Formal Analysis of Fractional Order Systems

in Higher-Order Logic

By

Muhammad Umair Siddique

(2009-NUST-MS-M&S-15)

A Thesis Submitted in Partial Fulfillment of

the Requirements of the Degree of

Master of Science in Computational Science and Engineering

Research Centre for Modeling and Simulation

National University of Sciences and Technology

Islamabad, Pakistan

© Muhammad Umair Siddique, 2011

Formal Analysis of Fractional Order

Systems in Higher-order Logic

by

Muhammad Umair Siddique

2009-NUST-MS-M&S-15

A Thesis Submitted in Partial Fulfillment of the

Requirement for the Degree of

Master of Science

in

Computational Science and Engineering

Research Centre for Modeling and Simulation (RCMS)

National University of Sciences and Technology (NUST)

Pakistan

c© Muhammad Umair Siddique, 2011

ABSTRACT

In the last two decades fractional order systems, which involve integration and dif-

ferentiation of non-integer order, have become very popular in the fields of control

systems, robotics, signal processing and circuit theory. Traditionally, the analy-

sis of fractional order systems has been performed using paper-and-pencil based

proofs or computer algebra systems. These analysis techniques compromise the ac-

curacy of their results and thus are not recommended to be used for safety-critical

fractional order systems. In the past couple of decades, formal methods have been

successfully used for the precise analysis of a variety of hardware and software

systems. The rigorous exercise of developing a mathematical model for the given

system and analyzing this model using mathematical reasoning usually increases

the chances for catching subtle but critical design errors that are often ignored

by traditional techniques like numerical methods. Given the sophistication of the

present age fractional order systems and their extensive usage in safety-critical

applications, there is a dire need of using formal methods in this domain. This

thesis proposes to use higher-order logic theorem proving to analyze fractional or-

der systems; the main reason being the highly expressive nature of higher-order

logic, which can be leveraged upon to essentially model any system that can be

expressed in a closed mathematical form.

In particular, this thesis provides higher-order logic formalization of the Gamma

function along with the verification of its classical properties using the HOL theo-

rem prover. We build upon the Gamma function for the formalization of Fractional

Calculus based on the Riemann-Liouville approach and verify its key properties

such as identity, relation to integer order calculus and linearity. This formalization

framework can be used to formalize and verify the properties of fractional order

systems within the sound core of HOL theorem prover. In order to demonstrate

the usefulness of current work, we utilize it to formalize fractional electrical com-

ponent Resistoductance, fractional integrator and differentiator circuits and then

verify their output response to benchmark input signals.

ACKNOWLEDGMENTS

First and foremost, I would like to thank almighty ALLAH who is creator and

sustainer of all the creatures.

I would like to express my thanks and sincere gratitude to my supervisor, Dr.

Osman Hasan, for his strong support, encouragement and guidance through out my

research. He was always approachable and his insights about research and immense

knowledge in the field of formal methods have strengthened this work significantly.

His emphasis for excellence kept me well-directed and focused. His cheerful and

enthusiastic encouragement was a source of strength for me to complete this thesis.

I would also like to acknowledge his efforts for establishing System Analysis and

Verification (SAVE) Lab at SEECS, NUST which is a first formal Methods Lab in

Pakistan.

I would like to express my gratitude to Dr. Zahid Anwar for taking time out of

his busy schedule to serve as my external examiner. I sincerely thank Dr. Jamil

Ahmad and Dr. Meraj Mustafa Hashmi for serving on my thesis committee. I

would also like to thank Engr. Sikandar Hayat Mirza, Principal, (RCMS) for his

unconditional help and encouragement for doing my thesis in SAVE Lab.

I would like to thank Mr. Mujeeb, PhD student at Center of Advance Mathematics

and Physics (CAMP), and Mr. Muhammad Zubair, Research Associate at GIKI

for providing me useful books related to Fractional Calculus. Very special thanks

go out to my colleagues in RCMS, without their motivation and encouragement,

I would not have reached this point. I would like to thank Waqr, Aqib, Ammar

Mushtaq, Junaid, Usman Rauf, Aamir shahzad and Nafees for their encouragement

iii

during my thesis.

