The resource-bounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number ffi ? 0, any uniformly polynomial-time computable sequence ~ fi = (fi 0 ; fi 1 ; fi 2 ; : : : ) of real numbers (biases) fi i 2 [ffi; 1 \Gamma ffi], and any complexity class C (such as P, NP, BPP, P/Poly, PH, PSPACE, etc.) that is closed under positive, polynomial-time, truth-table reductions with queries of at most linear length, it is shown that the following two conditions are equivalent. (1) C has p-measure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the coin-toss probability measure given by the sequence ~ fi. (2) C has p-measure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the uniform probability measure. The proof introduces three techniques that may be useful in other contexts, namely, (i) the transformation of an efficient martingale for one probability measu...

In this paper we extend a key result of Nisan and Wigderson [17] to the nondeterministic setting: for all ff ? 0 we show that if there is a language in E = DTIME(2 O(n) ) that is hard to approximate by nondeterministic circuits of size 2 ffn , then there is a pseudorandom generator that can be used to derandomize BP \Delta NP (in symbols, BP \Delta NP = NP). By applying this extension we are able to answer some open questions in [14] regarding the derandomization of the classes BP \Delta \Sigma P k and BP \Delta \Theta P k under plausible measure theoretic assumptions. As a consequence, if \Theta P 2 does not have p-measure 0, then AM " coAM is low for \Theta P 2 . Thus, in this case, the graph isomorphism problem is low for \Theta P 2 . By using the NisanWigderson design of a pseudorandom generator we unconditionally show the inclusion MA ` ZPP NP and that MA " coMA is low for ZPP NP . 1 Introduction In recent years, following the development of resource-bounded meas...

We consider the question whether two queries to SAT are as powerful as one query. We show that if P NP�℄� P NP�℄then Locally either NP�coNP or NP has polynomial-size circuits.

Resource-bounded measure theory [7] is a study of complexity classes via an adaptation of the probabilistic method. The central hypothesis in this theory is the assertion that NP does not have measure zero in Exponential Time. This is a quantitative strengthening of NP 6= P.

. We construct an oracle relative to which NP has p-measure 0 but D p has measure 1 in EXP. This gives a strong relativized negative answer to a question posed by Lutz [Lut96]. Secondly, we give strong evidence that BPP is small. We show that BPP has p-measure 0 unless EXP = MA and thus the polynomial-time hierarchy collapses. This contrasts with the work of Regan et. al. [RSC95], where it is shown that P=poly does not have p-measure 0 if exponentially strong pseudorandom generators exist. 1 Introduction Since the introduction of resource-bounded measure by Lutz [Lut92], many researchers investigated the size (measure) of complexity classes in exponential time (EXP). A particular point of interest is the hypothesis that NP does not have p-measure 0. Recent results have shown that many reasonable conjectures in computational complexity theory follow from the hypothesis that NP is not small (i.e., ¯ p (NP) 6= 0), and hence it seems to be a plausible scientific hypothesis [LM96, Lut96...

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 ...

We construct an oracle relative to which NP has p-measure 0 but D p has measure 1 in EXP. This gives a strong relativized negative answer to a question posed by Lutz [Lut96]. Secondly, we give strong evidence that BPP is small. We show that BPP has p-measure 0 unless EXP = MA and the polynomial-time hierarchy collapses. This contrasts the work of Regan et al. [RSC95], where it is shown that P=poly does not have p-measure 0 unless strong pseudorandom generators do not exist. 1 Introduction Since the introduction of resource-bounded measure by Lutz [Lut92], many researchers investigated the size (measure) of complexity classes in exponential time (EXP). A particular point of interest is the hypothesis that NP does not have p-measure 0. Recent results have shown that many reasonable conjectures in computational complexity theory follow from the hypothesis that NP is not small (i.e., ¯ p (NP) 6= 0), and hence it seems to be a plausible scientific hypothesis [LM96, Lut96]. In [Lut96], L...