Formalization of Transform Methods using HOL Light

Abstract

Algebraic techniques based on transform methods are widely used for solving differential equations and evaluating transfer function, and frequency response of signals while analyzing physical aspects of many safety-critical systems. To facilitate formal analysis of these systems, we present the formalization of transform methods (Laplace and Fourier transforms) using the multivariable calculus theories of HOL Light. In particular, we use integral, differential, transcendental and topological theories of multivariable calculus to formally define transform methods in higher-order logic and reason about the correctness of their properties, such as existence, linearity, frequency shifting, modulation, time shifting, time scaling, differentiation and integration in time domain, relationship of Laplace transform with Fourier transform and relationship of Fourier transform with Fourier Cosine and Fourier Sine transforms. In order to demonstrate the practical effectiveness of this formalization, we use it to formally verify some commonly used electrical circuits, an automobile suspension system, an audio equalizer, a MEMs accelerometer, controllers and compensators, and pitch control of an unmanned free swimming submersible vehicle.

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Publications

  1. A. Rashid and O. Hasan, “Formal Analysis of Linear Control Systems using Theorem proving”, International Conference on Formal Engineering Methods (ICFEM-2017), Springer, LNCS 10610, Xian, China, pp. 345-361.

  2. A. Rashid and O. Hasan, “Formal Analysis of Continuous-time Systems using Fourier Transform”, Journal of Symbolic Computation, Elsevier, (Accepted in June 2017).

  3. A. Rashid and O. Hasan, “Formalization of Transform Methods using HOL Light”, Conference on Intelligent Computer Mathematics (CICM-2017), Springer, LNAI 10383, Edinburgh, UK, pp. 319-332.

  4. S.H. Taqdees and O. Hasan, “Formally Verifying Transfer Functions of Linear Analog Circuits”, IEEE Design & Test, doi:10.1109/MDAT.2017.2713388, 2017.

  5. A. Rashid and O. Hasan, “On the Formalization of Fourier Transform in Higher-order Logic”, Interactive Theorem Proving (ITP 2016), Springer, LNCS 9807, Nancy, France, pp. 483-490. (Rank A, CORE)

  6. S.H. Taqdees and O. Hasan, “Formalization of Laplace Transform Using the Multivariate Calculus Theory of HOL Light”, Logic for Programing Artificial Intelligence and Reasoning (LPAR 2013), Springer LNCS 8312, Stellenbosch, South Africa, pp. 744–758. (Rank A, CORE)

  7. S.H. Taqdees, “Formalization of Laplace Transform using the Multivariable Calculus Theory of HOL Light”, Masters Thesis, National University of Sciences and Technology (NUST), Islamabad, Pakistan. 2013.