DMTCS Proceedings, Discrete Models: Combinatorics, Computation, and Geometry, DM-CCG 2001

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DMTCS Conference vol AA (2001), pp. 95-110

DMTCS

Discrete Models: Combinatorics, Computation, and Geometry, DM-CCG 2001

Robert Cori and Jacques Mazoyer and Michel Morvan and Rémy Mosseri (eds.)

DMTCS Conference Volume AA (2001), pp. 95-110


author: Christopher L. Barrett, III Hunt, Madhav V. Marathe, S. S. Ravi, Daniel J. Rosenkrantz, Richard E. Stearns and Predrag T. Tosic
title: Gardens of Eden and Fixed Points in Sequential Dynamical Systems
keywords: Discrete Dynamical Systems, Cellular Automata, Computational Complexity
abstract: A class of finite discrete dynamical systems, called Sequential Dynamical Systems (SDSs), was proposed in [BMR99,BR99] as an abstract model of computer simulations. Here, we address some questions concerning two special types of the SDS configurations, namely Garden of Eden and Fixed Point configurations. A configuration C of an SDS is a Garden of Eden (GE) configuration if it cannot be reached from any configuration. A necessary and sufficient condition for the non-existence of GE configurations in SDSs whose state values are from a finite domain was provided in [MR00]. We show this condition is sufficient but not necessary for SDSs whose state values are drawn from an infinite domain. We also present results that relate the existence of GE configurations to other properties of an SDS. A configuration C of an SDS is a fixed point if the transition out of C is to C itself. The FIXED POINT EXISTENCE (or FPE) problem is to determine whether a given SDS has a fixed point. We show that the FPE problem is NP-complete even for some simple classes of SDSs (e.g., SDSs in which each local transition function is from the set {NAND, XNOR}). We also identify several classes of SDSs (e.g., SDSs with linear or monotone local transition functions) for which the FPE problem can be solved efficiently.
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reference: Christopher L. Barrett and III Hunt and Madhav V. Marathe and S. S. Ravi and Daniel J. Rosenkrantz and Richard E. Stearns and Predrag T. Tosic (2001), Gardens of Eden and Fixed Points in Sequential Dynamical Systems, in Discrete Models: Combinatorics, Computation, and Geometry, DM-CCG 2001, Robert Cori and Jacques Mazoyer and Michel Morvan and Rémy Mosseri (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AA, pp. 95-110
bibtex: For a corresponding BibTeX entry, please consider our BibTeX-file.
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