Turning machines with few accepting computations and low sets for PP

The authors study two different ways to restrict the power of NP. They consider languages accepted by nondeterministic polynomial-time machines with a small number of accepting paths in case of acceptance, and they investigate three subclasses of NP that are low for complexity classes not known to be in the polynomial-time hierarchy. The subclasses, UP, FewP, and Few, are all defined in terms of nondeterministic machines with a bounded number of accepting paths for every input string, but for the last two classes this number is not known beforehand and can range over a space of polynomial size. The authors prove lowness properties of the class Few and some other interesting sets that are low for the class PP. The lowness results are used to obtain positive relativizations of complexity classes