Undecidable Problems in Fractal Geometry.

TLDR: Results show that for every Turing machine there exists a fractal-based geometrical model of computation which is computationally universal and can be viewed, in a certain sense, as geomet rically encoding the complement of the language accepted by t he machine.

Abstract: In this paper , a relationship between the classical theory of computat ion and fract al geometry is est ablish ed. It erated Function Syst ems are used as tool s to define fractals. It is shown t hat two questions about Iterat ed Function Systems are und ecidable: to t est if t he attractor of a given It erat ed Fun ction Syst em and a given line segment int ersect and to test if a given It erated Funct ion Syst em is totally disconnected. T he proofs are very simple and are obtained by reducing the Post Correspondence P roblem and by interp ret ing strings as numb ers and conca tenation operat ions as compositions of affine t ransformations. These results show that for every Turing machine there exists a fract al set which can be viewed, in a certain sense, as geomet rically encoding the complement of the language accepted by t he machine. One can build a fractal-based geometrical model of computation which is computationally universal.