On the power of generalized MOD-classes

The computational power of the counting class ModP, which generalizes the classes Mod_pP, p prime, is investigated. It is shown that ModP is truth-table equivalent in power to MP, and that ModP is contained in the class AmpMP. As a consequence, the lowness of AmpMP or of ModP for MP would imply the collapse of the counting hierarchy (CH) to MP. Further, every set in C=P is shown to be reducible to a ModP set via a random many-one reduction that uses only logarithmically many random bits. Therefore, ModP and AmpMP are not closed under such reductions unless CH collapses. Finally, ModP is generalized to the class Mod_*P, which turns out to be a superclass of C=P