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A Theory of Program Size Formally Identical to Information Theory
, 1975
"... A new definition of programsize complexity is made. H(A;B=C;D) is defined to be the size in bits of the shortest selfdelimiting program for calculating strings A and B if one is given a minimalsize selfdelimiting program for calculating strings C and D. This differs from previous definitions: (1) ..."
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Cited by 333 (16 self)
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A new definition of programsize complexity is made. H(A;B=C;D) is defined to be the size in bits of the shortest selfdelimiting program for calculating strings A and B if one is given a minimalsize selfdelimiting program for calculating strings C and D. This differs from previous definitions: (1) programs are required to be selfdelimiting, i.e. no program is a prefix of another, and (2) instead of being given C and D directly, one is given a program for calculating them that is minimal in size. Unlike previous definitions, this one has precisely the formal 2 G. J. Chaitin properties of the entropy concept of information theory. For example, H(A;B) = H(A) + H(B=A) + O(1). Also, if a program of length k is assigned measure 2 \Gammak , then H(A) = \Gamma log 2 (the probability that the standard universal computer will calculate A) +O(1). Key Words and Phrases: computational complexity, entropy, information theory, instantaneous code, Kraft inequality, minimal program, probab...
Kolmogorov Complexity and Hausdorff Dimension
 Inform. and Comput
, 1989
"... this paper we are mainly interested in the first order approximation (i.e. the linear growth) of K(fi=\Delta). We consider the functions ..."
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Cited by 67 (20 self)
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this paper we are mainly interested in the first order approximation (i.e. the linear growth) of K(fi=\Delta). We consider the functions
The Kolmogorov Complexity of Liouville Numbers
, 1999
"... We consider for a real number a the Kolmogorov complexities of its expansions with respect to different bases. In the paper it is shown that, for usual and selfdelimiting Kolmogorov complexity, the complexity of the prefixes of their expansions with respect to different bases r and b are related ..."
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Cited by 2 (0 self)
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We consider for a real number a the Kolmogorov complexities of its expansions with respect to different bases. In the paper it is shown that, for usual and selfdelimiting Kolmogorov complexity, the complexity of the prefixes of their expansions with respect to different bases r and b are related in a way which depends only on the relative information of one base with respect to the other. More precisely, we show that the complexity of the length l # log r b prefix of the base r expansion of a is the same (up to an additive constant) as the log r bfold complexity of the length l prefix of the base b expansion of a. Then we use this fact to derive complexity theoretic proofs for the base independence of the randomness of real numbers and for some properties of Liouville numbers. Kolmogorov Complexity is mainly attributed to finite strings over a finite alphabet. As a function or, more coarsely, as a limit it measures the complexity of infinite strings. Real numbers are de...
New games related to old and new sequences
 Heinz (Eds.), Proc 10th Advances in Computer Games Conference (ACG10
, 2003
"... We define an infinite class of 2pile subtraction games, where the amount that can be subtracted from both piles simultaneously, is a function f of the size of the piles. Wythoff’s game is a special case. For each game, the 2nd player winning positions are a pair of complementary sequences, some of ..."
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Cited by 2 (2 self)
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We define an infinite class of 2pile subtraction games, where the amount that can be subtracted from both piles simultaneously, is a function f of the size of the piles. Wythoff’s game is a special case. For each game, the 2nd player winning positions are a pair of complementary sequences, some of which are related to wellknown sequences, but most are new. The main result is a theorem giving necessary and sufficient conditions on f so that the sequences are 2nd player winning positions. Sample games are presented, strategy complexity questions are discussed, and possible further studies are indicated.
On Oscillationfree εrandom Sequences
, 2008
"... In this paper we discuss three notions of partial randomness or εrandomness. εrandomness should display all features of randomness in a scaled down manner. However, as Reimann and Stephan [15] proved, Tadaki [22] and Calude et al. [3] proposed at least three different concepts of partial randomnes ..."
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Cited by 1 (1 self)
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In this paper we discuss three notions of partial randomness or εrandomness. εrandomness should display all features of randomness in a scaled down manner. However, as Reimann and Stephan [15] proved, Tadaki [22] and Calude et al. [3] proposed at least three different concepts of partial randomness. We show that all of them satisfy the natural requirement that any εnonnull set contains an εrandom infinite word. This allows us to focus our investigations on the strongest one which is based on a priori complexity. We investigate this concept of partial randomness and show that it allows—similar to the random infinite words—oscillationfree (w.r.t. to a priori complexity) εrandom infinite words if only ε is a computable number. The proof uses the dilution principle. Alternatively, for certain sets of infinite words (ωlanguages) we show that their most complex infinite words are oscillationfree εrandom. Here the parameter ε is also computable and depends on the set chosen.