Towards the Formalization of Fractional Calculus in Higher-Order Logic
TTowards the Formalization of FractionalCalculus in Higher-Order Logic (cid:63)
Umair Siddique*, Osman Hasan**, and Sofi`ene Tahar* *Department of Electrical and Computer Engineering,Concordia University, Montreal, Quebec, Canada**School of Electrical Engineering and Computer Science,National University of Sciences and Technology, Islamabad, Pakistan { muh sidd,tahar } @[email protected]://save.seecs.nust.edu.pk/projects/fc.html Abstract.
Fractional calculus is a generalization of classical theories ofintegration and differentiation to arbitrary order (i.e., real or complexnumbers). In the last two decades, this new mathematical modeling ap-proach has been widely used to analyze a wide class of physical systemsin various fields of science and engineering. In this paper, we describean ongoing project which aims at formalizing the basic theories of frac-tional calculus in the HOL Light theorem prover. Mainly, we present themotivation and application of such formalization efforts, a roadmap toachieve our goals, current status of the project and future milestones.
Keywords:
Fractional Calculus, Higher-Order Logic, Theorem Proving
Physical and engineering systems are classified as continuous, discrete or hybriddepending upon the nature of underlying system parameters. The rich theories ofmathematics provide the necessary tools to study the behaviour of such systemsranging from very small biological organisms to the modern Quantum mechani-cal phenomenons. Generally, differential equations [39] and difference equations[12] are used to characterize the dynamics of these systems. Consequently, theconcept of higher-order differentiation and integration are widely studied in di-verse disciplines of science and engineering. For example, it is well understoodthat the first derivative ( ddt f ( t )) and second derivative ( d dt f ( t )) of a functiondescribe the rate of change and measure of concavity, respectively. However, werarely think what if the order ( n ) of higher-order derivative ( d n dt n ) becomes a real,complex or an irrational number? One immediate question arises in our mindsis the existence or possibility of such a concept in mathematics. Interestingly,this seemingly new concept dates back to 1695 when L’Hˆopital asked Leibniz (cid:63) This is authors’ version of CICM-2015 paper. The final publication is available at http://link.springer.com a r X i v : . [ c s . L O ] M a y egarding his notation d n ydx n : “ what if n is ”. In reply, Leibniz [20] prophesiedin his letter, “. . . Thus it follows that d x will be equal to x √ dx : x . This is anapparent paradox from which, one day, useful consequences can be drawn . . . ”.Leibniz’s initial work on the problem of defining the derivative of arbitrary ordergave birth to a new field of research in mathematics (called fractional calculus )and attracted the attention of many physicists, engineers and geometers. Someof the great mathematicians and physicists who touched the field of fractionalcalculus are Riemann, Liouville, Laurent, Heaviside and Riesz [24].The concept of fractional calculus has great potential to change the way wemodel and analyze the systems. It provides good opportunity to scientists andengineers for revisiting the origins. We briefly outline some of the the main ap-plications of fractional calculus in Table 1. The importance of fractional calculuscan be realized by the following quote from Miller and Ross [24]. They stated: “. . . The fractional calculus finds use in many fields of science and en-gineering, including fluid flow, rheology, diffusive transport akin todiffusion, electrical networks, electromagnetic theory, and probabil-ity. . . . It seems that hardly a field of science or engineering has re-mained untouched by this topic . . . ” Field Applications
Control Engineering - System identification [17]- Biomimetic (bionics) control [7]- Trajectory control [11]- Temperature control [30]- Fractional
P I α controller [23]Signal Processing - Fractional order integrator [19]- Fractional order FIR differentiator [37]- IIR-type fractional order differentiator [38]- Modeling of speech signals [18]Image Processing - Image restoration and edge detection [29]- Satellite image classification [6]Electromagnetics - Fractional curl operators [13,25]- Fractional Rectangular waveguides [14]Communication - Secure chaotic communication [1]- Informational network traffic modeling [40]Biology - Neuron modeling [5]- Biophysical processes [8]- Modeling of complex dynamics of tissues [22] Table 1: Applications of Fractional CalculusNowadays engineering systems exhibiting fractional order dynamics are in-creasingly used in some safety-critical applications such as control systems, sig-nal processing, electromagnetics and electrical networks (as listed in Table 1).or example, fractional meta-materials based devices are used to build sensitivemilitary and defence equipments and electromagnetic stealth technology [21].Considering these facts, it is quite interesting and important to build a