Garden-of-Eden states and fixed points of monotone dynamical systems

TL;DR

This paper investigates the properties of Garden-of-Eden states and fixed points in monotone dynamical systems, revealing bounds on limit cycle sizes and distinctions between sequential and parallel systems.

Contribution

It characterizes GoE states and fixed points in monotone SDS, establishes bounds on limit cycle sizes, and shows that some monotone PDS cannot be represented as SDS.

Findings

Maximum limit cycle size is less than binomial coefficient

Identifies a set of states that are either GoE or reach fixed points

Monotone PDS cannot always be expressed as monotone SDS

Abstract

In this paper we analyze Garden-of-Eden (GoE) states and fixed points of monotone, sequential dynamical systems (SDS). For any monotone SDS and fixed update schedule, we identify a particular set of states, each state being either a GoE state or reaching a fixed point, while both determining if a state is a GoE state and finding out all fixed points are generally hard. As a result, we show that the maximum size of their limit cycles is strictly less than ${n\choose \lfloor n/2 \rfloor}$. We connect these results to the Knaster-Tarski theorem and the LYM inequality. Finally, we establish that there exist monotone, parallel dynamical systems (PDS) that cannot be expressed as monotone SDS, despite the fact that the converse is always true.