Formalization of Normal Random Variables in HOL

Many components of engineering systems exhibit random and uncertain behaviors that are normally distributed. In order to conduct the analysis of such systems within the trusted kernel of a higher-order-logic theorem prover, in this paper, we provide a higher-order-logic formalization of Lebesgue measure and Normal random variables along with the proof of their classical properties. To illustrate the usefulness of our formalization, we present a formal analysis of the probabilistic clock synchronization in wireless sensor networks.

[1]  Sofiène Tahar,et al.  Formalization of Finite-State Discrete-Time Markov Chains in HOL , 2011, ATVA.

[2]  Sofiène Tahar,et al.  Formalization of Continuous Probability Distributions , 2007, CADE.

[3]  P. Spreij.  Probability and Measure , 1996 .

[4]  Amit Kumar Saha,et al.  Adaptive clock synchronization in sensor networks , 2004, Third International Symposium on Information Processing in Sensor Networks, 2004. IPSN 2004.

[5]  Sofiène Tahar,et al.  Evaluation of anonymity and confidentiality protocols using theorem proving , 2015, Formal Methods Syst. Des..

[6]  Feng Zhu,et al.  Modelling Clock Synchronization in the Chess gMAC WSN Protocol , 2009, QFM.

[7]  Sofiène Tahar,et al.  Towards the Formal Reliability Analysis of Oil and Gas Pipelines , 2014, CICM.

[8]  Sofiène Tahar,et al.  Formal Probabilistic Analysis of Stuck-at Faults in Reconfigurable Memory Arrays , 2009, IFM.

[9]  Johannes Hölzl.  Analyzing Discrete-Time Markov Chains with Countable State Space in Isabelle / HOL , 2013 .

[10]  J. Elson,et al.  Fine-grained network time synchronization using reference broadcasts , 2002, OSDI '02.

[11]  Johannes Hölzl,et al.  Three Chapters of Measure Theory in Isabelle/HOL , 2011, ITP.

[12]  Sofiène Tahar,et al.  Using Theorem Proving to Verify Expectation and Variance for Discrete Random Variables , 2008, Journal of Automated Reasoning.

[13]  Sofiène Tahar,et al.  Performance Analysis and Functional Verification of the Stop-and-Wait Protocol in HOL , 2008, Journal of Automated Reasoning.

[14]  Jianhua Zhao,et al.  Modeling and Evaluation of Wireless Sensor Network Protocols by Stochastic Timed Automata , 2013 .

[15]  Sofiène Tahar,et al.  Formalization of Entropy Measures in HOL , 2011, ITP.

[16]  R. Goldberg.  Methods of Real Analysis , 1964 .

[17]  J. Rice.  Mathematical Statistics and Data Analysis , 1988 .

[18]  Sofiène Tahar,et al.  Formal Analysis of Memory Contention in a Multiprocessor System , 2013, SBMF.

[19]  Sofiène Tahar,et al.  On the Formalization of the Lebesgue Integration Theory in HOL , 2010, ITP.

[20]  Sofiène Tahar,et al.  Formal probabilistic analysis of detection properties in wireless sensor networks , 2014, Formal Aspects of Computing.

[21]  Allan I. McInnes.  Model-checking the Flooding Time Synchronization Protocol , 2009, 2009 IEEE International Conference on Control and Automation.