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Rodney G Downey - One of the best experts on this subject based on the ideXlab platform.
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kolmogorov Complexity and solovay Functions
Symposium on Theoretical Aspects of Computer Science, 2009Co-Authors: Laurent Bienvenu, Rodney G DowneyAbstract:Solovay (1975) proved that there exists a computable upper bound~$f$ of the prefix-free Kolmogorov Complexity Function~$K$ such that $f(x)=K(x)$ for infinitely many~$x$. In this paper, we consider the class of computable Functions~$f$ such that $K(x) \leq f(x)+O(1)$ for all~$x$ and $f(x) \leq K(x)+O(1)$ for infinitely many~$x$, which we call Solovay Functions. We show that Solovay Functions present interesting connections with randomness notions such as Martin-L\"of randomness and K-triviality.
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kolmogorov Complexity and solovay Functions
arXiv: Computational Complexity, 2009Co-Authors: Laurent Bienvenu, Rodney G DowneyAbstract:Solovay proved that there exists a computable upper bound f of the prefix-free Kolmogorov Complexity Function K such that f (x) = K(x) for infinitely many x. In this paper, we consider the class of computable Functions f such that K(x) <= f (x)+O(1) for all x and f (x) <= K(x) + O(1) for infinitely many x, which we call Solovay Functions. We show that Solovay Functions present interesting connections with randomness notions such as Martin-L\"of randomness and K-triviality.
Julien Cassaigne - One of the best experts on this subject based on the ideXlab platform.
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On the Morse-Hedlund Complexity gap
arXiv: Formal Languages and Automata Theory, 2009Co-Authors: Julien Cassaigne, François NicolasAbstract:In 1938, Morse and Hedlund proved that the subword Complexity Function of a two-sided infinite word is either bounded or at least linearly growing. In 1982, Ehrenfeucht and Rozenberg proved that this gap property holds for the subword Complexity Function of any language. Their result was then sharpened in 2005 by
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Properties of the Complexity Function for finite words
2004Co-Authors: Mira-cristiana Anisiu, Julien CassaigneAbstract:The subword Complexity Function \(p_{w}\) of a finite word \(w\) over a finite alphabet \(A\) with \(\operatorname*{card}A=q\geq1\) is defined by \(p_{w}(n)=\operatorname*{card}(F(w)\cap A^{n})\) for \(n\in\mathbb{N},\) where \(F(w)\) represents the set of all the subwords or factors of \(w\). The shape of the Complexity Function, especially its piecewise monotonicity, is studied in detail.The Function \(h\) defined as \(h(n)=\min\left\{ q^{n},N-n+1\right\} \) for \(n\in\{0,1,\) \(...,N\}\) has values greater than or equal to those of the Complexity Function \(p_{w}\) for any \(w\in A^{N}\), i.e., \(p_{w}(n)\leq h(n)\) for all \(n\in\{0,1,...,N\}\). As a first result regarding \(h\), it is proved that for each \(N\in\mathbb{N}\) there exist words of length \(N\) for which the maximum of their Complexity Function is equal to the maximum of the Function \(h\); a way to construct such words is described. This result gives rise to a further question: for a given \(N\), is there a word of length \(N\) whose Complexity Function coincides with \(h\) for each \(n\in\{0,1,...,N\}?\) The problem is answered in affirmative, with different constructive proofs for binary alphabets (\(q=2\)) and for those with \(q>2.\) This means that for each \(N\in\mathbb{N},\) there exist words \(w\) of length \(N\) whose Complexity Function is equal to the Function \(h\). Such words are constructed using the de Bruijn graphs.
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Developments in Language Theory - Constructing infinite words of intermediate Complexity
Developments in Language Theory, 2003Co-Authors: Julien CassaigneAbstract:We present two constructions of infinite words with a Complexity Function that grows faster than any polynomial, but slower than any exponential. The first one is rather simple but produces a word which is not uniformly recurrent. The second construction, more involved, produces uniformly recurrent words and allows to choose the growth of the Complexity Function in a large family.
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Developments in Language Theory - Relationally Periodic Sequences and Subword Complexity
Developments in Language Theory, 1Co-Authors: Julien Cassaigne, Tomi Kärki, Luca Q. ZamboniAbstract:By the famous theorem of Morse and Hedlund, a word is ultimately periodic if and only if it has bounded subword Complexity, i.e., for sufficiently large n, the number of factors of length nis constant. In this paper we consider relational periods and relationally periodic sequences, where the relation is a similarity relation on words induced by a compatibility relation on letters. We investigate what would be a suitable definition for a relational subword Complexity Function such that it would imply a Morse and Hedlund-like theorem for relationally periodic words. We consider strong and weak relational periods and two candidates for subword Complexity Functions.
Laurent Bienvenu - One of the best experts on this subject based on the ideXlab platform.
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kolmogorov Complexity and solovay Functions
Symposium on Theoretical Aspects of Computer Science, 2009Co-Authors: Laurent Bienvenu, Rodney G DowneyAbstract:Solovay (1975) proved that there exists a computable upper bound~$f$ of the prefix-free Kolmogorov Complexity Function~$K$ such that $f(x)=K(x)$ for infinitely many~$x$. In this paper, we consider the class of computable Functions~$f$ such that $K(x) \leq f(x)+O(1)$ for all~$x$ and $f(x) \leq K(x)+O(1)$ for infinitely many~$x$, which we call Solovay Functions. We show that Solovay Functions present interesting connections with randomness notions such as Martin-L\"of randomness and K-triviality.
