John D. Rogers

TCS Journal 2003 Journal Article

Foreword

• André Berthiaume
• John D. Rogers

I&C Journal 2003 Journal Article

Inverting onto functions

• Stephen A. Fenner
• Lance Fortnow
• Ashish V. Naik
• John D. Rogers

We look at the hypothesis that all honest onto polynomial-time computable functions have a polynomial-time computable inverse. We show this hypothesis equivalent to several other complexity conjectures including: • In polynomial time, one can find accepting paths of nondeterministic polynomial-time Turing machines that accept Σ*. • Every total multivalued nondeterministic function has a polynomial-time computable refinement. • In polynomial time, one can compute satisfying assignments for any polynomial-time computable set of satisfiable formulae. • In polynomial time, one can convert the accepting computations of any nondeterministic Turing machine that accepts SAT to satisfying assignments. We compare these hypotheses with several other important complexity statements. We also examine the complexity of these statements where we only require a single bit instead of the entire inverse.

TCS Journal 1998 Journal Article

A hierarchy based on output multiplicity

• Ashish V. Naik
• John D. Rogers
• James S. Royer
• Alan L. Selman

The class NPkV consists of those partial, multivalued functions that can be computed by a nondeterministic, polynomial time-bounded transducer that has at most k distinct values on any input. We define the output-multiplicity hierarchy to consist of the collection of classes NPkV for all positive integers k ≥ 1. In this paper we investigate the strictness of the output-multiplicity hierarchy and establish three main results pertaining to this: 1. 1. If for any k > 1, the class NPkV collapses into the class NP(k − 1)V, then the polynomial hierarchy collapses to Σ 2 P. 2. 2. If the converse of the above result is true, then any proof of this converse cannot relativize. We exhibit an oracle relative to which the polynomial hierarchy collapses to PNP, but the output-multiplicity hierarchy is strict. 3. 3. Relative to a random oracle, the output-multiplicity hierarchy is strict. This result is in contrast to the still open problem of the strictness of the polynomial hierarchy relative to a random oracle. In introducing the technique for the third result we prove a related result of interest: relative to a random oracle UP ≠ NP.