On the Size of Finite Rational Matrix Semigroups

TL;DR

This paper establishes an upper bound on the length of products needed to generate any matrix in a finite rational matrix semigroup, enabling efficient algorithms for automata finiteness and reachability problems.

Contribution

It provides a new exponential bound on matrix product length in finite rational matrix semigroups, leading to elementary algorithms for related decision problems.

Findings

Bound on product length is at most exponential in n^2 log n.

Algorithms for automata finiteness are now elementary.

Decidability of reachability in affine integer systems with finite monoids.

Abstract

Let $n$ be a positive integer and $\mathcal M$ a set of rational $n \times n$-matrices such that $\mathcal M$ generates a finite multiplicative semigroup. We show that any matrix in the semigroup is a product of matrices in $\mathcal M$ whose length is at most $2^{n (2 n + 3)} g(n)^{n+1} \in 2^{O(n^2 \log n)}$, where $g(n)$ is the maximum order of finite groups over rational $n \times n$-matrices. This result implies algorithms with an elementary running time for deciding finiteness of weighted automata over the rationals and for deciding reachability in affine integer vector addition systems with states with the finite monoid property.