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CALCULUS OF COST FUNCTIONS
"... Abstract. We study algebraic properties of cost functions. We give an application: building sets close to being Turing complete. 1. ..."
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Abstract. We study algebraic properties of cost functions. We give an application: building sets close to being Turing complete. 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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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.
STRONG JUMPTRACEABILITY II: KTRIVIALITY
, 2010
"... Abstract. We show that every strongly jumptraceable set is Ktrivial. Unlike other results, we do not assume that the sets in question are computably enumerable. 1. ..."
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Abstract. We show that every strongly jumptraceable set is Ktrivial. Unlike other results, we do not assume that the sets in question are computably enumerable. 1.
SUPERHIGHNESS AND STRONG JUMP TRACEABILITY
"... Abstract. Let A be c.e. Then A is strongly jump traceable if and only if A is Turing below each superhigh MartinLöfrandom set. 1. ..."
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Abstract. Let A be c.e. Then A is strongly jump traceable if and only if A is Turing below each superhigh MartinLöfrandom set. 1.
Counting the Changes of Random ∆ 0 2 Sets
"... Abstract. Consider a MartinLöf random ∆ 0 2 set Z. We give lower bounds for the number of changes of Zs ↾n for computable approximations of Z. We show that each nonempty Π 0 1 class has a low member Z with a computable approximation that changes only o(2 n) times. We prove that each superlow MLran ..."
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Abstract. Consider a MartinLöf random ∆ 0 2 set Z. We give lower bounds for the number of changes of Zs ↾n for computable approximations of Z. We show that each nonempty Π 0 1 class has a low member Z with a computable approximation that changes only o(2 n) times. We prove that each superlow MLrandom set already satisfies a stronger randomness notion called balanced randomness, which implies that for each computable approximation and each constant c, there are infinitely many n such that Zs ↾n changes more than c2 n times. 1