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Counting Results in Weak Formalisms

Authors: Arnaud Durand, Clemens Lautemann, and Malika More

Published in: Dagstuhl Seminar Proceedings, Volume 6451, Circuits, Logic, and Games (2007)

Abstract

The counting ability of weak formalisms is of interest as a measure of their expressive power. The question was investigated in the 1980's in several papers in complexity theory and in weak arithmetic. In each case, the considered formalism (AC$^0$--circuits, first--order logic, $Delta_0$, respectively) was shown to be able to count precisely up to a polylogarithmic number. An essential part of each of the proofs is the construction of a 1--1 mapping from a small subset of ${0,ldots,N-1}$ into a small initial segment. In each case the expressibility of such a mapping depends on some strong argument (group theoretic device or prime number theorem) or intricate construction. We present a coding device based on a collision-free hashing technique, leading to a completely elementary proof for the polylog counting capability of first--order logic (with built--in arithmetic), $AC^0$--circuits, rudimentary arithmetic, the Linear Hierarchy, and monadic--second order logic with addition.

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Arnaud Durand, Clemens Lautemann, and Malika More. Counting Results in Weak Formalisms. In Circuits, Logic, and Games. Dagstuhl Seminar Proceedings, Volume 6451, pp. 1-11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{durand_et_al:DagSemProc.06451.4,
  author =	{Durand, Arnaud and Lautemann, Clemens and More, Malika},
  title =	{{Counting Results in Weak Formalisms}},
  booktitle =	{Circuits, Logic, and Games},
  pages =	{1--11},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{6451},
  editor =	{Thomas Schwentick and Denis Th\'{e}rien and Heribert Vollmer},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/DagSemProc.06451.4},
  URN =		{urn:nbn:de:0030-drops-9767},
  doi =		{10.4230/DagSemProc.06451.4},
  annote =	{Keywords: Small complexity classes, logic, polylog counting}
}

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Counting Results in Weak Formalisms

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