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Decidability and Universality in Symbolic Dynamical Systems

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Języki publikacji
EN
Abstrakty
EN
Many different definitions of computational universality for various types of dynamical systems have flourished since Turing's work. We propose a general definition of universality that applies to arbitrary discrete time symbolic dynamical systems. Universality of a system is defined as undecidability of a model-checking problem. For Turing machines, counter machines and tag systems, our definition coincides with the classical one. It yields, however, a new definition for cellular automata and subshifts. Our definition is robust with respect to initial condition, which is a desirable feature for physical realizability. We derive necessary conditions for undecidability and universality. For instance, a universal system must have a sensitive point and a proper subsystem. We conjecture that universal systems have infinite number of subsystems. We also discuss the thesis according to which computation should occur at the `edge of chaos' and we exhibit a universal chaotic system.
Słowa kluczowe
Wydawca
Rocznik
Strony
463--490
Opis fizyczny
bibliogr. 40 poz., wykr.
Twórcy
autor
autor
  • Universite catholique de Louvain, Department of Mathematical Engineering, Avenue Georges Lemaitre 4, B-1348 Louvain-la-Neuve, Belgium, delvenne@inma.ucl.ac.be
Bibliografia
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  • [4] A. Ben-Hur, H. T. Siegelmann, and S. Fishman. A theory of complexity for continuous time systems. Journal of Complexity, 18(1):51-86, 2002.
  • [5] O. Bournez and M. Cosnard. On the computational power of dynamical systems and hybrid systems. Theoretical Computer Science, 168:417-459, 1996.
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  • [8] M. D. Davis. A note on universal Turing machines. In C.E. Shannon and J. McCarthy, editors, Automata Studies, pages 167-175. Princeton University Press, 1956.
  • [9] J.-C. Delvenne and V. D. Blondel. Quasiperiodic configurations and undecidable dynamics for tilings, infinite words and Turing machines. Theoretical Computer Science, 319:127-143, 2004.
  • [10] J.-Ch. Delvenne, P. K°urka, and V. D. Blondel. Computational universality in symbolic dynamical systems. In M. Margenstern, editor, MCU 2004, number 3354 in Lecture Notes in Computer Science, pages 104-115. Springer-Verlag, 2005.
  • [11] R. Devaney. An Introduction to Chaotic Dynamical Systems. Addison-Wesley, 1989.
  • [12] B. Durand and Z. Róka. The Game of Life: universality revisited. In M. Delorme and J. Mazoyer, editors, Cellular Automata: a ParallelModel, volume 460 of Mathematics and its Applications, pages 51-74. Kluwer Academic Publishers, 1999.
  • [13] Edward Fredkin and Tommaso Toffoli. Conservative logic. International Journal of Theoretical Physics, 21(3-4):219-253, 1981/82. Physics of computation, Part I (Dedham,Mass., 1981).
  • [14] Peter Gács. Reliable cellular automata with self-organization. In 38th Annual Symposium on Foundations of Computer Science, pages 90-99, Miami Beach, Florida, 20-22 October 1997. IEEE.
  • [15] E. M. Clarke Jr., O. Grumberg, and D. A. Peled. Model checking. MIT Press, 1999.
  • [16] H. Rogers Jr. Theory of recursive functions and effective computability. MIT Press, Cambridge,MA, second edition, 1987.
  • [17] R. Kaivola. Using automata to characterise fixed point temporal logics. PhD thesis, University of Edinburgh, 1996.
  • [18] P. Koiran. A family of universal recurrent networks. Theoretical Computer Science, 168(2):473-480, 1996. Universal machines and computations (Paris, 1995).
  • [19] P. Koiran,M. Cosnard, andM. Garzon. Computability with low-dimensional dynamical systems. Theoretical Computer Science, 132(1-2):113-128, 1994.
  • [20] P. Koiran and Cr. Moore. Closed-form analytic maps in one and two dimensions can simulate universal Turing machines. Theoretical Computer Science, 210(1):217-223, 1999.
  • [21] P. Kurka. On topological dynamics of Turing machines. Theoretical Computer Science, 174(1-2):203-216, 1997.
  • [22] P. Kurka. Zero-dimensional dynamical systems, formal languages, and universality. Theory of Computing Systems, 32(4):423-433, 1999.
  • [23] P. Kurka. Topological and Symbolic Dynamics, volume 11 of Cours Spécialisés. Société Mathématique de France, 2003.
  • [24] C. G. Langton. Computation at the edge of chaos. Physica D, 42:12-37, 1990.
  • [25] W. Maass and P. Orponen. On the effect of analog noise in discrete-time analog computations. Neural Computation, 10(5):1071-1095, 1998.
  • [26] M. L. Minsky. Recursive unsolvability of Post's problem of "tag" and other topics in theory of Turing machines. Annals of Mathematics (2), 74:437-455, 1961.
  • [27] M. Mitchell, P. T. Hraber, and J. P. Crutchfield. Dynamic computation, and the "edge of chaos": A reexamination. In G. Cowan, D. Pines, and D. Melzner, editors, Complexity: Metaphors, Models, and Reality, Santa Fe Institute Proceedings, Volume 19, pages 497-513. Addison-Wesley, 1994. Santa Fe InstituteWorking Paper 93-06-040.
  • [28] Cr.Moore. Unpredictability and undecidability in dynamical systems. Physical Review Letters, 64(20):2354-2357, 1990.
  • [29] Cr. Moore. Generalized shifts: Unpredictability and undecidability in dynamical systems. Nonlinearity, 4:199-230, 1991.
  • [30] Cr. Moore. Dynamical recognizers: real-time language recognition by analog computers. Theoretical Computer Science, 201:99-136, 1998.
  • [31] Cr. Moore. Finite-dimensional analog computers: Flows, maps, and recurrent neural networks. In C.S. Calude, J. Casti, and M. J. Dinneen, editors, UnconventionalModels of Computation. Springer-Verlag, 1998.
  • [32] P. Orponen. A survey of continuous-time computation theory. In D.-Z. Du and K.-I Ko, editors, Advances in Algorithms, Languages, and Complexity, pages 209-224. Kluwer Academic Publishers, 1997.
  • [33] D. Perrin and J.-E. Pin. Infinite Words (Automata, Semigroups, Logic and Games), volume 141 of Pure and Applied Mathematics. Elsevier, 2004.
  • [34] H. T. Siegelmann. Neural Networks and Analog Computation: Beyond the Turing Limit. Progress in Theoretical Computer Science. Springer-Verlag, 1999.
  • [35] H. T. Siegelmann and S. Fishman. Analog computation with dynamical systems. Physica D, 120:214-235, 1998.
  • [36] K. Sutner. Cellular automata and intermediate degrees. Theoretical Computer Science, 296(2):365-375, 2003.
  • [37] K. Sutner. Universality and cellular automata. In M. Margenstern, editor, MCU 2004, number 3354 in Lecture Notes in Computer Science, pages 50-59. Springer-Verlag, 2005.
  • [38] A. M. Turing. On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, 42(2):230-265, 1936.
  • [39] K. Weihrauch. Computable Analysis. Springer-Verlag, 2000.
  • [40] S. Wolfram. A new kind of science. Wolfram Media, Inc., Champaign, IL, 2002.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-BUS2-0015-0070
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