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

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Abstract

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.

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Acknowledgements

The authors would like to express their gratitude to the anonymous reviewer for explaining the previous work on closely related notions done by the French school to the authors, for attracting the authors’ attention to the shortcomings of the original submission, and for most helpful suggestions on the presentation of the results.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, St. Petersburg State University, 14th Line V. O., 29, Saint Petersburg, 199178, Russia

    Ekaterina Shemetova & Alexander Okhotin

  2. Department of Mathematics and Mechanics, St. Petersburg State University, 7/9 Universitetskaya nab, Saint Petersburg, 199034, Russia

    Semyon Grigorev

  3. St. Petersburg Academic University, ul. Khlopina, 8, Saint Petersburg, 194021, Russia

    Ekaterina Shemetova

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  1. Ekaterina Shemetova
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E. Shemetova, A. Okhotin, S. Grigorev wrote the main manuscript text and prepared figures 1-7. All authors reviewed the manuscript.

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Shemetova, E., Okhotin, A. & Grigorev, S. Rational Index of Languages Defined by Grammars with Bounded Dimension of Parse Trees. Theory Comput Syst (2023). https://doi.org/10.1007/s00224-023-10159-3

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