The Convenience of Tilings

Tiling problems provide for a very simple and transparent mechanism for encod ing machine computations This gives rise to rather simple master reductions showing various versions of the tiling problem complete for various complexity classes We in vestigate the potential for using these tiling problems in subsequent reductions showing hardness of the combinatorial problems that really matter We ilustrate our approach by means of three examples a short reduction chain to the Knapsack problem followed by a Hilbert reduction using similar ingredients Finally we reprove the Deterministic Exponential Time lowerbound for satis ablility in Propositional Dynamic Logic The resulting reductions are relatively simple they do however infringe on the principle of orthogonality of reductions since they abuse extra structure in the in stances of the problems reduced from which results from the fact that these instances were generated by a master reduction previously