Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the early seventies. In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: self-application is meaningful, but not necessarily total. It has turned out that theories with self-application provide a natural setting for studying notions of abstract computability, especially from a proof-theoretic perspective.

Abstract. We are concerned with first order theories of operations, based on combinatory logic and extended with the type W of binary words. The theories include forms of “positive ” and “bounded ” induction on W and naturally characterize primitive recursive and polytime functions (respectively). We prove that the recursive content of the theories under investigation (i.e. the associated class of provably total functions on W) is invariant under addition of 1. an axiom of choice for operations and a uniformity principle, restricted to positive conditions; 2. a (form of) self-referential truth, providing a fixed point theorem for predicates. As to the proof methods, we apply a kind of internal forcing semantics, non-standard variants of realizability and cut-elimination. §1. Introduction. In this paper, we deal with theories of abstract computable operations, underlying the so-called explicit mathematics, introduced by Feferman in the midseventies as a logical frame to formalize Bishop’s style constructive mathematics ([18], [19]). Following a common usage, these theories

This paper studies systems of explicit mathematics as introduced by Feferman [9, 11]. In particular, we propose weak explicit type systems with a restricted form of elementary comprehension whose provably terminating operations coincide with the functions on binary words that are computable in polynomial time. The systems considered are natural extensions of the first-order applicative theories introduced in

This is a survey paper on various weak systems of Feferman’s explicit mathematics and their proof theory. The strength of the systems considered in measured in terms of their provably terminating operations typically belonging to some natural classes of computational time or space complexity. Keywords: Proof theory, Feferman’s explicit mathematics, applicative theories, higher types, types and names, partial truth, feasible operations 1

We study weak theories of truth over combinatory logic and their relationship to weak systems of explicit mathematics. In particular, we consider two truth theories TPR and TPT of primitive recursive and feasible strength. The latter theory is a novel abstract truth-theoretic setting which is able to interpret expressive feasible subsystems of explicit mathematics. 1

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