Gardens of Eden and Fixed Points in Sequential Dynamical SystemsReport as inadecuate

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1 LANL - Los Alamos National Laboratory 2 CS - Department of Computer Science Albany 3 Department of Computer Science

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 thatthe 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.

Keywords : Computational Complexity Cellular Automata Discrete Dynamical Systems

Author: Christopher Barrett - Marry Hunt - Madhav Marathe - S. Ravi Daniel Rosenkrantz - Richard Stearns - Predrag Tosic -



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