Machine Computability

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Jacquette Dale - One of the best experts on this subject based on the ideXlab platform.

  • Computable Diagonalizations and Turing's Cardinality Paradox
    2019
    Co-Authors: Jacquette Dale
    Abstract:

    A. N. Turing's 1936 concept of Computability, computing Machines, and computable binary digital sequences, is subject to Turing's Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing Machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing's objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of a Turing Machine, Computability, computable sequences, and Turing's effort to prove the unsolvability of the Entscheidungsproblem, are explained in light of the paradox. A solution to Turing's Cardinality Paradox is proposed, positing a higher geometrical dimensionality of Machine symbol-editing information processing and storage media than is available to canonical Turing Machine tapes. The suggestion is to add volume to Turing's discrete two-dimensional Machine tape squares, considering them instead as similarly ideally connected massive three-dimensional Machine information cells. Three-dimensional computing Machine symbol-editing information processing cells, as opposed to Turing's two-dimensional Machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. A three-dimensional model of Machine information storage and processing cells is recommended on independent grounds as better representing the biological realities of digital information processing isomorphisms in the three-dimensional neural networks of living computers

Dale Jacquette - One of the best experts on this subject based on the ideXlab platform.

  • Computable Diagonalizations and Turing’s Cardinality Paradox
    Journal for General Philosophy of Science, 2014
    Co-Authors: Dale Jacquette
    Abstract:

    A. N. Turing’s 1936 concept of Computability, computing Machines, and computable binary digital sequences, is subject to Turing’s Cardinality Paradox . The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing Machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing’s objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of a Turing Machine, Computability, computable sequences, and Turing’s effort to prove the unsolvability of the Entscheidungsproblem , are explained in light of the paradox. A solution to Turing’s Cardinality Paradox is proposed, positing a higher geometrical dimensionality of Machine symbol-editing information processing and storage media than is available to canonical Turing Machine tapes. The suggestion is to add volume to Turing’s discrete two-dimensional Machine tape squares, considering them instead as similarly ideally connected massive three-dimensional Machine information cells. Three-dimensional computing Machine symbol-editing information processing cells, as opposed to Turing’s two-dimensional Machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. A three-dimensional model of Machine information storage and processing cells is recommended on independent grounds as better representing the biological realities of digital information processing isomorphisms in the three-dimensional neural networks of living computers.

Jean Mosconi - One of the best experts on this subject based on the ideXlab platform.

  • The Developments of the Concept of Machine Computability from 1936 to the 1960s
    Logic Epistemology and the Unity of Science, 2014
    Co-Authors: Jean Mosconi
    Abstract:

    From the 1940s to the 1960s, despite the significant work done on recursive functions (properly) and later on the lambda-calculus, the theory of calculability was developed more and more as a theory of computation by an idealized Machine, or in the form of a general theory of algorithms. I will only deal here with the former aspect, a development that stems from the concepts introduced in 1936 by Turing. I will try to show how Turing’s ideas were gradually adopted, developed and modified. The Turing Machine had an increasingly important role and was the object of systematic investigation. It was subsequently reworked to such an extent that a new model of Machine was fashioned, the program and register Machine. However, the initial model kept a significant place, and extensions of Turing’s analysis led, toward the end of the century, to profound reflections about the notion of a constructive object and the general notion of an algorithm.

Herbert B. Enderton - One of the best experts on this subject based on the ideXlab platform.

  • Programs and Machines
    Computability Theory, 2011
    Co-Authors: Herbert B. Enderton
    Abstract:

    This chapter discusses another way of formalizing the concept of effective calculability, namely register-Machine programs. The main aim is to show that all general recursive partial functions are also computable by register Machines. It provides half of a significant fact that formalizing the effective calculability concept by means of general recursiveness and formalizing the effective calculability concept by means of register Machines lead to exactly the same class of partial functions. The register Machines are capable of doing much more than might have been apparent initially from their very simple definition. The methods used here to obtain the equivalence of general recursiveness to register-Machine Computability are adaptable to obtaining equivalence between other formalizations of effective calculability.

  • Degrees of Unsolvability
    Computability Theory, 2011
    Co-Authors: Herbert B. Enderton
    Abstract:

    This chapter discusses the concept of relative Computability. The concept of relative Computability first appeared in a 1939 paper by Alan Turing. At first glance, it might seem strange to combine the rather constructive concept of Computability with the almost mystical idea of an oracle. It is to Turing's credit that he perceived that the combination, strange or not, would be a useful tool in classifying the noncomputable sets. The focus is on an informal description of the concept of effective calculability relative to a set. The plan is to make the concept into a genuine mathematical concept in two ways: general recursiveness relatives and register-Machine Computability relatives. Further, enumeration theorem and normal form theorem related to relative Computability is discussed followed by equivalence relations

Kantorovitzisaiah Pinchas - One of the best experts on this subject based on the ideXlab platform.