双线性映射的增长 III：可判定性
Growth of bilinear maps III: Decidability
AI总结：
研究双线性映射的增长率问题，证明检查增长率λ≤1的可判定性问题仍不可判定，即使初始向量非负，且引入更多双线性映射和初始向量不会使问题更难。
摘要：
以下关于增长率的概念可以看作是联合谱半径的推广：给定一个具有非负系数的双线性映射*：R^d×R^d→R^d和一个非负向量s∈R^d，令g(n)表示通过使用n次s和n-1次*运算得到的向量中最大的元素。令λ表示增长速率limsup_{n→∞}√[n]{g(n)}。Rosenfeld证明检查λ≤1的问题是不可判定的，通过将联合谱半径的问题进行归约。在本文中，我们提供了一个更简单的归约方法，利用观察到矩阵乘法实际上是一个双线性映射。此外，我们扩展了归约，证明即使s是正数，检查λ≤1的问题仍不可判定。如果不对符号进行限制，我们还可以通过将矩阵对的死亡性问题归约，证明检查系统能否产生零向量的问题是不可判定的，回答了Rosenfeld提出的问题。此外，我们确认了Rosenfeld的一个评论，即引入更多双线性映射和更多初始向量不会使问题更难。已知如果向量s严格正，则极限上界λ实际上是极限。然而，我们证明当s仅非负时，检查极限存在的问题仍不可判定。这同样回答了Rosenfeld提出的问题。我们提供了一个公式，将增长率λ表示为对应于称为“线性模式”的特殊结构的矩阵对角线的函数。给出了一个条件，使得极限λ存在。这实际上为s>0时极限λ的存在性提供了更简单的证明。公式的最重要推论是增长率的可计算性，……
英文摘要：
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,....
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