On &ggr;-reducibility versus polynomial time many-one reducibility(Extended Abstract)

We prove that a class of functions (denoted by NPCPt), whose graphs can be accepted in non-deterministic polynomial time, can be evaluated in deterministic polynomial time if and only if &ggr;-reducibility is equivalent to polynomial time many-one reducibility. We also modify the proof technique used to obtain part of this result to obtain the stronger result that if every &ggr;-reduction can be replaced by a polynomial time Turing reduction then every function in NPCPt can be evaluated in deterministic polynomial time.