this paper we are interested in absolute results and consider advice classes slightly smaller than P=poly and circuit classes smaller than polynomial-size. And we establish several new lower bounds for exponential time problems with respect to these classes. This is not a new tack to explore. In last years' Structures Bin Fu [Fu93] considered lower bounds for polynomial time reductions to sparse sets, where limits are placed on the number of queries to the sparse set. The main result of his paper was that there are sets in EXP which are not polynomial time Turing reducible to a sparse set when the reduction is restricted to querying the sparse set no more than n

We prove unlikely consequences of the existence of sparse hard sets for P under deterministic as well as one-sided error randomized truth-table reductions. Our main results are as follows. We establish that the existence of a polynomially dense hard set for P under (randomized) logspace bounded truth-table reductions implies that P ` (R)L, and that the collapse goes down to P ` (R)NC 1 in case of reductions computable in (R)NC 1 . We also prove that the existence of a quasipolynomially dense hard set for P under (randomized) polylog-space truth-table reductions using polylogarithmically many queries implies that P ` (R)SPACE[polylogn]. The randomized space complexity classes we consider are based on the multiple access randomness concept. 1 Introduction A lot of research effort in complexity theory has been spent on the sparse hard set problem for NP, i.e., the question whether there are sparse hard sets for NP under various polynomial-time reducibilities. Two major motivations ...

Derandomization techniques are used to show that at least one of the following holds regarding the size of the counting complexity class SPP. 1. µp(SPP) = 0. 2. PH ⊆ SPP. In other words, SPP is small by being a negligible subset of exponential time or large by containing the entire polynomial-time hierarchy. This addresses an open problem about the complexity of the graph isomorphism problem: it is not weakly complete for exponential time unless PH is contained in SPP. It is also shown that the polynomial-time hierarchy is contained in SPP NP if NP does not have p-measure 0. 1

The existence of cryptographically secure one-way functions is related to the measure of a subclass of NP. This subclass, called fiNP ("balanced NP"), contains 3SAT and other standard NP problems. The hypothesis that fiNP is not a subset of P is equivalent to the P 6= NP conjecture. A stronger hypothesis, that fiNP is not a measure 0 subset of E 2 = DTIME(2 polynomial ) is shown to have the following two consequences. 1. For every k, there is a polynomial time computable, honest function f that is (2 n k =n k )-one-way with exponential security. (That is, no 2 n k -time-bounded algorithm with n k bits of nonuniform advice inverts f on more than an exponentially small set of inputs. ) 2. If DTIME(2 n ) "separates all BPP pairs," then there is a (polynomial time computable) pseudorandom generator that passes all probabilistic polynomial-time statistical tests. (This result is a partial converse of Yao, Boppana, and Hirschfeld's theorem, that the existence of pseudorandom ge...

Assuming that k 2 and \Delta P k does not have p-measure 0, it is shown that BP \Delta \Delta P k = \Delta P k . This implies that the following conditions hold if \Delta P 2 does not have p-measure 0. (i) AM " co-AM is low for \Delta P 2 . (Thus BPP and the graph isomorphism problem are low for \Delta P 2 .) (ii) If \Delta P 2 6= PH, then NP does not have polynomial-size circuits. This research was supported in part by National Science Foundation Grant CCR9157382, with matching funds from Rockwell International, Microware Systems Corporation, and Amoco Foundation. 1 Introduction Many widely believed conjectures in computational complexity are "strong" in the sense that they are known to imply that P 6= NP, but are not known to follow from the P 6= NP hypothesis. Recent investigations have shown that a number of these conjectures do follow from the (apparently) stronger hypothesis that NP does not have p-measure 0. (This hypothesis, written ¯ p (NP) 6= 0, is defined in...

