Hypercomputing the Mandelbrot Set?

TL;DR

This paper surveys the known results on the computability and non-computability of the Mandelbrot set across various models of computation, including classical, hypercomputational, and rational models.

Contribution

It provides a comprehensive overview of the different models of decidability applied to the Mandelbrot set, highlighting the distinctions and open questions.

Findings

Mandelbrot set's computability varies across models

Hypercomputation offers different perspectives on set decidability

The set's properties depend on the underlying computational framework

Abstract

The Mandelbrot set is an extremely well-known mathematical object that can be described in a quite simple way but has very interesting and non-trivial properties. This paper surveys some results that are known concerning the (non-)computability of the set. It considers two models of decidability over the reals (which have been treated much more thoroughly and technically by Hertling (2005), Blum, Shub and Smale, Brattka (2003) and Weihrauch (1999 and 2003) among others), two over the computable reals (the Russian school and hypercomputation) and a model over the rationals.