Effective Computation by Humans and Machines

There is an intensive discussion nowadays about the meaning of effective computability, with implications to the status and provability of the Church–Turing Thesis (CTT). I begin by reviewing what has become the dominant account of the way Turing and Church viewed, in 1936, effective computability. According to this account, to which I refer as the Gandy–Sieg account, Turing and Church aimed to characterize the functions that can be computed by a human computer. In addition, Turing provided a highly convincing argument for CTT by analyzing the processes carried out by a human computer. I then contend that if the Gandy–Sieg account is correct, then the notion of effective computability has changed after 1936. Today computer scientists view effective computability in terms of finite machine computation. My contention is supported by the current formulations of CTT, which always refer to machine computation, and by the current argumentation for CTT, which is different from the main arguments advanced by Turing and Church. I finally turn to discuss Robin Gandy's characterization of machine computation. I suggest that there is an ambiguity regarding the types of machines Gandy was postulating. I offer three interpretations, which differ in their scope and limitations, and conclude that none provides the basis for claiming that Gandy characterized finite machine computation.

[1]  Dana,et al.  JSL volume 88 issue 4 Cover and Front matter , 1983, The Journal of Symbolic Logic.

[2]  Wilfried Sieg,et al.  Mechanical Procedures and Mathematical Experience , 1991 .

[3]  Martin Davis,et al.  The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable Problems and Computable Functions , 2004 .

[4]  S. Kleene.  General recursive functions of natural numbers , 1936 .

[5]  D. Deutsch.  Quantum theory, the Church–Turing principle and the universal quantum computer , 1985, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[6]  P. Bernays,et al.  Grundlagen der Mathematik , 1934 .

[7]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[8]  John Impagliazzo,et al.  Computer Science: A Breadth-First Approach with C , 1995 .

[9]  A. Turing.  On Computable Numbers, with an Application to the Entscheidungsproblem. , 1937 .

[10]  Wilfried Sieg,et al.  An abstract model for parallel computations : Gandy's thesis , 1999 .

[11]  Wolfram,et al.  Undecidability and intractability in theoretical physics. , 1985, Physical review letters.

[12]  Robin Gandy,et al.  The confluence of ideas in 1936 , 1988 .

[13]  Rolf Herken,et al.  The Universal Turing Machine: A Half-Century Survey , 1992 .

[14]  George Boolos,et al.  Computability and logic , 1974 .

[15]  S. Kleene,et al.  General Recursive Functions of Natural Numbers. , 1937 .

[16]  Kurt Gödel,et al.  On Formally Undecidable Propositions of Principia Mathematica and Related Systems , 1966 .

[17]  H T Siegelmann,et al.  Dating and Context of Three Middle Stone Age Sites with Bone Points in the Upper Semliki Valley, Zaire , 2007 .

[18]  Alonzo Church,et al.  A note on the Entscheidungsproblem , 1936, Journal of Symbolic Logic.

[19]  Emil L. Post.  Finite combinatory processes—formulation , 1936, Journal of Symbolic Logic.

[20]  Owen Astrachan,et al.  A Computer Science Tapestry , 1996 .

[21]  Carol E. Cleland.  Is the Church-Turing thesis true? , 1993, Minds and Machines.

[22]  Stephen C. Kleene,et al.  Turing's analysis of computability, and major applications of it , 1988 .

[23]  A. Church.  Review: A. M. Turing, On Computable Numbers, with an Application to the Entscheidungsproblem , 1937 .

[24]  A. Church.  An Unsolvable Problem of Elementary Number Theory , 1936 .

[25]  Owen L. Astrachan.  Computer Science Tapestry: Exploring Programming and Computer Science , 1998 .

[26]  Rolf Herken,et al.  Alan Turing and the Turing Machine.Turing's Analysis of Computability, and Major Applications of it.The Confluence of Ideas in 1936.Turing in the Land of O.Mathematical Logic and the Origin of Modern Computers , 1991 .

[27]  Wilfried Sieg.  Calculations by Man and Machine: Mathematical Presentation , 2000 .

[28]  Paolo Mancosu.  Between Russell and Hilbert: Behmann on the foundations of mathematics , 1999, Bull. Symb. Log..

[29]  Kurt Godel.  Remarks before the Princeton Bicentennial Conference on problems in mathematics , 1990 .

[30]  L. M.-T..  Grundzüge der theoretischen Logik , 1929, Nature.

[31]  Wilfried Sieg,et al.  Calculations by Man and Machine: Conceptual Analysis , 2000 .

[32]  Wilfried Sieg.  Step by recursive step: Church's analysis of effective calculability , 1997, Bull. Symb. Log..

[33]  Carol E. Cleland.  On Effective Procedures , 2004, Minds and Machines.

[34]  Oron Shagrir.  A Neural Net With Self-Inhibiting Units For The N-Queens Problem , 1992, Int. J. Neural Syst..

[35]  Christos H. Papadimitriou,et al.  Elements of the Theory of Computation , 1997, SIGA.

[36]  Oron Shagrir,et al.  Physical Hypercomputation and the Church–Turing Thesis , 2003, Minds and Machines.

[37]  Robin Gandy,et al.  Church's Thesis and Principles for Mechanisms , 1980 .

[38]  Yishai A. Feldman,et al.  Algorithmics: The Spirit of Computing , 1987 .

[39]  John C. Shepherdson,et al.  Mechanism for computing over arbitrary structures , 1988 .

[40]  Oron Shagrir.  Two Dogmas of Computationalism , 2004, Minds and Machines.

[41]  M. Hogarth.  Non-Turing Computers and Non-Turing Computability , 1994, PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association.