We obtain several results that distinguish self-reducibility of a language L with the question of whether search reduces to decision for L. These include: (i) If NE 6= E, then there exists a set L in NP \Gamma P such that search reduces to decision for L, search does not nonadaptively reduces to decision for L, and L is not self-reducible. Funding for this research was provided by the National Science Foundation under grant CCR9002292. y Department of Computer Science, State University of New York at Buffalo, 226 Bell Hall, Buffalo, NY 14260 z Department of Computer Science, State University of New York at Buffalo, 226 Bell Hall, Buffalo, NY 14260 x Research performed while visiting the Department of Computer Science, State University of New York at Buffalo, Jan. 1992--Dec. 1992. Current address: Department of Computer Science, University of Electro-Communications, Chofu-shi, Tokyo 182, Japan. -- Department of Computer Science, State University of New York at Buffalo, 226...

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 ffl In polynomial time, one can find accepting paths of nondeterministic polynomial-time Turing machines that accept \Sigma . ffl Every total multivalued nondeterministic function has a polynomial-time computable refinement. ffl In polynomial time, one can compute satisfying assignments for any polynomial-time computable set of satisfiable formulae. ffl 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. 1 Introduction Understanding the power of nondeterminism has been one of the pri...

A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and non-deterministic double-exponential time are unequal) we construct a language in NP for which search does not reduce to decision. These ideas extend in a natural way to interactive proofs and program checking. Under similar assumptions we present languages in NP for which it is harder to prove membership interactively than it is to decide this membership, and languages in NP which are not checkable. Keywords: NP-completeness, self-reducibility, interactive proofs, program checking, sparse sets,

We show that any p-selective and self-reducible set is in P . As the converse is also true, we obtain a new characterization of the class P . A generalization and several consequences of this theorem are discussed. Among other consequences, we show that under reasonable assumptions auto-reducibility and self-reducibility differ on NP , and that there are non-p-T -mitotic sets in NP . 1 Introduction Separating complexity classes is a very popular, but rarely won game in complexity theory. Frustrated by misfortune, computer scientists have often turned to attempts of characterizing complexity classes in a different way. The hopes are, that the new characterization of the complexity class may provide new insights and a `handle' to force the separation where earlier attempts have failed. Well-known examples of this are the many ways to define the class of sets for which there exist small circuits [Pip79], and the identification of various forms of interactive proof systems with stan...

We settle all relativized questions of the relationships between the following ve propositions: P = NP P = UP P = NP \ coNP All disjoint pairs of NP sets are P-separable.

Can easy sets only have easy certificate schemes? In this paper, we study the class of sets that, for all NP certificate schemes (i.e., NP machines), always have easy acceptance certificates (i.e., accepting paths) that can be computed in polynomial time. We also study the class of sets that, for all NP certificate schemes, infinitely often have easy acceptance certificates. We give structural conditions that control the size of these classes. 1 Introduction Borodin and Demers [BD76] proved the following result. Theorem 1.1 [BD76] If NP " coNP 6= P, then there exists a set L such that 1. L 2 P, 2. L ` SAT, and 3. For no polynomial-time computable function f does it hold that: for each F 2 L, f(F ) outputs a satisfying assignment of F . That is, under a hypothesis most theoreticians would guess to be true, it follows that there is a set of satisfiable formulas for which it is trivial to determine they are satisfiable, yet it is hard to determine why (i.e., via what satisfying assignm...

this paper is a proof that NP-hard

This paper furthers the study of quasilinear time complexity initiated by Schnorr and Gurevich and Shelah. We show that the fundamental properties of the polynomial-time hierarchy carry over to the quasilinear-time hierarchy.

Recently, Watanabe proposed a new framework for testing the correctness and average case behavior of algorithms that purport to solve a given NP search problem efficiently on average. The idea is to randomly generate certified instances in a way that resembles the underlying distribution ¯. We discuss this approach and show that test instances can be generated for every NP search problem with non-adaptive queries to an NP oracle. Further, we introduce Las Vegas as well as Monte Carlo types of test instance generators and show that these generators can be used to find out whether an algorithm is correct and efficient on average under ¯. In fact, it is not hard to construct Monte Carlo generators for all RP search problems as well as Las Vegas generators for all ZPP search problems. On the other hand, we prove that Monte Carlo generators can only exist for problems in NP " co-AM. 1 Introduction The class NP plays a central role in computational complexity theory since it contains a larg...

. We study the sparse set conjecture for sets with low density. The sparse set conjecture states that P = NP if and only if there exists a sparse Turing hard set for NP . In this paper we study a weaker variant of the conjecture. We are interested in the consequences of NP having Turing hard sets of density f(n), for (unbounded) functions f(n), that are sub-polynomial, for example log(n). We establish a connection between Turing hard sets for NP with density f(n) and bounded nondeterminism: We prove that if NP has a Turing hard set of density f(n), then satisfiability is computable in polynomial time with O(log(n) f(n c )) many nondeterministic bits for some constant c. As a consequence of the proof technique we obtain absolute results about the density of Turing hard sets for EXP . We show that no Turing hard set for EXP can have sub-polynomial density. On the other hand we show that these results are optimal w.r.t. relativizing computations. For unbounded functions f(...