Comments on some theories of fuzzy computation

In classical computability theory, there are several (equivalent) definitions of computable function, decidable subset and semi-decidable subset. This paper is devoted to the discussion of some proposals for extending these definitions to the framework of fuzzy set theory. The paper mainly focuses on the notions of fuzzy Turing machine and fuzzy computability by limit processes. The basic idea of this paper is that the presence of real numbers in the interval [0,1] forces us to refer to endless approximation processes (as in recursive analysis) and not to processes terminating after a finite number of steps and giving the exact output (as in recursive arithmetic). In accordance with such a point of view, an extension of the famous Church thesis is proposed.