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arXiv 1709.08359cs.DB
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矩阵查询语言的表达能力

On the expressive power of query languages for matrices

Robert Brijder, Floris Geerts, Jan Van den Bussche, Timmy Weerwag

摘要:

我们调查的表达能力的$\mathsf{MATLANG}$,一个形式语言的基础上常见的矩阵运算和线性代数的矩阵操作。该语言可以用矩阵求逆的操作$\mathsf{inv}$来扩展。在$\mathsf{MATLANG}+\mathsf{inv}$中,我们可以计算有向图的传递闭包,而我们表明,这是不可能的,没有反转。事实上,我们表明,基本的语言可以模拟在关系代数的算术运算,分组和求和。我们还考虑了对角化矩阵的操作$\mathsf{eigen}$,它被定义为使得对于同一特征值返回的不同特征向量正交。我们发现$\mathsf{inv}$可以在$\mathsf{MATLANG}+\mathsf{eigen}$中表达。我们提出了一个开放的问题,是否有布尔查询矩阵,或通用查询图,在$\mathsf{MATLANG} + \mathsf{eigen}$,但不能在$\mathsf{MATLANG}+\mathsf{inv}$中表达。$\mathsf{MATLANG} + \mathsf{eigen}$的评价问题是完整的复杂性类$\exists \mathbf{R}$。

英文摘要:

We investigate the expressive power of $\mathsf{MATLANG}$, a formal language for matrix manipulation based on common matrix operations and linear algebra. The language can be extended with the operation $\mathsf{inv}$ of inverting a matrix. In $\mathsf{MATLANG}+\mathsf{inv}$ we can compute the transitive closure of directed graphs, whereas we show that this is not possible without inversion. Indeed we show that the basic language can be simulated in the relational algebra with arithmetic operations, grouping, and summation. We also consider an operation $\mathsf{eigen}$ for diagonalizing a matrix, which is defined so that different eigenvectors returned for a same eigenvalue are orthogonal. We show that $\mathsf{inv}$ can be expressed in $\mathsf{MATLANG}+\mathsf{eigen}$. We put forward the open question whether there are boolean queries about matrices, or generic queries about graphs, expressible in $\mathsf{MATLANG} + \mathsf{eigen}$ but not in $\mathsf{MATLANG}+\mathsf{inv}$. The evaluation problem for $\mathsf{MATLANG} + \mathsf{eigen}$ is shown to be complete for the complexity class $\exists \mathbf{R}$.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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