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Response to letter by E. V. Krishnamurthy
848
E. V. Krishnamurthy
are not Turing solvable. This does not imply that a "guessing machine" is a super-Turing machine, or that its existence can be considered as a counter example for Turing computability. E. V. Krishnamurthy Computer Sciences Lab., Australian National University Canberra, ACT 0200, Australia
REFERENCES Bishop, E., & Bridges, D. (1985). Constructive analysis. New York: Springer-Verlag. Garzon, M. (1990). Cellularautomata and discrete neural networks. Physica 1)45, 431-440. Judd, K. T., & Aihara, K. (1993). Pulse propagation networks: A neural network model that uses temporal coding by action potentials. Neural Networks, 6, 203-215. Troelstra, A. S., & van Dalen, D. (1989). Constructivism in mathematics. Amsterdam: North-Holland.
Response to Letter by E. V. Krishnamurthy Krishnamurthy criticises a line of research and references as an example a paper by ourselves (Judd & Aihara, 1993). We feel that he misunderstands our work and misrepresents us. In particular, we make no claims of having constructed super-Turing machines. In fact, we explicitly state in the paper that we do not address this question. Krishnamurthy desires a formal axiomatic foundation to the idea of computation over the reals; this foundation exists in the work of Herman and Isard (1970), Eilenberg (1974), Blum, Shub, and Smale (1989), and Smale (1990), all of whom are referenced in our paper. Some of Krishnamurthy's objections would be better answered by these people, but we will try to explain briefly their work and its part in our argument, referring readers to the original papers for details. Krishnamurthy lists nine pertinent and profound questions that all point to the heart of the problem: a finite alphabet system cannot deal with the continu u m - o r in concrete t e r m s b a Turing machine cannot deal with the set of reals. This follows from a result of Chaitin ( 1987 ), which shows that almost all real numbers cannot be reduced to G6del numbers and, consequently, cannot be manipulated by a Turing machine. So why not, in the spirit of mathematical invention, postulate the existence of computing machines that have for their alphabet the real numbers? A lucid formulation of this idea is found in Blum et al. (1989). The work of Blum et al. does not, as Krishnamurthy implies, claim to overthrow the foundations of logic. Rather, it generalises and extends logic into a void it has never before penetrated. Our statement that R-machines (machines that have the reals as an alphabet) are more powerful than Z-machines (Turing machines) is a simple observation that, by the construction of Rmachines in Blum et al., the Z-machines are a proper subset of the R-machines and there are R-machines that can do things Z-machines cannot, because a Zmachine cannot manipulate real numbers.
The contribution of our paper, in the light of this work, is by observing that a real number is defined by the time interval between two instantaneous pulses, so a special type of "network" can model a class of Rmachine on the basis of generation and propagation of such idealized action potentials. We think that our work follows knowledge of biological neural networks and leads in a direction of neural computation that has not been adequately explored. We would like, if we may, to make an assertion to the neural network community that is contrary to Krishnamurthy's prescription of what one may and may not say in the future. The processes of science and discovery are complex; let us not make the mistake of believing that new ideas come into being in the pristine form that we learn of them. Browsing through the history of science and mathematics soon reveals that there are many mistakes and wrong turns--by the giants too. We do not believe that science and mathematics are well-served by disallowing certain lines of enquiry. There is no obvious reason to ignore possibilities that living things have discovered, through their history over billions of years, computational principles different from Turing machines. Exploring the principles of biological computation will not diminish the achievements of the 100-year history of logical computation. Kevin T. Judd Department of Mathematics The University of Western Australia Nedlands, WA 6009 Australia Kazuyuki Aihara Department of Mathematical Engineering and Information Physics Faculty of Engineering The University of Tokyo 7-3-1 Hongo, Bunkyo-ku Tokyo 113 Japan
Letters to the Editor
849
REFERENCES Blum, L., Shub, M., & Smale, S. (1989). On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bulletin of the American Mathematical Society, 21 ( 1), 1-46. Chaitin, G. J. (1987). Algorithmic information theory, Cambridge tracts m theoretical computer science 1. Cambridge: Cambridge University Press.
Eilenberg, S. (1974). Automata, languages and machines (Vol. A). New York: Academic Press. Herman, G. T., & Isard, S. D. (1970). Computability over arbitrary fields. Journal of the London Mathematical Society, 2 (2), 73-79. Judd, K. T., & Aihara, K. (1993). Pulse propagation networks: A neural network model that uses temporal coding by action potentials. Neural Networks, 6 (2), 203-215. Smale, S. (1990). Some remarks on the foundations of numerical analysis. SIAM Review, 32 (2), 211-220.
