Abstract
A class of automata which build other automata is defined. These automata are called Turing machine automata because each one contains a Turing machine which acts as its computer-brain and which completely determines what its offspring, if any, will be. We show that for the descendants of an arbitrary progenitor Turing machine automaton there are exactly three possibilities: (1) there is a sterile descendant after an arbitrary number of generations, (2) after a delay of an arbitrary number of generations, the descendants repeat in generations with an arbitrary period, or (3) the descendants are aperiodic. We also show what sort of computing ability may be realized by the descendants in each of the possibilities. Furthermore, it is determined whether there are effective procedures for distinguishing between the various possibilities, and the exact degree of unsolvability is computed for those decision problems for which there is no effective procedure. Lastly, we discuss the relevance of the results to biology and pose several questions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. A. Arbib, Automata theory and development: Part I,J. Theoret. Biol. 14 (1967), 131–156.
M. A. Arbib, Self-reproducing automata—some implications for theoretical biology, inTowards a Theoretical Biology, II, C. H. Waddington, Ed., Aldine Publishing Company, Chicago, 1969, pp. 204–226.
M. A. Arbib,Theories of Abstract Automata, Prentice-Hall, Englewood Cliffs, N.J., 1969.
F. W. Baker andG. T. Herman, Simulation of organisms using a developmental model, Parts I and II,Int. J. Biomed. Comp., to appear.
E. R. Banks, Information processing and transmission in cellular automata, MAC TR-81, MIT, 1970.
A. W. Burks, Ed.,Essays on Cellular Automata, University of Illinois Press, Urbana, Ill., 1970.
J. Case, A note on degrees of self-describing Turing machines,J. Assoc. Comput. Mach. 18 (1971), 329–338.
E. F. Codd,Cellular Automata, Academic Press, New York, 1968.
G. T. Herman, The computing ability of a developmental model for filamentous organisms,J. Theoret. Biol. 25 (1970), 421–424.
G. T. Herman, Models for cellular interactions in development without polarity of individual cells, Parts I and II,Int. J. Systems Sci., to appear.
N. M. Jessop,Biosphere: A Study of Life, Prentice-Hall, Englewood Cliffs, N.J., 1970.
L. A. Laing, Asexual and sexual reproduction expressed in the von Neumann cellular system, T.R. 01251-3-T, University of Michigan, Ann Arbor, 1970.
C. Y. Lee, A Turing machine which prints its own code script,Proc. Symp. on the Mathematical Theory of Automata, Polytechnic Press, Brooklyn, N.Y., 1963, pp. 155–164.
A. Lindenmayer, Developmental systems without cellular interactions, their languages and grammars,J. Theoret. Biol. 30 (1971), 455–484.
A. Lindenmayer, Mathematical models for cellular interactions in development, Parts I and II,J. Theoret. Biol. 18 (1968), 280–315.
J. Myhill, Abstract theory of self-reproduction, inViews on General Systems Theory, M. D. Mesarović, Ed., Wiley, New York, 1964, pp. 106–118.
H. H. Pattee, Physical problems of heredity and evolution, inTowards a Theoretical Biology, II, C. H. Waddington, Ed., Aldine Publishing Company, Chicago, 1969, pp. 204–226.
H. H. Pattee, Physical theories, automata, and the origin of life, inNatural Automata and Useful Simulations, H. H. Pattee, et al., Eds. Macmillan, London, 1956.
H. Rogers, Jr.,Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York, 1967.
O. A. Schjeide andJ. de Vellis, Eds.,Cellular Differentiation, Van Nostrand-Reinhold, New York, 1970.
J. M. Smith, What use is sex,J. Theoret. Biol. 30 (1971), 319–335.
J. M. Thatcher, The construction of a self-describing Turing machine, inProc. Symp. on the Mathematical Theory of Automata, Polytechnic Press, Brooklyn, N.Y., 1963, pp. 165–171.
R. Tsanev andBl. Sendov, Possible molecular mechanism for cell differentiation in multicellular organisms,J. Theoret. Biol. 30 (1971), 337–393.
A. M. Turing, The chemical basis of morphogenesis,Phil. Trans. Royal Soc. London, Ser. B,237 (1953), 37–72.
D. van Dalen, A note on some systems of Lindenmayer,Math. Systems Theory 5 (1971), 128–140.
P. M. B. Vitányi, Sexually reproducing cellular automata, (unpublished).
J. von Neumann,Theory of Self-Reproducing Automata (edited and completed by A. W. Burks), University of Illinois Press, Urbana, Ill., 1966.
J. D. Watson,Molecular Biology of the Gene, W. A. Benjamin, New York, 1965.
E. P. Wigner, The probability of the existence of a self-reproducing unit, inThe Logic of Personal Knowledge: Essays Presented to Michael Polanyi, The Free Press, Glencoe, Ill., 1961, pp. 231–238.
Author information
Authors and Affiliations
Additional information
Department of Computer Science. The research for this paper was supported in part by Kansas General Research Grant 3683-5038.
Rights and permissions
About this article
Cite this article
Case, J. Periodicity in generations of automata. Math. Systems Theory 8, 15–32 (1974). https://doi.org/10.1007/BF01761704
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01761704