logicalreasoning framework which can be used to formally verify such sophisticatedapplications within the sound core of a proof assistant. In fact, proof assistantshave been successfully used to formalize and verify some challenging and para-doxical mathematical results, e.g., the formal proofs of the Kepler Conjecture(Flyspeck project) [16] and the Odd Order Theorem [15].In this paper, we present details of an ongoing project to develop a formalreasoning support for fractional calculus in higher-order-logic theorem prover.This project was originally started at the System Analysis and Verification(SAVe) lab in 2010. Earlier formalization was done in the HOL4 theorem proverwith the main focus on fractional operators for real-valued functions and theverification of fractional order electrical components. Later on, the scope of theproject was expanded to formalize fractional calculus involving complex-valuedfunctions due to its various engineering applications (as listed in Table 1). Cur-rently, we are using the HOL Light theorem prover due to the availability ofrich multivariate analysis libraries including Harrison’s recent formalization ofcomplex-valued Gamma function as well as the interesting related projects likeFlyspeck [16] and the formalization of optics theories (i.e., ray, wave, electro-magnetic and quantum) [2].The rest of the paper is organized as follows: In Section 2, we briefly reviewsome commonly used notations and definitions of fractional order operators.We provide an outline of the proposed formalization framework in Section 3.Consequently, the current status of the formalization and future milestones arediscussed in Section 4. Finally, we conclude the paper in Section 5. There are different notations available for fractional derivatives and integrals.We use J va f ( x ) and D v f ( x ) for fractional integral and fractional derivative, re-spectively. In these notations, v is the order of integration or differentiation and a is the lower limit of integration.For every function ( f : C → C ); and for every number v ∈ R or C , J va and D v should be related to f by the following criteria [10].1. If f ( x ) is an analytic function, then J va f ( x ) and D v f ( x ) must also be ananalytic function of the variable x and of the order v of integration or dif-ferentiation.2. The operations J va f ( x ) and D v f ( x ) must produce the same result as ordinaryintegration/differentiation when v is a positive integer. http://save.seecs.nust.edu.pk/projects/fc.html http://save.seecs.nust.edu.pk https://code.google.com/p/hol-light/source/browse/trunk/Multivariate/gamma.ml . The fractional operators must be linear. J va [ αf ( x ) + βg ( x )] = αJ va f ( x ) + βJ va g ( x ) (1) D v [ αf ( x ) + βg ( x )] = αD v f ( x ) + βD v g ( x ) (2)4. The operation of order zero must leave the function unchanged. J a f = f and D f = f (3)5. The law of exponents must hold for integration and differentiation of arbi-trary order under sufficient conditions on function f . J ua ( J va f ) = J u + va f and D u ( D v f ) = D u + v f (4)Fractional integrals and fractional derivatives are also referred to as Differinte-grals [27] and there are more than ten well-known definitions for Differintegrals[9]. We describe here two of them, which are most widely used in analyzingreal-world problems. These are the Riemann-Liouville and Gr¨unwald-Letnikovdefinitions, which are also equivalent for a wide class of functions [31]. Riemann-Liouville (RL) Definition: J va f ( x ) = 1 Γ ( v ) (cid:90) xa ( x − t ) v − f ( t ) dt (5)where J va f ( x ) represents fractional integration with order v and lower integrationlimit a . The parameter a = 0 gives the Riemann definition and a = −∞ givesthe Liouville definition of fractional integration. Indeed Equation (5) is the gen-eralization of Cauchy’s repeated integration formula to non-integer v [32]. Where Γ (.) in the above definition denotes the Gamma function which is defined usingthe well-known improper integral as follows: Γ ( z ) = (cid:90) ∞ t z − e − t dt (6)for Re ( z ) > D v f ( x ) = ( ddx ) m J m − va f ( x ) (7)where m represents the ceiling of v , i.e., (cid:100) v (cid:101) . Gr¨unwald-Letnikov (GL) Definition: c D vx f ( x ) = lim h → h − v [ x − ch ] (cid:88) k =0 ( − k (cid:18) vk (cid:19) f ( x − kh ) (8)Gr¨unwald-Letnikov definition caters for both fractional differentiation and inte-gration, as positive values of v give fractional differentiation and negative valuesof v give fractional integration. Here, (cid:0) vk (cid:1) represents the binomial coefficients,which are described in terms of the Gamma function. Formal Analysis Framework
The proposed framework, given in Figure 1, outlines the main ideas and roadmapto formalize the basic theory behind fractional calculus. The whole framework
Gamma Function
Riemann–Liouville
Integral
Finite Fractional
Difference Laplace Transform Z-Transform R-Transform
Fractional Order
Control Systems
Fractional
Electromagnetics
Fractional Signal
Processing
Electrical Networks
HOL LightMultivariate Libraries
ComplexesIntegrals Derivatives Vectors
Fig. 1.
