Gardens of Eden and Fixed Points in Sequential Dynamical Systems

• Published in: Discrete Mathematics and Theoretical Computer Science Vol. DMTCS Proceedings vol. AA,...; no. Proceedings; pp. 95 - 110
• Main Authors: Barrett, Christopher, Hunt, Marry, Marathe, Madhav, Ravi, S., Rosenkrantz, Daniel, Stearns, Richard, Tosic, Predrag
• Format: Journal Article Conference Proceeding
• Language: English
• Published: DMTCS 01.01.2001; Discrete Mathematics and Theoretical Computer Science; Discrete Mathematics & Theoretical Computer Science
• Series: DMTCS Proceedings
• Subjects: [info.info-cg] computer science [cs]/computational geometry [cs.cg]; [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]; [info.info-hc] computer science [cs]/human-computer interaction [cs.hc]; [info] computer science [cs]; [math.math-co] mathematics [math]/combinatorics [math.co]; cellular automata; Combinatorics; computational complexity; Computational Geometry; Computer Science; discrete dynamical systems; Discrete Mathematics; Human-Computer Interaction; Mathematics; Discrete Dynamical Systems; Cellular Automata; Computational Complexity
• ISSN: 1365-8050; 1462-7264; 1365-8050
• DOI: 10.46298/dmtcs.2294

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.