Osman Hasan

On the formalization of the lebesgue integration theory in HOL

Lebesgue integration is a fundamental concept in many mathematical theories, such as real analysis, probability and information theory. Reported higher-order-logic formalizations of the Lebesgue integral either do not include, or have a limited support for the Borel algebra, which is the canonical sigma algebra used on any metric space over which the Lebesgue integral is defined. In this paper, we overcome this limitation by presenting a formalization of the Borel sigma algebra that can be used on any metric space, such as the complex numbers or the n-dimensional Euclidean space. Building on top of this framework, we have been able to prove some key Lebesgue integral properties, like its linearity and monotone convergence. Furthermore, we present the formalization of the “almost everywhere” relation and prove that the Lebesgue integral does not distinguish between functions which differ on a null set as well as other important results based on this concept. As applications, we present the verification of Markov and Chebyshev inequalities and the Weak Law of Large Numbers theorem.

Survey of fall detection and daily activity monitoring techniques

The risk of sustaining heavy injuries through accidental falls creates a major medical problem for elderly people. This paper conducts a survey of the various automatic techniques and methods proposed to detect falls and anomalies in movements of the elderly, through monitoring of their daily life activities. These methods can be broadly divided into three main categories: 1) Video Analysis Based; 2) Acoustic and Ambience Sensor Based; and 3) Kinematic Sensor Based. This paper critically analyzes the various proposed methodologies, comparing their strengths and weaknesses. We further propose our own technique for fall detection and monitoring of common daily life activities (walking, running, sitting, standing, and lying down), through a novel approach that provides a low cost solution and ensures the safety and security of the elderly without restricting them to confined surroundings.

Formalization of entropy measures in HOL

Information theory is widely used in a very broad class 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 and thus cannot be completely relied upon. To facilitate the formal reasoning about information theoretic aspects, this paper presents a rigorous higher-order logic formalization of some of the most widely used information theoretic principles. Building on fundamental formalizations of measure and Lebesgue integration theories for extended reals, we formalize the Radon-Nikodym derivative and prove some of its properties using the HOL theorem prover. This infrastructure is then used to formalize information theoretic fundamentals like Shannon entropy and relative entropy. We discuss potential applications of the proposed formalization for the analysis of data compression and security protocols.

Formal Reasoning about Expectation Properties for Continuous Random Variables

Expectation (average) properties of continuous random variables are widely used to judge performance characteristics in engineering and physical sciences. This paper presents an infrastructure that can be used to formally reason about expectation properties of most of the continuous random variables in a theorem prover. Starting from the relatively complex higher-order-logic definition of expectation, based on Lebesgue integration, we formally verify key expectation properties that allow us to reason about expectation of a continuous random variable in terms of simple arithmetic operations. In order to illustrate the practical effectiveness and utilization of our approach, we also present the formal verification of expectation properties of the commonly used continuous random variables: Uniform, Triangular and Exponential.

Formalization of Continuous Probability Distributions

Continuous probability distributions are widely used to mathematically describe random phenomena in engineering and physical sciences. In this paper, we present a methodology that can be used to formalize any continuous random variable for which the inverse of the cumulative distribution function can be expressed in a closed mathematical form. Our methodology is primarily based on the Standard Uniform random variable, the classical cumulative distribution function properties and the Inverse Transform method. The paper includes the higher-order-logic formalization details of these three components in the HOL theorem prover. To illustrate the practical effectiveness of the proposed methodology, we present the formalization of Exponential, Uniform, Rayleigh and Triangular random variables.

Formal verification of cyber-physical systems: coping with continuous elements

The formal verification of cyber-physical systems is a challenging task mainly because of the involvement of various factors of continuous nature, such as the analog components or the surrounding environment. Traditional verification methods, such as model checking or automated theorem proving, usually deal with these continuous aspects by using abstracted discrete models. This fact makes cyber-physical system designs error prone, which may lead to disastrous consequences given the safety and financial critical nature of their applications. Leveraging upon the high expressiveness of higher-order logic, we propose to use higher-order-logic theorem proving to analyze continuous models of cyber-physical systems. To facilitate this process, this paper presents the formalization of the solutions of second-order homogeneous linear differential equations. To illustrate the usefulness of our foundational cyber-physical system analysis formalization, we present the formal analysis of a damped harmonic oscillator and a second-order op-amp circuit using the HOL4 theorem prover.