Formalization Framework for Fractional Calculus can be decomposed mainly into three major parts which are the formalizationof the core definitions of fractional order operators, formalization of supportingtransformations (i.e., Laplace transform [26], Z-transform [28] and R-Transform[3]) and engineering applications. The first part heavily relies upon the Gammafunction as mentioned in Section 2. So the core step is to formalize the Gammafunction in higher-order logic (HOL) and verify its important properties. Conse-quently, any definition of fractional order operators can be formalized in HOL.However, our focus is two main definitions, i.e., Riemann-Liouville (RL) andfractional difference which indeed represent continuous and discrete versions offractional order operators, respectively. This step also involves the validationof all the properties mentioned in Section 2. This requires some important re-sults of multivariate calculus such as the notion of Lebesgue measurability andFubini’s theorem which provides the reasoning support for iterated and doubleintegrals. Interestingly, both of these requirements are available in the multi-variate analysis libraries of HOL Light. The second step is the formalizationof important integral transforms which are necessary to analytically solve lin-ear fractional differential and difference equations. We mainly focus on threetransforms, namely the Laplace transform, the Z-transform, and the recentlyintroduced R-transform [4]. All of these transformations are used to transformcomplicated fractional differential (or difference) equations to algebraic equationswhich are easier to manipulate and to deduce interesting properties. Buildingpon these fundamentals, our ultimate goal is to formally verify a variety of en-gineering systems including control systems, signal processing, electromagneticsand electrical networks. All formalization steps make use of different multivariatetheories of HOL Light, e.g., derivatives, integrals, complex vectors and measurespaces. Finally, the developed libraries of this project will become part of theexisting HOL Light libraries.
As mentioned earlier, the project initially considered only real-order fractionaloperators in the HOL4 theorem prover. The main difficulty was the little supportto handle improper integrals which was required to formalize the real-valuedGamma function. Therefore, we extended the integration theory of HOL4 byformalizing a variant of improper integrals using sequential limits. This wasthen used to formalize the Gamma function and verify some of its main proper-ties, such as the pseudo-recurrence relation ( Γ ( z + 1) = zΓ ( z )), the functionalequation ( Γ (1) = 1) and the factorial generalization ( Γ ( k + 1) = k !) [34]. Weutilized these foundations to formalize Differintegrals, given in Equations (5)and (7), which in turn can be used to represent the dynamics of fractional or-der systems in higher-order logic. We also verified theorems corresponding tosome commonly used properties of Differintegrals namely Identity and Linearity.Consequently, we conducted the formal analysis of a fractional order electricalcomponent namely resistoductor, a fractional integrator and a fractional differ-entiator circuit [33]. Later on, the scope of the project was revised to includecomplex-valued functions and complex order fractional operators in HOL Light.The main requirement was to formalize the complex-valued Gamma function,Laplace and Z-transforms. However, the Gamma function was formalized byHarrison in early 2014. In the meantime, we formalized the basic theories of theLaplace transform [36] and the Z-transform [35]. Currently, we are working onthree main topics which include: 1) formal proofs of the uniqueness of Laplaceand Z-transforms which are required to formally verify the inverses of thesetransforms; 2) vectorial Z-transform, which extend the simple Z-transform overcomplex vectors; and 3) fractional difference equations, which are mainly basedon Gamma function, infinite summations and products over complex functions.Finally, we outline the major tasks to achieve the future milestones as follows: – Formalization of R-Transform. – Formalization of Differintegrals for complex-valued functions. This is mainlythe generalization of the formalization which was developed in HOL4. – Formalization of linear fractional differential and difference equations withsupport to analytical solutions using the transform methods.During the course of this project, two master and two PhD students havecontributed to the formalization. Interestingly, all of them are mainly electrical https://code.google.com/p/hol-light/source/browse/trunk/Multivariate/gamma.ml ngineers without prior background of formal methods and higher-order-logictheorem