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Lowness Properties and Randomness
 ADVANCES IN MATHEMATICS
"... The set A is low for MartinLof random if each random set is already random relative to A. A is Ktrivial if the prefix complexity K of each initial segment of A is minimal, namely K(n)+O(1). We show that these classes coincide. This implies answers to questions of AmbosSpies and Kucera [2 ..."
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Cited by 79 (21 self)
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The set A is low for MartinLof random if each random set is already random relative to A. A is Ktrivial if the prefix complexity K of each initial segment of A is minimal, namely K(n)+O(1). We show that these classes coincide. This implies answers to questions of AmbosSpies and Kucera [2], showing that each low for MartinLof random set is # 2 . Our class induces a natural intermediate # 3 ideal in the r.e. Turing degrees (which generates the whole class under downward closure). Answering
Hierarchies Of Generalized Kolmogorov Complexities And Nonenumerable Universal Measures Computable In The Limit
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
, 2000
"... The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with oneway writeonly output tape. This naturally leads to the universal enumerable SolomonoLevin measure. Here we introduce more general, nonenumerable but cumulatively enumerable m ..."
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Cited by 38 (20 self)
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The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with oneway writeonly output tape. This naturally leads to the universal enumerable SolomonoLevin measure. Here we introduce more general, nonenumerable but cumulatively enumerable measures (CEMs) derived from Turing machines with lexicographically nondecreasing output and random input, and even more general approximable measures and distributions computable in the limit. We obtain a natural hierarchy of generalizations of algorithmic probability and Kolmogorov complexity, suggesting that the "true" information content of some (possibly in nite) bitstring x is the size of the shortest nonhalting program that converges to x and nothing but x on a Turing machine that can edit its previous outputs. Among other things we show that there are objects computable in the limit yet more random than Chaitin's "number of wisdom" Omega, that any approximable measure of x is small for any x lacking a short description, that there is no universal approximable distribution, that there is a universal CEM, and that any nonenumerable CEM of x is small for any x lacking a short enumerating program. We briey mention consequences for universes sampled from such priors.
Randomness in effective descriptive set theory
 London. Math. Soc
"... Abstract. An analog of MLrandomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π1 1 counterparts. We prove the analogs of the KraftChaitin Theorem and Schnorr’s Theorem. In the new setting, while Ktrivial sets exist that are not hyper ..."
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Cited by 9 (3 self)
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Abstract. An analog of MLrandomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π1 1 counterparts. We prove the analogs of the KraftChaitin Theorem and Schnorr’s Theorem. In the new setting, while Ktrivial sets exist that are not hyperarithmetical, each low for random set is. Finally we study a very strong yet effective randomness notion: Z is strongly random if Z is in no null Π1 1 set of reals. We show that there is a greatest Π1 1 null set, that is, a universal test for this notion. 1.
Eliminating concepts
 Proceedings of the IMS workshop on computational prospects of infinity
, 2008
"... Four classes of sets have been introduced independently by various researchers: low for K, low for MLrandomness, basis for MLrandomness and Ktrivial. They are all equal. This survey serves as an introduction to these coincidence results, obtained in [24] and [10]. The focus is on providing backdo ..."
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Cited by 5 (2 self)
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Four classes of sets have been introduced independently by various researchers: low for K, low for MLrandomness, basis for MLrandomness and Ktrivial. They are all equal. This survey serves as an introduction to these coincidence results, obtained in [24] and [10]. The focus is on providing backdoor access to the proofs. 1. Outline of the results All sets will be subsets of N unless otherwise stated. K(x) denotes the prefix free complexity of a string x. A set A is Ktrivial if, within a constant, each initial segment of A has minimal prefix free complexity. That is, there is c ∈ N such that ∀n K(A ↾ n) ≤ K(0 n) + c. This class was introduced by Chaitin [5] and further studied by Solovay (unpublished). Note that the particular effective epresentation of a number n by a string (unary here) is irrelevant, since up to a constant K(n) is independent from the representation. A is low for MartinLöf randomness if each MartinLöf random set is already MartinLöf random relative to A. This class was defined in Zambella [28], and studied by Kučera and Terwijn [17]. In this survey we will see that the two classes are equivalent [24]. Further concepts have been introduced: to be a basis for MLrandomness (Kučera [16]), and to be low for K (Muchnik jr, in a seminar at Moscow State, 1999). They will also be eliminated, by showing equivalence with Ktriviality. All
Superhighness and strong jump traceability
"... Abstract. Let A be a c.e. set. Then A is strongly jump traceable if and only if A is Turing below each superhigh MartinLöf random set. The proof combines priority with measure theoretic arguments. 1 ..."
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Cited by 2 (0 self)
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Abstract. Let A be a c.e. set. Then A is strongly jump traceable if and only if A is Turing below each superhigh MartinLöf random set. The proof combines priority with measure theoretic arguments. 1
Interactions of Computability and Randomness
"... We survey results relating the computability and randomness aspects of sets of natural numbers. Each aspect corresponds to several mathematical properties. Properties originally defined in very different ways are shown to coincide. For instance, lowness for MLrandomness is equivalent to Ktrivialit ..."
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Cited by 2 (0 self)
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We survey results relating the computability and randomness aspects of sets of natural numbers. Each aspect corresponds to several mathematical properties. Properties originally defined in very different ways are shown to coincide. For instance, lowness for MLrandomness is equivalent to Ktriviality. We include some interactions of randomness with computable analysis. Mathematics Subject Classification (2010). 03D15, 03D32. Keywords. Algorithmic randomness, lowness property, Ktriviality, cost function.
RANDOMNESS VIA EFFECTIVE DESCRIPTIVE SET THEORY
"... An analog of MLrandomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π1 1 counterparts. We prove the analogs of the KraftChaitin Theorem and Schnorr’s Theorem. In the new setting, while Ktrivial sets exist that are not hyperarithmetica ..."
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Cited by 1 (1 self)
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An analog of MLrandomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π1 1 counterparts. We prove the analogs of the KraftChaitin Theorem and Schnorr’s Theorem. In the new setting, while Ktrivial sets exist that are not hyperarithmetical, each low for random set is. Finally, we begin to study a very strong yet effective randomness notion: Z is Π1 1 random if Z is in no null Π1 1 class. There is a greatest Π1 1 null class, that is, a universal test for this notion.
Open Problems in Reverse Mathematics
, 1999
"... The basic reference for reverse mathematics is my recently published book Subsystems of Second Order Arithmetic [32]. The web site for the book is www.math.psu.edu/simpson/sosoa/. This article is a writeup of some representative open problems in reverse mathematics. It was originally a handout ..."
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The basic reference for reverse mathematics is my recently published book Subsystems of Second Order Arithmetic [32]. The web site for the book is www.math.psu.edu/simpson/sosoa/. This article is a writeup of some representative open problems in reverse mathematics. It was originally a handout