On a theorem of Fran\c{c}ois Robert

TL;DR

This paper extends François Robert's theorem by identifying a broad family of updating modes in Boolean models that also lead all states to a single fixed point when the regulatory graph lacks circuits.

Contribution

It introduces a large family of updating modes that generalize Robert's theorem for Boolean models without circuits.

Findings

All states converge to a fixed point under these new updating modes.

The family of updating modes includes several previously known modes as special cases.

The results deepen understanding of stability in Boolean dynamical systems.

Abstract

A well-known theorem by Fran\c{c}ois Robert expresses the degenerated character of a synchronous Boolean finite dynamical system, in the case where the associated regulatory graph does not contain any circuit: all states of the system go towards a single fixed point. We present a large family of updating modes of Boolean models with the same particularity.