非交换电路的有理恒等式检验属于多项式空间
Rational Identity Testing for Noncommutative Circuits is in Polynomial Space
- Institute of Mathematical Sciences, Chennai, India（金奈数学科学研究所）
- Vishwakarma Institute of Technology, Pune, India（浦那维斯瓦卡尔玛理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明有理恒等式检验问题（RIT）对于有理电路属于PSPACE，通过简洁线性铅笔表示和NC秩计算，并推论非交换PIT也属于PSPACE。
AI中文摘要：
有理恒等式检验问题（RIT）询问一个输入的有理电路是否计算自由斜域中的零元素。对于有理公式，该问题已知属于确定性多项式时间。对于规模为$s$的有理电路，相关的线性铅笔的维数为$2^{O(s)}$，已知算法使用指数时间和指数空间。我们证明，在有理数域$Q$和有限域上的有理电路的RIT属于PSPACE。作为推论，对于次数无限制的非交换电路，非交换PIT也属于PSPACE。证明基于以下三个观察：分析Hrubes-Wigderson为输入有理公式构造线性铅笔的构造，我们获得一个简洁表示的线性铅笔$A$，其规模为$2^{O(s)}$，用于规模为$s$的输入有理电路。更精确地说，给定线性铅笔的索引$i$和$j$，我们可以在空间$s$和$i,j$长度的多项式内计算$A_{i,j}$。该算法本质上给出了通过归约获得的线性铅笔$A$的简洁表示。根据Chatterjee、Ghosh、Gurjar、Raj和Thierauf的最新定理，判定符号矩阵$\sum_i A_ix_i$是否具有满非交换秩属于NC。我们注意到他们的算法是对数空间均匀的NC，我们将其作为黑盒用于计算简洁表示的铅笔$A$的秩。最后，我们注意到一个民间模拟：如果一个问题由对数空间均匀的确定性布尔电路族以多对数深度解决，并且其输入不是写下来的，而是由一个多对数空间子程序呈现，该子程序返回任何请求的输入位，那么该电路可以在多对数空间内求值。因此，对于长度为$M = 2^{\Theta(s)}$的简洁呈现输入，模拟深度为$O(\log^{i}M)$的电路产生一个多项式空间算法。
英文摘要：
The rational identity testing problem, RIT, asks whether an input \emph{rational circuit}, computes the zero element of the free skew field. For rational \emph{formulas} the problem is known to be in deterministic polynomial time. For rational circuits of size $s$ the associated linear pencil has dimension $2^{O(s)}$, and the known algorithms use exponential time and exponential space. We show that RIT for rational circuits over $Q$ and finite fields is in PSPACE. As a consequence, noncommutative PIT for degree unrestricted noncommutative circuits is also in PSPACE. The proof is based on the following three observations Analyzing the Hrubes-Wigderson construction of a linear pencil for an input rational formula, we obtain a succinctly represented linear pencil $A$ of size $2^{O(s)}$ for the input rational \emph{circuit} of size $s$. More precisely, given indices $i$ and $j$ for the linear pencil we can compute $A_{i,j}$ in space polynomial in $s$ and length of $i,j$. This algorithm essentially gives a succinct presentation of the linear pencil $A$ obtained by the reduction. By the recent theorem of Chatterjee, Ghosh, Gurjar, Raj and Thierauf that deciding whether a symbolic matrix $\sum_i A_ix_i$ has full noncommutative rank is in NC. We note that their algorithm is logspace-uniform NC and we use it as a black-box for computing rank of succinctly represented pencil $A$. Finally, we note a folklore simulation: If a problem is solved by a logspace-uniform family of \emph{deterministic} Boolean circuits of polylogarithmic depth, and its input is not written down but is presented by a polylogspace subroutine that returns any requested input bit, then the circuit can be evaluated in polylogarithmic space. Hence, simulating a depth $O(\log^{i}M)$ circuit for succinctly presented inputs of length $M = 2^{Θ(s)}$ yields a polynomial space algorithm.