Finally, I would like to thank my parents for their prayers and support they always

provide me with. Most importantly, I wish to thank all my family members, spe-

cially my elder brother Muhammad Tariq Siddique for his motivation and support

for my higher studies.

iv

To My Parents

Late brother

Muhammad Mujahid Siddique

and

Muhammad Aarfeen Tariq

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

CHAPTER

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Fractional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.1 A Historical Note: . . . . . . . . . . . . . . . . . . . . . . 5

1.2.2 Mathematical Framework . . . . . . . . . . . . . . . . . . 10

1.2.3 Applications of Fractional Calculus . . . . . . . . . . . . 12

1.3 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Proposed Framework . . . . . . . . . . . . . . . . . . . . . . . . 17

1.5 Thesis Contributions . . . . . . . . . . . . . . . . . . . . . . . . 20

1.6 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . 21

2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1 Theorem Proving . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 HOL Theorem Prover . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.1 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.2 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.3 Inference Rules . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.4 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.5 Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

vi

Page

vii

2.2.6 Proofs in HOL . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.7 HOL Notations . . . . . . . . . . . . . . . . . . . . . . . 27

2.3 Harrison’s Formalization of Integer order Calculus . . . . . . . . 28

2.3.1 Formalization of Derivative . . . . . . . . . . . . . . . . . 28

2.3.2 Formalization of Gauge Integral . . . . . . . . . . . . . . 29

2.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 31

3 HOL Formalization of Gamma Function . . . . . . . . . . . . . 32

3.1 Formalization of Gamma Function . . . . . . . . . . . . . . . . . 32

3.2 Formal Verification of the Properties of Gamma Function . . . . 34

3.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 41

4 HOL Formalization of Fractional Calculus . . . . . . . . . . . . 43

4.1 Formalization of Differintegrals . . . . . . . . . . . . . . . . . . 43

4.2 Formal Verification of the Properties of Differintegrals . . . . . . 45

4.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.1 Euler’s Generalized Rule . . . . . . . . . . . . . . . . . . . . . . 52

5.2 Resistoductance . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3 Fractional Differentiator and Integrator Circuits . . . . . . . . . 57

5.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 61

6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . 63

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

viii

LIST OF TABLES

Table Page

1 HOL Symbols and Functions . . . . . . . . . . . . . . . . . . . 27

2 Properties of Gauge integral . . . . . . . . . . . . . . . . . . . . 30

3 Properties of Gamma Function . . . . . . . . . . . . . . . . . . 41

4 Properties of Differintegrals . . . . . . . . . . . . . . . . . . . . 50

ix

LIST OF FIGURES

Figure Page

1 Proposed Framework . . . . . . . . . . . . . . . . . . . . . . . . 18

2 Resistoductance . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3 (a) Integrator (b) Differentiator . . . . . . . . . . . . . . . . . . 58

x

CHAPTER 1

Introduction

1.1 Motivation

In reality, many situations arise when integer order calculus is not sufficient to

model all kind of dynamics. For example, an electrical component Resistoductance

[1] exhibits an intermediate behavior between that of a resistor and inductor and

thus its accurate modeling involves the differentiation of order between 0 and 1.

Such systems that involve integration and differentiation of non integer order, or

fractional calculus [2], for their modeling are usually referred to as fractional order

systems. The idea of fractional calculus is as old as integer order calculus itself.

The question which gave birth to fractional calculus was about the interpretation

of dnydxn

, if n is not an integer or more broadly if n is any real, irrational or even a

complex number.

Accurate modeling of engineering and scientific systems have become imper-

ative these days due to their extensive usage in safety-critical domains, such as,

medicine and transportation. This fact has led to the widespread usage of frac-

tional calculus in modeling physical systems. For example, in control engineering

the concept of fractional operations is mostly used in fractional system identifi-

cation [3], biomimetic control [4], fractional PIα [5] and PDµ controllers [6]. In

signal processing, fractional operators are used in the design of fractional order

differentiators and integrators [7] and for modeling the speech signals [8]. Other

interesting applications of fractional calculus are in image processing [9], electro-

1

magnetic theory [10], chaotic communication [11], and circuit theory [1].

