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Counting Results in Weak Formalisms Durand, Arnaud Lautemann, Clemens More, Malika Small complexity classes logic polylog counting 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Arnaud Durand and Clemens Lautemann and Malika More 2007 Is Part Of Dagstuhl Seminar Proceedings, Volume 6451, Circuits, Logic, and Games (2007) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/DagSemProc.06451.4 urn:nbn:de:0030-drops-9767 https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06451.4 eng https://creativecommons.org/licenses/by/4.0/legalcode