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DOI: 10.4230/LIPIcs.MFCS.2021.63

The Simplest Non-Regular Deterministic Context-Free Language

Authors: Petr Jančar and Jiří Šíma

Published in: LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)

Abstract

We introduce a new notion of 𝒞-simple problems for a class 𝒞 of decision problems (i.e. languages), w.r.t. a particular reduction. A problem is 𝒞-simple if it can be reduced to each problem in 𝒞. This can be viewed as a conceptual counterpart to 𝒞-hard problems to which all problems in 𝒞 reduce. Our concrete example is the class of non-regular deterministic context-free languages (DCFL'), with a truth-table reduction by Mealy machines. The main technical result is a proof that the DCFL' language L_# = {0^n1^n ∣ n ≥ 1} is DCFL'-simple, and can be thus viewed as one of the simplest languages in the class DCFL', in a precise sense. The notion of DCFL'-simple languages is nontrivial: e.g., the language L_R = {wcw^R∣ w ∈ {a,b}^*} is not DCFL'-simple. By describing an application in the area of neural networks (elaborated in another paper), we demonstrate that 𝒞-simple problems under suitable reductions can provide a tool for expanding the lower-bound results known for single problems to the whole classes of problems.

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Petr Jančar and Jiří Šíma. The Simplest Non-Regular Deterministic Context-Free Language. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 63:1-63:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{jancar_et_al:LIPIcs.MFCS.2021.63,
  author =	{Jan\v{c}ar, Petr and \v{S}{\'\i}ma, Ji\v{r}{\'\i}},
  title =	{{The Simplest Non-Regular Deterministic Context-Free Language}},
  booktitle =	{46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)},
  pages =	{63:1--63:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-201-3},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{202},
  editor =	{Bonchi, Filippo and Puglisi, Simon J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.63},
  URN =		{urn:nbn:de:0030-drops-145037},
  doi =		{10.4230/LIPIcs.MFCS.2021.63},
  annote =	{Keywords: deterministic context-free language, truth-table reduction, Mealy automaton, pushdown automaton}
}

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The Simplest Non-Regular Deterministic Context-Free Language

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