BibTeX
@MISC{Beigel92np-hardsets,
author = {Richard Beigel},
title = {NP-hard Sets are P-Superterse Unless R = NP},
year = {1992}
}
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Abstract
A set A is p-terse (p-superterse) if, for all q, it is not possible to answer q queries to A by making only q \Gamma 1 queries to A (any set X). Formally, let PF A q-tt be the class of functions reducible to A via a polynomial-time truthtable reduction of norm q, and let PF A q-T be the class of functions reducible to A via a polynomial-time Turing reduction that makes at most q queries. A set A is p-terse if PF A q-tt 6` PF A (q\Gamma1)-T for all constants q. A is p-superterse if PF A q-tt 6` PF X q-T for all constants q and sets X . We show that all NP-hard sets (under p tt -reductions) are p-superterse, unless it is possible to distinguish uniquely satisfiable formulas from satisfiable formulas in polynomial time. Consequently, all NP-complete sets are psuperterse unless P = UP (one-way functions fail to exist), R = NP (there exist randomized polynomial-time algorithms for all problems in NP), and the polynomial-time hierarchy collapses. This mostly solves the main open...
