Formalization of Measure Theory and Lebesgue Integration for Probabilistic Analysis in HOL

Dynamic systems that exhibit probabilistic behavior represent a large class of man-made systems such as communication networks, air traffic control, and other mission-critical systems. Evaluation of quantitative issues like performance and dependability of these systems is of paramount importance. In this paper, we propose a generalized methodology to formally reason about probabilistic systems within a theorem prover. We present a formalization of measure theory in the HOL theorem prover and use it to formalize basic concepts from the theory of probability. We also use the Lebesgue integration to formalize statistical properties of random variables. To illustrate the practical effectiveness of our methodology, we formally prove classical results from the theories of probability and information and use them in a data compression application in HOL

Semantic Foundations of Higher-Order Probabilistic Programs in Isabelle/HOL

Higher-order probabilistic programs are used to describe statistical models and machine-learning mechanisms. The programming languages for them are equipped with three features: higher-order functions, sampling, and conditioning. In this paper, we propose an Isabelle/HOL library for probabilistic programs supporting all of those three features. We extend our previous quasi-Borel theory library in Isabelle/HOL. As a basis of the theory, we formalize s-finite kernels, which is considered as a theoretical foundation of first-order probabilistic programs and a key to support conditioning of probabilistic programs. We also formalize the Borel isomorphism theorem which plays an important role in the quasi-Borel theory. Using them, we develop the s-finite measure monad on quasi-Borel spaces. Our extension enables us to describe higher-order probabilistic programs with conditioning directly as an Isabelle/HOL term whose type is that of morphisms between quasi-Borel spaces. We also implement the qbs prover for checking well-typedness of an Isabelle/HOL term as a morphism between quasi-Borel spaces. We demonstrate several verification examples of higher-order probabilistic programs with conditioning

Formalized Haar Measure

We describe the formalization of the existence and uniqueness of Haar measure in the Lean theorem prover. The Haar measure is an invariant regular measure on locally compact groups, and it has not been formalized in a proof assistant before. We will also discuss the measure theory library in Lean's mathematical library \textsf{mathlib}, and discuss the construction of product measures and the proof of Fubini's theorem for the Bochner integral.Comment: 16 pages (excluding references

Information-Theoretic Analysis using Theorem Proving

Information theory is widely used for analyzing a wide range of scientific and engineering problems, including cryptography, neurobiology, quantum computing, plagiarism detection and other forms of data analysis. Despite the safety-critical nature of some of these applications, most of the information-theoretic analysis is done using informal techniques, mainly computer simulation and paper-and-pencil analysis, and thus cannot be completely relied upon. The unreliable nature of the produced results poses a serious problem in safety-critical applications and may result in heavy financial losses or even the loss of human life. In order to overcome the inaccuracy limitations of these techniques, this thesis proposes to conduct the analysis within the trusted kernel of a higher-order-logic (HOL) theorem prover. For this purpose, we provide HOL formalizations of the fundamental theories of measure, Lebesgue integration and probability and use them to formalize some of the most widely used information-theoretic principles. We use the Kullback-Leibler divergence as a unified measure of information which is in turn used to define the main measures of information like the Shannon entropy, mutual information and conditional mutual information. Furthermore, we introduce two new measures of information leakage, namely the information leakage degree and the conditional information leakage degree and compare them with existing measures. We illustrate the usefulness of the proposed framework by tackling various applications including the performance analysis of a communication encoder used in the proof of the Shannon source coding theorem, the quantitative analysis of privacy properties of a digital communications mixer and the one-time pad encryption system using information-theoretic measures

Formalization of Normal Random Variables

Engineering systems often have components that exhibit random behavior. This randomness in many cases is normally distributed. To verify such systems, proba- bilistic analysis is used. Such engineering systems have applications in domains like transportation, medicine and military. Despite the safety-critical nature of these ap- plications, most of the analysis is done using informal techniques like simulation and paper-and-pencil analysis, and thus cannot be completely relied upon. The unreliable results produced by such methods may result in heavy financial loss or even the loss of a human life. To overcome the limitation of traditional methods, we propose to conduct the analysis of such systems within the trusted kernel of a higher-order-logic theorem prover HOL4. The soundness and the deduction style of the theorem prover guarantee the validity of the analysis and the results of this type of analysis are generic and valid for any instance of the system. For this purpose, we provide HOL4 formalization of Lebesgue measure and normal random variables along with the proof of their classical properties. We also ported the theory of Gauge integral and other required foundational concepts from HOL Light and Isabelle/HOL theorem provers. To illustrate the usefulness of our formalization, we conducted the formal analysis of two applications, i.e., error probability of binary transmission in the presence of Gaussian noise and probabilistic clock synchronization in wireless sensor networks

Formal Reliability Analysis using Higher-order Logic Theorem Proving

Traditional techniques used in the reliability analysis of engineering systems have limitations. Paper-and-pencil based analysis is prone to human error and simulation based techniques cannot be computationally one hundred percent accurate. An alternative to these two traditional approaches is modeling and analysis of reliability of systems using formal methods based techniques such as probabilistic theorem proving. Probabilistic theorem proving using higher-order logic can be used for modeling and analysis of reliability of engineering systems provided a certain reasoning infrastructure is developed. The developed infrastructure can include random variables, their probabilistic and statistical properties, and basic reliability theory concepts such as survival and hazard functions. This thesis describes state-of-the-art research in reliability analysis using theorem proving. It also describes the main contributions of this thesis which include: the formalization of statistical properties of continuous random variables, the formalization of multiple continuous random variables and the formalization of the basic notions of reliability that can be applied to single and multiple component systems. Engineering applications of the formalization are presented that illustrate the usefulness of our formalization infrastructure. These applications include reliability analysis of electronic system components such as a capacitor and an underground power transmission cable. We also present the reliability analysis of an automobile transmission using our higher-order logic formalization. To the best of our knowledge, for the very first time, the use of theorem proving based infrastructure enables formal reliability analysis of engineering systems that is computationally one hundred percent accurate and sound. The analysis is performed using real and true random variables. We show that the results presented in this thesis are general and can be applied to many reliability engineering problems