scholarly journals Fractal Intersections and Products via Algorithmic Dimension

2021 ◽  
Vol 13 (3) ◽  
pp. 1-15
Author(s):  
Neil Lutz
Keyword(s):  
Kolmogorov Complexity ◽  
Fractal Dimensions ◽  
Euclidean Spaces ◽  
Cartesian Products ◽  
Information Density ◽  
Algorithmic Dimension ◽  
Packing Dimensions ◽  
Algorithmic Information ◽  
Analytic Sets

Algorithmic fractal dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions of intersections and Cartesian products of fractals in Euclidean spaces. This approach shows that two prominent, fundamental results about the dimension of Borel or analytic sets also hold for arbitrary sets.

Dimension spectra of lines1

Computability ◽  
2021 ◽  
pp. 1-28
Author(s):  
Neil Lutz ◽  
D.M. Stull
Keyword(s):  
Hausdorff Dimension ◽  
Kolmogorov Complexity ◽  
Euclidean Plane ◽  
Unit Interval ◽  
Packing Dimension ◽  
Hausdorff Dimensions ◽  
Dimension Spectrum ◽  
Algorithmic Dimension

This paper investigates the algorithmic dimension spectra of lines in the Euclidean plane. Given any line L with slope a and vertical intercept b, the dimension spectrum sp ( L ) is the set of all effective Hausdorff dimensions of individual points on L. We draw on Kolmogorov complexity and geometrical arguments to show that if the effective Hausdorff dimension dim ( a , b ) is equal to the effective packing dimension Dim ( a , b ), then sp ( L ) contains a unit interval. We also show that, if the dimension dim ( a , b ) is at least one, then sp ( L ) is infinite. Together with previous work, this implies that the dimension spectrum of any line is infinite.

scholarly journals A Survey on Using Kolmogorov Complexity in Cybersecurity

Entropy ◽  
2019 ◽  
Vol 21 (12) ◽  
pp. 1196 ◽  
Author(s):  
João S. Resende ◽  
Rolando Martins ◽  
Luís Antunes
Keyword(s):  
Kolmogorov Complexity ◽  
Security And Privacy ◽  
Algorithmic Information Theory ◽  
Specific Knowledge ◽  
Privacy Concerns ◽  
Web Interfaces ◽  
Algorithmic Information ◽  
Software Implementations ◽  
New Research ◽  
And Control

Security and privacy concerns are challenging the way users interact with devices. The number of devices connected to a home or enterprise network increases every day. Nowadays, the security of information systems is relevant as user information is constantly being shared and moving in the cloud; however, there are still many problems such as, unsecured web interfaces, weak authentication, insecure networks, lack of encryption, among others, that make services insecure. The software implementations that are currently deployed in companies should have updates and control, as cybersecurity threats increasingly appearing over time. There is already some research towards solutions and methods to predict new attacks or classify variants of previous known attacks, such as (algorithmic) information theory. This survey combines all relevant applications of this topic (also known as Kolmogorov Complexity) in the security and privacy domains. The use of Kolmogorov-based approaches is resource-focused without the need for specific knowledge of the topic under analysis. We have defined a taxonomy with already existing work to classify their different application areas and open up new research questions.

SETS OF K-INDEPENDENT STRINGS

2010 ◽  
Vol 21 (03) ◽  
pp. 321-327 ◽  
Author(s):  
YEN-WU TI ◽  
CHING-LUEH CHANG ◽  
YUH-DAUH LYUU ◽  
ALEXANDER SHEN
Keyword(s):  
Information Theory ◽  
Kolmogorov Complexity ◽  
Upper Bounds ◽  
The Other ◽  
Lower And Upper Bounds ◽  
Algorithmic Information Theory ◽  
Algorithmic Information ◽  
Conditional Complexity

A bit string is random (in the sense of algorithmic information theory) if it is incompressible, i.e., its Kolmogorov complexity is close to its length. Two random strings are independent if knowing one of them does not simplify the description of the other, i.e., the conditional complexity of each string (using the other as a condition) is close to its length. We may define independence of a k-tuple of strings in the same way. In this paper we address the following question: what is that maximal cardinality of a set of n-bit strings if any k elements of this set are independent (up to a certain constant)? Lower and upper bounds that match each other (with logarithmic precision) are provided.

scholarly journals Combinatorial proofs of two theorems of Lutz and Stull

2021 ◽  
pp. 1-12
Author(s):  
TUOMAS ORPONEN
Keyword(s):  
Information Theory ◽  
Euclidean Space ◽  
Pigeonhole Principle ◽  
Algorithmic Information Theory ◽  
Orthogonal Projections ◽  
Information Theoretic ◽  
Secondary Purpose ◽  
Packing Dimensions ◽  
Algorithmic Information ◽  
Projection Theorems

