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

Abstract

The rational index ρL of a language L is an integer function, where ρL(n) is the maximum length of the shortest string in L ∩ 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 tree dimension, and shows that it is polynomial in n. More precisely, it is proved that for a context-free grammar with tree dimension bounded by d, its rational index is at most O(n 2d ), and that this estimation is asymptotically tight, as there exists a grammar with rational index Θ(n 2d ). For a multi-component grammar of rank k and with tree dimension bounded by d, the rational index is bounded by O(n 2kd ), and there exists a grammar with rational index Ω(n 2kd ).