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On Fuzzy Turing Machines

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Theory of Fuzzy Computation

Part of the book series: IFSR International Series on Systems Science and Engineering ((IFSR,volume 31))

Abstract

The Turing machine is the archetypal conceptual computing device and as such it is almost always used to define new models of computations by augmenting the functionality of the machine. For example, as was explained in Sect. 2.2.2, a probabilistic model of computation is defined by introducing a probabilistic version of the Turing machine, that is, a machine that operates nondeterministically. Therefore, it was quite expected to see the emergence of fuzzy Turing machines as models of fuzzy computation, that is, computation that encompasses vagueness in the form of fuzziness as a basic ingredient of computation.

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Notes

  1. 1.

    Donald M. Kaplan [71] had used regular expressions to represent elementary programs in order to study program equivalence.

  2. 2.

    In fact, Santos, following Scott, had defined a program to be a set of commands. Note that typically an ordinary program is understood to be a sequence of commands to be executed in that particular order. But since there is no implicit order in the way one may pick up elements from any set, it is better to talk about sequences than sets.

  3. 3.

    This last clarification is actually a proposition; nevertheless, its proof is trivial and not necessary for what comes next.

  4. 4.

    The original text mentions that \(\mathcal{M}\) functions in essentially the same way as the Turner machine defined in [117]. Unfortunately, there is no mention of a Turner machine in [117]. Moreover, to the best of my knowledge, there is no abstract machine called Turner machine! Chances are that this was a typo and that the text refers to the Turing machine mentioned in [117].

  5. 5.

    A t-norm ∧ : [0, 1] ×[0, 1] → [0, 1] can be seen as a fuzzy relation on [0, 1]; thus, one can compute its transitive closure.

  6. 6.

    In the original paper, the second condition is “0 ≤ b < 1,” which is obviously a typo since in what follows the authors use “a” instead of “b”.

  7. 7.

    Gerla [49] uses instead the term recursively co-enumerable, which does not make any sense to this author.

  8. 8.

    The proofs are omitted since the original text is in Spanish and my knowledge of this language is nonexistent.

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  1. Xanthi, Greece

    Apostolos Syropoulos

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  1. Apostolos Syropoulos
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Syropoulos, A. (2014). On Fuzzy Turing Machines. In: Theory of Fuzzy Computation. IFSR International Series on Systems Science and Engineering, vol 31. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8379-3_4

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