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.

Abstract. We use hypotheses of structural complexity theory to separate various NP-completeness notions. In particular, we introduce an hypothesis from which we describe a set in NP that is ¡ P T-complete but not ¡ P tt-complete. We provide fairly thorough analyses of the hypotheses that we introduce. Key words. Turing completeness, truth-table completeness, many-one completeness, p-selectivity, p-genericity AMS subject classifications. 1. Introduction. Ladner, Lynch, and Selman [LLS75] were the first to compare the strength of polyno-), truth-), that mial-time reducibilities. They showed, for the common polynomial-time reducibilities, ( ¢ Turing P T ( ¢ table P tt), bounded truth-table ( ¢ P btt), and many-one ( ¢ P m

Abstract We consider hypotheses about nondeterministic computation that have been studied in dif-ferent contexts and shown to have interesting consequences: * The measure hypothesis: NP does not have p-measure 0.* The pseudo-NP hypothesis: there is an NP language that can be distinguished from anyDTIME(2 nffl) language by an NP refuter. * The NP-machine hypothesis: there is an NP machine accepting 0 * for which no 2n ffl-time machine can find infinitely many accepting computations. We show that the NP-machine hypothesis is implied by each of the first two. Previously, norelationships were known among these three hypotheses. Moreover, we unify previous work by showing that several derandomizations and circuit-size lower bounds that are known to followfrom the first two hypotheses also follow from the NP-machine hypothesis. In particular, the NPmachine hypothesis becomes the weakest known uniform hardness hypothesis that derandomizesAM. We also consider UP versions of the above hypotheses as well as related immunity and scaled dimension hypotheses. 1 Introduction The following uniform hardness hypotheses are known to imply full derandomization of ArthurMerlin games (NP = AM): * The measure hypothesis: NP does not have p-measure 0 [24].

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...

We show how to construct proof systems for NP languages where a deterministic polynomialtime verifier can check membership, given any N (2=3)+ffl bits of an N -bit witness of membership.

Under the assumption that NP does not have p-measure 0, we investigate reductions to NP-complete sets and prove the following: (1) Adaptive reductions are more powerful than nonadaptive reductions: there is a problem that is Turing-complete for NP but not truth-table-complete. (2) Strong nondeterministic reductions are more powerful than deterministic reductions: there is a problem that is SNP-complete for NP but not Turingcomplete. (3) Every problem that is many-one complete for NP is complete under lengthincreasing reductions that are computed by polynomial-size circuits. The first item solves one of Lutz and Mayordomo’s “Twelve Problems in Resource-Bounded Measure ” (1999). We also show that every many-one complete problem for NE is complete under one-to-one, length-increasing reductions that are computed by polynomial-size circuits. 1

We study several properties of sets that are complete for NP. We prove that if L is an NP-complete set and S � ⊇ L is a p-selective sparse set, then L − S is ≤p m-hard for NP. We demonstrate existence of a sparse set S ∈ DTIME(22n) such that for every L ∈ NP − P, L − S is not ≤p m-hard for NP. Moreover, we prove for every L ∈ NP − P, that there exists a sparse S ∈ EXP such that L − S is not ≤ p m-hard for NP. Hence, removing sparse information in P from a complete set leaves the set complete, while removing sparse information in EXP from a complete set may destroy its completeness. Previously, these properties were known only for exponential time complexity classes. We use hypotheses about pseudorandom generators and secure one-way permutations to derive consequences for longstanding open questions about whether NP-complete sets are immune. For example, assuming that pseudorandom generators and secure one-way permutations exist, it follows easily that NP-complete sets are not p-immune. Assuming only that secure one-way permutations exist, we prove that no NP-complete set is DTIME(2nɛ)-immune. Also, using these hypotheses we show that no NPcomplete set is quasipolynomial-close to P. We introduce a strong but reasonable hypothesis and infer from it that disjoint Turing-complete sets for NP are not closed under union. Our hypothesis asserts existence of a UP-machine M that accepts 0 ∗ such that for some 0 < ɛ < 1, no 2nɛ time-bounded machine can correctly compute infinitely many accepting computations of M. We show that if UP∩coUP contains DTIME(2nɛ)-bi-immune sets, then this hypothesis is true.

A desirable property of one-way functions is that they be total, one-to-one, and onto—in other words, that they be permutations. We prove that one-way permutations exist exactly if PaUP-coUP: This provides the first characterization of the existence of one-way permutations based on a complexity-class separation and shows that their existence is equivalent to a number of previously studied complexitytheoretic hypotheses. We study permutations in the context of witness functions of nondeterministic Turing machines. A language is in PermUP if, relative to some unambiguous, nondeterministic, polynomial-time Turingmachine acceptingthe language, the function mappingeach stringto its unique witness is a permutation of the members of the language. We show that, under standard complexity-theoretic assumptions, PermUP is a strict subset of UP. We study SelfNP, the set of all languages such that, relative to some nondeterministic, polynomial-time Turing machine that accepts the language, the set of all witnesses of strings in the language is identical to the language itself. We show that SATASelfNP; and, under standard complexity-theoretic assumptions, SelfNPaNP:

We show the following new lowness results for the probabilistic class ZPP NP . { The class AM \ coAM is low for ZPP NP . As a consequence it follows that Graph Isomorphism and several group-theoretic problems known to be in AM \ coAM are low for ZPP NP . { The class IP[P=poly], consisting of sets that have interactive proof systems with honest provers in P=poly, is also low for ZPP NP . We consider lowness properties of nonuniform function classes, namely, NPMV=poly, NPSV=poly, NPMV t =poly, and NPSV t =poly. Specifically, we show that { Sets whose characteristic functions are in NPSV=poly and that have program checkers (in the sense of Blum and Kannan [8]) are low for AM and ZPP NP . { Sets whose characteristic functions are in NPMV t =poly are low for p 2 .

We study whether one can prune solutions from NP functions. Though it is known that, unless surprising complexity class collapses occur, one cannot reduce the number of accepting paths of NP machines [OH93], we nonetheless show that it often is possible to reduce the number of solutions of NP functions. For finite cardinality types, we give a sufficient condition for such solution reduction. We also give absolute and conditional necessary conditions for solution reduction, and in particular we show that in many cases solution reduction is impossible unless the polynomial hierarchy collapses.