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Theory of Computing Systems

, Volume 36, Issue 4, pp 303–328 | Cite as

Arithmetic Complexity, Kleene Closure, and Formal Power Series

  • Eric Allender
  • V. Arvind
  • Meena Mahajan
Article

Abstract.

The aim of this paper is to use formal power series techniques to study the structure of small arithmetic complexity classes such as GapNC 1 and GapL. More precisely, we apply the formal power series operations of inversion and root extraction to these complexity classes. We define a counting version of Kleene closure and show that it is intimately related to inversion and root extraction within GapNC 1 and GapL. We prove that Kleene closure, inversion, and root extraction are all hard operations in the following sense: there is a language in AC 0 for which inversion and root extraction are GapL-complete and Kleene closure is NLOG-complete, and there is a finite set for which inversion and root extraction are GapNC 1 -complete and Kleene closure is NC 1 -complete, with respect to appropriate reducibilities.

The latter result raises the question of classifying finite languages so that their inverses fall within interesting subclasses of GapNC 1 , such as GapAC 0 . We initiate work in this direction by classifying the complexity of the Kleene closure of finite languages. We formulate the problem in terms of finite monoids and relate its complexity to the internal structure of the monoid.

Some results in this paper show properties of complexity classes that are interesting independent of formal power series considerations, including some useful closure properties and complete problems for GapL.

Keywords

Word Problem Complexity Class Formal Power Series Root Extraction Regular Language 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York 2003

Authors and Affiliations

  • Eric Allender
    • 1
  • V. Arvind
    • 2
  • Meena Mahajan
    • 2
  1. 1.Department of Computer Science, Rutgers University, Hill Center, allender@cs.rutgers.edu US
  2. 2.The Institute of Mathematical Sciences, C.I.T. Campus, arvind@imsc.res.in,meena@imsc.res.inIN

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