Abstract. The notion of p-selective sets, and tally languages, are used to study polynomial time reducibilities on NP. P-selectivity has the property that a set A belongs to the class P if and only if both A < e~A and A is p-selective. We prove that for every tally language set in NP there exists a polynomial time equivalent set in NP that is p-selective. From this result it follows that if NEXT=/= DEXT, then polynomial time Turing and many-one reducibilities differ on NP.
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