Ashish V. Naik

P-Selective Sets and Reducing Search to Decision vs Self-Reducibility

We distinguish self-reducibility of a languageLwith the question of whether search reduces to decision forL. Results include: (i) If NE?E, then there exists a setLin NP?P such that search reduces to decision forL, search doesnotnonadaptively reduce to decision forLandLis not self-reducible. (ii) If UE?E, then there exists a languageL?UP?P such that search nonadaptively reduces to decision for L, but L is not self-reducible. (iii) If UE?co-UE?E, then there is a disjunctive self-reducible languageL?UP?P for which search doesnotnonadaptively reduce to decision. We prove that if NE?BPE, then there is a languageL?NP?BPP such thatLis randomly self-reducible,notnonadaptively randomly self-reducible, andnotself-reducible. We obtain results concerning trade-offs in multiprover interactive proof systems and results that distinguish checkable languages from those that are nonadaptively checkable. Many of our results are proven by constructing p-selective sets. We obtain a p-selective set that isnot?Ptt-equivalent to any tally language, and we show that if P=PP, then every p-selective set is ?PT-equivalent to a tally language. Similarly, if P=NP, then every cheatable set is ?Pm-equivalent to a tally language. We construct a recursive p-selective tally set that isnotcheatable.

NONDETERMINISTICALLY SELECTIVE SETS

In this note, we study NP-selective sets (formally, sets that are selective via NPSVt functions) as a natural generalization of P-selective sets. We show that, assuming P≠NP∩coNP, the class of NP-selective sets properly contains the class of P-selective sets. We study several properties of NP-selective sets such as self-reducibility, hardness under various reductions, lowness, and nonuniform complexity. We prove many of our results via a “relativization technique,” by using the known properties of P-selective sets. Using this technique, we strengthen a result of Longpre and Selman on hard promise problems and show that the result “NP⊆(NP∩coNP)/poly⇒PH=NPNP” is implicit in Karp and Lipton’s seminal result on nonuniform classes.

On the Existence of Hard Sparse Sets under Weak Reductions

Recently a 1978 conjecture by Hartmanis was resolved by Cai and Sivakumar, following progress made by Ogihara. It was shown that there is no sparse set that is hard for P under logspace many-one reductions, unless P=LOGSPACE. We extend these results to the case of sparse sets that are hard under more general reducibilities. Furthermore, the proof technique can be applied to resolve open questions about hard sparse sets for NP as well. Using algebraic and probabilistic techniques, we show the following results.

Quasilinear Time Complexity Theory

This paper furthers the study of quasi-linear time complexity initiated by Schnorr [Sch76] and Gurevich and Shelah [GS89]. We show that the fundamental properties of the polynomial-time hierarchy carry over to the quasilineartime hierarchy. Whereas all previously known versions of the Valiant-Vazirani reduction from NP to parity run in quadratic time, we give a new construction using error-correcting codes that runs in quasilinear time. We show, however, that the important equivalence between search problems and decision problems in polynomial time is unlikely to carry over: if search reduces to decision for SAT in quasi-linear time, then all of NP is contained in quasi-polynomial time. Other connections to work by Stearns and Hunt [SH86, SH90, HS90] on “power indices” of NP languages are made.

Inverting onto functions

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.

P-selective sets, and reducing search to decision vs. self-reducibility

Several results that distinguish self-reducibility of a language L with the question of whether search reduces to decision for L are obtained. It is proved that if NE intersection co-NE not=E, then there exists a set L in NP-P such that search reduces to decision for L, search does not nonadaptively reduce to decision for L, and L is not self-reducible. Results that distinguish adaptively randomly self-reducible sets from nonadaptively randomly self-reducible sets, results concerning tradeoffs in multipower interactive proof systems, and results that distinguish checkable languages from those that are nonadaptively checkable are obtained. Many of the results depend on new techniques for constructing P-selective sets.<<ETX>>

A hierarchy based on output multiplicity

Abstract The class NP k V 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 NP k V 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 NP k V 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 P NP , 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.

On coherence, random-self-reducibility, and self-correction

Abstract. We study three types of self‐reducibility that are motivated by the theory of program verification. A set A is random‐self‐reducible if one can determine whether an input x is in A by making random queries to an A‐oracle. The distribution of each query may depend only on the length of x. A set B is self‐correctable over a distribution

Adaptive versus nonadaptive queries to NP and P-selective sets

{\cal D}

Computing Solutions Uniquely collapses the Polynomial Hierarchy

if one can convert a program that is correct on most of the probability mass of