Lattice Theory, Measures and Probability

Knuth, Kevin H.

doi:10.1063/1.2821268

Cited by 6 publications

(4 citation statements)

References 10 publications

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“…The second example, which was the original inspiration for this work is the derivation of probability theory [13,17,18,14]. By founding probability theory as a quantification of implication among logical statements, we obtain a theory that encompasses and generalizes both the Cox and Kolmogorov formulations.…”

Section: Applicationsmentioning

confidence: 99%

Information Physics: The New Frontier

Knuth

Mohammad‐Djafari²,

Bercher³

et al. 2011

AIP Conference Proceedings

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At this point in time, two major areas of physics, statistical mechanics and quantum mechanics, rest on the foundations of probability and entropy. The last century saw several significant fundamental advances in our understanding of the process of inference, which make it clear that these are inferential theories. That is, rather than being a description of the behavior of the universe, these theories describe how observers can make optimal predictions about the universe. In such a picture, information plays a critical role. What is more is that little clues, such as the fact that black holes have entropy, continue to suggest that information is fundamental to physics in general. In the last decade, our fundamental understanding of probability theory has led to a Bayesian revolution. In addition, we have come to recognize that the foundations go far deeper and that Cox's approach of generalizing a Boolean algebra to a probability calculus is the first specific example of the more fundamental idea of assigning valuations to partially-ordered sets. By considering this as a natural way to introduce quantification to the more fundamental notion of ordering, one obtains an entirely new way of deriving physical laws. I will introduce this new way of thinking by demonstrating how one can quantify partially-ordered sets and, in the process, derive physical laws. The implication is that physical law does not reflect the order in the universe, instead it is derived from the order imposed by our description of the universe. Information physics, which is based on understanding the ways in which we both quantify and process information about the world around us, is a fundamentally new approach to science.Comment: 17 pages, 6 figures. Knuth K.H. 2010. Information physics: The new frontier. J.-F. Bercher, P. Bessi\`ere, and A. Mohammad-Djafari (eds.) Bayesian Inference and Maximum Entropy Methods in Science and Engineering (MaxEnt 2010), Chamonix, France, July 201

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Section: Applicationsmentioning

confidence: 99%

Information Physics: The New Frontier

Knuth

Mohammad‐Djafari²,

Bercher³

et al. 2011

AIP Conference Proceedings

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“…Assuming the set of questions is seen as an ordered set, with the largest questions being the most relevant (since their answers carry potentially more cogent information), a cognitive device can decide which appropriate request to formulate to the transmitter. The work on relevance and questions is however still in its infancy, but we insist that those are fundamental needs to the cognitive radio field; for instance, interesting contributions are found in the works of Knuth et al [20], who uses lattice theory to create partial orders of finite sets of questions, which is seen as the dual (in the lattice theory terminology) of the set of answers to those questions.…”

Section: Relevancementioning

confidence: 99%

Mathematical Foundations of Cognitive Radios

Couillet,

Debbah

2009

JTIT

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Recently, much interest has been directed towards software defined radios and embedded intelligence in telecommunication devices. However, no fundamental basis for cognitive radios has ever been proposed. In this paper, we introduce a fundamental vision of cognitive radios from a physical layer viewpoint. Specifically, our motivation in this work is to embed human-like intelligence in mobile wireless devices, following the three century-old work on Bayesian probability theory, the maximum entropy principle and minimal probability update. This allows us to partially answer such questions as, what are the signal detection capabilities of a wireless device, when facing a situation in which most parameters are missing, how to react and so on. As an introductory example, we will present previous works from the same authors following the cognitive framework, and especially the multi-antenna channel modeling and signal sensing.

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“…In this paper, I will describe the important concepts behind this new foundation, and will leave many of the mathematical details to previous works published during the development of these ideas [1,2,3,4]. Although, it should be noted that in [3] equation 31 and the results that follow are incorrect.…”

Section: Introductionmentioning

confidence: 99%

The Origin of Probability and Entropy

Knuth¹

2008

AIP Conference Proceedings

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Abstract. Measuring is the quantification of ordering. Thus the process of ordering elements of a set is a more fundamental activity than measuring. Order theory, also known as lattice theory, provides a firm foundation on which to build measure theory. The result is a set of new insights that cast probability theory and information theory in a new light, while simultaneously opening the door to a better understanding of measures as a whole.

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Lattice Theory, Measures and Probability

Cited by 6 publications

References 10 publications

Information Physics: The New Frontier

Information Physics: The New Frontier

Mathematical Foundations of Cognitive Radios

The Origin of Probability and Entropy

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