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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.
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Gilda Ferreira, The counting hierarchy in binary notation. Port. Math. 66 (2009), no. 1, pp. 81–94DOI 10.4171/PM/1832