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URN: urn:nbn:de:0030-drops-9767

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Counting Results in Weak Formalisms



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.

BibTeX - Entry

  author =	{Arnaud Durand and Clemens Lautemann and Malika More},
  title =	{Counting Results in Weak Formalisms},
  booktitle =	{Circuits, Logic, and Games},
  year =	{2007},
  editor =	{Thomas Schwentick and Denis Th{\'e}rien and Heribert Vollmer },
  number =	{06451},
  series =	{Dagstuhl Seminar Proceedings},
  ISSN =	{1862-4405},
  publisher =	{Internationales Begegnungs- und Forschungszentrum f{\"u}r Informatik (IBFI), Schloss Dagstuhl, Germany},
  address =	{Dagstuhl, Germany},
  URL =		{},
  annote =	{Keywords: Small complexity classes, logic, polylog counting}

Keywords: Small complexity classes, logic, polylog counting
Seminar: 06451 - Circuits, Logic, and Games
Issue date: 2007
Date of publication: 2007

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