Formal Reliability Analysis Using Theorem Proving

Reliability analysis has become a tool of fundamental importance to virtually all electrical and computer engineers because of the extensive usage of hardware systems in safety and mission critical domains, such as medicine, military, and transportation. Due to the strong relationship between reliability theory and probabilistic notions, computer simulation techniques have been traditionally used to perform reliability analysis. However, simulation provides less accurate results and cannot handle large-scale systems due to its enormous CPU time requirements. To ensure accurate and complete reliability analysis and thus more reliable hardware designs, we propose to conduct a formal reliability analysis of systems within the sound core of a higher order logic theorem prover (HOL). In this paper, we present the higher order logic formalization of some fundamental reliability theory concepts, which can be built upon to precisely analyze the reliability of various engineering systems. The proposed approach and formalization is then utilized to analyze the repairability conditions for a reconfigurable memory array in the presence of stuck-at and coupling faults.

Applying Formal Methods to Networking: Theory, Techniques, and Applications

Despite its great importance, modern network infrastructure is remarkable for the lack of rigor in its engineering. The Internet, which began as a research experiment, was never designed to handle the users and applications it hosts today. The lack of formalization of the Internet architecture meant limited abstractions and modularity, particularly for the control and management planes, thus requiring for every new need a new protocol built from scratch. This led to an unwieldy ossified Internet architecture resistant to any attempts at formal verification and to an Internet culture where expediency and pragmatism are favored over formal correctness. Fortunately, recent work in the space of clean slate Internet design-in particular, the software defined networking (SDN) paradigm-offers the Internet community another chance to develop the right kind of architecture and abstractions. This has also led to a great resurgence in interest of applying formal methods to specification, verification, and synthesis of networking protocols and applications. In this paper, we present a self-contained tutorial of the formidable amount of work that has been done in formal methods and present a survey of its applications to networking.

Performance Analysis and Functional Verification of the Stop-and-Wait Protocol in HOL

Real-time systems usually involve a subtle interaction of a number of distributed components and have a high degree of parallelism, which makes their performance analysis quite complex. Thus, traditional techniques, such as simulation, or the state-based formal methods usually fail to produce reasonable results. In this paper, we propose to use higher-order-logic (HOL) theorem proving for the performance analysis of real-time systems. The idea is to formalize the real-time system as a logical conjunction of HOL predicates, whereas each one of these predicates define an autonomous component or process of the given real-time system. The random or unpredictable behavior found in these components is modeled as random variables. This formal specification can then be used in a HOL theorem prover to reason about both functional and performance related properties of the given real-time system. In order to illustrate the practical effectiveness of our approach, we present the analysis of the Stop-and-Wait protocol, which is a classical example of real-time systems. The functional correctness of the protocol is verified by proving that the protocol ensures reliable data transfers. Whereas, the average message delay relation is verified in HOL for the sake of performance analysis. The paper includes the protocol’s formalization details along with the HOL proof sketches for the major theorems.

RRT*-SMART: A Rapid Convergence Implementation of RRT*

Many sampling based algorithms have been introduced recently. Among them Rapidly Exploring Random Tree (RRT) is one of the quickest and the most efficient obstacle free path finding algorithm. Although it ensures probabilistic completeness, it cannot guarantee finding the most optimal path. Rapidly Exploring Random Tree Star (RRT*), a recently proposed extension of RRT, claims to achieve convergence towards the optimal solution thus ensuring asymptotic optimality along with probabilistic completeness. However, it has been proven to take an infinite time to do so and with a slow convergence rate. In this paper an extension of RRT*, called as RRT*-Smart, has been prposed to overcome the limitaions of RRT*. The goal of the proposecd method is to accelerate the rate of convergence, in order to reach an optimum or near optimum solution at a much faster rate, thus reducing the execution time. The novel approach of the proposed algorithm makes use of two new techniques in RRT*–Path Optimization and Intelligent Sampling. Simulation results presented in various obstacle cluttered environments along with statistical and mathematical analysis confirm the efficiency of the proposed RRT*-Smart algorithm.