proving. Given the complexity and interdisciplinarity of this researchproject, it is quite encouraging to see people with an engineering (or physics)background to use proof assistants as a complementary tool. The formalizationof the fractional calculus is quite challenging as it requires advanced mathe-matical concepts of vector integration and Lebesgue measurable functions, etc.So expertise in formal reasoning about these complex mathematical phenomenais required for this formalization, which is quite unique compared to reason-ing about software and digital hardware systems. The learning curve of HOLLight varies from student to student. Generally, students start proving basicmath equations after a couple of months and the pace of formalization increasesover time. Learning HOL Light libraries is not difficult once the basic conceptshave been grasped by the user. The formalization of the improper integrals,the Gamma function, the fractional calculus, the Z-transform and the Laplacetransform is approximately 15,000 lines of HOL Light code. One of the majorobstacles in the formalization was the identification of suitable mathematicaldefinitions and models. Sometimes, textbook proofs do not follow due to variousreasons (corner cases, or the proof steps are too abstract, etc.) and they neededto be re-proved on paper with subtle details. Consequently, we have to modifythe definitions and thus change the proofs. But now the current formalizationseems quite stable as most of the classical properties have been formally ver-ified for our definitions. We believe that future developments can be built onthe foundations that have been formalized as most of the work is for generalsystems. Finally, another important aspect of this project is the potential toapply developed theories to various applications other than fractional calculus.For example, we demonstrated the use of the Gamma function in probabilitytheory [34], the Z-transform in signal processing [35], and the Laplace transformin power electronics [36]. In this paper, we mainly presented the motivation and ongoing activities of ourlong term project about the formalization of fractional calculus in the HOLLight theorem prover. The main contribution of this project is a comprehensiveframework of formal definitions and theorems about fractional calculus whichcan be used to verify modern control, signal processing and electromagneticsystems. Some future directions and recommendations for HOL Light are theimprovements in the visualization of proofs, better automation and more acces-sible tutorials with examples from different engineering/physics topics.
References
1. N. Pariz A. Kiani-B, K. Fallahi and H. Leung. A Chaotic Secure Communica-tion Scheme Using Fractional Chaotic Systems Based on an Extended FractionalKalman Filter.
Communications in Nonlinear Science and Numerical Simulation ,14:863–879, 2009.. S. K. Afshar, U. Siddique, M. Y. Mahmoud, V. Aravantinos, O. Seddiki, O. Hasan,and S. Tahar. Formal Analysis of Optical Systems.
Mathematics in ComputerScience , 8(1):39–70, 2014.3. F. M. Atici. A Transform Method in Discrete Fractional Calculus.
InternationalJournal of Difference Equations , 2(2):165–176, 2007.4. F. M. Atici and P. W. Eloe. Initial Value Problems in Discrete Fractional Calculus.
Proceeding of the American Mathematical Society , 137(3):981–989, 2009.5. T. J. Auastasio. The Fractional-Order Dynamics of Brainstem Vestibulo-Oculomotor Neurons .
Biological Cybernetics , 72(1):69–79, 1994.6. E. Cuestab C. Quintanoa. Improving Satellite Image Classification by Using Frac-tional Type Convolution Filtering.
International Journal of Applied Earth Obser-vation and Geoinformation , 12(4):298–301, 2010.7. Y. Q. Chen, D. Xue, and H. Dou. Fractional Calculus and Biomimetic Control. In
Robotics and Biomimetics , pages 901 –906. IEEE, 2004.8. M. Mar´ın D. M. Dom´ınguez and M. Camacho. Macrophage Ion Currents are Fitby a Fractional Model and Therefore are a Time Series with Memory .
EuropeanBiophysics Journal , 38(4):457–464, 2009.9. M. Dalir and M. Bashour. Application of Fractional Calculus.
Applications ofFractional Calculus in Physics , 4(21):12, 2010.10. S. Das.
Functional Fractional Calculus for System Identification and Controls .Springer, 2007.11. Fernando B. M. Duarte and Jos´e Ant´onio Tenreiro Machado. Pseudoinverse Tra-jectory Control of Redundant Manipulators: A Fractional Calculus Perspective.In
International Conference on Robotics and Automation , pages 2406–2411. IEEE,2002.12. S. Elaydi.
An Introduction to Difference Equations . Springer, 2005.13. N. Engheta. Fractional Curl Operator in Electromagnetics.
Microwave OpticsTechnology Letters , 17(2):86–91, 1998.14. M. Faryad and Q. A. Naqvi. Fractional Rectangular Waveguide.
