Admissible closures of polynomial time computable arithmetic

We propose two admissible closures A(PTCA) and A(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) A(PTCA) is conservative over PTCA with respect to ∀∃Σ1 sentences, and (ii) A(PHCA) is conservative over full bounded arithmetic PHCA for ∀∃Σ∞ sentences. This yields that (i) the Σ1 definable functions of A(PTCA) are the polytime functions, and (ii) the Σ∞ definable functions of A(PHCA) are the functions in the polynomial time hierarchy.