CICM 2017: Intelligent Computer Mathematics pp 319-332

# Formalization of Transform Methods Using HOL Light

• Osman Hasan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10383)

## Abstract

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.

## Keywords

Laplace transform Fourier transform Interactive theorem proving HOL Light

## Notes

### Acknowledgements

This work was supported by the National Research Program for Universities grant (number 1543) of Higher Education Commission (HEC), Pakistan.

## References

1. 1.
Abad, G.: Power Electronics and Electric Drives for Traction Applications. Wiley, Hoboken (2016)
2. 2.
Akbarpour, B., Tahar, S.: A methodology for the formal verification of FFT algorithms in HOL. In: Hu, A.J., Martin, A.K. (eds.) FMCAD 2004. LNCS, vol. 3312, pp. 37–51. Springer, Heidelberg (2004). doi:
3. 3.
Beerends, R.J., Morsche, H.G., Van den Berg, J.C., Van de Vrie, E.M.: Fourier and Laplace Transforms. Cambridge University Press, Cambridge (2003)
4. 4.
Bogart, T.F.: Laplace Transforms and Control Systems Theory for Technology: Including Microprocessor-Based Control Systems. Wiley, New York (1982)Google Scholar
5. 5.
Born, M., Wolf, E.: Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light. Elsevier, Amsterdam (1980)
6. 6.
Boyce, W.E., DiPrima, R.C., Haines, C.W.: Elementary Differential Equations and Boundary Value Problems, vol. 9. Wiley, New York (1969)
7. 7.
Bracewell, R.N.: The Fourier Transform and its Applications. McGraw-Hill, New York (1978)
8. 8.
Capretta, V.: Certifying the fast fourier transform with Coq. In: Boulton, R.J., Jackson, P.B. (eds.) TPHOLs 2001. LNCS, vol. 2152, pp. 154–168. Springer, Heidelberg (2001). doi:
9. 9.
Chapin, L.: Communication Systems (1978)Google Scholar
10. 10.
Chau, C.K., Kaufmann, M., Hunt Jr., W.A.: Fourier Series Formalization in ACL2 (r). arXiv preprint arXiv:1509.06087 (2015)
11. 11.
Chu, E.: Discrete and Continuous Fourier Transforms: Analysis, Applications and Fast Algorithms. CRC Press, Boca Raton (2008)
12. 12.
Davidson, D.B.: Computational Electromagnetics for RF and Microwave Engineering. Cambridge University Press, Cambridge (2005)
13. 13.
Devasahayam, S.R.: Signals and Systems in Biomedical Engineering: Signal Processing and Physiological Systems Modeling. Springer Science & Business Media, New York (2012)Google Scholar
14. 14.
Dorf, R.C., Bishop, R.H.: Modern Control Systems. Prentice Hall, Eindhoven (1998)
15. 15.
Dougherty, G.: Digital Image Processing for Medical Applications. Cambridge University Press, Cambridge (2009)Google Scholar
16. 16.
Du, K.L., Swamy, M.N.S.: Wireless Communication Systems: From RF Subsystems to 4G Enabling Technologies. Cambridge University Press, Cambridge (2010)
17. 17.
Fortmann, T.E., Hitz, K.L.: An introduction to linear control systems. CRC Press, Boca Raton (1977)
18. 18.
Gamboa, R.A.: Mechanically verifying the correctness of the fast fourier transform in ACL2. In: Rolim, J. (ed.) IPPS 1998. LNCS, vol. 1388, pp. 796–806. Springer, Heidelberg (1998). doi:
19. 19.
Gamboa, R.A.: The correctness of the fast fourier transform: a structured proof in ACL2. Formal Methods Syst. Des. 20(1), 91–106 (2002)
20. 20.
Gaskill, J.D.: Linear Systems, Fourier Transforms, and Optics, 1st edn. Wiley, New York (1978)Google Scholar
21. 21.
Gaydecki, P.: Foundations of Digital Signal Processing: Theory, Algorithms and Hardware Design. Institution of Engineering and Technology, Stevenage (2004)
22. 22.
