用快速矩阵乘法分析线性上下文无关重写系统
Parsing Linear Context-Free Rewriting Systems with Fast Matrix Multiplication
摘要：
我们描述了一个矩阵乘法识别算法的一个子集的二进制线性上下文无关重写系统（LCFRS）与运行时间$O(n^{ωd})$其中$M(m) = O(m^ω)$是运行时间为$m \times m$矩阵乘法和$d$是“接触秩”的LCFRS -最大数量的组合和非组合点出现在语法规则。我们还表明，该算法可以作为一个子程序，得到一个识别算法的一般二进制LCFRS与运行时间$O(n^{ωd + 1})$。目前最著名的$ω$小于$2.38$。我们的结果提供了另一个证明的最有名的结果，分析温和的上下文敏感的形式主义，如组合范畴文法，头文法，线性索引文法，和树邻接文法，这可以在时间$O(n^{4.76})$。它还表明，反转转换语法可以在时间$O(n^{5.76})$中解析。此外，二进制LCFRS包含许多其他形式主义和类型的语法，其中一些我们还提高了解析的渐近复杂性。
英文摘要：
We describe a matrix multiplication recognition algorithm for a subset of binary linear context-free rewriting systems (LCFRS) with running time $O(n^{ωd})$ where $M(m) = O(m^ω)$ is the running time for $m \times m$ matrix multiplication and $d$ is the "contact rank" of the LCFRS -- the maximal number of combination and non-combination points that appear in the grammar rules. We also show that this algorithm can be used as a subroutine to get a recognition algorithm for general binary LCFRS with running time $O(n^{ωd + 1})$. The currently best known $ω$ is smaller than $2.38$. Our result provides another proof for the best known result for parsing mildly context sensitive formalisms such as combinatory categorial grammars, head grammars, linear indexed grammars, and tree adjoining grammars, which can be parsed in time $O(n^{4.76})$. It also shows that inversion transduction grammars can be parsed in time $O(n^{5.76})$. In addition, binary LCFRS subsumes many other formalisms and types of grammars, for some of which we also improve the asymptotic complexity of parsing.
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