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42
Dimension in Complexity Classes
 SIAM Journal on Computing
, 2000
"... A theory of resourcebounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound (a parameter of the theory) is unrestricted, the resulting dimension is precisely the classical Hausdorff dimension (sometimes called "fractal dimension&qu ..."
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Cited by 102 (16 self)
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A theory of resourcebounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound (a parameter of the theory) is unrestricted, the resulting dimension is precisely the classical Hausdorff dimension (sometimes called "fractal dimension"). Other choices of the parameter yield internal dimension theories in E, E 2 , ESPACE, and other complexity classes, and in the class of all decidable problems. In general, if C is such a class, then every set X of languages has a dimension in C, which is a real number dim(X j C) 2 [0; 1]. Along with the elements of this theory, two preliminary applications are presented: 1. For every real number 0 1 2 , the set FREQ( ), consisting of all languages that asymptotically contain at most of all strings, has dimension H()  the binary entropy of  in E and in E 2 . 2. For every real number 0 1, the set SIZE( 2 n n ), consisting of all languages decidable by Boolean circuits of at most 2 n n gates, has dimension in ESPACE.
Resourcebounded measure
 In Proceedings of the 13th IEEE Conference on Computational Complexity
, 1998
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Computational Randomness and Lowness
, 2001
"... . We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0 0 . This contrasts with a result of Kucera and Terwijn [5] on sets tha ..."
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Cited by 30 (1 self)
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. We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0 0 . This contrasts with a result of Kucera and Terwijn [5] on sets that are low for the class of MartinLof random reals. The Cantor space 2 ! is the set of infinite binary sequences; these are called reals and are identified with subsets of !. If oe 2 2 !! , that is, oe is a finite binary sequence, we denote by [oe] the set of reals that extend oe. These form a basis of clopen sets for the usual discrete topology on 2 ! . Write joej for the length of oe 2 2 !! . The Lebesgue measure on 2 ! is defined by stipulating that [oe] = 2 \Gammajoej . With every set U ` 2 !! we associate the open set S oe2U [oe]. When it is convenient, we confuse U with the open set associated to it, in particular we write U for the measure of the open set correspondi...
Algorithmic Randomness and Lowness
, 1997
"... . We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0 0 . This contrasts with a result of Kucera and Terwijn [2] on sets tha ..."
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Cited by 17 (2 self)
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. We prove that there are uncountably many sets that are low for the class of Schnorr random reals. We give a purely recursion theoretic characterization of these sets and show that they all have Turing degree incomparable to 0 0 . This contrasts with a result of Kucera and Terwijn [2] on sets that are low for the class of MartinLof random reals. The Cantor space 2 ! is the set of infinite binary sequences; these are called reals and are identified with subsets of !. If oe 2 2 !! , that is, oe is a finite binary sequence, we denote by [oe] the set of reals that extend oe. These form a basis of clopens for the usual discrete topology on 2 ! . Write joej for the length of oe 2 2 !! . The Lebesgue measure ¯ on 2 ! is defined by stipulating that ¯[oe] = 2 \Gammajoej . With every set U ` 2 !! we associate the open set S oe2U [oe]. When it is convenient, we confuse U with the open set associated to it, in particular we write ¯U for the measure of the open set correspondin...
Turing’s unpublished algorithm for normal numbers
"... In an unpublished manuscript Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he gave an algorithm for computing instances in this set. We complete his manuscript by giving full proofs and correcting minor er ..."
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Cited by 16 (4 self)
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In an unpublished manuscript Alan Turing gave a computable construction to show that absolutely normal real numbers between 0 and 1 have Lebesgue measure 1; furthermore, he gave an algorithm for computing instances in this set. We complete his manuscript by giving full proofs and correcting minor errors. While doing this, we recreate Turing’s ideas as accurately as possible. One of his original lemmas remained unproved but we have replaced it with a weaker lemma that still allows us to maintain Turing’s proof idea and obtain his result. 1
On the Autoreducibility of Random Sequences
, 2001
"... A binary sequence A = A(0)A(1) ... is called i.o. Turingautoreducible if A is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either A(x) or a don'tknow symbol on any given input x, and outputs A(x) for infinitely many x. If in addi ..."
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Cited by 16 (1 self)
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A binary sequence A = A(0)A(1) ... is called i.o. Turingautoreducible if A is reducible to itself via an oracle Turing machine that never queries its oracle at the current input, outputs either A(x) or a don'tknow symbol on any given input x, and outputs A(x) for infinitely many x. If in addition the oracle Turing machine terminates on all inputs and oracles, A is called i.o. truthtableautoreducible.
The KolmogorovLoveland stochastic sequences are not closed under selecting subsequences
 Journal of Symbolic Logic
, 2002
"... It is shown that the class of KolmogorovLoveland stochastic sequences is not closed under selecting subsequences by monotonic computable selection rules. This result gives a strong negative answer to the notorious open problem whether the KolmogorovLoveland stochastic sequences are closed unde ..."
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Cited by 15 (5 self)
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It is shown that the class of KolmogorovLoveland stochastic sequences is not closed under selecting subsequences by monotonic computable selection rules. This result gives a strong negative answer to the notorious open problem whether the KolmogorovLoveland stochastic sequences are closed under selecting subsequences by KolmogorovLoveland selection rules, i.e., by not necessarily monotonic partially computable selection rules. As a corollary, we obtain an easy proof for the previously known result that the KolmogorovLoveland stochastic sequences form a proper subclass of the MisesWaldChurch stochastic sequences.
Hardness hypotheses, derandomization, and circuit complexity
"... We consider hypotheses about nondeterministic computation that have been studied in different contexts and shown to have interesting consequences: • The measure hypothesis: NP does not have pmeasure 0. • The pseudoNP hypothesis: there is an NP language that can be distinguished from any DTIME(2nǫ) ..."
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Cited by 14 (5 self)
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We consider hypotheses about nondeterministic computation that have been studied in different contexts and shown to have interesting consequences: • The measure hypothesis: NP does not have pmeasure 0. • The pseudoNP hypothesis: there is an NP language that can be distinguished from any DTIME(2nǫ) language by an NP refuter. • The NPmachine hypothesis: there is an NP machine accepting 0 ∗ for which no 2nǫtime machine can find infinitely many accepting computations. We show that the NPmachine hypothesis is implied by each of the first two. Previously, no relationships were known among these three hypotheses. Moreover, we unify previous work by showing that several derandomizations and circuitsize lower bounds that are known to follow from the first two hypotheses also follow from the NPmachine hypothesis. In particular, the NPmachine hypothesis becomes the weakest known uniform hardness hypothesis that derandomizes AM. We also consider UP versions of the above hypotheses as well as related immunity and scaled dimension hypotheses.