Transcending Turing Computability
 B.J. Maclennan
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
It has been argued that neural networks and other forms of analog computation may transcend the limits of Turingmachine computation; proofs have been offered on both sides, subject to differing assumptions. In this article I argue that the important comparisons between the two models of computation are not so much mathematical as epistemological. The Turingmachine model makes assumptions about information representation and processing that are badly matched to the realities of natural computation (information representation and processing in or inspired by natural systems). This points to the need for new models of computation addressing issues orthogonal to those that have occupied the traditional theory of computation.
 Ballard, D.H. (1997), An Introduction to Natural Computation, Cambridge, MA: MIT Press.
 Blum, L., Shub, M. and Smale, S. (1988), 'On a Theory of Computation and Complexity over the Real Numbers: NP Completeness, Recursive Functions and Universal Machines', The Bulletin of the American Mathematical Society 21, pp. 1–46.
 Copeland, B.J. (1997), 'Vague Identity and Fuzzy Logic', The Journal of Philosophy 94, pp. 514–534.
 Copeland, B.J. (2000), 'Narrow Versus Wide Mechanism', The Journal of Philosophy 97, pp. 5–32.
 Copeland, B.J. and Proudfoot, D. (1999), 'Alan Turing's Forgotten Ideas in Computer Science', Scientific American 280(4), pp. 77–81.
 Dreyfus, H.L. (1979), What Computers Can't Do: The Limits of Artificial Intelligence, NewYork, NY: Harper & Row, revised edition.
 Gandy, R. (1980), 'Church's Thesis and Principles for Mechanisms' in J. Barwise, H.J. Keisler and K. Kunen, eds., The Kleene Symposium, Amsterdam, The Netherlands: NorthHolland, pp. 123–148.
 Goodman, N. (1968), Languages of Art: An Approach to a Theory of Symbols, Indianapolis, IN and New York, NY: BobbsMerrill.
 Lakoff, G. and Johnson, M. (1999), Philosophy in the Flesh: The Embodied Mind and Its Challenge to Western Thought, New York, NY: Basic Books.
 Maass, W. and Sontag, E.D. (1999), 'Analog Neural Nets with Gaussian or Other Common Noise Distributions Cannot Recognize Arbitrary Regular Languages', Neural Computation 11(3), pp. 771–782.
 MacLennan, B.J. (1987), 'TechnologyIndependent Design of Neurocomputers: The Universal Field Computer' in M. Caudill and C. Butler, eds., Proceedings of the IEEE First International Conference on Neural Networks, Vol. 3, IEEE Press, pp. 39–49.
 MacLennan, B.J. (1988), 'Logic for the New AI' in J.H. Fetzer, ed., Aspects of Artificial Intelligence, Dordrecht, The Netherlands: Kluwer Academic Publishers, pp. 163–192.
 MacLennan, B.J. (1990a), 'The Discomforts of Dualism', Behavioral and Brain Sciences 13, pp. 673–674. Also available from www.cs.utk.edu/~mclennan.
 MacLennan, B.J. (1990b), 'Field Computation: A Theoretical Framework for Massively Parallel Analog Computation, Parts I IV', Technical Report CS90 100, Department of Computer Science, University of Tennessee, Knoxville. Also available from www.cs.utk.edu/~mclennan.
 MacLennan, B.J. (1991), 'Gabor Representations of Spatiotemporal Visual Images', Technical Report CS91 144, Department of Computer Science, University of Tennessee, Knoxville. Also available from www.cs.utk.edu/~mclennan.
 MacLennan, B.J. (1993a), 'Characteristics of Connectionist Knowledge Representation', Information Sciences 70, pp. 119–143. Also available from www.cs.utk.edu/~mclennan.
 MacLennan, B.J. (1993b), 'Field Computation in the Brain' in K. Pribram, ed., Rethinking Neural Networks: Quantum Fields and Biological Data, Hillsdale, NJ: Lawrence Erlbaum, pp. 199–232. Also available from www.cs.utk.edu/~mclennan.
 MacLennan, B.J. (1993c), 'Grounding Analog Computers', Think 2, pp. 48–51. Also available from www.cs.utk.edu/~mclennan and at cogprints.soton.ac.uk/abs/comp/199906003.
 MacLennan, B.J. (1993d), 'Information Processing in the Dendritic Net' in K.H. Pribram, ed., Rethinking Neural Networks: Quantum Fields and Biological Data, Hillsdale, NJ: Lawrence Erlbaum, pp. 161–197. Also available from www.cs.utk.edu/~mclennan.
 MacLennan, B.J. (1994a), 'Continuous Computation and the Emergence of the Discrete' in K.H. Pribram, ed., Origins: Brain & SelfOrganization, Hillsdale, NJ: Lawrence Erlbaum, pp. 121–151. Also available from www.cs.utk.edu/~mclennan and at cogprints. soton.ac.uk/abs/comp/199906001.
