A Tight Linear Bound on the Neighborhood of Inverse Cellular Automata
Reversible cellular automata (RCA) are models of massively parallel computation that preserve information. They consist of an array of identical finite state machines that change their states synchronously according to a local update rule. By selecting the update rule properly the system has been made information preserving, which means that any computation process can be traced back step-by-step using an inverse automaton. We investigate the maximum range in the array that a cell may need to see in order to determine its previous state. We provide a tight upper bound on this inverse neighborhood size in the one-dimensional case: we prove that in a RCA with n states the inverse neighborhood is not wider than n–1, when the neighborhood in the forward direction consists of two consecutive cells. Examples are known where range n–1 is needed, so the bound is tight. If the forward neighborhood consists of m consecutive cells then the same technique provides the upper bound n m − 1–1 for the inverse direction.
KeywordsCellular Automaton Cellular Automaton Neighborhood Range Consecutive Cell Minimal Neighborhood
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- 3.Kari, J.: On the Inverse Neighborhood of Reversible Cellular Automata. In: Rosenberg, G., Salomaa, A. (eds.) Lindenmayer Systems, Impact in Theoretical Computer Science, Computer Graphics and Developmental Biology, pp. 477–495. Springer, Heidenberg (1989)Google Scholar
- 6.Moore, E.F.: Machine Models of Self-reproduction. In: Proceedings of the Symposium in Applied Mathematics, vol. 14, pp. 17–33 (1962)Google Scholar
- 7.Morita, K., Harao, M.: Computation Universality of one-dimensional reversible (injective) cellular automata. IEICE Transactions E72, 758–762 (1989)Google Scholar
- 8.Myhill, J.: The Converse to Moore’s Garden-of-Eden Theorem. In: Proceedings of the American Mathematical Society, vol. 14, pp. 685–686 (1963)Google Scholar
- 12.Wolfram, S.: A New Kind of Science. Wolfram Media (2002)Google Scholar