Durand, Arnaud ;
Lautemann, Clemens ;
More, Malika
Counting Results in Weak Formalisms
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.
BibTeX - Entry
@InProceedings{durand_et_al:DSP:2007:976,
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 = {http://drops.dagstuhl.de/opus/volltexte/2007/976},
annote = {Keywords: Small complexity classes, logic, polylog counting}
}
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Keywords: |
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Small complexity classes, logic, polylog counting |
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Seminar: |
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06451 - Circuits, Logic, and Games
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Issue date: |
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2007 |
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Date of publication: |
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23.04.2007 |
