33 Pages

The Convenience of Tilings

WithPeter van Emde Boas

Tiling problems provide for a very simple and transparent mechanism for encoding machine computations. This gives rise to rather simple master reductions showing various versions of the tiling problem complete for various complexity classes. We investigate the potential for using these tiling problems in subsequent reductions showing hardness of the combinatorial problems that really matter.

We illustrate our approach by means of three examples: a short reduction chain to the Knapsack problem followed by a Hilbert 10 reduction using similar ingredients. Finally we reprove the Deterministic Exponential Time lowerbound for satisfiablility 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 instances of the problems which results from the fact that these instances were previously generated by a master reduction.