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Subsequence Matching and Analysis Problems for Formal Languages

Authors Szilárd Zsolt Fazekas , Tore Koß , Florin Manea , Robert Mercaş , Timo Specht

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Author Details

Szilárd Zsolt Fazekas
  • Akita University, Japan
Tore Koß
  • University of Göttingen, Germany
Florin Manea
  • University of Göttingen, Germany
Robert Mercaş
  • Loughborough University, UK
Timo Specht
  • University of Göttingen, Germany

Funding

  • Fazekas, Szilárd Zsolt: Supported by the JSPS Kakenhi Grant Number 23K10976.
  • Manea, Florin: Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) in the framework of the Heisenberg Programme project number 466789228.
  • Mercaş, Robert: Supported by a Return Fellowship from the Alexander von Humboldt Foundation for a research stay at University of Göttingen, Germany.

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Szilárd Zsolt Fazekas, Tore Koß, Florin Manea, Robert Mercaş, and Timo Specht. Subsequence Matching and Analysis Problems for Formal Languages. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 28:1-28:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.28

Abstract

In this paper, we study a series of algorithmic problems related to the subsequences occurring in the strings of a given language, under the assumption that this language is succinctly represented by a grammar generating it, or an automaton accepting it. In particular, we focus on the following problems: Given a string w and a language L, does there exist a word of L which has w as subsequence? Do all words of L have w as a subsequence? Given an integer k alongside L, does there exist a word of L which has all strings of length k, over the alphabet of L, as subsequences? Do all words of L have all strings of length k as subsequences? For the last two problems, efficient algorithms were already presented in [Adamson et al., ISAAC 2023] for the case when L is a regular language, and efficient solutions can be easily obtained for the first two problems. We extend that work as follows: we give sufficient conditions on the class of input-languages, under which these problems are decidable; we provide efficient algorithms for all these problems in the case when the input language is context-free; we show that all problems are undecidable for context-sensitive languages. Finally, we provide a series of initial results related to a class of languages that strictly includes the regular languages and is strictly included in the class of context-sensitive languages, but is incomparable to the of class context-free languages; these results deviate significantly from those reported for language-classes from the Chomsky hierarchy.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Stringology
  • String Combinatorics
  • Subsequence
  • Formal Languages
  • Context-Free Languages
  • Context-Sensitive Languages

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