The LIMITS of MATHEMATICS

Chaitin, Gregory J.

doi:10.1007/978-1-4471-0015-7

Cited by 26 publications

(29 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Those interested in that equivalence may read, for example, the second section (pp. 129-133), "Proof that Martin-Löf randomness is equivalent to Chaitin randomness" in Part III of Chaitin's book cited above [1].…”

Section: Numerical Results Of Particular Interestmentioning

confidence: 99%

See 1 more Smart Citation

Indices of regularity and indices of randomness for m-ary strings

Skliar

Monge²,

Oviedo

et al. 2009

RMTA

View full text Add to dashboard Cite

The notions "regularity index" and "randomness index" previously introduced for binary strings (2-ary) have been modified slightly and generalized for m-ary strings (m = 2, 3, 4, . . .). These notions are complementary and the regular/random dichotomy has been replaced by a gradation of values of regularity and of randomness. With this approach, the more regular an m-ary string, the less random it is, and vice versa. The distributions of frequencies of different length strings -2-ary and 3-ary strings-according to their indices of randomness, are shown by histograms.Keywords: regularity index, randomness index, m-ary strings.Resumen Las nociones deíndice de regularidad y deíndice de aleatoriedad previamente introducidas para cadenas binarias (2-arias) son modificadas ligeramente y generalizadas para cadenas m-arias (m = 2, 3, 4, . . .). Dichas nociones resultan complementarias y la dicotomía regular-aleatorio es sustituida por una gradación de valores de regularidad y de aleatoriedad. Con el enfoque utilizado, cuanto más regular es una cadena m-aria menos aleatoria debe ser considerada y viceversa. Las distribuciones de frecuencias de cadenas -de diversas longitudes-2-arias y 3-arias en función de susíndices de aleatoriedad son presentadas mediante histogramas.Palabras clave:índice de regularidad,índice de aleatoriedad, cadenas m-arias.Mathematics Subject Classification: 49J55.

show abstract

Section: Numerical Results Of Particular Interestmentioning

confidence: 99%

“…Similarly, as seen below, if an m-ary string is considered to be highly random (in naïve but intuitive terms, for now), it will not be too regular. For an alternative and clearly dichotomic approach, see [2] and [1], for example.…”

Section: Introductionmentioning

confidence: 99%

Indices of regularity and indices of randomness for m-ary strings

Skliar

Monge²,

Oviedo

et al. 2009

RMTA

View full text Add to dashboard Cite

show abstract

“…Truth as a control mechanism is arrived at first in the interaction, based on propositional knowledge, between several agents (inter-subjective consensus about knowledge) or in the relationship between different pieces of propositional knowledge that an agent possess and can reason about. In the sense of Chaitin's "truth islands" [88], some well-defined parts of reality can be organized and systematized in such a way that truth may be well-defined within those sets, via inter-agent communication. For an agent, meaning is a more fundamental phenomenon than truth, and both must be possible to express in terms of models [83]:…”

Section: Søren Brier In His the Cybersemiotic Framework As A Means Tomentioning

confidence: 99%

Floridi’s “Open Problems in Philosophy of Information”, Ten Years Later

Dodig-Crnković

Hofkirchner

2011

Information

View full text Add to dashboard Cite

In his article Open Problems in the Philosophy of Information [1] LucianoFloridi presented a Philosophy of Information research program in the form of eighteen open problems, covering the following fundamental areas: Information definition, information semantics, intelligence/cognition, informational universe/nature and values/ethics. We revisit Floridi's program, highlighting some of the major advances, commenting on unsolved problems and rendering the new landscape of the Philosophy of Information (PI) emerging at present. As we analyze the progress of PI we try to situate Floridi's program in the context of scientific and technological development that have been made last ten years. We emphasize that Philosophy of Information is a huge and vibrant research field, with its origins dating before Open Problems, and its domains extending even outside their scope. In this paper, we have been able only to sketch some of the developments during the past ten years. Our hope is that, even if fragmentary, this review may serve as a contribution to the effort of understanding the present state of the art and the paths of development of Philosophy of Information as seen through the lens of Open Problems.

show abstract

“…The Kolmogorov complexity (also known as Kolmogorov-Chaitin complexity, stochastic complexity, and algorithmic entropy) of an object is a measure of the computational resources needed to specify the object [19,26,8]. In other words, the complexity of a string is the length of the string's shortest description in some fixed description language.…”

Section: Example 3 (Stochastic Signals Coding and Kolmogorov Complementioning

confidence: 99%

Gauge theory for finite-dimensional dynamical systems

Gurfil

2007

Chaos: An Interdisciplinary Journal of Nonlinear Science

View full text Add to dashboard Cite

Gauge theory is a well-established concept in quantum physics, electrodynamics, and cosmology. This theory has recently proliferated into new areas, such as mechanics and astrodynamics. In this paper, we discuss a few applications of gauge theory in finite-dimensional dynamical systems with implications to numerical integration of differential equations. We distinguish between rescriptive and descriptive gauge symmetry. Rescriptive gauge symmetry is, in essence, re-scaling of the independent variable, while descriptive gauge symmetry is a Yang-Mills-like transformation of the velocity vector field, adapted to finite-dimensional systems. We show that a simple gauge transformation of multiple harmonic oscillators driven by chaotic processes can render an apparently "disordered" flow into a regular dynamical process, and that there exists a remarkable connection between gauge transformations and reduction theory of ordinary differential equations. Throughout the discussion, we demonstrate the main ideas by considering examples from diverse engineering and scientific fields, including quantum mechanics, chemistry, rigid-body dynamics and information theory.

show abstract

The LIMITS of MATHEMATICS

Cited by 26 publications

References 0 publications

Indices of regularity and indices of randomness for m-ary strings

Indices of regularity and indices of randomness for m-ary strings

Floridi’s “Open Problems in Philosophy of Information”, Ten Years Later

Gauge theory for finite-dimensional dynamical systems

Contact Info

Product

Resources

About