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Equivalence of Measures of Complexity Classes
"... The resourcebounded 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 polynomialtime computable sequence ~ fi = (fi 0 ; fi 1 ; fi 2 ; : : : ) of real numbers (biases ..."
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Cited by 70 (21 self)
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The resourcebounded 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 polynomialtime 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, polynomialtime, truthtable reductions with queries of at most linear length, it is shown that the following two conditions are equivalent. (1) C has pmeasure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the cointoss probability measure given by the sequence ~ fi. (2) C has pmeasure 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...
On Pseudorandomness and ResourceBounded Measure
 Theoretical Computer Science
, 1997
"... 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 u ..."
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Cited by 46 (3 self)
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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 pmeasure 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 resourcebounded meas...
Two queries
 In CCC
, 1999
"... 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 polynomialsize circuits. ..."
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Cited by 32 (7 self)
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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 polynomialsize circuits.
Constant Depth Circuits and the Lutz Hypothesis
"... Resourcebounded 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. ..."
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Cited by 7 (2 self)
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Resourcebounded 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.
Results on ResourceBounded Measure
, 1997
"... . We construct an oracle relative to which NP has pmeasure 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 pmeasure 0 unless EXP = MA and thus the polyn ..."
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. We construct an oracle relative to which NP has pmeasure 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 pmeasure 0 unless EXP = MA and thus the polynomialtime hierarchy collapses. This contrasts with the work of Regan et. al. [RSC95], where it is shown that P=poly does not have pmeasure 0 if exponentially strong pseudorandom generators exist. 1 Introduction Since the introduction of resourcebounded 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 pmeasure 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...
Complete Sets under NonAdaptive Reductions are Scarce
, 1997
"... We investigate the frequency of complete sets for various complexity classes within EXP under nonadaptive 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 ..."
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We investigate the frequency of complete sets for various complexity classes within EXP under nonadaptive 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 polynomialtime 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 \Deltalevels of the polynomialtime hierarchy have pmeasure 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 ...
Category, Measure, Inductive Inference: A Triality Theorem and its Applications
, 1997
"... The famous SierpinskiErdos Duality Theorem [Sie34b, Erd43] states, informally, that any theorem about effective measure 0 and/or first category sets is also true when all occurrences of "effective measure 0" are replaced by "first category" and vice versa. This powerful and nice ..."
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The famous SierpinskiErdos Duality Theorem [Sie34b, Erd43] states, informally, that any theorem about effective measure 0 and/or first category sets is also true when all occurrences of "effective measure 0" are replaced by "first category" and vice versa. This powerful and nice result shows that "measure" and "category" are equally useful notions neither of which can be preferred to the other one when making formal the intuitive notion "almost all sets." Effective versions of measure and category are used in recursive function theory and related areas, and resourcebounded versions of the same notions are used in Theory of Computation. Again they are dual in the same sense. We show that in the world of recursive functions there is a third equipotent notion dual to both measure and category. This new notion is related to learnability (also known as inductive inference or identifiability). We use the term "triality" to describe this threeparty duality. 1 Introduction Mathematicians h...
Two Results on ResourceBounded Measure
, 1996
"... We construct an oracle relative to which NP has pmeasure 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 pmeasure 0 unless EXP = MA and the polynomialti ..."
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We construct an oracle relative to which NP has pmeasure 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 pmeasure 0 unless EXP = MA and the polynomialtime hierarchy collapses. This contrasts the work of Regan et al. [RSC95], where it is shown that P=poly does not have pmeasure 0 unless strong pseudorandom generators do not exist. 1 Introduction Since the introduction of resourcebounded 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 pmeasure 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...