A Prime Decomposition of Probabilistic Automata

TL;DR

This paper introduces a prime decomposition framework for probabilistic automata based on Krohn-Rhodes theorem, linking local structure to global properties through holonomy decomposition and finite group representations.

Contribution

It extends the Krohn-Rhodes theorem to probabilistic automata, providing a new structural analysis method and connecting local and global automaton properties.

Findings

Prime decomposition derived from Krohn-Rhodes theorem

Representation theory linked to finite groups in holonomy decomposition

Framework for studying global structure of probabilistic automata

Abstract

A definition of a probabilistic automaton is formulated in which its prime decomposition follows as a direct consequence of Krohn-Rhodes theorem. We first characterize the local structure of probabilistic automata. The prime decomposition is presented as a framework to study the global structure of probabilistic automata. We prove that the representation theory of a probabilistic automaton is determined by that of the finite groups in its holonomy decomposition.