In this article we study applications of the bounded functional interpretation to theories of feasible arithmetic and analysis. The main results show that the novel interpretation is sound for considerable generalizations of weak König’s Lemma, even in the presence of very weak induction. Moreover, when this is combined with Cook and Urquhart’s variant of the functional interpretation, one obtains effective versions of conservation results regarding weak König’s Lemma which have been so far only obtained non-constructively.

In this paper we study (self)-applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full self-application and whose provably total functions on W = {0, 1} * are exactly the polynomial time computable functions. Our treatment of PTO is proof-theoretic and very much in the spirit of reductive proof theory.

This paper develops the very basic notions of analysis in a weak second-order theory of arithmetic BTFA whose provably total functions are the polynomial time computable functions. We formalize within BTFA the real number system and the notion of a continuous real function of a real variable. The theory BTFA is able to prove the intermediate value theorem, wherefore it follows that the system of real numbers is a real closed ordered field. In the last section of the paper, we (...) show how to interpret the theory BTFA in Robinson's theory of arithmetic Q. This fact entails that the elementary theory of the real closed ordered fields is interpretable in Q. (shrink)

We prove two conservation results involving a generalization of the principle of strict ${\Pi^1_1}$ -reflection, in the context of bounded arithmetic. In this context a separation between the concepts of bounded set and binary sequence seems to emerge as fundamental.

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. (shrink)