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First Order Extensions of Residue Classes and Uniform Circuit Complexity

  • Argimiro Arratia
  • Carlos E. Ortiz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8071)

Abstract

The first order logic \(\mathcal{R}ing(0,+,*,<)\) for finite residue class rings with order is presented, and extensions of this logic with generalized quantifiers are given. It is shown that this logic and its extensions capture DLOGTIME-uniform circuit complexity classes ranging from AC 0 to TC 0. Separability results are obtained for the hierarchy of these logics when order is not present, and for \(\mathcal{R}ing(0,+,*,<)\) from the unordered version. These separations are obtained using tools from class field theory, adapting notions as the spectra of polynomials over finite fields to sets of sentences in this logic of finite rings, and studying asymptotic measures of these sets such as their relative densities. This framework of finite rings with order provides new algebraic tools and a novel perspective for descriptive complexity.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Argimiro Arratia
    • 1
  • Carlos E. Ortiz
    • 2
  1. 1.LSIUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Arcadia UniversityGlensideU.S.A.

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