E. V. Krishnamurthy
are not Turing solvable. This does not imply that a "guessing machine" is a super-Turing machine, or that its existence can be considered as a counter example for Turing computability. E. V. Krishnamurthy Computer Sciences Lab., Australian National University Canberra, ACT 0200, Australia
REFERENCES Bishop, E., & Bridges, D. (1985). Constructive analysis. New York: Springer-Verlag. Garzon, M. (1990). Cellularautomata and discrete neural networks. Physica 1)45, 431-440. Judd, K. T., & Aihara, K. (1993). Pulse propagation networks: A neural network model that uses temporal coding by action potentials. Neural Networks, 6, 203-215. Troelstra, A. S., & van Dalen, D. (1989). Constructivism in mathematics. Amsterdam: North-Holland.
Response to Letter by E. V. Krishnamurthy Krishnamurthy criticises a line of research and references as an example a paper by ourselves (Judd & Aihara, 1993). We feel that he misunderstands our work and misrepresents us. In particular, we make no claims of having constructed super-Turing machines. In fact, we explicitly state in the paper that we do not address this question. Krishnamurthy desires a formal axiomatic foundation to the idea of computation over the reals; this foundation exists in the work of Herman and Isard (1970), Eilenberg (1974), Blum, Shub, and Smale (1989), and Smale (1990), all of whom are referenced in our paper. Some of Krishnamurthy's objections would be better answered by these people, but we will try to explain briefly their work and its part in our argument, referring readers to the original papers for details. Krishnamurthy lists nine pertinent and profound questions that all point to the heart of the problem: a finite alphabet system cannot deal with the continu u m - o r in concrete t e r m s b a Turing machine cannot deal with the set of reals. This follows from a result of Chaitin ( 1987 ), which shows that almost all real numbers cannot be reduced to G6del numbers and, consequently, cannot be manipulated by a Turing machine. So why not, in the spirit of mathematical invention, postulate the existence of computing machines that have for their alphabet the real numbers? A lucid formulation of this idea is found in Blum et al. (1989). The work of Blum et al. does not, as Krishnamurthy implies, claim to overthrow the foundations of logic. Rather, it generalises and extends logic into a void it has never before penetrated. Our statement that R-machines (machines that have the reals as an alphabet) are more powerful than Z-machines (Turing machines) is a simple observation that, by the construction of Rmachines in Blum et al., the Z-machines are a proper subset of the R-machines and there are R-machines that can do things Z-machines cannot, because a Zmachine cannot manipulate real numbers.
The contribution of our paper, in the light of this work, is by observing that a real number is defined by the time interval between two instantaneous pulses, so a special type of "network" can model a class of Rmachine on the basis of generation and propagation of such idealized action potentials. We think that our work follows knowledge of biological neural networks and leads in a direction of neural computation that has not been adequately explored. We would like, if we may, to make an assertion to the neural network community that is contrary to Krishnamurthy's prescription of what one may and may not say in the future. The processes of science and discovery are complex; let us not make the mistake of believing that new ideas come into being in the pristine form that we learn of them. Browsing through the history of science and mathematics soon reveals that there are many mistakes and wrong turns--by the giants too. We do not believe that science and mathematics are well-served by disallowing certain lines of enquiry. There is no obvious reason to ignore possibilities that living things have discovered, through their history over billions of years, computational principles different from Turing machines. Exploring the principles of biological computation will not diminish the achievements of the 100-year history of logical computation. Kevin T. Judd Department of Mathematics The University of Western Australia Nedlands, WA 6009 Australia Kazuyuki Aihara Department of Mathematical Engineering and Information Physics Faculty of Engineering The University of Tokyo 7-3-1 Hongo, Bunkyo-ku Tokyo 113 Japan
Letters to the Editor
849
REFERENCES Blum, L., Shub, M., & Smale, S. (1989). On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bulletin of the American Mathematical Society, 21 ( 1), 1-46. Chaitin, G. J. (1987). Algorithmic information theory, Cambridge tracts m theoretical computer science 1. Cambridge: Cambridge University Press.
Eilenberg, S. (1974). Automata, languages and machines (Vol. A). New York: Academic Press. Herman, G. T., & Isard, S. D. (1970). Computability over arbitrary fields. Journal of the London Mathematical Society, 2 (2), 73-79. Judd, K. T., & Aihara, K. (1993). Pulse propagation networks: A neural network model that uses temporal coding by action potentials. Neural Networks, 6 (2), 203-215. Smale, S. (1990). Some remarks on the foundations of numerical analysis. SIAM Review, 32 (2), 211-220.