On Kolmogorov complexity in the real Turing machine setting

1998 ◽  
Vol 67 (2) ◽  
pp. 81-86 ◽  
Author(s):  
J.L. Montaña ◽  
Luis M. Pardo
Keyword(s):  
Turing Machine ◽  
Kolmogorov Complexity ◽  
The Real ◽  
Machine Setting

scholarly journals Fuzzy Kolmogorov Complexity Based on a Classical Description

Entropy ◽  
2020 ◽  
Vol 22 (1) ◽  
pp. 66
Author(s):  
Songsong Dai
Keyword(s):  
Turing Machine ◽  
Kolmogorov Complexity ◽  
Classical Description ◽  
Fuzzy Language ◽  
Classical Setting

In this paper, we give a definition for fuzzy Kolmogorov complexity. In the classical setting, the Kolmogorov complexity of a single finite string is the length of the shortest program that produces this string. We define the fuzzy Kolmogorov complexity as the minimum classical description length of a finite-valued fuzzy language through a universal finite-valued fuzzy Turing machine that produces the desired fuzzy language. The classical Kolmogorov complexity is extended to the fuzzy domain retaining classical descriptions. We show that our definition is robust, that is to say, the complexity of a finite-valued fuzzy language does not depend on the underlying finite-valued fuzzy Turing machine.

scholarly journals Undecidable, Unrecognizable, and Quantum Computing

2020 ◽  
Vol 2 (3) ◽  
pp. 337-342
Author(s):  
Michael Siomau
Keyword(s):  
Quantum Computing ◽  
Turing Machine ◽  
Quantum Computer ◽  
Practical Realization ◽  
Real Numbers ◽  
Theoretical Limit ◽  
Halting Problem ◽  
The Real ◽  
Computing Model ◽  
Successful Design

Quantum computing allows us to solve some problems much faster than existing classical algorithms. Yet, the quantum computer has been believed to be no more powerful than the most general computing model—the Turing machine. Undecidable problems, such as the halting problem, and unrecognizable inputs, such as the real numbers, are beyond the theoretical limit of the Turing machine. I suggest a model for a quantum computer, which is less general than the Turing machine, but may solve the halting problem for any task programmable on it. Moreover, inputs unrecognizable by the Turing machine can be recognized by the model, thus breaking the theoretical limit for a computational task. A quantum computer is not just a successful design of the Turing machine as it is widely perceived now, but is a different, less general but more powerful model for computing, the practical realization of which may need different strategies than those in use now.

THE SECOND QUANTIZED QUANTUM TURING MACHINE AND KOLMOGOROV COMPLEXITY

2008 ◽  
Vol 22 (12) ◽  
pp. 1203-1210 ◽  
Author(s):  
CAROLINE ROGERS ◽  
VLATKO VEDRAL
Keyword(s):  
Turing Machine ◽  
Kolmogorov Complexity ◽  
Physical State ◽  
Physical Reality ◽  
Physical Resources ◽  
Quantum Turing Machine

The Kolmogorov complexity of a physical state is the minimal physical resources required to reproduce that state. We define a second quantized quantum Turing machine and use it to define second quantized Kolmogorov complexity. There are two advantages to our approach — our measure of the second quantized Kolmogorov complexity is closer to physical reality and unlike other quantum Kolmogorov complexities, it is continuous. We give examples where the second quantized and quantum Kolmogorov complexity differ.

Quantum information distance

2021 ◽  
Author(s):  
Songsong Dai
Keyword(s):  
Quantum Information ◽  
Turing Machine ◽  
Kolmogorov Complexity ◽  
Transition Program ◽  
Information Distance ◽  
Universal Turing Machine ◽  
Classical Setting ◽  
Universal Quantum ◽  
Definition Of ◽  
Quantum Turing Machine

In this paper, we give a definition for quantum information distance. In the classical setting, information distance between two classical strings is developed based on classical Kolmogorov complexity. It is defined as the length of a shortest transition program between these two strings in a universal Turing machine. We define the quantum information distance based on Berthiaume et al.’s quantum Kolmogorov complexity. The quantum information distance between qubit strings is defined as the length of the shortest quantum transition program between these two qubit strings in a universal quantum Turing machine. We show that our definition of quantum information distance is invariant under the choice of the underlying quantum Turing machine.

