On the Complexity of Real Functions

TL;DR

This paper introduces a new framework for understanding the computability and complexity of real-valued functions, extending existing models to include some discontinuities and multiple values, providing a more natural measure of their difficulty.

Contribution

It develops a novel notion of real function computability and complexity that generalizes BSS and bit computability, accommodating discontinuities and multiple values.

Findings

Defines a natural measure of function difficulty over reals

Extends existing models to include discontinuous functions

Provides a unified framework for real function complexity

Abstract

We develop a notion of computability and complexity of functions over the reals, which seems to be very natural when one tries to determine just how "difficult" a certain function is. This notion can be viewed as an extension of both BSS computability [Blum, Cucker, Shub, Smale 1998], and bit computability in the tradition of computable analysis [Weihrauch 2000] as it relies on the latter but allows some discontinuities and multiple values.