Portugaliae Mathematica

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Volume 66, Issue 1, 2009, pp. 81–94

DOI: 10.4171/PM/1832

The counting hierarchy in binary notation

Gilda Ferreira (1)

(1) Department of Computer Science, Queen Mary, University of London, E1 4NS, LONDON, UNITED KINGDOM

We present a new recursion-theoretic characterization of FCH, the hierarchy of counting functions, in binary notation. Afterwards we introduce a theory of bounded arithmetic, TCA, that can be seen as a reformulation, in the binary setting, of Jan Johannsen and Chris Pollett's system D⁰₂. Using the previous inductive characterization of FCH, we show that a strategy similar to the one applied to D⁰₂ can be used in order to characterize FCH as the class of functions provably total in TCA.

Keywords: Counting hierarchy, bounded arithmetic, complexity theory