Shortest paths in one-counter systems

Chistikov, Dmitry; Czerwiński, Wojciech; Hofman, Piotr; Pilipczuk, Michał; Wehar, Michael

doi:10.23638/lmcs-15(1:19)2019

Cited by 3 publications

(1 citation statement)

References 21 publications

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“…This can be proved using the standard hill-cutting argument (see, e.g., [20, Lemma 5]; cf. [26, Proposition 7] as well as [9] and references therein): in scenario (i) one can apply it to reduce the maximum value of the counter whilst retaining the minimum value; and in scenario (ii) one can increase the minimum value whilst retaining the maximum one.…”

Section: Lemma 67mentioning

confidence: 99%

Rational subsets of Baumslag-Solitar groups

Cadilhac,

Chistikov,

Zetzsche

2020

Preprint

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We consider the rational subset membership problem for Baumslag-Solitar groups. These groups form a prominent class in the area of algorithmic group theory, and they were recently identified as an obstacle for understanding the rational subsets of GL(2, Q).We show that rational subset membership for Baumslag-Solitar groups BS(1, q) with q ≥ 2 is decidable and PSPACE-complete. To this end, we introduce a word representation of the elements of BS(1, q): their pointed expansion (PE), an annotated q-ary expansion. Seeing subsets of BS(1, q) as word languages, this leads to a natural notion of PE-regular subsets of BS(1, q): these are the subsets of BS(1, q) whose sets of PE are regular languages. Our proof shows that every rational subset of BS(1, q) is PE-regular.Since the class of PE-regular subsets of BS(1, q) is well-equipped with closure properties, we obtain further applications of these results. Our results imply that (i) emptiness of Boolean combinations of rational subsets is decidable, (ii) membership to each fixed rational subset of BS(1, q) is decidable in logarithmic space, and (iii) it is decidable whether a given rational subset is recognizable. In particular, it is decidable whether a given finitely generated subgroup of BS(1, q) has finite index. ACM Subject ClassificationTheory of computation → Problems, reductions and completeness; Theory of computation → Formal languages and automata theory; Theory of computation → Models of computation Keywords and phrases Rational subsets, Baumslag-Solitar groups, decidability, regular languages, pointed expansion Acknowledgements The research for this work was carried out in part at the Autobóz Research Camp in 2019 in Firbush, Scotland. The authors are very grateful to Piotr Hofman for comments that led to important insights and to Michael Blondin as well as to all other participants for discussions.

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Section: Lemma 67mentioning

confidence: 99%

Rational subsets of Baumslag-Solitar groups

Cadilhac,

Chistikov,

Zetzsche

2020

Preprint

Self Cite

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Rational Index of Languages with Bounded Dimension of Parse Trees

Shemetova

Okhotin

Grigorev

2022

Developments in Language Theory

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

Shemetova,

Okhotin,

Grigorev

2023

Theory Comput Syst

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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 ).

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Shortest paths in one-counter systems

Cited by 3 publications

References 21 publications

Rational subsets of Baumslag-Solitar groups

Rational subsets of Baumslag-Solitar groups

Rational Index of Languages with Bounded Dimension of Parse Trees

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

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