The rational index \(\rho _L\) of a language L is an integer function, where \(\rho _L(n)\) is the maximum length of the shortest string in \(L \cap R\), over all regular languages R recognized by n-state nondeterministic finite automata (NFA). This paper investigates the rational index of languages defined by grammars with bounded parse tree dimension: this is a numerical measure of the amount of branching in a tree (with trees in a linear grammar having dimension 1). For context-free grammars, a grammar with tree dimension bounded by d has rational index at most \(O(n^{2d})\), and it is known from the literature that there exists a grammar with rational index \(\Theta (n^{2d})\). In this paper, it is shown that for multi-component grammars with at most k components (k-MCFG) and with a tree dimension bounded by d, the rational index is at most \(O(n^{2kd})\), where the constant depends on the grammar, and there exists such a grammar with rational index \(\frac{k}{2^{kd^2 - kd -2k -1} \cdot (8k+1)^{2kd}} n^{2kd}\). Also, for the case of ordinary context-free grammars, a more precise lower bound \(\frac{1}{2^{d^2 + d - 3} 3^{2d}} n^{2d}\) is established.

语言的有理索引\(\rho_L\)L 是一个整数函数，其中 \(\rho _L(n)\) 是 k成立。\(\frac{1}{2^{d^2 + d - 3} 3^{2d} } n^{2d}\)。另外，对于普通上下文无关语法的情况，更精确的下界 \ (\frac{k}{2^{kd^2 - kd -2k -1} \cdot (8k+1)^{2kd}} n^{2kd}\)，其中常数取决于语法，存在这样一个有理数索引的语法\(O(n^{2kd})\) 为界，有理索引最多为 d-MCFG) 且树维度以 个组件的多组件语法（k。本文表明，对于最多具有 \(\Theta (n^{ 2d})\)，由文献可知存在有理索引的文法\(O( n^{2d})\) 为界的文法最多具有有理索引 d-状态非确定性有限自动机 (NFA)。本文研究了由具有有界解析树维度的语法定义的语言的理性索引：这是树中分支数量的数值度量（线性语法中的树具有维度 1）。对于上下文无关文法，树维数以 n 识别R，在所有常规语言中 \(L \cap R\)