Gorini, V., Frigerio, A.: Fundamental Aspects of Quantum Theory, vol. 144. Springer Science & Business Media, USA (2012)
23. 23.
24. 24.
Harrison, J.: HOL Light Multivariate Calculus (2017). https://github.com/jrh13/hol-light/tree/master/Multivariate
25. 25.
Harrison, J.: Integration Theory in HOL Light (2017). https://github.com/jrh13/hol-light/blob/master/Multivariate/integration.ml
26. 26.
Harrison, J.: Real Vectors in Euclidean Space (2017). http://github.com/jrh13/hol-light/blob/master/Multivariate/vectors.ml
27. 27.
Hasan, O., Tahar, S.: Formal Verification Methods. Encyclopedia of Information Science and Technology, pp. 7162–7170. IGI Global Pub., Hershey (2015)Google Scholar
28. 28.
Hilbe, J.M.: Astrostatistical Challenges for the New Astronomy, vol. 1. Springer Science & Business Media, New York (2012)Google Scholar
29. 29.
Jancewicz, B.: Trivector fourier transformation and electromagnetic field. J. Math. Phys. 31(8), 1847–1852 (1990)
30. 30.
Jin, J.M.: Theory and Computation of Electromagnetic Fields. Wiley, Hoboken (2011)Google Scholar
31. 31.
Kriezis, E.E., Chrissoulidis, D., Papagiannakis, A.: Electromagnetics and Optics. World Scientific, Singapore (1992)
32. 32.
Madhow, U.: Introduction to Communication Systems. Cambridge University Press, Cambridge (2014)
33. 33.
McLachlan, N.W.: Laplace Transforms and their Applications to Differential Equations. Courier Corporation, Cedar City (2014)Google Scholar
34. 34.
Nise, N.S.: Control Systems Engineering. Wiley, New York (2007)
35. 35.
Ogata, K., Yang, Y.: Modern Control Engineering. Prentice-Hall, Englewood Cliffs (1970)Google Scholar
36. 36.
Oppenheim, A.V., Willsky, A.S., Hamid Nawab, S.: Signals and Systems. Prentice Hall Processing Series, 2nd edn. Prentice Hall Inc., Upper Saddle River (1996)Google Scholar
37. 37.
Papoulis, A.: Signal Analysis, vol. 2. McGraw-Hill, New York (1977)
38. 38.
Pytel, A., Kiusalaas, J.: Engineering Mechanics: Dynamics. Nelson Education, Scarborough (2016)Google Scholar
39. 39.
Rashid, A., Hasan, O.: On the formalization of fourier transform in higher-order logic. In: Blanchette, J.C., Merz, S. (eds.) ITP 2016. LNCS, vol. 9807, pp. 483–490. Springer, Cham (2016). doi:
40. 40.
Rashid, M.H.: Power Electronics: Circuits, Devices, and Applications. Pearson Education India, Delhi (2009)Google Scholar
41. 41.
Siddique, U., Mahmoud, M.Y., Tahar, S.: On the formalization of Z-transform in HOL. In: Klein, G., Gamboa, R. (eds.) ITP 2014. LNCS, vol. 8558, pp. 483–498. Springer, Cham (2014). doi: Google Scholar
42. 42.
Siebert, W.M.: Circuits, Signals, and Systems, vol. 2. MIT press, Cambridge (1986)Google Scholar
43. 43.
Stacey, W.M.: Nuclear Reactor Physics. Wiley, New York (2007)
44. 44.
Stark, H.: Application of Optical Fourier Transforms. Elsevier, Burlington (2012)Google Scholar
45. 45.
Taqdees, S.H., Hasan, O.: Formalization of laplace transform using the multivariable calculus theory of HOL-light. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR 2013. LNCS, vol. 8312, pp. 744–758. Springer, Heidelberg (2013). doi:
46. 46.
Taqdees, S.H., Hasan, O.: Formally verifying transfer functions of linear analog circuits. IEEE Des. Test (2017). http://save.seecs.nust.edu.pk/pubs/2017/DTnA_2017.pdf
47. 47.
Thomas, R.E., Rosa, A.J., Toussaint, G.J.: The Analysis and Design of Linear Circuits, Binder Ready Version. Wiley, New York (2016)Google Scholar
48. 48.
Ziemer, R., Tranter, W.H.: Principles of Communications: System Modulation and Noise. Wiley, Chichester (2006)Google Scholar