License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2020.16
URN: urn:nbn:de:0030-drops-118770
URL: https://drops.dagstuhl.de/opus/volltexte/2020/11877/
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Gawrychowski, Paweł ; Lange, Martin ; Rampersad, Narad ; Shallit, Jeffrey ; Szykuła, Marek

Existential Length Universality

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Abstract

We study the following natural variation on the classical universality problem: given a language L(M) represented by M (e.g., a DFA/RE/NFA/PDA), does there exist an integer 𝓁 ≥ 0 such that Σ^𝓁 ⊆ L(M)? In the case of an NFA, we show that this problem is NEXPTIME-complete, and the smallest such 𝓁 can be doubly exponential in the number of states. This particular case was formulated as an open problem in 2009, and our solution uses a novel and involved construction. In the case of a PDA, we show that it is recursively unsolvable, while the smallest such 𝓁 is not bounded by any computable function of the number of states. In the case of a DFA, we show that the problem is NP-complete, and e^{√{n log n} (1+o(1))} is an asymptotically tight upper bound for the smallest such 𝓁, where n is the number of states. Finally, we prove that in all these cases, the problem becomes computationally easier when the length 𝓁 is also given in binary in the input: it is polynomially solvable for a DFA, PSPACE-complete for an NFA, and co-NEXPTIME-complete for a PDA.

BibTeX - Entry

@InProceedings{gawrychowski_et_al:LIPIcs:2020:11877,
  author =	{Paweł Gawrychowski and Martin Lange and Narad Rampersad and Jeffrey Shallit and Marek Szykuła},
  title =	{{Existential Length Universality}},
  booktitle =	{37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)},
  pages =	{16:1--16:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-140-5},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{154},
  editor =	{Christophe Paul and Markus Bl{\"a}ser},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/11877},
  URN =		{urn:nbn:de:0030-drops-118770},
  doi =		{10.4230/LIPIcs.STACS.2020.16},
  annote =	{Keywords: decision problem, deterministic automaton, nondeterministic automaton, pushdown automaton, regular expression, regular language, universality}
}

Keywords: decision problem, deterministic automaton, nondeterministic automaton, pushdown automaton, regular expression, regular language, universality
Collection: 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)
Issue Date: 2020
Date of publication: 04.03.2020


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