Abstract
In the Cellular Automata (CA) literature, discrete lines inside (discrete) space-time diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special purposes. In this article, we present a parallel analog model of computation corresponding to this idealization: dimensionless signals are moving on a continuous space in continuous time generating Euclidean lines on (continuous) space-time diagrams. Like CA, this model is parallel, synchronous, uniform in space and time, and uses local updating. The main difference is that space and time are continuous and not discrete (ie ℝ instead of ℤ). In this article, the model is restricted to ℚ in order to remain inside Turing-computation theory. We prove that our model can carry out any Turing-computation through two-counter automata simulation and provide some undecidability results.
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Ilachinski, A.: Cellular automata –a discrete universe. World Scientific, Singapore (2001)
Boccara, N., Nasser, J., Roger, M.: Particle-like structures and interactions in spatio-temporal patterns generated by one-dimensional deterministic cellular automaton rules. Phys. Rev. A 44, 866–875 (1991)
Durand-Lose, J.: Parallel transient time of one-dimensional sand pile. Theoret. Comp. Sci. 205, 183–193 (1998)
Hordijk, W., Shalizi, C.R., Crutchfield, J.P.: An upper bound on the products of particle interactions in cellular automata. Phys. D 154, 240–258 (2001)
Jakubowsky, M.H., Steiglitz, K., Squier, R.: Computing with solitons: a review and prospectus. In: [35], pp. 277–297 (2002)
Fischer, P.C.: Generation of primes by a one-dimensional real-time iterative array. J. ACM 12, 388–394 (1965)
Goto, E.: Ōtomaton ni kansuru pazuru [Puzzles on automata]. In: Kitagawa, T. (ed.) Jōhōkagaku eno michi [The Road to information science], pp. 67–92. Kyoristu Shuppan Publishing Co., Tokyo (1966)
Varshavsky, V.I., Marakhovsky, V.B., Peschansky, V.A.: Synchronization of interacting automata. Math. System Theory 4, 212–230 (1970)
Lindgren, K., Nordahl, M.G.: Universal computation in simple one-dimensional cellular automata. Complex Systems 4, 299–318 (1990)
Mazoyer, J.: On optimal solutions to the Firing squad synchronization problem. Theoret. Comp. Sci. 168, 367–404 (1996)
Durand-Lose, J.: Intrinsic universality of a 1-dimensional reversible cellular automaton. In: Reischuk, R., Morvan, M. (eds.) STACS 1997. LNCS, vol. 1200, pp. 439–450. Springer, Heidelberg (1997)
Durand-Lose, J.: Reversible space-time simulation of cellular automata. Theoret. Comp. Sci. 246, 117–129 (2000)
Delorme, M., Mazoyer, J.: Signals on cellular automata. In: [35], pp. 234–275 (2002)
Mazoyer, J., Terrier, V.: Signals in one-dimensional cellular automata. Theoret. Comp. Sci. 217, 53–80 (1999)
Durand-Lose, J.: Calculer géométriquement sur le plan – machines à signaux. Habilitation à diriger des recherches, École Doctorale STIC, Université de Nice-Sophia Antipolis (2003)
Jacopini, G., Sontacchi, G.: Reversible parallel computation: an evolving space-model. Theoret. Comp. Sci. 73, 1–46 (1990)
Durand-Lose, J.: Abstract geometrical computation for black hole computation (extended abstract). In: Margenstern, M. (ed.) MCU 2004. LNCS, vol. 3354, pp. 175–186. Springer, Heidelberg (2005)
Asarin, E., Maler, O.: Achilles and the Tortoise climbing up the arithmetical hierarchy. In: Thiagarajan, P.S. (ed.) FSTTCS 1995. LNCS, vol. 1026, pp. 471–483. Springer, Heidelberg (1995)
Bournez, O.: Some bounds on the computational power of piecewise constant derivative systems. In: Degano, P., Gorrieri, R., Marchetti-Spaccamela, A. (eds.) ICALP 1997. LNCS, vol. 1256, pp. 143–153. Springer, Heidelberg (1997)
Bournez, O.: Achilles and the Tortoise climbing up the hyper-arithmetical hierarchy. Theoret. Comp. Sci. 210, 21–71 (1999)
Weihrauch, K.: Introduction to computable analysis. Texts in Theoretical computer science. Springer, Berlin (2000)
Moore, C.: Recursion theory on the reals and continuous-time computation. Theoret. Comp. Sci. 162, 23–44 (1996)
Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and real computation. Springer, New York (1998)
Siegelmann, H.T., Sontag, E.D.: On the computational power of neural nets. J. Comput. System Sci. 50, 132–150 (1995)
Orponen, P.: A survey of continuous-time computation theory. In: Du, D.Z., Ko, K.J. (eds.) Advances in Algorithms, languages and complexity, pp. 209–224. Kluwer Academic Publisher, Dordrecht (1994)
Šíma, J., Orponen, P.: Computing with continuous-time Liapunov systems. In: STOC 2001, pp. 722–731. ACM Press, New York (2001)
Pour-El, M.B.: Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations and analog computers). Trans. Amer. Math. Soc. 199, 1–28 (1974)
Branicky, M.S.: Universal computation and other capabilities of hybrid and continuous dynamical systems. Theoret. Comp. Sci. 138, 67–100 (1995)
Minsky, M.: Finite and infinite machines. Prentice-Hall, Englewood Cliffs (1967)
Eberbach, E., Wegner, P.: Beyond Turing machines. Bull. EATCS 81, 279–304 (2003)
Hamkins, J.D., Lewis, A.: Infinite time turing machines. J. Symb. Log. 65, 567–604 (2000)
Hamkins, J.D.: Infinite time Turing machines: Supertask computation. Minds and Machines 12, 521–539 (2002)
Earman, J., Norton, J.D.: Forever is a day: supertasks in Pitowsky and Malament-Hogarth spacetimes. Philosophy of Science 60, 22–42 (1993)
Etesi, G., Nemeti, I.: Non-Turing computations via Malament-Hogarth space-times. Int. J. Theor. Phys. 41, 341–370 (2002)
Adamatzky, A. (ed.): Collision based computing. Springer, Heidelberg (2002)
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Durand-Lose, J. (2005). Abstract Geometrical Computation: Turing-Computing Ability and Undecidability. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds) New Computational Paradigms. CiE 2005. Lecture Notes in Computer Science, vol 3526. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11494645_14
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DOI: https://doi.org/10.1007/11494645_14
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