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DOI: 10.4230/LIPIcs.FSTTCS.2025.26

Characterizing NC¹ with Typed Monoids

Authors: Anuj Dawar and Aidan T. Evans

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract

Krebs et al. (2007) gave a characterization of the complexity class TC⁰ as the class of languages recognized by a certain class of typed monoids. The notion of typed monoid was introduced to extend methods of algebraic automata theory to infinite monoids and hence characterize classes beyond the regular languages. We advance this line of work beyond TC⁰ by giving a characterization of NC¹. This is obtained by first showing that NC¹ can be defined as the languages expressible in an extension of first-order logic using only unary quantifiers over regular languages. The expressibility result is a consequence of a general result showing that finite monoid multiplication quantifiers of higher dimension can be replaced with unary quantifiers in the context of interpretations over strings, which also answers a question of Lautemann et al. (2001). We estblish this collapse result for a much more general class of interpretations using results on interpretations due to Bojańczyk et al. (2019), which may be of independent interest.

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Anuj Dawar and Aidan T. Evans. Characterizing NC¹ with Typed Monoids. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 26:1-26:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dawar_et_al:LIPIcs.FSTTCS.2025.26,
  author =	{Dawar, Anuj and Evans, Aidan T.},
  title =	{{Characterizing NC¹ with Typed Monoids}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{26:1--26:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.26},
  URN =		{urn:nbn:de:0030-drops-251070},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.26},
  annote =	{Keywords: algebraic automata theory, circuit complexity, descriptive complexity, typed monoids, semigroups, generalized quantifiers}
}

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Characterizing NC¹ with Typed Monoids

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