构造公式的小树文法和小回路
Constructing small tree grammars and small circuits for formulas
摘要：
它表明，每一个树的大小$n$在一个固定的一组$σ$不同的排名符号可以分解（在线性时间以及在对数空间）成$O\big(\frac{n}{\log_σn}\big) = O\big(\frac{n \log σ}{\log n}\big)$许多层次定义的片。形式上，这种分层分解具有大小为$O\big(\frac{n}{\log_σn}\big)$的直线线性上下文无关树文法的形式，其可以用作输入树的压缩表示。这推广了字符串的类似结果。以前的基于语法的树压缩器没有分析计算语法的最坏情况大小，除了Bille等人的顶部dag，其中，只有较弱的上界$O\big(\frac{n}{\log_σ^{0.19} n}\big)$（最近由Hübschle-Schneider和Raman改进为$O\big(\frac{n \cdot \log \log_σn}{\log_σn}\big)$）的未排名和未标记的树已经得到。本文的主要结果表明，每一个$n$大小的算术公式（其中只有$m \leq n$不同的变量出现）都可以变换成（在线性时间和对数空间中）一个大小为$O\big(\frac{n \cdot \log m}{\log n}\big)$和深度为$O(\log n)$的算术电路。这改进了Brent在1974年的一个经典结果，根据这个结果，一个长度为$n$的算术公式可以转化为一个长度为$O(n)$的对数深度回路.
英文摘要：
It is shown that every tree of size $n$ over a fixed set of $σ$ different ranked symbols can be decomposed (in linear time as well as in logspace) into $O\big(\frac{n}{\log_σn}\big) = O\big(\frac{n \log σ}{\log n}\big)$ many hierarchically defined pieces. Formally, such a hierarchical decomposition has the form of a straight-line linear context-free tree grammar of size $O\big(\frac{n}{\log_σn}\big)$, which can be used as a compressed representation of the input tree. This generalizes an analogous result for strings. Previous grammar-based tree compressors were not analyzed for the worst-case size of the computed grammar, except for the top dag of Bille et al., for which only the weaker upper bound of $O\big(\frac{n}{\log_σ^{0.19} n}\big)$ (which was very recently improved to $O\big(\frac{n \cdot \log \log_σn}{\log_σn}\big)$ by Hübschle-Schneider and Raman) for unranked and unlabelled trees has been derived. The main result is used to show that every arithmetical formula of size $n$, in which only $m \leq n$ different variables occur, can be transformed (in linear time as well as in logspace) into an arithmetical circuit of size $O\big(\frac{n \cdot \log m}{\log n}\big)$ and depth $O(\log n)$. This refines a classical result of Brent from 1974, according to which an arithmetical formula of size $n$ can be transformed into a logarithmic depth circuit of size $O(n)$.
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。
