In this paper we prove conservation theorems for theories of classical first-order arithmetic over their intuitionistic version. We also prove generalized conservation results for intuitionistic theories when certain weak forms of the principle of excluded middle are added to them. Members of two families of subsystems of Heyting arithmetic and Buss-Harnik’s theories of intuitionistic bounded arithmetic are the intuitionistic theories we consider. For the first group, we use a method described by Leivant based on the negative translation combined with a (...) variant of Friedman’s translation. For the second group, we use Avigad’s forcing method. (shrink)

In this paper we naturally define when a theory has bounded quantifier elimination, or is bounded model complete. We give several equivalent conditions for a theory to have each of these properties. These results provide simple proofs for some known results in the model theory of the bounded arithmetic theories like CPV and PV1. We use the mentioned results to obtain some independence results in the context of intuitionistic bounded arithmetic. We show that, if the intuitionistic theory of polynomial induction (...) on strict $\Pi_2^{b}$ formulas proves decidability of $\Sigma_1^b$ formulas, then P=NP. We also prove that, if the mentioned intuitionistic theory proves ${\rm LMIN}(\Sigma_{1}^{b})$ , then P=NP. (shrink)

We use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra PV ω defines exactly the PTIME-functions. As a byproduct we obtain a syntax-free generalisation of PTIME-computability to higher types. By restricting to sheaves for a suitable topology we obtain a model for intuitionistic predicate logic with ∑ 1 b -induction over PV ω and use this to re-establish that the provably total functions in this system are polynomial time computable. Finally, we (...) apply the category-theoretic approach to a new higher-order extension of Bellantoni-Cook's system BC of safe recursion. (shrink)

This paper presents a new method - which does not rely on the cut-elimination theorem - for characterizing the provably total functions of certain intuitionistic subsystems of arithmetic. The new method hinges on a realizability argument within an infinitary language. We illustrate the method for the intuitionistic counterpart of Buss's theory Smath image, and we briefly sketch it for the other levels of bounded arithmetic and for the theory IΣ1.

A notion of feasible function of finite type based on the typed lambda calculus is introduced which generalizes the familiar type 1 polynomial-time functions. An intuitionistic theory IPVω is presented for reasoning about these functions. Interpretations for IPVω are developed both in the style of Kreisel's modified realizability and Gödel's Dialectica interpretation. Applications include alternative proofs for Buss's results concerning the classical first-order system S12 and its intuitionistic counterpart IS12 as well as proofs of some of Buss's conjectures concerning IS12, (...) and a proof that IS12 cannot prove that extended Frege systems are not polynomially bounded. (shrink)