Growth of bilinear maps III: Decidability
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909724245491712 |
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| author | Bui, Vuong |
| author_facet | Bui, Vuong |
| contents | 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 $λ$ denote the growth rate $\limsup_{n\to\infty} \sqrt[n]{g(n)}$. Rosenfeld showed that the problem of checking $λ\le 1$ 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 $λ\le 1$ 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,.... |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_09850 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Growth of bilinear maps III: Decidability Bui, Vuong Combinatorics 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 $λ$ denote the growth rate $\limsup_{n\to\infty} \sqrt[n]{g(n)}$. Rosenfeld showed that the problem of checking $λ\le 1$ 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 $λ\le 1$ 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,.... |
| title | Growth of bilinear maps III: Decidability |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2201.09850 |