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Representations of measurable sets in computable measure theory

2014
*
Logical Methods in Computer Science
*

This article is a fundamental study

doi:10.2168/lmcs-10(3:7)2014
fatcat:ynfkfscmfbcvdfxluhnbbripja
*in**computable**measure**theory*. ... As a basic*computability*structure we consider a*computable**measure*on a*computable*σ-algebra. We introduce and compare w.r.t. reducibility several natural*representations**of**measurable**sets*. ... Introduction*Measure**theory*is a fundament*of*modern analysis.*In*particular,*computable**measure**theory*is a fundament*of**computable*analysis. ...##
###
Difficulty and Interest in Mathematical Representation of Human Heeling -Practical Role of Fuzzy Sets and Fuzzy Measure in Soft Computing -

人の気持ちを数理的に表現する「むずかしさ」と「おもしろさ」 -ソフトコンピューティングにおけるファジィ集合・ファジィ測度の実用的役割-

2015
*
Journal of Japan Society for Fuzzy Theory and Intelligent Informatics
*

人の気持ちを数理的に表現する「むずかしさ」と「おもしろさ」 -ソフトコンピューティングにおけるファジィ集合・ファジィ測度の実用的役割-

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Computability on the probability measures on the Borel sets of the unit interval

1999
*
Theoretical Computer Science
*

*In*this contribution we introduce

*computability*on the

*set*M

*of*probability

*measures*on the Bore1 subsets

*of*the unit interval [O; 11. ...

*In*this approach,

*computability*is defined on finite and infinite sequences

*of*symbols explicitly by Turing machines and on other

*sets*by means

*of*notations and

*representations*. ... less natural

*representations*and hence

*computability*

*theories*for the

*set*M

*of*probability

*measures*on ([O; l],B) can be introduced. ...

##
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Multivariate stress scenarios and solvency

2012
*
Insurance, Mathematics & Economics
*

*Measure*

*Theory*for Linear Portfolios Stress Test

*Representations*for Standard Risk

*Measures*Value-at-Risk Expected Shortfall Reverse Stress Tests The Case

*of*ES for Non-Elliptical Distributions The

*set*...

*In*the stress test Q is a

*set*

*of*Dirac

*measures*{δ x : x ∈ S} which place all the probability on each scenario

*in*S

*in*turn. ...

##
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Computability on the probability measures on the Borel sets of the unit interval
[chapter]

1997
*
Lecture Notes in Computer Science
*

*In*this contribution we introduce a natural

*computability*concept on the

*set*M

*of*probability

*measures*on the Borel subsets

*of*the unit interval 0; 1]. ... As background we consider TTE, Type 2

*Theory*

*of*E ectivity, KW84, KW85], where

*computability*is de ned on nite and

*in*nite sequences

*of*symbols explicitly by Turing machines and on other

*sets*by means ... However, a systematic study

*of*

*computability*

*in*integration and

*measure*

*theory*does not yet exist.

*In*this paper we extend TTE, Type 2

*Theory*

*of*E ectivity, to

*measure*

*theory*. ...

##
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Posterior distributions are computable from predictive distributions

2010
*
Journal of machine learning research
*

*In*addition to providing a brief survey

*of*

*computable*probability

*theory*geared towards the A.I. and statistics community, we give a new result characterizing when conditioning is

*computable*

*in*the

*setting*... for

*computing*the posterior distribution

*of*the directing random

*measure*. ...

*Computable*probability

*theory*provides a framework for exploring these questions. We present aspects

*of*this

*theory*

*in*Section 2. ...

##
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Complexity distortion theory

2003
*
IEEE Transactions on Information Theory
*

This closes the circle

doi:10.1109/tit.2002.808135
fatcat:lvwiax43x5as5mjna5vifsd2ny
*of**representation*models, from probabilistic models*of*information proposed by Shannon*in*information and rate distortion*theories*, to deterministic algorithmic models, proposed ... The key component*of*this*theory*is the substitution*of*the decoder*in*Shannon's classical communication model with a universal Turing machine. ... They also would like to thank the anonymous reviewers*of*this manuscript for their constructive comments. ...##
###
The descriptive theory of represented spaces
[article]

