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LIMIT COMPLEXITIES REVISITED
"... Abstract. The main goal of this paper is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result from [7] saying that lim supn C(xn) (here C(xn) is conditional (plain) Kolmogorov complexity of x when n is known) equals C 0′ (x), the ..."
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Abstract. The main goal of this paper is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result from [7] saying that lim supn C(xn) (here C(xn) is conditional (plain) Kolmogorov complexity of x when n is known) equals C 0′ (x), the plain Kolmogorov complexity with 0 ′oracle. Then we use the same argument to prove similar results for prefix complexity (and also improve results of [4] about limit frequencies), a priori probability on binary tree and measure of effectively open sets. As a byproduct, we get a criterion of 0 ′ MartinLöf randomness (called also 2randomness) proved in [3]: a sequence ω is 2random if and only if there exists c such that any prefix x of ω is a prefix of some string y such that C(y) � y  − c. (In 1960ies this property was suggested in [1] as one of possible randomness definitions; its equivalence to 2randomness was shown in [3] while proving another 2randomness criterion (see also [5]): ω is 2random if and only if C(x) � x  − c for some c and infinitely many prefixes x of ω. Finally, we show that lowbasis theorem can be used to get alternative proofs for these results and to improve the result about effectively open sets; this stronger version implies the 2randomness criterion mentioned in the previous sentence. 1. Plain complexity By C(x) we mean the plain complexity of a binary string x (the length of the shortest description of x when an optimal description method is fixed, see [2]; no requirements about prefixes). By C(xn) we mean conditional complexity of x when n is given [2]. Superscript 0 ′ in C0 ′ means that we consider the relativized (with oracle 0 ′ , the universal enumerable set) version of complexity.
unknown title
, 2008
"... Abstract. The main goal of this paper is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result from [7] saying that lim supn C(xn) (here C(xn) is conditional (plain) Kolmogorov complexity of x when n is known) equals C 0′ (x), the ..."
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Abstract. The main goal of this paper is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result from [7] saying that lim supn C(xn) (here C(xn) is conditional (plain) Kolmogorov complexity of x when n is known) equals C 0′ (x), the plain Kolmogorov complexity with 0 ′oracle. Then we use the same argument to prove similar results for prefix complexity (and also improve results of [4] about limit frequencies), a priori probability on binary tree and measure of effectively open sets. As a byproduct, we get a criterion of 0 ′ MartinLöf randomness (called also 2randomness) proved in [3]: a sequence ω is 2random if and only if there exists c such that any prefix x of ω is a prefix of some string y such that C(y) � y  − c. (In the 1960ies this property was suggested in [1] as one of possible randomness definitions; its equivalence to 2randomness was shown in [3] while proving another 2randomness criterion (see also [5]): ω is 2random if and only if C(x) � x  − c for some c and infinitely many prefixes x of ω. Finally, we show that the lowbasis theorem can be used to get alternative proofs for these results and to improve the result about effectively open sets; this stronger version implies the 2randomness criterion mentioned in the previous sentence. Key words and phrases: Kolmogorov complexity, limit complexities, limit frequencies, 2randomness, low basis. c○