HIERARCHIES OF GENERALIZED KOLMOGOROV COMPLEXITIES AND NONENUMERABLE UNIVERSAL MEASURES COMPUTABLE IN THE LIMIT

2002 ◽  
Vol 13 (04) ◽  
pp. 587-612 ◽  
Author(s):  
JÜRGEN SCHMIDHUBER
Keyword(s):  
Information Content ◽  
Turing Machine ◽  
Kolmogorov Complexity ◽  
Short Description ◽  
Traditional Theory ◽  
Turing Machines ◽  
Random Input ◽  
Algorithmic Probability ◽  
Output Tape

The traditional theory of Kolmogorov complexity and algorithmic probability focuses on monotone Turing machines with one-way write-only output tape. This naturally leads to the universal enumerable Solomonoff-Levin measure. Here we introduce more general, nonenumerable but cumulatively enumerable measures (CEMs) derived from Turing machines with lexicographically nondecreasing output and random input, and even more general approximable measures and distributions computable in the limit. We obtain a natural hierarchy of generalizations of algorithmic probability and Kolmogorov complexity, suggesting that the "true" information content of some (possibly infinite) bitstring x is the size of the shortest nonhalting program that converges to x and nothing but x on a Turing machine that can edit its previous outputs. Among other things we show that there are objects computable in the limit yet more random than Chaitin's "number of wisdom" Omega, that any approximable measure of x is small for any x lacking a short description, that there is no universal approximable distribution, that there is a universal CEM, and that any nonenumerable CEM of x is small for any x lacking a short enumerating program. We briefly mention consequences for universes sampled from such priors.

scholarly journals Teorias da Aleatoriedade

2004 ◽  
Vol 11 (2) ◽  
pp. 75-98
Author(s):  
Carlos A. P. Campani ◽  
Paulo Blauth Menezes
Keyword(s):  
Turing Machine ◽  
Kolmogorov Complexity ◽  
Random Sequence ◽  
The Other ◽  
Turing Machines ◽  
Intuitive Concept ◽  
Theory Of Information ◽  
Definition Of

This work is a survey about the definition of “random sequence”. We emphasize the definition of Martin-Löf and the definition based on incompressibility (Kolmogorov complexity). Kolmogorov complexity is a profound and sofisticated theory of information and randomness based on Turing machines. These two definitions solve all the problems of the other approaches, satisfying our intuitive concept of randomness, and both are mathematically correct. Furthermore, we show the Schnorr’s approach, that includes a requisite of effectiveness (computability) in his definition. We show the relations between all definitions in a critical way. Keywords: randomness, Kolmogorov complexity, Turing machine, computability, probability.

scholarly journals EFFICIENT TESTING OF FORECASTS

2010 ◽  
Vol 21 (01) ◽  
pp. 61-72
Author(s):  
CHING-LUEH CHANG ◽  
YUH-DAUH LYUU
Keyword(s):  
Computational Complexity ◽  
Prior Knowledge ◽  
Turing Machine ◽  
High Probability ◽  
Weather Data ◽  
Arbitrary Distribution ◽  
The Real ◽  
The Family ◽  
Deterministic Turing Machine ◽  
High Computational Complexity

Each day a weather forecaster predicts a probability for each type of weather for the next day. After n days, all the predicted probabilities and the real weather data are sent to a test which decides whether to accept the forecaster as having prior knowledge about the distribution of nature. Consider tests that accept with high probability forecasters who know the distribution of nature. Sandroni shows that any such test can be passed with high probability by a forecaster who has no prior knowledge about the distribution of nature, provided that the duration n is revealed to the forecaster in advance [14]. However, Fortnow and Vohra show that Sandroni's result requires forecasters with high computational complexity [6]. Consider the family [Formula: see text] of forecasters who select a deterministic Turing-machine forecaster according to an arbitrary distribution and then use that machine for all future forecasts. We show that Sandroni's result requires forecasters even more powerful than those in [Formula: see text]. We also show that Sandroni's result does not apply when the duration n is not revealed to the forecaster in advance.

Scanning electron microscopy of human iliac bone by a simple preparation method

1990 ◽  
Vol 48 (3) ◽  
pp. 226-227
Author(s):  
Toshihiko Takita ◽  
Tomonori Naguro ◽  
Toshio Kameie ◽  
Akihiro Iino ◽  
Kichizo Yamamoto
Keyword(s):  
Electron Microscopy ◽  
Scanning Electron Microscopy ◽  
Preparation Method ◽  
Specimen Preparation ◽  
Real Surface ◽  
Public Attention ◽  
Preparation Methods ◽  
The Real ◽  
Simple Preparation ◽  
Scanning Electron

Recently with the increase in advanced age population, the osteoporosis becomes the object of public attention in the field of orthopedics. The surface topography of the bone by scanning electron microscopy (SEM) is one of the most useful means to study the bone metabolism, that is considered to make clear the mechanism of the osteoporosis. Until today many specimen preparation methods for SEM have been reported. They are roughly classified into two; the anorganic preparation and the simple preparation. The former is suitable for observing mineralization, but has the demerit that the real surface of the bone can not be observed and, moreover, the samples prepared by this method are extremely fragile especially in the case of osteoporosis. On the other hand, the latter has the merit that the real information of the bone surface can be obtained, though it is difficult to recognize the functional situation of the bone.

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