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On the gap between trivial and nontrivial initial segment prefixfree complexity
, 2010
"... Abstract. An infinite sequence X is said to have trivial (prefixfree) initial segment complexity if K(X ↾n) ≤ + K(0n) for all n, where K is the prefixfree complexity and ≤ + denotes inequality modulo a constant. In other words, if the information in any initial segment of it is merely the informa ..."
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Abstract. An infinite sequence X is said to have trivial (prefixfree) initial segment complexity if K(X ↾n) ≤ + K(0n) for all n, where K is the prefixfree complexity and ≤ + denotes inequality modulo a constant. In other words, if the information in any initial segment of it is merely the information in a sequence of 0s of the same length. We study the gap between the trivial complexity K(0n) and the complexity of a nontrivial sequence, i.e. the functions f such that (⋆) K(X ↾n) ≤ + K(0 n) + f(n) for all n for a nontrivial (in terms of initial segment complexity) sequence X. We show that given any ∆0 2 unbounded nondecreasing function f there exist uncountably many sequences X which satisfy (⋆). On the other hand there exists a ∆0 3 unbounded nondecreasing function f which does not satisfy (⋆) for any X with nontrivial initial segment complexity. This improves the bound ∆0 4 that was known from [CM06]. Finally we give some applications of these results. 1.
KOLMOGOROV COMPLEXITY OF INITIAL SEGMENTS OF SEQUENCES AND ARITHMETICAL DEFINABILITY
"... Abstract. The structure of the Kdegrees provides a way to classify sets of natural numbers or infinite binary sequences with respect to the level of randomness of their initial segments. In the Kdegrees of infinite binary sequences, X is below Y if the prefixfree Kolmogorov complexity of the firs ..."
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Abstract. The structure of the Kdegrees provides a way to classify sets of natural numbers or infinite binary sequences with respect to the level of randomness of their initial segments. In the Kdegrees of infinite binary sequences, X is below Y if the prefixfree Kolmogorov complexity of the first n bits of X is less than the complexity of the first n bits of Y, for each n. Identifying infinite binary sequences with subsets of N, we study the Kdegrees of arithmetical sets and explore the interactions between arithmetical definability and prefix free Kolmogorov complexity. We show that in the Kdegrees, for each n> 1 there exists a Σ0 n nonzero degree which does not bound any ∆0 n nonzero degree. An application of this result is that in the Kdegrees there exists a Σ0 2 degree which forms a minimal pair with all Σ0 1 degrees. This extends work of Csima/Montalbán [CM06] and Merkle/Stephan [MS07]. Our main result is that, given any ∆0 2 family C of sequences, there is a ∆0 2 sequence of nontrivial initial segment complexity which is not larger than the initial segment complexity of any nontrivial member of C. This general theorem has the following surprising consequence. There is a 0 ′computable sequence of nontrivial initial segment complexity which is not larger than the initial segment complexity of any nontrivial computably enumerable set. Our analysis and results demonstrate that, examining the extend to which arithmetical definability interacts with the K reducibility (and in general any ‘weak reducibility’) is a fruitful way of studying the induced structure. 1.
ALGORITHMIC RANDOMNESS AND MEASURES OF COMPLEXITY
"... Abstract. We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress s ..."
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Abstract. We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on (weak) reducibilities that measure (a) the initial segment complexity of reals and (b) the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability. 1.