Abstract

The class UP is the class of languages in NP that are accepted by machines with at most one accepting path on each input. We define U k(n) P as the class of languages in NP that are accepted by machines with at most k(n) accepting paths on each input of length n. Obviously P ` UP ` U k(n) P ` U k(n)+1 P ` FewP ` NP for every polynomial k(n) 2, where FewP is the class of languages in NP that are accepted by machines with a polynomial-bounded number of accepting paths on each input. It is an open question whether any of the containments is proper. A proper containment would imply P 6= NP. We consider the relativized class U k(n) P A , and we construct an oracle A such that P A ae UP A ae U k(n) P A ae U k(n)+1 P A ae FewP A ae NP A for every polynomial k(n) 2. Relative to a random oracle A, we prove that P A ae UP A ae U k P A ae U k+1 P A ae FewP A for every constant k 2, and we prove similar separations when k(n) is not a constant. Previously, Hartmanis and H...

Citations