Abstract Recently, Lutz and Stull used methods from algorithmic information theory to prove two new Marstrand-type projection theorems, concerning subsets of Euclidean space which are not assumed to be Borel, or even analytic. One of the theorems states that if \[K \subset {\mathbb{R}^n}\] is any set with equal Hausdorff and packing dimensions, then \begin{equation} \[{\dim _{\text{H}}}{\pi _e}(K) = \min \{ {\dim _{\text{H}}}{\text{ }}K{\text{, 1}}\} \] \end{equation} for almost every \[e \in {S^{n - 1}}\] . Here \[{\pi _e}\] stands for orthogonal projection to span ( \[e\] ). The primary purpose of this paper is to present proofs for Lutz and Stull’s projection theorems which do not refer to information theoretic concepts. Instead, they will rely on combinatorial-geometric arguments, such as discretised versions of Kaufman’s “potential theoretic” method, the pigeonhole principle, and a lemma of Katz and Tao. A secondary purpose is to generalise Lutz and Stull’s theorems: the versions in this paper apply to orthogonal projections to m-planes in \[{\mathbb{R}^n}\] , for all \[0 < m < n\] .

scholarly journals Generic algorithms for halting problem and optimal machines revisited

2017 ◽  
Author(s):  
Laurent Bienvenu ◽  
Damien Desfontaines ◽  
Alexander Shen
Keyword(s):  
Kolmogorov Complexity ◽  
Asymptotic Properties ◽  
Probabilistic Algorithms ◽  
Algorithmic Information Theory ◽  
Halting Problem ◽  
Negative Results ◽  
Long Time ◽  
Algorithmic Information ◽  
Random Reals ◽  
Generic Algorithms

The halting problem is undecidable — but can it be solved for “most” inputs? This natural question was considered in a number of papers, in different settings. We revisit their results and show that most of them can be easily proven in a natural framework of optimal machines (considered in algorithmic information theory) using the notion of Kolmogorov complexity. We also consider some related questions about this framework and about asymptotic properties of the halting problem. In particular, we show that the fraction of terminating programs cannot have a limit, and all limit points are Martin-L¨of random reals. We then consider mass problems of finding an approximate solution of halting problem and probabilistic algorithms for them, proving both positive and negative results. We consider the fraction of terminating programs that require a long time for termination, and describe this fraction using the busy beaver function. We also consider approximate versions of separation problems, and revisit Schnorr’s results about optimal numberings showing how they can be generalized.

DESCRIPTIONAL COMPLEXITY IN ENCODED BLUM STATIC COMPLEXITY SPACES

2014 ◽  
Vol 25 (07) ◽  
pp. 917-932
Author(s):  
CEZAR CÂMPEANU
Keyword(s):  
Kolmogorov Complexity ◽  
Regular Language ◽  
Formal Language ◽  
Function Spaces ◽  
Language Theory ◽  
Algorithmic Information Theory ◽  
Descriptional Complexity ◽  
Algorithmic Information ◽  
Minimal Description ◽  
Minimal Description Length

Algorithmic Information Theory is based on the notion of descriptional complexity known as Chaitin-Kolmogorov complexity, defined in the '60s in terms of minimal description length. Blum Static Complexity spaces defined using Blum axioms, and Encoded Function spaces defined using properties of the complexity function, were introduced in 2012 to generalize the concept of descriptional complexity. In formal language theory we also use the concept of descriptional complexity for the number of states, or the number of transitions in a minimal finite automaton accepting a regular language, and apparently, this number has no connection to the general case of descriptional complexity. In this paper we prove that all the definitions of descriptional complexity, including complexity of operations, can be defined within the framework of Encoded Blum Static Complexity spaces, which extend both Blum Static Complexity spaces and Encoded Function spaces.

scholarly journals Computational Creativity and Aesthetics with Algorithmic Information Theory

Entropy ◽  
2021 ◽  
Vol 23 (12) ◽  
pp. 1654
Author(s):  
Tiasa Mondol ◽  
Daniel G. Brown
Keyword(s):  
Information Theory ◽  
Kolmogorov Complexity ◽  
Creative Work ◽  
Computational Creativity ◽  
Algorithmic Information Theory ◽  
Information Theoretic ◽  
Related Research ◽  
Computational Aesthetics ◽  
Algorithmic Information ◽  
Randomness Deficiency

We build an analysis based on the Algorithmic Information Theory of computational creativity and extend it to revisit computational aesthetics, thereby, improving on the existing efforts of its formulation. We discuss Kolmogorov complexity, models and randomness deficiency (which is a measure of how much a model falls short of capturing the regularities in an artifact) and show that the notions of typicality and novelty of a creative artifact follow naturally from such definitions. Other exciting formalizations of aesthetic measures include logical depth and sophistication with which we can define, respectively, the value and creator’s artistry present in a creative work. We then look at some related research that combines information theory and creativity and analyze them with the algorithmic tools that we develop throughout the paper. Finally, we assemble the ideas and their algorithmic counterparts to complete an algorithmic information theoretic recipe for computational creativity and aesthetics.

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