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Characterizing NC¹ with Typed Monoids

Authors Anuj Dawar , Aidan T. Evans

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LIPIcs.FSTTCS.2025.26.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Regular languages
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Circuit complexity
  • Theory of computation → Finite Model Theory
Keywords
  • algebraic automata theory
  • circuit complexity
  • descriptive complexity
  • typed monoids
  • semigroups
  • generalized quantifiers

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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.

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.FSTTCS.2025.26

Author Details

Anuj Dawar
  • Department of Computer Science and Technology, University of Cambridge, UK
Aidan T. Evans
  • Department of Computer Science and Technology, University of Cambridge, UK

Funding

  • Dawar, Anuj: Research funded in part by UK Research and Innovation (UKRI) under the UK government’s Horizon Europe funding guarantee: grant number EP/X028259/1.
  • Evans, Aidan T.: Researched supported by Huawei HiSilicon Studentship.

References

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