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Roots in the semiring of finite deterministic dynamical systems

François Doré, Kévin Perrot, Antonio E. Porreca, Sara Riva, Marius Rolland

TL;DR

This work studies polynomial equations over the semiring of finite discrete-time dynamical systems (FDDS) by leveraging the unroll construction to separate transient from cyclic behavior. It establishes that division by a connected quotient and extraction of connected $k$-th roots are polynomial-time tasks, enabling solutions to equations of the form $A X^k = B$ and, more generally, $A X = B$ under connectivity constraints. The results hinge on translating FDDS to unroll forests, proving a cancellation property, and solving a finite-tree forest division problem on depth-cut representations, with complexity bounds such as $O(m^9)$. These contributions advance the algorithmic toolkit for FDDS polynomial equations and provide a foundation for broader equation-solving pipelines in the semiring of finite dynamical systems.

Abstract

Finite discrete-time dynamical systems (FDDS) model phenomena that evolve deterministically in discrete time. It is possible to define sum and product operations on these systems (disjoint union and direct product, respectively) giving a commutative semiring. This algebraic structure led to several works employing polynomial equations to model hypotheses on phenomena modelled using FDDS. To solve these equations, algorithms for performing the division and computing $k$-th roots are needed. In this paper, we propose two polynomial algorithms for these tasks, under the condition that the result is a connected FDDS. This ultimately leads to an efficient solution to equations of the type $AX^k=B$ for connected $X$. These results are some of the important final steps for solving more general polynomial equations on FDDS.

Roots in the semiring of finite deterministic dynamical systems

TL;DR

This work studies polynomial equations over the semiring of finite discrete-time dynamical systems (FDDS) by leveraging the unroll construction to separate transient from cyclic behavior. It establishes that division by a connected quotient and extraction of connected -th roots are polynomial-time tasks, enabling solutions to equations of the form and, more generally, under connectivity constraints. The results hinge on translating FDDS to unroll forests, proving a cancellation property, and solving a finite-tree forest division problem on depth-cut representations, with complexity bounds such as . These contributions advance the algorithmic toolkit for FDDS polynomial equations and provide a foundation for broader equation-solving pipelines in the semiring of finite dynamical systems.

Abstract

Finite discrete-time dynamical systems (FDDS) model phenomena that evolve deterministically in discrete time. It is possible to define sum and product operations on these systems (disjoint union and direct product, respectively) giving a commutative semiring. This algebraic structure led to several works employing polynomial equations to model hypotheses on phenomena modelled using FDDS. To solve these equations, algorithms for performing the division and computing -th roots are needed. In this paper, we propose two polynomial algorithms for these tasks, under the condition that the result is a connected FDDS. This ultimately leads to an efficient solution to equations of the type for connected . These results are some of the important final steps for solving more general polynomial equations on FDDS.
Paper Structure (3 sections, 2 theorems, 3 equations, 1 figure)

This paper contains 3 sections, 2 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions
  3. Complexity of FDDS division with connected quotient

Key Result

lemma thmcounterlemma

Let $\mathbf{A}$, $\mathbf{X}$, and $\mathbf{B}$ be finite tree forests. Then, $\mathbf{A}\mathbf{X} = \mathbf{B}$ if and only if $\mathcal{R}(\mathbf{A})\mathcal{R}(\mathbf{X}) = \mathcal{R}(\mathbf{B})$.

Figures (1)

  • Figure 1: The unroll $\mathcal{U}(A)$ of a disconnected FDDS $A$. Only the first $6$ levels of $\mathcal{U}(A)$ are shown. Both the FDDS and its unroll have depth $2$.

Theorems & Definitions (6)

  • definition thmcounterdefinition: Unroll
  • definition thmcounterdefinition: Product of trees
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof