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Counting Results in Weak Formalisms

Authors Arnaud Durand, Clemens Lautemann, Malika More



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Arnaud Durand
Clemens Lautemann
Malika More

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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)
https://doi.org/10.4230/DagSemProc.06451.4

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.
Keywords
  • Small complexity classes
  • logic
  • polylog counting

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