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

Outline n Introduction and Motivation n Formalization n Case Study: Clock Synchronization in WSNs n Conclusions Formalization of Normal Random Variables O. Hasan 2

Motivation Ag Agin ing Noise ise Ph Phenome mena Enviro En vironme mental l Condit itio ions s Unpre redict ictable le Inputs Inputs Formalization of Normal Random Variables O. Hasan 3

Probabilistic Analysis R andom ¡Variables Random (Discrete/ Components C ontinuous) System Syst m Mo Model l Probabilistic and Hardware Software Statistical Properties Computer Based Analysis Framework Property Satisfied? Formalization of Normal Random Variables O. Hasan 4

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]) Formalization of Normal Random Variables O. Hasan 5

Probabilistic Analysis Basics – Probabilistic Properties Pro Propert rty Descrip scriptio ion Exa Examp mple les s Discre iscrete Contin inuous Probability Mass Probability that the Function (PMF) random variable is equal to some number n Cumulative Probability that the Distribution random variable is less Function (CDF) than or equal to some number n Probability Slope of CDF for Density Function continuous random (PDF) variables Formalization of Normal Random Variables O. Hasan 6

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 Formalization of Normal Random Variables O. Hasan 7

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 Approximate Probabilistic Probabilistic Random Comp mponents random variable State Machine State Machine variable functions dgsd functions Analysis An lysis Observing Mathematical Exhaustive some test cases Reasoning Verification Accu Accura racy cy û ü ü Expre Exp ressive ssiveness ss ü û ü Au Automa matio ion ü ü û Ma Maturit rity û ü û Formalization of Normal Random Variables O. Hasan 8

Probabilistic Analysis using Theorem Proving Syst System m Descrip scriptio ion s) les) Higher-order logic Formalization of Probability Theory les) s) riable riable Varia Random Varia Discrete Random Continuous Random m Va Components m Va Variables Variables inuous Random iscrete Random System Model Statistical Probabilistic Probabilistic Statistical Properties Properties Properties Properties s (Discre PMF Expectation CDF Expectation s (Contin ies CDF Variance PDF Variance ies rtie rtie Propert Probabilistic Propert Analysis m Pro m Pro Proof Goals System System Syst Syst Theorem Prover Formal Proofs of Properties • [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 Formalization of Normal Random Variables O. Hasan 9

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 Formalization of Normal Random Variables O. Hasan 10

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] Formalization of Normal Random Variables O. Hasan 11

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 Definition: Probability Density Function Formalization of Normal Random Variables O. Hasan 12

Normal Random Variable n Normal PDF Definition: Normal Random Variable 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 Formalization of Normal Random Variables O. Hasan 13

Normal Random Variable – Properties Theorem: PDF of a Normal random variable is non-negative Theorem: PDF over the whole space is 1 Formalization of Normal Random Variables O. Hasan 14

Normal Random Variable – Properties Theorem: PDF of a Normal random variable is symmetric around its mean Theorem: PDF of a Normal random variable is symmetric around its mean Formalization of Normal Random Variables O. Hasan 15

Normal Random Variable – Properties Theorem: Summation of Normal Random Variables n The proofs of these properties not only ensure the correctness of our definitions but also facilitate the formal reasoning process about the Normal Random Variable Formalization of Normal Random Variables O. Hasan 16

Application: Probabilistic Clock Synchronization in WSNs n Synchronizing receivers with one another n Randomness in Message delivery latency n Probabilistic bounds Wireless on clock Sensor synchronization error Network n single hop n & multi-hop networks Formalization of Normal Random Variables O. Hasan 17

Capturing the Randomness in the Latency n Multiple pulses are sent from the sender to the set of receivers n The difference in R1 reception time at the R2 receivers is plotted S R3 R5 R3 – R4 R4 Pairwise difference in packet reception time – Normally Distributed with mean = 0 Formalization of Normal Random Variables O. Hasan 18

Error Bounds – Single Hop Theorem: Probability of synchronization error for single hop network Formalization of Normal Random Variables O. Hasan 19

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 Formalization of Normal Random Variables O. Hasan 20

Thank you! Formalization of Normal Random Variables O. Hasan 21