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Formalization of Continuous Probability Distributions
 In Conference on Automated Deduction, volume 4603 of LNAI
, 2007
"... In order to overcome the limitations of stateoftheart simulation based probabilistic analysis, we propose to perform probabilistic analysis within the environment of a higherorderlogic theorem prover. The foremost requirement for conducting such analysis is the formalization of probability dist ..."
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Cited by 6 (6 self)
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In order to overcome the limitations of stateoftheart simulation based probabilistic analysis, we propose to perform probabilistic analysis within the environment of a higherorderlogic theorem prover. The foremost requirement for conducting such analysis is the formalization of probability distributions. In this report, we present a methodology for the formalization of continuous probability distributions for which the inverse of the cumulative distribution function can be expressed in a closed mathematical form. Our methodology is primarily based on the formalization of the Standard Uniform random variable, cumulative distribution function properties and the Inverse Transform method. The report presents all this formalization using the HOL theorem prover. In order to illustrate the practical effectiveness of our methodology, the formalization of a few continuous probability distributions has also been included. 1 1
Probabilistic Analysis using Theorem Proving
"... Abstract. Traditionally, computer simulation techniques are used to perform probabilistic analysis. However, they provide less accurate results and cannot handle largescale problems due to their enormous CPU time requirements. Recently, a significant amount of formalization has been done in the HOL ..."
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Abstract. Traditionally, computer simulation techniques are used to perform probabilistic analysis. However, they provide less accurate results and cannot handle largescale problems due to their enormous CPU time requirements. Recently, a significant amount of formalization has been done in the HOL theorem prover that allows us to conduct precise probabilistic analysis using theorem proving and thus overcome the limitations of the simulation based probabilistic analysis approach. Some major contributions include the formalization of both discrete and continuous random variables and the verification of some of their corresponding probabilistic and statistical properties. This paper presents a concise description of the infrastructures behind these capabilities and the utilization of these features to conduct the probabilistic analysis of realworld systems. For illustration purposes, the paper describes the theorem proving based probabilistic analysis of three examples, i.e., the roundoff error of a digital processor, the Coupon Collector’s problem and the StopandWait protocol. 1
Standard uniform distribution theory in HOL4
, 2007
"... In this report, we present the complete HOL theory for the Standard Uniform random variable. The Standard Uniform random variable is defined as the limit value of the sequence of its discrete approximations. The report also contains all the theorems leading to the verification of the probability dis ..."
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In this report, we present the complete HOL theory for the Standard Uniform random variable. The Standard Uniform random variable is defined as the limit value of the sequence of its discrete approximations. The report also contains all the theorems leading to the verification of the probability distribution properties of the Standard Uniform random variable. More details about this formalization can be found in [3]. 1 1 HOL Theorem Prover The HOL theorem prover [2] is an interactive theorem prover which is capable of conducting proofs in higherorder logic. It utilizes the simple type theory of Church [1] along with HindleyMilner polymorphism [4] to implement higherorder logic. HOL has been successfully used as a verification framework for both software and hardware as well as a platform for the formalization of pure mathematics. It supports the formalization of various mathematical theories including sets, natural numbers, real numbers, measure and probability. HOL is an interactive theorem prover with access to many proof assistants and automatic
Probabilistic Analysis of Wireless Systems using Theorem Proving
"... Probabilistic techniques play a major role in the design and analysis of wireless systems as they contain a significant amount of random or unpredictable components. Traditionally, computer simulation techniques are used to perform probabilistic analysis of wireless systems but they provide inaccura ..."
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Probabilistic techniques play a major role in the design and analysis of wireless systems as they contain a significant amount of random or unpredictable components. Traditionally, computer simulation techniques are used to perform probabilistic analysis of wireless systems but they provide inaccurate results and usually require enormous amount of CPU time in order to attain reasonable estimates. To overcome these limitations, we propose to use a higherorderlogic theorem prover (HOL) for the analysis of wireless systems. The paper presents a concise description of the formal foundations required to conduct the analysis of a wireless system in a theorem prover, such as, the higherorderlogic modeling of random variables and the verification of their corresponding probabilistic and statistical properties in a theorem prover. In order to illustrate the utilization and effectiveness of the proposed idea for handling realworld wireless system analysis problems, we present an analysis of the automated repeat request (ARQ) mechanism at the logic link control (LLC) layer of the General Packet Radio Service (GPRS), which is a packet oriented mobile data service available to the users of Global System for Mobile Communications (GSM).
Time Systems
, 2011
"... any form or by any means, electronic or mechanical, including photocopying, without written permission from the publisher. Product or company names used in this set are for identification purposes only. Inclusion of the names of the products or companies does not indicate a claim of ownership by IG ..."
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any form or by any means, electronic or mechanical, including photocopying, without written permission from the publisher. Product or company names used in this set are for identification purposes only. Inclusion of the names of the products or companies does not indicate a claim of ownership by IGI Global of the trademark or registered trademark. Library of Congress CataloginginPublication Data Reconfigurable embedded control systems: applications for flexibility and agility / Mohamed Khalgui and HansMichael Hanisch, editors. p. cm. Includes bibliographical references and index. ISBN 9781609600860 (hardcover) ISBN 9781609600884 (ebook) 1. Programmable controllers. 2. Embedded computer systems. 3. Digital control systems. I. Khalgui, Mohamed. II. Hanisch, HansMichael.