TITLE: Is the distance function from a point to the Mandelbrot set computable?
QUESTION [5 upvotes]: There is at least one result saying that the Mandelbrot set is undecidable, and there might be more, but I think it (or they all) use real computation rather than Turing machines.  This makes some sense, as $\mathbb{C}$ is connected, so the only decidable subsets of it are $\{\}$ and $\mathbb{C}$ itself.  However, I've been reading about reverse mathematics, and I was wondering if the set is computable in the representations used.  The Mandelbrot set is easily seen to be the complement of an effectively open set, and I'd guess asking whether it is separably closed would run into the open problem of are the hyperbolic components dense.  This still leaves located closed, so the question follows:
Let $f : \mathbb{Q}[i] \to \mathbb{R} \hspace{.05 in}$ be given by $f(q) := \operatorname{min}(\{d(q,z) : z\in (\operatorname{Mandelbrot set})\})$.  Is $f$ computable?

REPLY [4 votes]: I just want to add one small piece of information to Bjørn's answer. The distance function for the Mandelbrot set $M$ is not known to be computable as a map into the reals, but it is computable as a map into the upper reals. This means that we can calculate upper bounds for the distance: given $z \in \mathbb{C}$, the set of all rational upper bounds for $d(x,M)$ is computably enumerable. In many cases (such as drawing algorithms) this is already quite useful.