ''s'' and its complement –''s'' are recursively enumerable. An extension of such a theory to the general case of the L-subsets is possible (see Gerla 2006).
The proposed definitions are well related to fuzzy logic. Indeed, the following theorem holds true (provided that the deduction apparatus of the considered fuzzy logic satisfies some obvious effectiveness property).

Any "axiomatizable" fuzzy theory is recursively enumerable. In particular, the [[fuzzy set]] of logically true formulas is recursively enumerable in spite of the fact that the crisp set of valid formulas is not recursively enumerable, in general. Moreover, any axiomatizable and complete theory is decidable.

It is an open question to give support for a "Church thesis" for [[fuzzy mathematics]], the proposed notion of recursive enumerability for fuzzy subsets is the adequate one. In order to solve this, an extension of the notions of fuzzy grammar and fuzzy [[Turing machine]] are necessary. Another open question is to start from this notion to find an extension of [[Gödel]]'s theorems to fuzzy logic.