My question/conjecture comes from this post: Configurations whose convex hull contains the origin I wrote:
"This is an interesting example of a geometric problem, for which it seems that the best one can do is to come up with an algorithm which would decide (in some reasonable time), given n points, whether or not the origin lies in the convex hull of these points or not. In some sense, this is like describing algorithmically the characteristic function of a subset we are interested in. This leads me to conjecture that there may be some subsets of $\mathbb{R}^n$ whose characteristic function is given as the output of an algorithm, and can not be described in any "closed form" way. I am not sure how to make this statement precise."
So my question, which is vague, is whether or not there are some subsets of $\mathbb{R}^n$ whose characteristic function is given by the output of an algorithm, and such that this is the best one can do, in the sense that there is no "closed form" description of such a subset. One needs to define what "closed form" means, in precise terms. I am completely ignorant of complexity theory and foundational Mathematics, so please bear this in mind, and explain to me whether what I wrote can be made precise, and whether or not it is correct. I apologize for the vagueness part of my question, but hope I conveyed the intended meaning in a sufficiently clear way.
Would something like the Mandelbrot set count?
Points $p\in\mathbb{C}$ belong to the set if the sequence $$z_i = \begin{cases} p, & i = 0\\z_{i-1}^2 + p, & i > 0\end{cases}$$ remains bounded as $i\to\infty$. As far as I know, there is no "closed form" indicator function for this set: in practice you compute the iterates algorithmically until $z_i$ lands in a region where you can prove that the sequence must diverge or remain bounded.