Polynomial Time Samplable Distributions

This paper studies distributions which can be \approximated" by sampling algorithms in time polynomial in the length of their outputs. First, it is known that if polynomial-time samplable distributions are polynomial-time computable, then NP collapses to P. This paper shows by a simple counting argument that every polynomial-time samplable distribution is computable in polynomial time if and only if so is every #P function. By this result, the class of polynomially samplable distributions contains no universal distributions if FP = #P. Second, it is also known that there exists polynomially samplable distributions which are not polynomially dominated by any polynomial-time computable distribution if strongly one-way functions exist. This paper strengthens this statement and shows that an assumption, namely, NP 6 6 BPP leads to the same consequence. Third, this paper shows that P = NP follows from the assumption that every polynomial-time samplable distribution is polynomially equivalent to some polynomial-time computable distribution.