Fuzzy Kolmogorov Complexity Based on a Classical Description

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.

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Quantum information distance

International Journal of Quantum Information ◽

10.1142/s0219749921500313 ◽

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.

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On the complexity of Turing machine accepting fuzzy language

2012 9th International Conference on Fuzzy Systems and Knowledge Discovery ◽

10.1109/fskd.2012.6233747 ◽

2012 ◽

Author(s):

Yingjun Pan ◽

Tie Zou

Keyword(s):

Turing Machine ◽

Fuzzy Language

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THE SECOND QUANTIZED QUANTUM TURING MACHINE AND KOLMOGOROV COMPLEXITY

Modern Physics Letters B ◽

10.1142/s021798490801464x ◽

2008 ◽

Vol 22 (12) ◽

pp. 1203-1210 ◽

Cited By ~ 1

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.

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

International Journal of Foundations of Computer Science ◽

10.1142/s0129054102001291 ◽

2002 ◽

Vol 13 (04) ◽

pp. 587-612 ◽

Cited By ~ 45

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.

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On Kolmogorov complexity in the real Turing machine setting

Information Processing Letters ◽

10.1016/s0020-0190(98)00089-1 ◽

1998 ◽

Vol 67 (2) ◽

pp. 81-86 ◽

Cited By ~ 3

Author(s):

J.L. Montaña ◽

Luis M. Pardo

Keyword(s):

Turing Machine ◽

Kolmogorov Complexity ◽

The Real ◽

Machine Setting

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Teorias da Aleatoriedade

Revista de Informática Teórica e Aplicada ◽

10.22456/2175-2745.5983 ◽

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.

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