Isomorphisms of NP complete problems on random instances

Polynomial isomorphisms are defined for NP-complete sets on random instances. Not only are polynomial-time computable and invertible bijections among complete sets considered, but also it is required that these bijections preserve distributions on random instances of these sets. Sufficient conditions for randomized decision problems to be polynomially isomorphic are shown. It is then proved that all the known average-case NP-complete problems under many-one reductions are polynomially isomorphic. These problems include the randomized tiling problem, the randomized halting problem, the randomized Post correspondence problem, and the randomized word problem for Thue systems