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DOI: 10.4230/LIPIcs.MFCS.2016.61
URN: urn:nbn:de:0030-drops-64738
URL: http://drops.dagstuhl.de/opus/volltexte/2016/6473/
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Krötzsch, Markus ; Masopust, Tomás ; Thomazo, Michaël

On the Complexity of Universality for Partially Ordered NFAs

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Abstract

Partially ordered nondeterminsitic finite automata (poNFAs) are NFAs whose transition relation induces a partial order on states, i.e., for which cycles occur only in the form of self-loops on a single state. A poNFA is universal if it accepts all words over its input alphabet. Deciding universality is \PSpace-complete for poNFAs, and we show that this remains true even when restricting to a fixed alphabet. This is nontrivial since standard encodings of alphabet symbols in, e.g., binary can turn self-loops into longer cycles. A lower coNP-complete complexity bound can be obtained if we require that all self-loops in the poNFA are deterministic, in the sense that the symbol read in the loop cannot occur in any other transition from that state. We find that such restricted poNFAs (rpoNFAs) characterise the class of R-trivial languages, and we establish the complexity of deciding if the language of an NFA is R-trivial. Nevertheless, the limitation to fixed alphabets turns out to be essential even in the restricted case: deciding universality of rpoNFAs with unbounded alphabets is PSPACE-complete. Our results also prove the complexity of the inclusion and equivalence problems, since universality provides the lower bound, while the upper bound is mostly known or proved in the paper.

BibTeX - Entry

@InProceedings{krtzsch_et_al:LIPIcs:2016:6473,
  author =	{Markus Kr{\"o}tzsch and Tom{\'a}s Masopust and Micha{\"e}l Thomazo},
  title =	{{On the Complexity of Universality for Partially Ordered NFAs}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{61:1--61:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-016-3},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{58},
  editor =	{Piotr Faliszewski and Anca Muscholl and Rolf Niedermeier},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6473},
  URN =		{urn:nbn:de:0030-drops-64738},
  doi =		{10.4230/LIPIcs.MFCS.2016.61},
  annote =	{Keywords: automata, nondeterminism, partial order, universality}
}

Keywords: automata, nondeterminism, partial order, universality
Seminar: 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)
Issue Date: 2016
Date of publication: 19.08.2016


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