The widespread use of fractional order systems demands accurate, precise and

scalable analysis techniques. The analysis of fractional order systems is normally

conducted by paper-and-pencil based proof methods. These kinds of traditional

methods suffer from many problems, such as, expertise to deal with complex math-

ematical theories and scalability, as it is difficult to maintain the correctness of large

proofs which is influenced by the risk of human error. The second commonly used

analysis method for fractional order systems is computer based techniques, which

can be divided into two main streams, i.e., the simulations based methods and

the computer algebra systems (CAS). The availability of high speed computers

attracted the attention of researchers working in this field to perform simulation

based analysis using numerical algorithms. The main idea of simulation based

methods is to construct a discretized system model and then simulate the output

of the system for different input patterns. The second alternative, i.e., computer

algebra systems is also becoming popular for the fractional order system analysis.

In computer algebra systems, the mathematical computations are done using sym-

bolic algorithms, and hence they are better than simulation based analysis. But

computer simulations as well as analysis based on computer algebra systems cannot

provide 100% percent accurate results. In computer simulation, analysis is based

on different approximations which leads to the erroneous analysis. On the other

hand computer algebra systems, which are very efficient for mathematical compu-

tation, are not sound because the computed results are not always mathematically

2

correct. For example, in Maple [12], if we write :

x2 − 1

x− 1(1)

the result will be x+1, which is an over simplification, x = 1 gives an indeterminate

value 00.

The precision and accuracy is the main concern of system analysis because the

erroneous analysis leads to faulty systems, which may result in the loss of human

lives. Some consequences of erroneous simulations based analysis include, the Intel

hardware error in floating point unit in 1994, which resulted in the monetary loss

of 500 million US$, the Arian 5 crash in 1996 due to data conversion error, resulted

in the loss of more than 500 million US$ and the air France flight 447 crash due

to inaccurate air speed measurement by sensors, resulted in the loss of 228 human

lives. Due to above mentioned limitations, the traditional analysis techniques

cannot be relied upon for the analysis of fractional order systems.

Formal methods [13] allow accurate and precise analysis and provide a mean

to overcome the above mentioned limitations of traditional approaches. The main

idea behind formal methods is to develop a mathematical model for the given sys-

tem and analyzing this model using mathematical reasoning which in turn increases

the chances for catching subtle but critical design errors that are often ignored by

traditional techniques like numerical methods. There are two most commonly used

formal methods techniques, i.e., model checking [14] and higher-order logic theo-

rem proving [15]. Model checking is an automatic verification technique for the

systems that can be expressed as finite-state machine. On the other hand, higher-

order logic theorem proving is an interactive verification technique, which is more

3

flexible and can handle variety of systems.

In the past couple of decades, model checking and theorem proving have been

successfully used for the precise analysis of a variety of hardware and software

systems. Given the sophistication of the present age fractional order systems and

their extensive usage in safety-critical applications, there is a dire need of using

formal techniques in this domain. In fact, the applicability of model checking and

theorem proving for fractional order system analysis is somewhat limited. Due

to the continuous nature of the analysis and the involvement of transcendental

functions, automatic state-based approaches, like model checking [14], cannot be

used in this domain. On the other hand, the main limitation of theorem proving is

the lack of mathematical foundations to conduct fractional order system analysis

related formal proofs.

This thesis takes steps to fill this gap as it presents mathematical foundations

that provide a novel platform for the formal analysis of fractional order systems

using higher-order logic theorem proving. Besides achieving 100% accurate results,

another motivation of using formal methods for fractional order systems analysis

is the fact that fractional order systems generalizes the integer order systems and

thus establishment of a formal framework for fractional order systems equally fa-

cilitates the formal analysis of integer order systems. The ability to accurately

conduct fractional order system analysis may prove to be a very useful feature

for the systems used in safety-critical domains, such as, military, medicine and

transportation.

4

1.2 Fractional Calculus

In this section, we provide a short history of fractional calculus along with its

mathematical frame work. We also provide the two most commonly used defini-

tions of fractional calculus and discuss the choice of their usage in this thesis. At

the end of this section, we provide a brief survey of the applications of fractional

calculus in science and engineering. This information is used in the next section

to compare capabilities of the available computer based fractional order system

analysis techniques.

