Abstract
The chapter studies relations between billiard ball model, reversible cellular automata, conservative and reversible logics and Turing machines. At first we introduce block cellular automata and consider the automata reversibility and simulation dependencies between the block cellular automata and classical cellular automata. We prove that there exists a universal, i.e. simulating any Turing machine, block cellular automaton with eleven states, which is geometrically minimal. Basics of the billiard ball model and presentation of an information in the model are discussed then. We demonstrate how to implement ball movement, reflection of a signal, delays and cycles, collision of signals in configurations of the cellular automaton with Margolus neighborhood. Realizations of Fredkin gate and NOT gate with dual signal encoding are offered. The rest of the chapter deals with a Turing and an intrinsic universality, and uncomputable properties of the billiard ball model. The Turing universality is proved via simulation of a two-counter automaton, which itself is Turing universal. We demonstrate that the billiard ball model is intrinsically universal, or complete, in a class of reversible cellular automata, i.e. the model can simulate any reversible automaton over finite or infinite configurations. A novel notion of space-time simulation, that employs whole space-time diagrams of automaton evolution, is brought up. It is proved that the billiard ball model is also able to space-time simulate any (ir)reversible cellular automaton. Since the billiard ball model possesses the Turing computation power we can project a Turing machine’s halting problem to development of cellular automaton simulating the billiard ball model. Namely, we uncover a connection between undecidability of computation and high unpredictability of configurations of the billiard ball model.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Albert J. and Culik II K. A simple universal cellular automaton and its one-way and totalistic versionComplex Systems 1 (1987) 1–16
Bennett C.H. Logical reversibility of computation IBM Journal of Research and Development 6 (1973) 525–532
Burks AEssays on Cellular Automata(Univ. of Illinois Press, 1970).
Durand-Lose J. Reversible cellular automaton able to simulate any other reversible one using partitioning automata, In: R. Baeza-Yates and E. Goles and P. Poblete (Eds.), LATIN ’95, Lecture Notes in Computer Sciences 911 (1995) 230–244
Durand-Lose J. Intrinsic universality of a 1-dimensional reversible cellular automaton, In: R. Reischuk and M. Morvan (Eds.), STAGS ’97, Lecture Notes in Computer Sciences 1200 (1997) 439–450.
Durand-Lose J. Reversible space-time simulation of cellular automata Theoretical Computer Science 246 (2000) 117–129
Durand-Lose J. Representing reversible cellular automata with reversible block cellular automata In: R. Cori, J. Mazoyer, M. Morvan, and R. Mosery (Eds.) Discrete models, combinatorics, computation and geometry (DM-CCG’01), volume AA. Discrete Mathematics and Theoretical Computer Science, 2001
Fredkin E. and Toffoli T. Conservative logic International Journal of Theoretical Physics 21 (1982) 219–253
Hedlund G.A. Endomorphism and automorphism of the shift dynamical system Mathematical System Theory 3 (1969) 320–375
Kari J. Reversibility of 2D cellular automata is undecidable Physica D 45 (1990) 379–385
Kari J. Representation of reversible cellular automata with block permutations Mathematical System Theory 29 (1996) 47–61
Lecerf Y. Machines de Turing r¨¦versibles. R¨¦cursive insolubilit¨¦ en n E IN de l’¨¦quation u = 6r ou O est un “isomorphism de codes” Comptes Rendus de L’Academie Francaise des Sciences 257 (1963) 2597–2600
Margolus N. Physics-like models of computation Physica D 10 (1984) 81–95
Minsky M. Finite and Infinite Machines (Prentice Hall, 1967).
Morita K. and Harao M. Computation universality of one-dimensional reversible (injective) cellular automata Transactions of the IEICE E 72 (1989) 758–762
Morita K. A simple construction method of a reversible finite automaton out of Fredkin gates, and its related problem Transactions of the IEICE E 73 (1990) 978–984
Morita K. Any irreversible cellular automaton can be simulated by a reversible one having the same dimension (on finite configurations) Technical Report of the IEICE, Comp. 92–45 (1992) 55–64
Morita K. Computation-universality of one-dimensional one-way reversible cellular automata Information Processing Letters 42 (1992) 325–329
Morita K. Reversible simulation of one-dimensional irreversible cellular automata Theoretical Computer Science 148 (1995) 157–163
Morita K.. and Ueno S. Computation-universal models of two-dimensional 16-state reversible automata IEICE Transactions on Informations and Systems E75-D (1992) 141–147
Morita K. and Ueno S. Parallel generation and parsing of array languages using reversible cellular automata Lecture Notes in Computer Science 654 (1992) 213–230
Richardson D. Tessellations with local transformations Journal of Computer and System Sciences 6 (1972) 373–388
Rogozhin Yu.V. Seven universal Turing machines In Systems and Theoretical Programming, number 63, Matematicheskije Issledovanija (Academia Nauk Moldayskoi SSR) (1992) 76–90; in Russian
Rogozhin Yu.V. Small universal Turing machines Theoretical Computer Science 168 (1996) 215–240
Sipser M. Introduction to the Theory of Computation (PWS Publishing Co., Boston, Massachusetts, 1997)
Smith III A.R. Simple computation-universal cellular spaces Journal of the Association for Computing Machinery 18 (1971) 339–353
Sutner K. On the complexity of finite cellular automata Journal of Computer and System Sciences 50 (1995) 87–97
Toffoli T. and Margolus N. Cellular Automata Machine A New Environment for Modeling (MIT Press, Cambridge, MA, 1987)
Toffoli T. and Margolus N. Invertible cellular automata: A review Physica D 45 (1990) 229–253
Toffoli T. Computation and construction universality of reversible cellular automata Journal of Computer and System Sciences 15 (1977) 213–231
Wolfram S. Universality and complexity in cellular automata Physica D 10 (1984) 1–35
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag London
About this chapter
Cite this chapter
Durand-Lose, J. (2002). Computing Inside the Billiard Ball Model. In: Adamatzky, A. (eds) Collision-Based Computing. Springer, London. https://doi.org/10.1007/978-1-4471-0129-1_6
Download citation
DOI: https://doi.org/10.1007/978-1-4471-0129-1_6
Publisher Name: Springer, London
Print ISBN: 978-1-85233-540-3
Online ISBN: 978-1-4471-0129-1
eBook Packages: Springer Book Archive