Fractal geometry, Turing machines and divide-and-conquer recurrences

Language Theoretic and Algorithmic Properties ofd-dimensional Collages and Patterns in a Grid

“…The main results concerning m-grid IFSs say that one can decide whether the fractals defined by two of them intersect and, for the subclass of non-hierarchical m-grid IFSs, whether the generated patterns are connected. These results are in contrast to the mostly negative results for related questions found so far (see the papers by Dassow and Hinz [10], Dassow, Habel, and Taubenberger [9], Drewes and Kreowski [14], and, in particular, [15] by Dube, who showed that the questions studied in the present paper are undecidable for a more general class of IFSs).…”

“…In particular, the decidability results shown are in sharp contrast to certain undecidability results known from the literature on picture generating devices (cf. the work by Dassow and Hinz [10], or by Drewes and Kreowski [14], by Dassow, Habel, and Taubenberger [9], but, in particular, the paper by Dube [15], where questions like those considered here are proved to be undecidable for more general classes of IFSs). The``grid'' restriction used to obtain the positive results is on the one hand quite a strong requirement that rules out many cases one would probably also like to have decision algorithms for.…”

Algorithms to test open set condition for self-similar set related to P.V. numbers

“…(1) Based on the undecidability of Post correspondence problem (PCP), Dube [7,8] discussed the undecidability of the problem on the invariant fractals of iterated function systems. For example, Theorem 5 of [7] actually shows that given a self-affine set in the plane, it is undecidable to test if it satisfies the SSC.…”