Adnan Rashid

On the Formalization of Fourier Transform in Higher-order Logic

Fourier transform based techniques are widely used for solving differential equations and to perform the frequency response analysis of signals in many safety-critical systems. To perform the formal analysis of these systems, we present a formalization of Fourier transform using higher-order logic. In particular, we use the HOL-Light’s differential, integral, transcendental and topological theories of multivariable calculus to formally define Fourier transform and reason about the correctness of its classical properties, such as existence, linearity, frequency shifting, modulation, time reversal and differentiation in time-domain. In order to demonstrate the practical effectiveness of the proposed formalization, we use it to formally verify the frequency response of an automobile suspension system.

Formalization of Transform Methods Using HOL Light

Transform methods, like Laplace and Fourier, are frequently used for analyzing the dynamical behaviour of engineering and physical systems, based on their transfer function, and frequency response or the solutions of their corresponding differential equations. In this paper, we present an ongoing project, which focuses on the higher-order logic formalization of transform methods using HOL Light theorem prover. In particular, we present the motivation of the formalization, which is followed by the related work. Next, we present the task completed so far while highlighting some of the challenges faced during the formalization. Finally, we present a roadmap to achieve our objectives, the current status and the future goals for this project.

Formal analysis of a ZigBee-based routing protocol for smart grids using UPPAAL

Smart grid is an emerging technology which integrates the modern communication network to the traditional power grids. The performance and efficiency of the smart grid mainly depends on reliable communication between its different components and in turn on the routing protocols that establish this communication network. ZigBee protocol is a widely used routing protocol in the home area networks of the smart grids. Traditionally, these protocols are analysed using computer simulations and net testing. All these methods are error-prone and thus cannot provide an accurate analysis, which poses a serious threat to the safety-critical domain of smart grids. To guarantee the correctness of analysis, we propose to use model checking for the verification of the ZigBee routing protocol. We used UPPAAL model checker to formally model the ZigBee routing protocol and verified it using the collision avoidance and liveness properties.

Formal reasoning about systems biology using theorem proving

System biology provides the basis to understand the behavioral properties of complex biological organisms at different levels of abstraction. Traditionally, analysing systems biology based models of various diseases have been carried out by paper-and-pencil based proofs and simulations. However, these methods cannot provide an accurate analysis, which is a serious drawback for the safety-critical domain of human medicine. In order to overcome these limitations, we propose a framework to formally analyze biological networks and pathways. In particular, we formalize the notion of reaction kinetics in higher-order logic and formally verify some of the commonly used reaction based models of biological networks using the HOL Light theorem prover. Furthermore, we have ported our earlier formalization of Zsyntax, i.e., a deductive language for reasoning about biological networks and pathways, from HOL4 to the HOL Light theorem prover to make it compatible with the above-mentioned formalization of reaction kinetics. To illustrate the usefulness of the proposed framework, we present the formal analysis of three case studies, i.e., the pathway leading to TP53 Phosphorylation, the pathway leading to the death of cancer stem cells and the tumor growth based on cancer stem cells, which is used for the prognosis and future drug designs to treat cancer patients.

Formal analysis of continuous-time systems using Fourier transform

Abstract To study the dynamical behavior of the engineering and physical systems, we often need to capture their continuous behavior, which is modeled using differential equations, and perform the frequency-domain analysis of these systems. Traditionally, Fourier transform methods are used to perform this frequency domain analysis using paper-and-pencil based analytical techniques or computer simulations. However, both of these methods are error prone and thus are not suitable for analyzing systems used in safety-critical domains, like medicine and transportation. In order to provide an accurate alternative, we propose to use higher-order-logic theorem proving to conduct the frequency domain analysis of these systems. For this purpose, the paper presents a higher-order-logic formalization of Fourier transform using the HOL-Light theorem prover. In particular, we use the higher-order-logic based formalizations of differential, integral, transcendental and topological theories of multivariable calculus to formally define Fourier transform and reason about the correctness of its classical properties, such as existence, linearity, time shifting, frequency shifting, modulation, time scaling, time reversal and differentiation in time domain, and its relationships with Fourier Cosine, Fourier Sine and Laplace transforms. We use our proposed formalization for the formal verification of the frequency response of a generic n-order linear system, an audio equalizer and a MEMs accelerometer, using the HOL-Light theorem prover.

Formal Verification of Platoon Control Strategies

Recent developments in autonomous driving, vehicle-to-vehicle communication and smart traffic controllers have provided a hope to realize platoon formation of vehicles. The main benefits of vehicle platooning include improved safety, enhanced highway utility, efficient fuel consumption and reduced highway accidents. One of the central components of reliable and efficient platoon formation is the underlying control strategies, e.g., constant spacing, variable spacing and dynamic headway. In this paper, we provide a generic formalization of platoon control strategies in higher-order logic. In particular, we formally verify the stability constraints of various strategies using the libraries of multivariate calculus and Laplace transform within the sound core of HOL Light proof assistant. We also illustrate the use of verified stability theorems to develop runtime monitors for each controller, which can be used to automatically detect the violation of stability constraints in a runtime execution or a logged trace of the platoon controller. Our proposed formalization has two main advantages: 1) it provides a framework to combine both static (theorem proving) and dynamic (runtime) verification approaches for platoon controllers, and 2) it is inline with the industrial standards, which explicitly recommend the use of formal methods for functional safety, e.g., automotive ISO 26262.

Formal Analysis of Linear Control Systems Using Theorem Proving

Control systems are an integral part of almost every engineering and physical system and thus their accurate analysis is of utmost importance. Traditionally, control systems are analyzed using paper-and-pencil proof and computer simulation methods, however, both of these methods cannot provide accurate analysis due to their inherent limitations. Model checking has been widely used to analyze control systems but the continuous nature of their environment and physical components cannot be truly captured by a state-transition system in this technique. To overcome these limitations, we propose to use higher-order-logic theorem proving for analyzing linear control systems based on a formalized theory of the Laplace transform method. For this purpose, we have formalized the foundations of linear control system analysis in higher-order logic so that a linear control system can be readily modeled and analyzed. The paper presents a new formalization of the Laplace transform and the formal verification of its properties that are frequently used in the transfer function based analysis to judge the frequency response, gain margin and phase margin, and stability of a linear control system. We also formalize the active realizations of various controllers, like Proportional-Integral-Derivative (PID), Proportional-Integral (PI), Proportional-Derivative (PD), and various active and passive compensators, like lead, lag and lag-lead. For illustration, we present a formal analysis of an unmanned free-swimming submersible vehicle using the HOL Light theorem prover.