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.

We provide a natural characterization of the type two Mehlhorn-CookUrquhart basic feasible functionals as the provably total type two functionals of our (classical) applicative theory PT introduced in [27], thus providing a proof of a result claimed in the conclusion of [27].

Abstract We propose an alternative notion of asymptotic behaviors for the study of type-2 computational complexity. Since the classical asymptotic notion (for all but finitely many) is not acceptable in type-2 context, we alter the notion of “small sets ” from “finiteness ” to topological “compactness ” for type-2 complexity theory. A natural reference for type-2 computations is the standard Baire topology. However, we point out some serious drawbacks of this and introduce an alternative topology for describing compact sets. Following our notion explicit type-2 complexity classes can be defined in terms of resource bounds. We show that such complexity classes are recursively representable; namely, every complexity class has a programming system. We also prove type-2 analogs of Rabin’s Theorem, Recursive Relatedness Theorem, and Gap Theorem to provide evidence that our notion of type-2 asymptotic is workable. We speculate that our investigation will give rise to a possible approach in examining the complexity structure at type-2 along the line of the classical complexity theory. Keywords: Type-2 Complexity, Type-2 Asymptotic Notation, Baire Topology. 1.

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

Abstract. In [12] we defined a class of functions called Type-2 Time Bounds (henceforth T2TB) for clocking the Oracle Turing Machine (OTM) in order to capture the long missing notion of complexity classes at type-2. In the present paper we further advance the type-2 complexity theory under our notion of type-2 complexity classes. We have learned that the theory is highly sensitive to how the oracle answers are handled. We present a reasonable alternative called unit cost model, and examine how this model shapes the outlook of the type-2 complexity theory. Under the unit cost model we prove two theorems opposite to the classical union theorem and gap theorem. We also investigate some properties of T2TB including a very useful theorem stating that there is an effective operator to convert any β ∈ T2TB into an equivalent one that is locking-detectable. The existence of such operator allows us to simplify many proofs without loss of generality. 1 1

This paper has explored three examples of good semantical analyses of programming structures. The three examples share two characteristics: the semantic models are abstract enough to be applicable in many situations, and the models lead to proofs of non-computability. Other examples of programming structures have been omitted from this short essay: foundations for object-oriented languages, descriptions of languages with local variables, and the theory of database query languages. Each of these examples have corresponding semantical theories that enjoy the two characteristics above. The richness of programming structure suggests a corollary: it is folly to look for one universal model to explain all programming structures. Of course, as a theoretical subject, semantics benefits from the reduction of many concepts to a primitive, common level. Nevertheless, reduction must often be resisted. We have seen how computability theory loses all kinds of relevant distinctions. Another example is the naive semantics of PCF based on dcpos: the model is not abstract enough,

A well-known theorem of type-two recursion theory states that a functional is continuous if and only if it is computable relative to some oracle. We show that a feasible analogue of this theorem holds, using techniques originally developed in the study of Boolean decision tree complexity. 1 Introduction Type-two computability theory deals with the computability of functionals, which take functions and numbers as input, and produce numbers as output. A surprising and pleasing aspect of type-two computability is its close connections with topology on Baire space. Notions of relative typetwo computability (that is, computability with respect to some oracle,) can be characterized using purely topological notions. In particular, a type-two functional is computable relative to an oracle if and only if it is continuous. While the theory of type-two computability has been widely successful, relatively little work has been done on the development of a complexity theory for type-two functionals...