2014
*
arXiv
*
pre-print

This is a survey on the ongoing development

arXiv:1408.5329v1
fatcat:rb43guaodnex5llkn5szxpmvv4
*of*a descriptive*theory**of*represented spaces, which is intended as an extension*of*both classical and effective descriptive*set**theory*to deal with both*sets*... Most material is from work-*in*-progress, and thus there may be a stronger focus on projects involving the author than an objective survey would merit. ... Schröder and Victor Selivanov for discussions conductive to the inception*of*the research programme outlined here. ...##
###
Computability of probability measures and Martin-Lof randomness over metric spaces
[article]

2008
*
arXiv
*
pre-print

We show that any

arXiv:0709.0907v2
fatcat:ftwryuijj5gsjnoabdbzwxlhxu
*computable*metric space with a*computable*probability*measure*is isomorphic to the Cantor space*in*a*computable*and*measure*-theoretic sense. ...*In*this paper we investigate algorithmic randomness on more general spaces than the Cantor space, namely*computable*metric spaces. ... use*representation**theory**in*its general*setting*. ...##
###
Theories and Ordinals: Ordinal Analysis
[chapter]

2007
*
Lecture Notes in Computer Science
*

The connection between ordinal

doi:10.1007/978-3-540-73001-9_65
fatcat:jtij35ctirffdahd7hvyh6oesm
*representation*systems and*theories*is established*in*ordinal analysis, a central area*of*proof*theory*. Ordinal-theoretic proof*theory*came into existence*in*1936. ... How do ordinals*measure*the strength and*computational*power*of*formal*theories*and what information can be gleaned from this correlation? This will be the guiding question*of*this talk. ... How do ordinals*measure*the strength and*computational*power*of*formal*theories*and what information can be gleaned from this correlation? This will be the guiding question*of*this talk. ...##
###
Computability of the Radon-Nikodym derivative
[article]

2011
*
arXiv
*
pre-print

We study the

arXiv:1112.2838v1
fatcat:hx75yunljba4nopyb4us7tspiy
*computational*content*of*the Radon-Nokodym theorem from*measure**theory**in*the framework*of*the*representation*approach to*computable*analysis. ... We define*computable**measurable*spaces and canonical*representations**of*the*measures*and the integrable functions on such spaces. ... For the special application*in**measure**theory*we introduce*computable**measurable*spaces and*representations**of**measures*and*of*integrable functions. ...##
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Page 607 of Mathematical Reviews Vol. , Issue 87b
[page]

1987
*
Mathematical Reviews
*

The

*theory**of**representations*as a foundation for a unified type 2*computability**theory*is developed*in*the second paper under review. ...*Computability*on other*sets*S can be derived from computabil- ity on F by means*of**representations*, i.e. partial mappings from F onto S. ...##
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Computability of the Radon-Nikodym Derivative

2012
*
Computability - The Journal of the Assosiation
*

We study the

doi:10.3233/com-2012-005
fatcat:6v7us3i2kncrtedsuywyjqaqme
*computational*content*of*the Radon-Nokodym theorem from*measure**theory**in*the framework*of*the*representation*approach to*computable*analysis. ... We define*computable**measurable*spaces and canonical*representations**of*the*measures*and the integrable functions on such spaces. ... All rights reserved For the special application*in**measure**theory*we introduce*computable**measurable*spaces and*representations**of**measures*and*of*integrable functions. ...##
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Computability of probability measures and Martin-Löf randomness over metric spaces

2009
*
Information and Computation
*

We show that any

doi:10.1016/j.ic.2008.12.009
fatcat:2ccg4og7bfguhm43i2htej5ryq
*computable*metric space with a*computable*probability*measure*is isomorphic to the Cantor space*in*a*computable*and*measure*-theoretic sense. ...*In*this paper, we investigate algorithmic randomness on more general spaces than the Cantor space, namely*computable*metric spaces. ... not use*representation**theory**in*its general*setting*. ...##
###
Negative quasi-probability as a resource for quantum computation

2012
*
New Journal of Physics
*

We establish a remarkable connection between the potential for quantum speed-up and the onset

doi:10.1088/1367-2630/14/11/113011
fatcat:saf656xf5rfi3dqaosoemmvoiy
*of*negative values*in*a distinguished quasi-probability*representation*, a discrete analog*of*the Wigner function ... A central problem*in*quantum information is to determine the minimal physical resources that are required for quantum*computational*speedup and,*in*particular, for fault-tolerant quantum*computation*. ... We also allow arbitrary product*measurements*with positive discrete Wigner*representation*. This simulation is ecient(linear)*in*the number*of*input registers to the quantum circuit. ...
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