5,338 research outputs found
Quantum Kolmogorov Complexity
In this paper we give a definition for quantum Kolmogorov complexity. In the
classical setting, the Kolmogorov complexity of a string is the length of the
shortest program that can produce this string as its output. It is a measure of
the amount of innate randomness (or information) contained in the string.
We define the quantum Kolmogorov complexity of a qubit string as the length
of the shortest quantum input to a universal quantum Turing machine that
produces the initial qubit string with high fidelity. The definition of Vitanyi
(Proceedings of the 15th IEEE Annual Conference on Computational Complexity,
2000) measures the amount of classical information, whereas we consider the
amount of quantum information in a qubit string. We argue that our definition
is natural and is an accurate representation of the amount of quantum
information contained in a quantum state.Comment: 14 pages, LaTeX2e, no figures, \usepackage{amssymb,a4wide}. To appear
in the Proceedings of the 15th IEEE Annual Conference on Computational
Complexit
Kolmogorov Complexity of Categories
Kolmogorov complexity theory is used to tell what the algorithmic
informational content of a string is. It is defined as the length of the
shortest program that describes the string. We present a programming language
that can be used to describe categories, functors, and natural transformations.
With this in hand, we define the informational content of these categorical
structures as the shortest program that describes such structures. Some basic
consequences of our definition are presented including the fact that equivalent
categories have equal Kolmogorov complexity. We also prove different theorems
about what can and cannot be described by our programming language.Comment: 16 page
Kolmogorov Complexity and Solovay Functions
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
Kolmogorov complexity
In dieser Dissertation werden neue Ergebnisse über Kolmogorovkomplexität diskutiert. Ihr erster Teil konzentriert sich auf das Studium von Kolmogorovkomplexität ohne Zeitschranken. Hier beschäftigen wir uns mit dem Konzept nicht-monotoner Zufälligkeit, d.h. Zufälligkeit, die von Martingalen charakterisiert wird, die in nicht-monotoner Reihenfolge wetten dürfen. Wir werden in diesem Zusammenhang eine Reihe von Zufälligkeitsklassen einführen, und diese dann von einander separieren. Wir präsentieren auß erdem einen systematischen überblick über verschiedene Traceability-Begriffe und charakterisieren diese durch (Auto-)Komplexitätsbegriffe. Traceabilities sind eine Gruppe von Begriffen, die ausdrücken, dass eine Menge beinahe berechenbar ist. Der zweite Teil dieses Dokuments beschäftigt sich mit dem Thema zeitbeschränkter Kolmogorovkomplexität. Zunächst untersuchen wir den Unterschied zwischen zwei Arten, ein Wort zu beschreiben: Die Komplexität, es genau genug zu beschreiben, damit es von anderen Wörter unterschieden werden kann; sowie die Komplexität, es genau genug zu beschreiben, damit das Wort aus der Beschreibung tatsächlich generiert werden kann. Diese Unterscheidung ist im Falle zeitunbeschränkter Kolmogorovkomplexität nicht von Bedeutung; sobald wir jedoch Zeitschranken einführen, wird sie essentiell. Als nächstes führen wir den Begriff der Tiefe ein und beweisen ein ihn betreffendes Dichotomieresultat, das in seiner Struktur an Kummers bekanntes Gap-Theorem erinnert. Zu guter Letzt betrachten wir den wichtigen Begriff der Solovayfunktionen. Hierbei handelt es sich um berechenbare obere Schranken der Kolmogorovkomplexität, die unendlich oft scharf sind. Wir benutzen sie, um in einem gewissen Zusammenhang Martin-Löf-Zufälligkeit zu charakterisieren, und um eine Charakterisierung von Jump-Traceability anzugeben
Shannon Information and Kolmogorov Complexity
We compare the elementary theories of Shannon information and Kolmogorov
complexity, the extent to which they have a common purpose, and where they are
fundamentally different. We discuss and relate the basic notions of both
theories: Shannon entropy versus Kolmogorov complexity, the relation of both to
universal coding, Shannon mutual information versus Kolmogorov (`algorithmic')
mutual information, probabilistic sufficient statistic versus algorithmic
sufficient statistic (related to lossy compression in the Shannon theory versus
meaningful information in the Kolmogorov theory), and rate distortion theory
versus Kolmogorov's structure function. Part of the material has appeared in
print before, scattered through various publications, but this is the first
comprehensive systematic comparison. The last mentioned relations are new.Comment: Survey, LaTeX 54 pages, 3 figures, Submitted to IEEE Trans
Information Theor
Topological arguments for Kolmogorov complexity
We present several application of simple topological arguments in problems of
Kolmogorov complexity. Basically we use the standard fact from topology that
the disk is simply connected. It proves to be enough to construct strings with
some nontrivial algorithmic properties.Comment: Extended versio
Game interpretation of Kolmogorov complexity
The Kolmogorov complexity function K can be relativized using any oracle A,
and most properties of K remain true for relativized versions. In section 1 we
provide an explanation for this observation by giving a game-theoretic
interpretation and showing that all "natural" properties are either true for
all sufficiently powerful oracles or false for all sufficiently powerful
oracles. This result is a simple consequence of Martin's determinacy theorem,
but its proof is instructive: it shows how one can prove statements about
Kolmogorov complexity by constructing a special game and a winning strategy in
this game. This technique is illustrated by several examples (total conditional
complexity, bijection complexity, randomness extraction, contrasting plain and
prefix complexities).Comment: 11 pages. Presented in 2009 at the conference on randomness in
Madison
- …