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Representations of measurable sets in computable measure theory

Klaus Weihrauch, Nazanin Tavana-Roshandel, Ning Zhong
2014 Logical Methods in Computer Science  
This article is a fundamental study 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.  ... 
doi:10.2168/lmcs-10(3:7)2014 fatcat:ynfkfscmfbcvdfxluhnbbripja

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-

Hisao SHIIZUKA
2015 Journal of Japan Society for Fuzzy Theory and Intelligent Informatics  
doi:10.3156/jsoft.27.1_2 fatcat:ngbkex2wwrcahdnl42xc3vnhfm

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

Klaus Weihrauch
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.  ... 
doi:10.1016/s0304-3975(98)00298-9 fatcat:hgdwmcemyjellomxqqdotk22gq

Multivariate stress scenarios and solvency

Alexander J. McNeil, Andrew D. Smith
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.  ... 
doi:10.1016/j.insmatheco.2011.12.005 fatcat:x7sw3e43ofbqbc4hoy4ihcwqii

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

Klaus Weihrauch
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.  ... 
doi:10.1007/3-540-63165-8_174 fatcat:e362uxbfszg3dmv7nuxbeaf4ku

Posterior distributions are computable from predictive distributions

Cameron E. Freer, Daniel M. Roy
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.  ... 
dblp:journals/jmlr/FreerR10 fatcat:cnhuixsuebdkxma7zk7hw2ywai

Complexity distortion theory

D.M. Sow, A. Eleftheriadis
2003 IEEE Transactions on Information Theory  
This closes the circle 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.  ... 
doi:10.1109/tit.2002.808135 fatcat:lvwiax43x5as5mjna5vifsd2ny

The descriptive theory of represented spaces [article]

Arno Pauly
2014 arXiv   pre-print
This is a survey on the ongoing development 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.  ... 
arXiv:1408.5329v1 fatcat:rb43guaodnex5llkn5szxpmvv4

Computability of probability measures and Martin-Lof randomness over metric spaces [article]

Mathieu Hoyrup, Cristobal Rojas
2008 arXiv   pre-print
We show that any 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.  ... 
arXiv:0709.0907v2 fatcat:ftwryuijj5gsjnoabdbzwxlhxu

Theories and Ordinals: Ordinal Analysis [chapter]

Michael Rathjen
2007 Lecture Notes in Computer Science  
The connection between ordinal 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.  ... 
doi:10.1007/978-3-540-73001-9_65 fatcat:jtij35ctirffdahd7hvyh6oesm

Computability of the Radon-Nikodym derivative [article]

Mathieu Hoyrup, Cristobal Rojas, Klaus Weihrauch
2011 arXiv   pre-print
We study the 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.  ... 
arXiv:1112.2838v1 fatcat:hx75yunljba4nopyb4us7tspiy

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.  ... 

Computability of the Radon-Nikodym Derivative

Mathieu Hoyrup, Cristóbal Rojas, Klaus Weihrauch
2012 Computability - The Journal of the Assosiation  
We study the 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.  ... 
doi:10.3233/com-2012-005 fatcat:6v7us3i2kncrtedsuywyjqaqme

Computability of probability measures and Martin-Löf randomness over metric spaces

Mathieu Hoyrup, Cristóbal Rojas
2009 Information and Computation  
We show that any 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.  ... 
doi:10.1016/j.ic.2008.12.009 fatcat:2ccg4og7bfguhm43i2htej5ryq

Negative quasi-probability as a resource for quantum computation

Victor Veitch, Christopher Ferrie, David Gross, Joseph Emerson
2012 New Journal of Physics  
We establish a remarkable connection between the potential for quantum speed-up and the onset 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.  ... 
doi:10.1088/1367-2630/14/11/113011 fatcat:saf656xf5rfi3dqaosoemmvoiy
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