We investigate the frequency of complete sets for various complexity classes within EXP under non-adaptive reductions in the sense of resource bounded measure. We show that these sets are rare: ffl The sets that are complete under 6 p n ff \Gammatt -reductions for NP, the levels of the polynomial-time hierarchy, PSPACE, and EXP have p 2 - measure zero for any constant ff ! 1. ffl Assuming MA 6= EXP, the 6 p tt -complete sets for PSPACE and the \Delta-levels of the polynomial-time hierarchy have p-measure zero. A key ingredient is the Small Span Theorem, which states that for any set A in EXP at least one of its lower span (i.e., the sets that reduce to A) or its upper span (i.e., the sets that A reduces to) has p 2 -measure zero. Previous to our work, the theorem was only known to hold for 6 p k\Gammatt -reductions for any constant k. We establish it for 6 p n o(1) \Gammatt -reductions. 1 Introduction Lutz introduced resource bounded measure [Lut90] to formalize the notions ...

Let PSel be the class of all P-selective sets. We show that DTIME(2 n k+a ) 6` P n k \GammaT (PSel) for all k; a ? 0. It implies EXP 6` P n c \GammaT (PSel) for all c ? 0. This greatly improves Toda's result that EXP 6` P tt (PSel) since P tt (PSel) is equal to P O(logn)\GammaT (PSel). We construct an oracle A such that DTIME(2 n k ) A ` P A n k+1+a -T (PSel). We also show that the symmetric difference of a E- P m -hard set and a P-selective set is a E- P 2\GammaT -hard. This generalizes a result by Rao [18] who showed the symmetric difference of a E- P m -hard set and a P-selective set is exponentially dense. Symbols used in the paper: a; b; c; d; e; f; g; h; i; j; k; l; m; n; o; p; q; r; s; t; u; v; w; x; y; z A; B;C;D;E;F; G; H; I; J; K; L; M;N;O;P; Q; R; S; T; U; V; W;X;Y;Z [; "; \Phi; \Gamma; \Sigma; ffi; oe; fi ; ø; 9; 8; ; ß; 4;k; [; ](; ); f:g; ();=); \Gamma!; 6=; =; 2; `; ;; ; 6`; 8; 1. Introduction The study of reducibilities of complexity classes t...

Results of the kind "Almost every oracle in exponential space separates P from NP" or "Almost every set in exponential time is P-bi-immune" can be precisely formulated via a new approach in Structural Complexity recently introduced by Lutz. He defines a resource-bounded measure in exponential time and space classes that generalizes Lebesgue measure, a powerful mathematical tool. We investigate here the possibility of extending this resource-bounded measure to other classes, mainly PSPACE. We prove here that the natural candidate of a resource bound for measuring in PSPACE is not valid unless some unlikely consequences are true. We then obtain a weaker way of measuring in PSPACE that lacks a property that resource-bounded measure has in bigger classes. 1 Introduction Resource-bounded measure was introduced by Lutz in [3] and [4]. It deals with complexity classes within exponential time or space, distinguishing between "large" and "small" classes. This method generalizes a powerful math...

Introduction Investigation of the measure-theoretic structure of complexity classes began with the development of resource-bounded measure in 1991 [56]. Since that time, a growing body of research by more than forty scientists around the world has shown resource-bounded measure to be a powerful tool that sheds new light on many aspects of computational complexity. Recent survey papers by Lutz [60], Ambos-Spies and Mayordomo [3], and Buhrman and Torenvliet [22] describe many of the achievements of this line of inquiry. In this column, we give a more recent snapshot of resource-bounded measure, focusing not so much on what has been achieved to date as on what we hope will be achieved in the near future. Section 2 below gives a brief, nontechnical overview of resource-bounded measure in terms of its motivation and principal ideas. Sections 3, 4, and 5 describe twelve specific open problems in the area. We have used the following three criteria in choosing these problems. 1. Their

We extend Lutz's measure to probabilistic classes, and obtain notions of measure on probabilistic complexity classes C such as BPP, BPE and BPEXP. Unlike former attempts, all our measure notions satisfy all three Lutz's measure axioms, that is every singleton fLg has measure zero in C, the whole space C has measure one in C, and "easy infinite unions" of measure zero sets have measure zero. Finally we prove a conditional time hierarchy theorem for probabilistic classes, and show that under the same assumption, both the class of T -autoreducible sets and the class of T -complete sets for EXP have measure zero in BPE.