Admissible closures of polynomial time computable arithmetic

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