Factorisation in the semiring of finite dynamical systems

TL;DR

This paper explores the algebraic structure of finite dynamical systems (FDSs), focusing on their factorisation, division, and root problems within a semiring framework, and introduces unrolling to analyze their properties.

Contribution

It provides new characterizations of cancellative FDSs, proves the uniqueness of roots, and identifies classes of FDS monoids with unique factorisation, using unrolling techniques.

Findings

FDSs are cancellative iff they have a fixpoint.

k-th roots of FDSs are unique when they exist.

Certain FDS monoids exhibit unique factorisation properties.

Abstract

Finite dynamical systems (FDSs) are commonly used to model systems with a finite number of states that evolve deterministically and at discrete time steps. Considered up to isomorphism, those correspond to functional graphs. As such, FDSs have a sum and product operation, which correspond to the direct sum and direct product of their respective graphs; the collection of FDSs endowed with these operations then forms a semiring. The algebraic structure of the product of FDSs is particularly interesting. For instance, an FDS can be factorised if and only if it is composed of two sub-systems running in parallel. In this work, we further the understanding of the factorisation, division, and root finding problems for FDSs. Firstly, an FDS $A$ is cancellative if one can divide by it unambiguously, i.e. $AX = AY$ implies $X = Y$. We prove that an FDS $A$ is cancellative if and only if it has a fixpoint. Secondly, we prove that if an FDS $A$ has a $k$-th root (i.e. $B$ such that $B^k = A$), then it is unique. Thirdly, unlike integers, the monoid of FDS product does not have unique factorisation into irreducibles. We instead exhibit a large class of monoids of FDSs with unique factorisation. To obtain our main results, we introduce the unrolling of an FDS, which can be viewed as a space-time expansion of the system. This allows us to work with (possibly infinite) trees, where the product is easier to handle than its counterpart for FDSs.