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On a theorem of François Robert

Brigitte Mossé, Sasha Pignol, Elisabeth Remy

TL;DR

The paper investigates how the topology of regulatory graphs governs the dynamics of Boolean finite systems, focusing on the circuit-free case. It recalls François Robert's theorem, which guarantees that a circuit-free regulatory graph yields a degenerate synchronous dynamics converging to a unique fixed point, and then extends this insight to a broad family of updating modes. The authors provide a unified demonstration—via Boolean distance, a transposed adjacency matrix $B(S)$, and topological sorts—that, under the circuit-free condition, these updating schemes are simple and drive any initial state to the fixed point in at most $n$ steps (and $n+1$ in the Most Permissive setting). They also introduce a $\mathcal{P}$-asynchronous framework, proving similar convergence properties and showing pruning cannot generate multi-state cycles. The results offer a robust link between graph topology and dynamical behavior across updating schemes, with implications for modeling biological networks and potential extensions to multi-valued and differential systems.

Abstract

A well-known theorem by François 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.

On a theorem of François Robert

TL;DR

The paper investigates how the topology of regulatory graphs governs the dynamics of Boolean finite systems, focusing on the circuit-free case. It recalls François Robert's theorem, which guarantees that a circuit-free regulatory graph yields a degenerate synchronous dynamics converging to a unique fixed point, and then extends this insight to a broad family of updating modes. The authors provide a unified demonstration—via Boolean distance, a transposed adjacency matrix , and topological sorts—that, under the circuit-free condition, these updating schemes are simple and drive any initial state to the fixed point in at most steps (and in the Most Permissive setting). They also introduce a -asynchronous framework, proving similar convergence properties and showing pruning cannot generate multi-state cycles. The results offer a robust link between graph topology and dynamical behavior across updating schemes, with implications for modeling biological networks and potential extensions to multi-valued and differential systems.

Abstract

A well-known theorem by François 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.

Paper Structure

This paper contains 7 sections, 7 theorems, 1 figure.

Table of Contents

  1. Abstract
  2. Introduction
  3. Basics on Boolean finite dynamical systems
  4. Robert's theorem
  5. Asynchronous version of Robert's theorem
  6. Extension to a family of updating modes
  7. Conclusion

Key Result

Theorem 1

Let $S$ be a Boolean model on $X = \{0,1\}^n$. If $\mathcal{RG}(S)$ does not contain any circuit, then the synchronous dynamics defined by $S$ is simple. Moreover, for all $x\in X$, the sequence $(S^k(x))_{k\in\mathbb{N}}$ converges in at most $n$ iterations to the single fixed point of $S$.

Figures (1)

  • Figure 1: Example of Boolean model with simple synchronous STG and not simple asynchronous STG

Theorems & Definitions (16)

  • Theorem 1: Robert's theorem
  • proof
  • Corollary 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Proposition 1
  • proof
  • Theorem 3
  • proof
  • ...and 6 more