All Natural NPC Problems Have Average-Case Complete Versions

In 1984 Levin put forward a suggestion for a theory of average case complexity. In this theory a problem, called a distributional problem, is defined as a pair consisting of a decision problem and a probability distribution over the instances. Introducing adequate notions of simple distributions and average case preserving reductions, Levin developed a theory analogous to the theory of NP-completeness. In particular, he showed that there exists a simple distributional problem that is complete under these reductions. But since then very few distributional problems were shown to be complete in this sense. In this paper we show a very simple sufficient condition for an NP-complete decision problem to have a distributional version that is complete under these reductions. Apparently all known NP-complete decision problems meet this condition.