Growth of bilinear maps III: Decidability

TL;DR

This paper investigates the decidability of growth rates in bilinear maps, extending previous results, and provides new reductions and formulas to analyze the complexity and computability of these growth problems.

Contribution

It simplifies the reduction showing undecidability of growth rate problems and extends results to cases with positive initial vectors and multiple bilinear maps.

Findings

Checking if the growth rate λ ≤ 1 is undecidable.

Decidability remains undecidable even with positive initial vectors.

The problem of determining the existence of the limit is undecidable for nonnegative vectors.

Abstract

The following notion of growth rate can be seen as a generalization of joint spectral radius: Given a bilinear map $*:\mathbb R^d\times\mathbb R^d\to\mathbb R^d$ with nonnegative coefficients and a nonnegative vector $s\in\mathbb R^d$, denote by $g(n)$ the largest possible entry of a vector obtained by combining $n$ instances of $s$ using $n-1$ applications of $*$. Let $\lambda$ denote the growth rate $\limsup_{n\to\infty} \sqrt[n]{g(n)}$. Rosenfeld showed that the problem of checking $\lambda\le 1$ is undecidable by reducing the problem of joint spectral radius. In this article, we provide a simpler reduction using the observation that matrix multiplication is actually a bilinear map. Moreover, we extend the reduction to show that checking $\lambda\le 1$ is still undecidable even if $s$ is positive. If there is no restriction on the signs, we can also show that the problem of checking if the system can produce a zero vector is undecidable by reducing the problem of checking the mortality of a pair of matrices. This answers a question asked by Rosenfeld. Beside that, we confirm a remark of Rosenfeld that the problem does not become harder when we introduce more bilinear maps and more starting vectors. It is known that if the vector $s$ is strictly positive, then the limit superior $\lambda$ is actually a limit. However, we show that when $s$ is only nonnegative, the problem of checking the existence of the limit is undecidable. This also answers a question asked by Rosenfeld. We provide a formula for the growth rate $\lambda$ in terms of the diagonals of matrices corresponding to a special structure called ``linear pattern''. A condition is given so that the limit $\lambda$ exists. This actually provides a simpler proof for the existence of the limit $\lambda$ when $s>0$. An important corollary of the formula is the computability of the growth rate,....