Linear sequential dynamical systems, incidence algebras, and M\"{o}bius functions

TL;DR

This paper develops explicit formulas for linear sequential dynamical systems (SDS), connects them with incidence algebras and Möbius functions of posets, and introduces a cut theorem for these functions.

Contribution

It provides a closed-form expression for linear SDS as synchronous systems and links SDS with incidence algebras and Möbius functions of posets.

Findings

Explicit formula for linear SDS as a synchronous system

Representation of any linear system as a linear SDS

Möbius function computation via SDS on Hasse diagrams

Abstract

A sequential dynamical system (SDS) consists of a graph, a set of local functions and an update schedule. A linear sequential dynamical system is an SDS whose local functions are linear. In this paper, we derive an explicit closed formula for any linear SDS as a synchronous dynamical system. We also show constructively, that any synchronous linear system can be expressed as a linear SDS, i.e. it can be written as a product of linear local functions. Furthermore, we study the connection between linear SDS and the incidence algebras of partially ordered sets (posets). Specifically, we show that the M\"{o}bius function of any poset can be computed via an SDS, whose graph is induced by the Hasse diagram of the poset. Finally, we prove a cut theorem for the M\"{o}bius functions of posets with respect to certain chain decompositions.