催化计算的寄存器程序的一种算子方法

中文摘要 AI 辅助

在一项开创性工作中，Buhrman等人（STOC 2014）引入了催化计算，并证明了均匀$TC^1$电路可在催化对数空间内计算，催化对数空间是指使用大小为$c$的额外催化带（其初始内容必须在计算结束时恢复）在空间$s$内可解的问题类别。他们证明的一个核心要素是寄存器程序模型。具体而言，他们构造了一个均匀的寄存器程序族，该程序使用$n$个寄存器和四次对$x$的访问来计算$x^n$。此后，确定计算给定次数的多项式所需的寄存器数量和输入访问次数已成为催化计算研究中的核心问题。一方面，我们证明了Buhrman等人的四次访问界限是最优的：每个计算次数大于三的多项式的被动输出寄存器程序至少需要四次输入访问，这与寄存器数量无关。另一方面，我们表明他们的寄存器界限并非最优。对于每个$t\geq2$和每个特征为$0$或大于$2t-1$的域$K$，我们构造了一个用于$x^{2t-1}$的寄存器程序，该程序具有四次输入访问和$t$个寄存器。我们的证明依赖于推导及其指数算子。这种方法将寄存器程序表示为一系列指数推导算子，将寄存器恢复简化为算子恒等式。最后，我们使用均匀的寄存器程序族来改进催化流算法和矩阵幂运算的寄存器程序的已知权衡。下界方法的推广和均匀寄存器程序族的构造是在ChatGPT 5.6的协助下开发的。

英文摘要

In a seminal work, Buhrman et al.\ (STOC 2014) introduced catalytic computation and proved that uniform $TC^1$ circuits are computable in catalytic logspace, the class of problems solvable in space $s$ with an additional catalytic tape of size $c$, a tape whose initial content must be restored at the end of the computation. A central ingredient of their proof is the register program model. Namely, they constructed a uniform family of register programs that computes $x^n$ using $n$ registers and four accesses to $x$. Since then, determining the number of registers and input accesses required to compute a polynomial of a given degree has become a central question in the study of catalytic computation. On one hand, we prove that the four-access bound of Buhrman et al.\ is optimal: every passive-output register program computing a polynomial of degree greater than three requires at least four input accesses, independently of the number of registers. On the other hand, we show that their register bound is not optimal. For every $t\geq2$ and every field $K$ of characteristic $0$ or greater than $2t-1$, we construct a register program for $x^{2t-1}$ with four input accesses and $t$ registers. Our proofs rely on derivations and their exponential operators. This approach represents a register program as a series of exponential derivation operators, reducing register restoration to an operator identity. Finally, we use the uniform family of register programs to improve known trade-offs for catalytic streaming algorithms and register programs for matrix powering. The generalization of the lower-bound methods and the construction of the uniform family of register programs were developed with assistance from ChatGPT 5.6.

发表机构

• Technion Israel Institute of Technology（以色列理工学院）

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