Growth of bilinear maps III: decidability

Abstract

The following notion of growth rate can be seen as a generalization of joint spectral radius: Given a bilinear map $⁎:R^d\times R^d\to R^d$ with nonnegative coefficients and a nonnegative vector $s\in R^d$, denote by $g\left(n\right)$ the largest possible entry of a vector obtained by combining n instances of s using $n-1$ applications of ⁎. Let λ denote the growth rate ${\lim \sup }_{n\to \infty }\sqrt[n]{g\left(n\right)}$. 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 λ 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 λ in terms of the diagonals of matrices corresponding to a special structure called “linear pattern”. A condition is given so that the limit λ exists. This actually provides a simpler proof for the existence of the limit λ when $s>0$. An important corollary of the formula is the computability of the growth rate, which answers another question by Rosenfeld. Another corollary is that the problem of checking $\lambda \le 1$ is reducible to the problem of joint spectral radius, via the halting problem, i.e., the two problems are Turing equivalent. Also, we relate the finiteness property of a set of matrices to the notion “linear pattern” of a bilinear system.