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, firstorder 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 11
mapping from a small subset of ${0,ldots,N1}$ 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 collisionfree hashing technique, leading to a completely elementary proof for the polylog counting capability of firstorder logic (with builtin arithmetic), $AC^0$circuits, rudimentary arithmetic, the Linear Hierarchy, and monadicsecond 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 = {18624405},
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}
}
Keywords: 

Small complexity classes, logic, polylog counting 
Seminar: 

06451  Circuits, Logic, and Games 
Issue Date: 

2007 
Date of publication: 

23.04.2007 