Formal verification of tail distribution bounds in the HOL theorem prover

Abstract

SUMMARYTail distribution bounds play a major role in the estimation of failure probabilities in performance and reliability analysis of systems. They are usually estimated using Markov's and Chebyshev's inequalities, which represent tail distribution bounds for a random variable in terms of its mean or variance. This paper presents the formal verification of Markov's and Chebyshev's inequalities for discrete random variables using a higher-order-logic theorem prover. The paper also provides the formal verification of mean and variance relations for some of the widely used discrete random variables, such as Uniform(m), Bernoulli( p), Geometric( p) and Binomial(m, p) random variables. This infrastructure allows us to precisely reason about the tail distribution properties and thus turns out to be quite useful for the analysis of systems used in safety-critical domains, such as space, medicine or transportation. For illustration purposes, we present the performance analysis of the coupon collector's problem, a well-known commercially used algorithm.

On the Formalization of the Lebesgue Integration Theory in HOL

“…All of this framework is not available in the work of Coble [3] because his formalization does not include the Borel sets so he cannot prove the Lebesgue properties and the theorems of this section. The Markov and Chebyshev inequalities were previously proven by Hasan and Tahar [10] but only for discrete random variables. Our formalization allows us to provide a proof valid for both the discrete and continuous cases.…”

Probabilistic Analysis Using Theorem Proving

“…3, are the formal verification of Markov's and Chebyshev's inequalities in the HOL the prover, respectively. The verification is based on the formal definitions of expectation and variance and their formally verified properties and is outlined in [12].…”

“…For illustration purposes, [12] presents the formal verification of the expectation and variance relations for four discrete random variables: Bernoulli, Uniform Binomial and Geometric.…”

“…For example, it is more interesting to find out the expected value of the runtime of an algorithm for an NP-hard problem, rather than the PMF or CDF of the runtime. In [15,12], we tackled the formalization of expectation and variance in HOL for the first time. We extended Hurd's formalization framework with a formal definition of expectation, which can be utilized to formalize and verify the expectation and variance characteristics associated with discrete random variables that attain values in positive integers only.…”

“…This description illustrates how the already formalized mathematical concepts of probability theory fit into the global picture of probabilistic analysis while highlighting some of the interesting future research directions in the area of theorem proving based probabilistic analysis. Then in Sections 3 to 5, we briefly describe the already developed HOL infrastructures for the formalization of discrete random variables [16], the formalization of continuous random variables [13] and the verification of statistical properties [15,12], respectively. In order to illustrate the practical effectiveness of the proposed approach, we then present the probabilistic analysis of three examples using the HOL theorem prover in Section 6.…”