Revising Type-2 Computation and Degrees of Discontinuity

TL;DR

This paper unifies various approaches to understanding discontinuous computable functions by analyzing the concept of the JUMP of a representation and exploring the computational power of nondeterminism in Type-2 computation.

Contribution

It introduces a unified framework for relaxed notions of computability of discontinuous functions using the concept of the JUMP of a representation.

Findings

The JUMP of a representation generalizes classical notions of hypercomputation.

Type-2 nondeterminism matches the first level of the Analytical Hierarchy.

Characterization of Markov and Banach/Mazur oracle computations for discontinuous functions.

Abstract

By the sometimes so-called MAIN THEOREM of Recursive Analysis, every computable real function is necessarily continuous. Weihrauch and Zheng (TCS'2000), Brattka (MLQ'2005), and Ziegler (ToCS'2006) have considered different relaxed notions of computability to cover also discontinuous functions. The present work compares and unifies these approaches. This is based on the concept of the JUMP of a representation: both a TTE-counterpart to the well known recursion-theoretic jump on Kleene's Arithmetical Hierarchy of hypercomputation: and a formalization of revising computation in the sense of Shoenfield. We also consider Markov and Banach/Mazur oracle-computation of discontinuous fu nctions and characterize the computational power of Type-2 nondeterminism to coincide with the first level of the Analytical Hierarchy.