 MacLennan, B.J. (1994b), 'Continuous Symbol Systems: The Logic of Connectionism' in D.S. Levine and M. Aparicio IV, eds., Neural Networks for Knowledge Representation and Inference, Hillsdale, NJ: Lawrence Erlbaum, pp. 83–120. Also available from www.cs.utk.edu/~mclennan.
 MacLennan, B.J. (1994c), 'Image and Symbol: Continuous Computation and the Emergence of the Discrete', in V. Honavar & L. Uhr, eds., Artificial Intelligence and Neural Networks: Steps Toward Principled Integration, Volume I: Basic Paradigms; Learning Representational Issues; and Integrated Architectures, New York, NY: Academic Press, pp. 207–240.
 MacLennan, B.J. (1994d), 'Words Lie in our Way', Minds and Machines 4(4), pp. 421–437. Also available from www.cs.utk.edu/~mclennan and at cogprints.soton.ac.uk/abs/phil/199906001.
 MacLennan, B.J. (1995), 'Continuous Formal Systems: A Unifying Model in Language and Cognition' in Proceedings of the IEEE Workshop on Architectures for Semiotic Modeling and Situation Analysis in Large Complex Systems, Monterey, CA, pp. 161–172. Also available from www.cs.utk.edu/~mclennan and at cogprints.soton.ac.uk/abs/comp/199906002.
 MacLennan, B.J. (1997), 'Field Computation in Motor Control' in P.G. Morasso and V. Sanguineti, eds., SelfOrganization, Computational Maps and Motor Control, Amsterdam, The Netherlands: Elsevier, pp. 37–73. Also available from www.cs.utk.edu/~mclennan.
 MacLennan, B.J. (1999), 'Field Computation in Natural and Artificial Intelligence', Information Sciences 119, pp. 73–89. Also available from www.cs.utk.edu/~mclennan.
 MacLennan, B.J. (2001), 'Can Differential Equations Compute?', Technical Report UTCS–01–459, Department of Computer Science, University of Tennessee, Knoxville. Also available from www.cs.utk.edu/~mclennan.
 Markov, A.A. (1961), Theory of Algorithms, Jerusalem, Israel: Israel Program for Scientific Translation. US Dept. of Commerce Office of Technical Service OTS 60–51085, transl. by Jacques J. SchorrKon and PST Staff. Translation of Teoriya Algorifmov, Moscow, USSR: Academy of Sciences of the USSR, 1954.
 Parkinson, L.L. (1966), Leibniz Logical Papers: A Selection, Oxford, UK: Oxford University Press.
 Penrose, R. (1989), The Emperor's New Mind: Concerning Computers, Minds, and The Laws of Physics, Oxford, UK: Oxford University Press.
 Penrose, R. (1990), 'Précis of: The Emperor's New Mind', Behavioral and Brain Sciences 13, pp. 643–705.
 PourEl, M.B. (1974), 'Abstract Computability and its Relation to The General Purpose Analog Computer (Some Connections Between Logic, Differential Equations and Analog Computers)', Transactions of the American Mathematical Society 199, pp. 1–29.
 Rubel, L.A. (1981), 'A Universal Differential Equation', Bulletin (New Series) of the American Mathematical Society 4(3), pp. 345–349.
 Rubel, L.A. (1993), 'The Extended Analog Computer', Advances in Applied Mathematics 14, pp. 39–50.
 Rumelhart, D.E., McClelland, J.L. and the PDP Research Group (1986), Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Cambridge, MA: MIT Press.
 Shannon, C.E. (1941), 'Mathematical Theory of the Differential Analyzer', Journal of Mathematics and Physics of the Massachusetts Institute of Technology 20, pp. 337–354.
 Simon, H.A. (1969), The Sciences of the Artificial, Cambridge, MA: MIT Press.
 Tarski, A. (1936), 'Der Wahrheitsbegriff in den formalisierten Sprachen', Studia Philosophica 1, pp. 261–405.
 Turing, A.M. (1936), 'On Computable Numbers, with an Application to the Entscheidungsproblem', Proceedings of the London Mathematical Society, Series 2, 42 (1936 37), pp. 230–265; corrections, ibid., 43 (1937), pp. 544 546.
 A.M. (1950), 'Computing Machinery and Intelligence', Mind 59, pp. 433–460.
 Title
 Transcending Turing Computability
 Journal

Minds and Machines
Volume 13, Issue 1 , pp 322
 Cover Date
 20030201
 DOI
 10.1023/A:1021397712328
 Print ISSN
 09246495
 Online ISSN
 15728641
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 analog computation
 analog computer
 biocomputation
 computability
 continuous computation
 hypercomputation
 natural computation
 Turing machine
 Industry Sectors
 Authors

 B.J. Maclennan ^{(1)}
 Author Affiliations

 1. Department of Computer Science, University of Tennessee, Knoxville, TN, USA