Formalization of Normal Random Variables M. Qasim, O. Hasa san, M. - - PowerPoint PPT Presentation

Formalization of Normal Random Variables

• M. Qasim, O. Hasa

san, M. Elleuch, S. Tahar

Hardware Verification Group ECE Department, Concordia University, Montreal, Canada

CICM M 16

July 28, 2016

• O. Hasan

2 Formalization of Normal Random Variables

Outline

n Introduction and Motivation n Formalization n Case Study: Clock Synchronization in

WSNs

n Conclusions

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3 Formalization of Normal Random Variables

Motivation

En Enviro vironme mental l Condit itio ions s Ag Agin ing Ph Phenome mena Unpre redict ictable le Inputs Inputs Noise ise

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4 Formalization of Normal Random Variables

Probabilistic Analysis

Hardware Software Syst System m Mo Model l Property Satisfied? Random Components Probabilistic and Statistical Properties Computer Based Analysis Framework

R andom ¡Variables (Discrete/ C ontinuous)

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5 Formalization of Normal Random Variables

Probabilistic Analysis Basics –

Random Variables n Discrete Random Variables

n Attain a countable number of values n Example

n Dice[1, 6]

n Continuous Random Variables

n Attain an uncountable number of values n Examples

n Uniform (all real numbers in an interval [a,b])

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6 Formalization of Normal Random Variables

Probabilistic Analysis Basics –

Probabilistic Properties

Pro Propert rty Descrip scriptio ion Exa Examp mple les s

Discre iscrete Contin inuous

Probability Mass Function (PMF) Probability that the random variable is equal to some number n Cumulative Distribution Function (CDF) Probability that the random variable is less than or equal to some number n Probability Density Function (PDF) Slope of CDF for continuous random variables

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7 Formalization of Normal Random Variables

Probabilistic Analysis Basics –

Statistical Properties

Pro Propert rty Descrip scriptio ion Illu llust stra ratio ion

Expectation Long-run average value of a random variable Variance Measure of dispersion of a random variable

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8 Formalization of Normal Random Variables

Probabilistic Analysis Approaches

Simu Simula latio ion Forma rmal l Me Methods Mo Model l Checkin cking Theore rem m Pro Provin ving Random m Comp mponents Probabilistic State Machine dgsd An Analysis lysis Accu Accura racy cy Exp Expre ressive ssiveness ss Au Automa matio ion Ma Maturit rity Approximate random variable functions Observing some test cases

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Probabilistic State Machine Exhaustive Verification

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Random variable functions Mathematical Reasoning

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9 Formalization of Normal Random Variables

Probabilistic Analysis using Theorem Proving

Syst System m Descrip scriptio ion

Syst System m Pro Propert rtie ies s (Discre iscrete Random m Va Varia riable les) s) Syst System m Pro Propert rtie ies s (Contin inuous Random m Va Varia riable les) s) System Model Probabilistic Analysis Proof Goals Discrete Random Variables Continuous Random Variables

Random Components Probabilistic Properties Statistical Properties PMF CDF Expectation Variance Probabilistic Properties Statistical Properties CDF PDF Expectation Variance

Theorem Prover Formal Proofs of Properties

Higher-order logic Formalization of Probability Theory

• [Hurd, 2002]: Probability Theory, Discrete Random Variables (RVs), PMF
• [Hasan, 2007]: Statistical Properties for Discrete RVs, CDF, Continuous RVs
• [Mhamdi, 2011] Probability (Arbitrary space) Lebesgue Integration, Multiple Continuous RVs Statistical Properties
• [Hölzl, 2012] Isabelle/HOL: Probability, Measure and Lebesgue Integration, Markov, Central Limit Theorem

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10 Formalization of Normal Random Variables

Paper Contributions

n Formalization of Probability Density Function

(PDF)

n Formalization of Normal Random Variable

n Enormous Applications n Sample mean of most distributions can be

treated as Normally Distributed

n Case Study: Clock Synchronization in WSNs

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Probability Density Function

n PDF p(x) of a random variable x is used to define its

distribution

n The PDF of a random variable is formally defined as the

Radon-Nikodym (RN) derivative of the probability measure with respect to the Lebesgue-Borel measure

n RN derivative and probability measure was available in HOL4 n Lebesgue-Borel measure

n Ported from Isabelle/HOL [Hölzl, 2012] n Some theorems and tactics (e.g. SET_TAC) also ported from the Lebesgue

measure theory of HOL-Light [Harrison, 2013]

11 Formalization of Normal Random Variables

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Probability Density Function

n The PDF of a random variable is formally defined as the

Radon-Nikodym (RN) derivative of the probability measure with respect to the Lebesgue-Borel measure

12 Formalization of Normal Random Variables

Definition: Probability Density Function

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Normal Random Variable

n Normal PDF

n X is a real random variable, i.e., it is measurable from the

probability space (p) to Borel space

n The distribution of X is that of the Normal random variable

13 Formalization of Normal Random Variables

Definition: Normal Random Variable

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Normal Random Variable – Properties

14 Formalization of Normal Random Variables

Theorem: PDF of a Normal random variable is non-negative Theorem: PDF over the whole space is 1

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Normal Random Variable – Properties

15 Formalization of Normal Random Variables

Theorem: PDF of a Normal random variable is symmetric around its mean Theorem: PDF of a Normal random variable is symmetric around its mean

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Normal Random Variable – Properties

16 Formalization of Normal Random Variables

Theorem: Summation of Normal Random Variables n The proofs of these properties not only ensure the correctness

• f our definitions but also facilitate the formal reasoning

process about the Normal Random Variable

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Wireless Sensor Network

Application: Probabilistic Clock Synchronization in WSNs

n Synchronizing

receivers with one another

n Randomness in

Message delivery latency

n Probabilistic bounds

• n clock

synchronization error

n single hop n & multi-hop networks 17 Formalization of Normal Random Variables

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Capturing the Randomness in the Latency

n Multiple pulses are

sent from the sender to the set of receivers

n The difference in

reception time at the receivers is plotted

18 Formalization of Normal Random Variables

Pairwise difference in packet reception time – Normally Distributed with mean = 0

S R1 R2 R3 R4 R5 R3 – R4

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Error Bounds – Single Hop

19 Formalization of Normal Random Variables

Theorem: Probability of synchronization error for single hop network

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20 Formalization of Normal Random Variables

Conclusions

n Probabilistic Theorem Proving

n Exact Answers n Useful for the analysis of Safety critical application

n Our Contributions

n Formalization of Probability Density Functions and

Normal random variables

n Case Study

n Clock Synchronization in WSNs

n Future Work

n More Applications – Probabilistic Round off Error

Bounds in Computer Arithmetic

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21 Formalization of Normal Random Variables

Thank you!