Church Without Dogma: Axioms for Computability
Church’s and Turing’s theses assert dogmatically that an informal notion of effective calculability is captured adequately by a particular mathematical concept of computabilty. I present analyses of calculability that are embedded in a rich historical and philosophical context, lead to precise concepts, and dispense with theses.
To investigate effective calculability is to analyze processes that can in principle be carried out by calculators. This is a philosophical lesson we owe to Turing. Drawing on that lesson and recasting work of Gandy, I formulate boundedness and locality conditions for two types of calculators, namely, human computing agents and mechanical computing devices (or discrete machines). The distinctive feature of the latter is that they can carry out parallel computations.
Representing human and machine computations by discrete dynamical systems, the boundedness and locality conditions can be captured through axioms for Turing computors and Gandy machines; models of these axioms are all reducible to Turing machines. Cellular automata and a variety of artificial neural nets can be shown to satisfy the axioms for machine computations.
KeywordsTuring Machine Discrete Dynamical System Symbolic Process Calculable Function Machine Computation
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- Church, A. (1936) An unsolvable problem of elementary number theory; American Journal of Mathematics 58, 345-363; reprinted in (Davis 1965).Google Scholar
- — (1937) Review of (Turing 1936); The Journal of Symbolic Logic 2, 40-41.Google Scholar
- Davis, M., (ed.) (1965) The Undecidable, Basic papers on undecidable propositions, unsolvable problems and computable functions; Raven Press, Hewlett, New York.Google Scholar
- De Pisapia, N. (2000) Gandy Machines: an abstract model of parallel computation for Turing Machines, the Game of Life, and Artificial Neural Networks; M.S. Thesis, Carnegie Mellon University, Pittsburgh.Google Scholar
- Gandy, R. (1980) Church’s Thesis and principles for mechanisms; in: The Kleene Symposium (edited by J. Barwise, H. J. Keisler and K. Kunen, North-Holland, 123-148.Google Scholar
- Gödel, K. (1934) On undecidable propositions of formal mathematical systems; in: Collected Works I, 346-369.Google Scholar
- — (1936) Über die Länge von Beweisen; in: Collected Works I, 396-399.Google Scholar
- — (1946) Remarks before the Princeton bicentennial conference on problems in mathematics; in: Collected Works II, 150-153.Google Scholar
- [1986-2003]— (1986-2003) Collected Works, volumes I-V; Oxford University Press.Google Scholar
- Hilbert, D. and P. Bernays (1939) Die Grundlagen der Mathematik II ; Springer Verlag, Berlin.Google Scholar
- Kolmogorov, A. N and V. A. Uspensky (1958) On the definition of an algorithm; Us-pekhi Mat. Nauk 13 (Russian), 1958; English translation in: AMS Translations, 2, 21 (1963), 217-245.Google Scholar
- — (1943) Formal reductions of the general combinatorial decision problem; American Journal of Mathematics, 65 (2), 197-215.Google Scholar
- — (1947) Recursive unsolvability of a problem of Thue; The Journal of Symbolic Logic 12, 1-11.Google Scholar
- Sieg, W. (1994) Mechanical procedures and mathematical experience; in: Mathematics and Mind (A. George, ed.), Oxford University Press, 71-117.Google Scholar
- — (1997) Step by recursive step: Church’s analysis of effective calculability; The Bulletin of Symbolic Logic 3 (2), 154-180.Google Scholar
- [2002a]— (2002a) Calculations by man and machine: conceptual analysis; Lecture Notes in Logic 15, 390-409.Google Scholar
- [2002b]— (2002b) Calculations by man and machine: mathematical presentation; in: In the Scope of Logic, Methodology and Philosophy of Science, volume one of the 11th Inter-national Congress of Logic, Methodology and Philosophy of Science, Cracow, August 1999 (P. Gärdenfors, J. Wolenski and K. Kijania-Placek, eds.), Synthese Library volume 315, Kluwer, 247-262.Google Scholar
- Sieg, W. and J. Byrnes (1996) K-Graph machines: generalizing Turing’s machines and arguments; in: Gödel ’96 (P. Hajek, ed.), Lecture Notes in Logic 6, Springer Verlag, 98-119.Google Scholar
- — (1939) Systems of logic based on ordinals; Proceedings of the London Mathemat-ical Society (Series 2) 45, 161-228; reprinted in (Davis 1965).Google Scholar
- — (1950) The word problem in semi-groups with cancellation; Ann. of Math. 52, 491-505.Google Scholar
- — (1954) Solvable and unsolvable problems; Science News 31, 7-23; reprinted in Collected Works of A. M. Turing: Mechanical intelligence, (D. C. Ince, ed.), North-Holland, 1992.Google Scholar