Comparison of max-plus automata and joint spectral radius of tropical matrices

TL;DR

This paper explores the computational limits of max-plus automata and tropical matrices, proving certain quantities are uncomputable but approximable, and identifying cases where they are computable.

Contribution

It establishes the undecidability of comparing functions in max-plus automata and links this to the non-computability of joint spectral radius and ultimate rank.

Findings

Joint spectral radius and ultimate rank are not computable in general.

Approximation algorithms exist for the joint spectral radius.

Computability is possible in restricted cases.

Abstract

Weighted automata over the max-plus semiring S are closely related to finitely generated semigroups of matrices over S. In this paper, we use results in automata theory to study two quantities associated with sets of matrices: the joint spectral radius and the ultimate rank. We prove that these two quantities are not computable over the tropical semiring, i.e. there is no algorithm that takes as input a finite set of matrices M and provides as output the joint spectral radius (resp. the ultimate rank) of M. On the other hand, we prove that the joint spectral radius is nevertheless approximable and we exhibit restricted cases in which the joint spectral radius and the ultimate rank are computable. To reach this aim, we study the problem of comparing functions computed by weighted automata over the tropical semiring. This problem is known to be undecidable and we prove that it remains undecidable in some specific subclasses of automata.