International audience
A class of finite discrete dynamical systems, called <b>Sequential Dynamical Systems</b> (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 <b>NP</b>-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.
We study herewith some aspects related to predictability of the long-term global behavior of the star topology based infrastructures when all the nodes, including the central node, are assumed to function reliably, faultlessly and synchronously. In particular, we use the nonlinear complex systems concepts and methodology, coupled with those of computational complexity, to show that, simple as the star topology is, determining and predicting the longterm global behavior of the star-based infrastructures are computationally challenging tasks. More formally, determining various configuration space properties of the appropriate star network abstractions is shown to be hard in general. We particularly focus herein on the computational (in)tractability of counting the "fixed point" configurations of a class of formal discrete dynamical systems defined over the star interconnection topology.
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