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The quantitative structure of exponential time
 Complexity theory retrospective II
, 1997
"... ABSTRACT Recent results on the internal, measuretheoretic structure of the exponential time complexity classes E and EXP are surveyed. The measure structure of these classes is seen to interact in informative ways with biimmunity, complexity cores, polynomialtime reductions, completeness, circuit ..."
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Cited by 90 (13 self)
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ABSTRACT Recent results on the internal, measuretheoretic structure of the exponential time complexity classes E and EXP are surveyed. The measure structure of these classes is seen to interact in informative ways with biimmunity, complexity cores, polynomialtime reductions, completeness, circuitsize complexity, Kolmogorov complexity, natural proofs, pseudorandom generators, the density of hard languages, randomized complexity, and lowness. Possible implications for the structure of NP are also discussed. 1
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 (19 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...