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Directed Regular and Context-Free Languages

Authors Moses Ganardi , Irmak Sağlam , Georg Zetzsche

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Author Details

Moses Ganardi
  • Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany
Irmak Sağlam
  • Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany
Georg Zetzsche
  • Max Planck Institute for Software Systems (MPI-SWS), Kaiserslautern, Germany

Funding

Funded by the European Union (ERC, FINABIS, 101077902). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.

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Moses Ganardi, Irmak Sağlam, and Georg Zetzsche. Directed Regular and Context-Free Languages. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 36:1-36:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.STACS.2024.36

Abstract

We study the problem of deciding whether a given language is directed. A language L is directed if every pair of words in L have a common (scattered) superword in L. Deciding directedness is a fundamental problem in connection with ideal decompositions of downward closed sets. Another motivation is that deciding whether two directed context-free languages have the same downward closures can be decided in polynomial time, whereas for general context-free languages, this problem is known to be coNEXP-complete. We show that the directedness problem for regular languages, given as NFAs, belongs to AC¹, and thus polynomial time. Moreover, it is NL-complete for fixed alphabet sizes. Furthermore, we show that for context-free languages, the directedness problem is PSPACE-complete.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Concurrency
  • Theory of computation → Formal languages and automata theory
Keywords
  • Subword
  • ideal
  • language
  • regular
  • context-free
  • equivalence
  • downward closure
  • compression

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