1. Rational Index of Languages Defined by Grammars with Bounded Dimension of Parse Trees.
- Author
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Shemetova, Ekaterina, Okhotin, Alexander, and Grigorev, Semyon
- Subjects
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GRAMMAR , *FINITE state machines , *PARSING (Grammar) , *TREES , *TREE branches - Abstract
The rational index ρ L of a language L is an integer function, where ρ L (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 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 2 d) , and it is known from the literature that there exists a grammar with rational index Θ (n 2 d) . 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 2 k d) , where the constant depends on the grammar, and there exists such a grammar with rational index k 2 k d 2 - k d - 2 k - 1 · (8 k + 1) 2 k d n 2 k d . Also, for the case of ordinary context-free grammars, a more precise lower bound 1 2 d 2 + d - 3 3 2 d n 2 d is established. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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