nLab Type Two Theory of Effectivity

Contents

Idea

What is called Weihrauch’s Type-2 Theory (Weihrauch 87) is a measure for the complexity of computable functions on non-discrete topological spaces such as the real line, the topic of computable analysis. While some approaches to computability of continuous functions take pointwise addition as an elementary operation, Weihrauch’s Type-2 theory is instead “bit oriented” and takes into account the finitary aspects of computation.

A quick survey is in the beginning of (Schröder 04).

Essentially the type-II theory considers functions between topological spaces which may be encoded by infinite sequences of natural numbers, this is discussed in detail at computable function (analysis). As such this is a mathematical theory of computability closely related to exact real computer arithmetic.

	type I computability	type II computability
typical domain	natural numbers $\mathbb{N}$	Baire space of infinite sequences $\mathbb{B} = \mathbb{N}^{\mathbb{N}}$
computable functions	partial recursive function	computable function (analysis)
type of computable mathematics	recursive mathematics	computable analysis, Type Two Theory of Effectivity
type of realizability	number realizability	function realizability
partial combinatory algebra	Kleene's first partial combinatory algebra	Kleene's second partial combinatory algebra

The original article is

Klaus Weihrauch, Computability, EATCS Monographs in Theoretical Computer ScienceSpringer Verlag, Berlin Heidelberg (1987)
Klaus Weihrauch, A foundation of computable analysis in

D.S. Bridges, C.S. Calude, J. Gibbons, S. Reeves, I.H. Witten (eds.), Combinatorics, complexity and logic, discrete mathematics and computer science, Proceedings of DMTCS 96, Springer Verlag, Singapore (1997), pp. 66–89

The equivalence of this to real PCF is shown in

Holger Schulz, Type Two Theory of Effectivity and Real PCF, Electronic Notes in Theoretical Computer Science Volume 35, 2000, Pages 191–203