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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 359 (17 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 65 (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 infinite words
 7TH WORKSHOP ”DESCRIPTIONAL COMPLEXITY OF FORMAL SYSTEMS"
, 2007
"... We present a brief survey of results on relations between the Kolmogorov complexity of infinite strings and several measures of information content (dimensions) known from dimension theory, information theory or fractal geometry. Special emphasis is laid on bounds on the complexity of strings in con ..."
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Cited by 5 (2 self)
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We present a brief survey of results on relations between the Kolmogorov complexity of infinite strings and several measures of information content (dimensions) known from dimension theory, information theory or fractal geometry. Special emphasis is laid on bounds on the complexity of strings in constructively given subsets of the Cantor space. Finally, we compare the Kolmogorov complexity to the subword complexity of infinite strings.
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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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.
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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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.
Theoretical Computer Science On Partial Randomness
, 2004
"... If x = x1x2 · · · xn · · · is a random sequence, then the sequence y = 0x10x2 · · · 0xn · · · is clearly not random; however, y seems to be “about half random”. Staiger [14, 15] and Tadaki [16] have studied the degree of randomness of sequences or reals by measuring their “degree of compress ..."
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If x = x1x2 · · · xn · · · is a random sequence, then the sequence y = 0x10x2 · · · 0xn · · · is clearly not random; however, y seems to be “about half random”. Staiger [14, 15] and Tadaki [16] have studied the degree of randomness of sequences or reals by measuring their “degree of compression”. This line of study leads to various definitions of partial randomness. In this paper we explore some relations between these definitions. Among other results we obtain
Annals of Pure and Applied Logic 138 (2006) 20–30 On partial randomness www.elsevier.com/locate/apal
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words ∗
, 2006
"... We present a brief survey of results on relations between the Kolmogorov complexity of infinite strings and several measures of information content (dimensions) known from dimension theory, information theory or fractal geometry. Special emphasis is laid on bounds on the complexity of strings in con ..."
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We present a brief survey of results on relations between the Kolmogorov complexity of infinite strings and several measures of information content (dimensions) known from dimension theory, information theory or fractal geometry. Special emphasis is laid on bounds on the complexity of strings in constructively given subsets of the Cantor space. Finally, we compare the Kolmogorov complexity to the subword complexity of infinite strings. ∗ This paper was presented at the 7th Workshop ”Descriptional Complexity of Formal