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arXiv · 2401.07106

Directed Regular and Context-Free Languages

Moses Ganardi·Irmak Saglam·Georg Zetzsche
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Abstract

We study the problem of deciding whether a given language is directed. A language $L$ is \emph{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 \emph{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 $AC^1$, 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.

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Moses Ganardi, Irmak Saglam, Georg Zetzsche. 2024-01-13. Directed Regular and Context-Free Languages. https://arxiv.org/abs/2401.07106

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