Admissible closures of polynomial time computable arithmetic

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Abstract

We propose two admissible closures \({\mathbb{A}({\sf PTCA})}\) and \({\mathbb{A}({\sf PHCA})}\) of Ferreira’s system PTCA of polynomial time computable arithmetic and of full bounded arithmetic (or polynomial hierarchy computable arithmetic) PHCA. The main results obtained are: (i) \({\mathbb{A}({\sf PTCA})}\) is conservative over PTCA with respect to \({\forall\exists\Sigma^b_1}\) sentences, and (ii) \({\mathbb{A}({\sf PHCA})}\) is conservative over full bounded arithmetic PHCA for \({\forall\exists\Sigma^b_{\infty}}\) sentences. This yields that (i) the \({\Sigma^b_1}\) definable functions of \({\mathbb{A}({\sf PTCA})}\) are the polytime functions, and (ii) the \({\Sigma^b_{\infty}}\) definable functions of \({\mathbb{A}({\sf PHCA})}\) are the functions in the polynomial time hierarchy.

Keywords

Polynomial time computable arithmetic Kripke Platek set theory Second order arithmetic 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Mathematisches InstitutLudwig-Maximilians-Universität MünchenMünchenGermany
  2. 2.Institut für Informatik und Angewandte MathematikUniversität BernBernSwitzerland

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