Authors: Richard Beigel
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
We show that all NP-hard sets (under ≤ptt-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 p-superterse 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 question of Amir & Gasarch (Structures '87).
Allowing the number of queries q to be a function of the input length n, we obtain nontrivial partial answers to a question posed by Krentel: Namely, for all NP-complete sets A and all sets X,
Finally, we define pbtt-cylinders, a natural class that includes all ≤pm-complete sets for NP, PSPACE, and exponential time. For pbtt-cylinders, we show that p-terseness is equivalent to p-superterseness. We also prove a translational lemma for non-p-terse sets; this allows us to extend the preceding result to ≤pbtt-complete sets for NP, PSPACE, and exponential time.
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