Rational Index of Languages Defined by Grammars with Bounded Dimension of Parse Trees

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.