Real Hypercomputation and Continuity

TL;DR

This paper explores various models of hypercomputation to determine if they can evaluate discontinuous real functions, finding that some models still require continuity while others can compute discontinuous functions like the sign function.

Contribution

It analyzes three super-Turing notions of real function computability and shows which models can or cannot evaluate discontinuous functions.

Findings

Relativized computation with oracles like the Halting Problem remains continuous.

Encoding methods related to the Arithmetic Hierarchy also preserve continuity.

Non-deterministic hypercomputation can evaluate discontinuous functions like the sign function.

Abstract

By the sometimes so-called 'Main Theorem' of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of HYPERcomputation allow for the effective evaluation of also discontinuous f:R->R. More precisely the present work considers the following three super-Turing notions of real function computability: * relativized computation; specifically given oracle access to the Halting Problem 0' or its jump 0''; * encoding real input x and/or output y=f(x) in weaker ways also related to the Arithmetic Hierarchy; * non-deterministic computation. It turns out that any f:R->R computable in the first or second sense is still necessarily continuous whereas the third type of hypercomputation does provide the required power to evaluate for instance the discontinuous sign function.