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kolmogorov Complexity and solovay Functions
arXiv: Computational Complexity, 2009Co-Authors: Laurent Bienvenu, Rodney G DowneyAbstract:Solovay proved that there exists a computable upper bound f of the prefix-free Kolmogorov Complexity Function K such that f (x) = K(x) for infinitely many x. In this paper, we consider the class of computable Functions f such that K(x) <= f (x)+O(1) for all x and f (x) <= K(x) + O(1) for infinitely many x, which we call Solovay Functions. We show that Solovay Functions present interesting connections with randomness notions such as Martin-L\"of randomness and K-triviality.
Carlos Gustavo Moreira - One of the best experts on this subject based on the ideXlab platform.
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Complexity and fractal dimensions for infinite sequences with positive entropy
Communications in Contemporary Mathematics, 2019Co-Authors: Christian Mauduit, Carlos Gustavo MoreiraAbstract:The Complexity Function of an infinite word w on a finite alphabet A is the sequence counting, for each non-negative n, the number of words of length n on the alphabet A that are factors of the inf...
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Entropy ratio for infinite sequences with positive entropy
arXiv: Dynamical Systems, 2018Co-Authors: Christian Mauduit, Carlos Gustavo MoreiraAbstract:The Complexity Function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. For any given Function $f$ with exponential growth, we introduced in [MM17] the notion of {\it word entropy} $E_W(f)$ associated to $f$ and we described the combinatorial structure of sets of infinite words with a Complexity Function bounded by $f$. The goal of this work is to give estimates on the word entropy $E_W(f)$ in terms of the limiting lower exponential growth rate of $f$.
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WORDS - The Word Entropy and How to Compute It
Lecture Notes in Computer Science, 2017Co-Authors: Sébastien Ferenczi, Christian Mauduit, Carlos Gustavo MoreiraAbstract:The Complexity Function of an infinite word counts the number of its factors. For any positive Function f, its exponential rate of growth \(E_0(f)\) is \(\lim \limits _{n\rightarrow \infty } \inf \frac{1}{n}\log f(n)\). We define a new quantity, the word entropy \(E_W(f)\), as the maximal exponential growth rate of a Complexity Function smaller than f. This is in general smaller than \(E_0(f)\), and more difficult to compute; we give an algorithm to estimate it. The quantity \(E_W(f)\) is used to compute the Hausdorff dimension of the set of real numbers whose expansions in a given base have Complexity bounded by f.
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Complexity and fractal dimensions for infinite sequences with positive entropy
2017Co-Authors: Carlos Gustavo Moreira, Christian MauduitAbstract:The Complexity Function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. One of the goals of this work is to estimate the number of words of length $n$ on the alphabet $A$ that are factors of an infinite word $w$ with a Complexity Function bounded by a given Function $f$ with exponential growth. We describe the combinatorial structure of such sets of infinite words and we introduce a real parameter, the $word\,\, entropy$ $E_W(f)$ associated to a given Function $f$. We give estimates on the word entropy $E_W(f)$ in terms of the exponential rate of growth of $f$ and we determine fractal dimensions of sets of infinite sequences with Complexity Function bounded by $f$ in terms of its word entropy.
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Complexity and fractal dimensions for infinite sequences with positive entropy
arXiv: Dynamical Systems, 2017Co-Authors: Carlos Gustavo Moreira, Christian MauduitAbstract:The Complexity Function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. The goal of this work is to estimate the number of words of length $n$ on the alphabet $A$ that are factors of an infinite word $w$ with a Complexity Function bounded by a given Function $f$ with exponential growth and to describe the combinatorial structure of such sets of infinite words. We introduce a real parameter, the {\it word entropy} $E_W(f)$ associated to a given Function $f$ and we determine the fractal dimensions of sets of infinite sequences with Complexity Function bounded by $f$ in terms of its word entropy. We present a combinatorial proof of the fact that $E_W(f)$ is equal to the topological entropy of the subshift of infinite words whose Complexity is bounded by $f$ and we give several examples showing that even under strong conditions on $f$, the word entropy $E_W(f)$ can be strictly smaller than the limiting lower exponential growth rate of $f$.
Juris Hartmanis - One of the best experts on this subject based on the ideXlab platform.
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on hausdorff and topological dimensions of the kolmogorov Complexity of the real line
Foundations of Computer Science, 1994Co-Authors: Juris HartmanisAbstract:We investigate the Kolmogorov Complexity of real numbers. Let K be the Kolmogorov Complexity Function; we determine the Hausdorff dimension and the topological dimension of the graph of K. Since these dimensions are different, the graph of the Kolmogorov Complexity Function of the real line forms a fractal in the sense of Mandelbrot. We also solve an open problem of Razborov using our exact bound on the topological dimension.
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Structure in Complexity Theory Conference - The Complexity of the real line is a fractal
[1989] Proceedings. Structure in Complexity Theory Fourth Annual Conference, 1Co-Authors: Jin-yi Cai, Juris HartmanisAbstract:The authors investigate the Kolmogorov Complexity of real numbers. They determine the Hausdorff dimension and the topological dimension of the graph of K, the Kolmogorov Complexity Function. They conclude that the Complexity graph is a fractal. >