Progress InElectromagnetics Research, PIER , 75:383–396, 2007.15. G. Gonthier, A. Asperti, J. Avigad, Y. Bertot, C. Cohen, F. Garillot, S. Le Roux,A. Mahboubi, R. OConnor, S. Ould Biha, I. Pasca, L. Rideau, A. Solovyev, E. Tassi,and L. Thry. A Machine-Checked Proof of the Odd Order Theorem. In
InteractiveTheorem Proving , volume 7998 of
LNCS , pages 163–179. Springer, 2013.16. T. C. Hales. Introduction to the Flyspeck Project. In
Mathematics, Algorithms,Proofs , volume 05021 of
Dagstuhl Seminar Proceedings , pages 1–11, 2005.17. T. T. Hartley and C. F. Lorenzo. Fractional System Identification: An ApproachUsing Continuous Order Distributions. Technical report, National Aeronautics andSpace Administration, Glenn Research Cente NASA TM, 1999.18. W.M. Ahmad K. Assaleh. Modeling of Speech Signals Using Fractional Calculus.In
International Symposium on Signal Processing and its Applications , pages 1–4.IEEE, 2007.19. B. T. Krishna and K. V. V. S. Reddy. Design of Digital Differentiators and Inte-grators of Order . World Journal of Modelling and Simulation , 4:182–187, 2008.20. G. W. Leibnitz. Leibnitzens Mathematische Schriften.
SIGDA News Letter , 2:301–302, 1962.21. K. A. Lurie.
An Introduction to the Mathematical Theory of Dynamic Materials .Springer, 2007.22. Richard L. Magin. Fractional Calculus Models of Complex Dynamics in BiologicalTissues.
Computers and Mathematics with Applications , 59:1586–1593, 2010.3. G. Maione and P. Lino. New Tuning Rules for Fractional pi α Controllers.
NonlinearDynamics , 49(1-2):pp 251–257, 2007.24. K. S. Miller and B. Ross.
An Introduction to Fractional Calculus and FractionalDifferential Equations . John Willey, 1993.25. Q. A. Naqvi and M. Abbas. Complex and Higher Order Fractional Curl Operatorin Electromagnetics .
Optics Communications , 241:349–355, 2004.26. K. Ogata.
Modern Control Engineering . Prentice Hall, 2010.27. K. B. Oldham and J. Spanier.
The Fractional Calculus . New York, AcademicPress, 1974.28. A. V. Oppenheim, R. W. Schafer, and J. R. Buck.
Discrete-Time Signal Processing .Prentice Hall, 1999.29. B. Mathieu, P. Melchior, A. Oustaloup and Ch. Ceyral. Fractional Differentiationfor Edge Detection.
Signal Processing , 83:2421–2432, 2003.30. I. Petr´as and B.M. Vinagre. Practical Application of Digital Fractional-OrderController to Temperature Control.
Acta Montanistica Slovaca , 7(2):131–137, 2002.31. I. Podlubny. Fractional Differential Equations, Academic Press. 1999.32. B. Ross. A Brief History And Exposition of The Fundamental Theory of FractionalCalculus. In
Fractional Calculus and Its Applications , volume 457 of
Lecture Notesin Mathematics , pages 1–36. Springer, 1975.33. U. Siddique and O. Hasan. Formal Analysis of Fractional Order Systems in HOL.In
Formal Methods in Computer Aided Design , pages 163–170. IEEE, 2011.34. U. Siddique and O. Hasan. On the Formalization of Gamma Function in HOL.
Journal of Automated Reasoning , 53(4):407–429, 2014.35. U. Siddique, M. Y. Mahmoud, and S. Tahar. On the Formalization of Z-Transformin HOL. In
Interactive Theorem Proving , volume 8558 of
LNCS , pages 483–498.Springer, 2014.36. S. H. Taqdees and O. Hasan. Formalization of Laplace Transform Using the Mul-tivariable Calculus Theory of HOL-Light. In
Logic for Programming, ArtificialIntelligence, and Reasoning , volume 8312 of
LNCS , pages 744–758. 2013.37. C. C. Tseng. Design of Fractional Order Digital FIR Differentiators.
IEEE Signalprocessing Letters , 8(3):77–79, 2001.38. B. M. Vinagreb Y. Q. Chena. Fractional Differentiation for Edge Detection.
SignalProcessing , 83:2359–2365, 2003.39. X. S. Yang.
Mathematical Modeling with Multidisciplinary Applications . JohnWiley, 2013.40. V. Zaborovsky and R. Meylanov. Informational Network Traffic Model Based onFractional Calculus . In