On ~-reducibility versus Polynomial Time Many-one Reducibility*

We prove that each element of a class of f,anctions (denoted by NPCtP), whose graphs can be accepted in nondeterministic polynomial time, can be evaluated in deterministic polynomial time if and only if '/-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 ,/-reduction can be replaced by a poiynomial time Turing reduction, then every function in NPCt p can be evaluated in deterministic polynomial time.