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Formal Grammars for Turn-Bounded Deterministic Context-Free Languages

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Theoretical Aspects of Computing – ICTAC 2022 (ICTAC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13572))

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Abstract

As its computation proceeds, a one-way deterministic pushdown automaton (or a 1dpda) changes the height (or volume) of its stack (or pushdown store) by switching between a non-decreasing phase and a decreasing phase. Such a changing of the two different phases is referred to as a “turn” of the machine. Languages that are recognized by k-turn 1dpda’s for each fixed number k are succinctly called k-turn deterministic context-free (dcf) languages. We first discuss closure properties of k-turn dcf languages. Such closure properties help us prove that finite-turn dcf languages are precisely characterized by a deterministic analogue of ultralinear grammars, called LR(1)-ultralinear grammars. In particular, when a 1dpda is further required to empty its stack at the beginning of each turn, the associated languages are characterized in terms of LR(1)-metalinear grammars. As an immediate application of these grammar characterizations, we prove a structural lemma, known as a pumping lemma, for finite-turn dcf languages.

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Notes

  1. 1.

    A tape is said to be read-once if its tape head never moves back to the left and, whenever it scans a non-blank symbol, it must move to the right.

  2. 2.

    Slightly differing from [8], in this paper, we define the number of “turns” by counting the times when a stack height changes from a nondecreasing phase to a decreasing phase (see [6, 13]). The details will be given in Sect. 2.2.

  3. 3.

    These languages coincide with quasi-rational languages and are also characterized by context-free grammars of finite “indices”, where the index of a derivation refers to the maximum number of occurrences of variables in the sentential forms used in the derivation (see, e.g., [1]).

  4. 4.

    The use of endmarkers and a halting set pair \((Q_{acc},Q_{rej})\) does not change the computational power of 1dpda’s. In particular, the right endmarker helps a machine empty its stack at the end of its computation.

  5. 5.

    This grammar is also categorized to an expansive grammar (see, e.g., [1]).

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Authors and Affiliations

  1. Faculty of Engineering, University of Fukui, 3-9-1, Bunkyo, Fukui, 910-8507, Japan

    Tomoyuki Yamakami

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  1. Tomoyuki Yamakami
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Correspondence to Tomoyuki Yamakami .

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  1. TU München, München, Germany

    Helmut Seidl

  2. Southwest University, Chongqing, China

    Zhiming Liu

  3. NASA Ames Research Center, Moffett Field, CA, USA

    Corina S. Pasareanu

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Yamakami, T. (2022). Formal Grammars for Turn-Bounded Deterministic Context-Free Languages. In: Seidl, H., Liu, Z., Pasareanu, C.S. (eds) Theoretical Aspects of Computing – ICTAC 2022. ICTAC 2022. Lecture Notes in Computer Science, vol 13572. Springer, Cham. https://doi.org/10.1007/978-3-031-17715-6_27

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  • DOI: https://doi.org/10.1007/978-3-031-17715-6_27

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