## 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 |
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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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