1.2.1 A Historical Note:

In 1695, L’Hopital asked to Leibnitz regarding his notation dnydxn

: “What if n is

12”. Leibnitiz [16] prophesied in his letter to L’Hopital,“. . . Thus it follows that

d12x will be equal to x

√dx : x. This is an apparent paradox from which, one day,

useful consequences can be drawn . . . ”. Leibnitz [17] continued with this idea and

discussed with Johann Bernouli about the derivatives of ‘general orders’. After

three years, in his correspondence with John Wallis, Leibnitz [18] discussed the

ways for using fractional derivatives in finding the infinite product for 12π. Euler,

a great mathematician and physicist, also contributed in the development of frac-

tional calculus. In 1730, he proposed in his dissertation, an integral expression for

infamous gamma function which generalizes factorial over non integer numbers.

The Gamma function plays the same role in the fractional calculus as factorial

in the integer order calculus. After Euler’s dissertation, Laplace worked on Frac-

tional Calculus. In 1812, Laplace discussed the fractional derivatives by means of

integrals [2].

5

In 1819, derivatives of arbitrary order appeared first the time in the texts.

S. F. Lacroix [19] dedicated about two pages of his 700 pages book to fractional

derivative. He developed the formula for nth order derivative using factorial and

then generalizes the concept using the Gamma function as follows:

dny

dxn=

m!

(m− n)!xm−n,m ≥ n (2)

and the generalization is:

dny

dxn=

Γ(m+ 1)

Γ(m− n+ 1)xm−n (3)

Then he took one example, i.e., y = x and n = 12

and obtained the following result.

d12y

dx12

=2√x√π

(4)

It is a very interesting to note that the result obtained by Lacroix was the same

as obtained by modern definitions of the fractional derivative, such as Riemann-

Liouville.

Fourier [20] was the next mathematician who worked on this idea. He defined

fractional operations using his integral definition of f(x), i.e.,

f(x) =1

2π

∫ ∞−∞

f(α)dα

∫ ∞−∞

cos p(x− α)dp (5)

Now

dn

dxncos p(x− α) = pn cos[p(x− α) +

1

2nπ] (6)

where n is an integer. Then, by replacing n with any arbitrary value u, he gener-

alized the idea as follows:

du

dxuf(x) =

1

2π

∫ ∞−∞

f(α)dα

∫ ∞−∞

pu cos[p(x− α) +1

2uπ]dp (7)

6

Fourier stated [20, 2], “. . . The number u that appears in the above expression will

be regarded as any quantity whatsoever, positive or negative. . . ”.

To this point, we have seen that mathematicians and physicist of that time

only tried to define fractional derivatives of arbitrary order. Neils Henrik Abel

[21], in 1823, was the first who made the use of fractional operations in finding the

solutions of famous Tautochrone problem (i.e. the problem of finding the shape of

the curve such that a bead placed anywhere would fall under the action of gravity

independent of starting point).

The classical work of Fourier and Abel inspired Joseph Liouville to work in this

area. In 1823, he published three long memoirs. Liouville applied his definitions to

the problems of potential theory [2]. He started from the known results of integral

order derivatives.

Dmeax = ameax (8)

and then replaced m with v (arbitrary) as:

Dveax = aveax (9)

Then, he used a very intuitive approach and assumed that the function f(x) may

be written in the series form as:

f(x) =∞∑n=0

Cneanx, Re an > 0 (10)

Using Equation (8) and (9)

Dvf(x) =∞∑n=0

Cnavneanx (11)

The last formula is sometimes also termed as Liouville’s first formula. This is the

generalization from ordinary to the derivative of arbitrary order v, where v can be

7

rational, irrational or complex. But this formula has its limitations as it is suitable

for only those functions which can be expressed in the series form (Equation (9)).

Liouville might be aware of the limitations of the above formula for fractional

derivative and thus came up with a second definition of fractional derivative. This

time, he started with an integral of the form,

K =

∫ ∞0

ua−1e−xudu, a > 0, x > 0 (12)

He played with the above expression by changing variables t = xu and operating

both sides with Dv to obtain the following result.

Dvx−v =(−1)vΓ(a+ v)

Γ(a)x−a−v, a > 0 (13)

Where v is any number rational, irrational or complex. This formula is termed as

Liouville’s second formula for fractional derivative. Liouville was the first one who

solved differential equations with fractional derivatives but could not found correct

results. His both formulae for fractional derivatives have certain limitations as first

one is only suitable for functions that can be expanded in series form (Equation

(9)) and second one is only capable of dealing with the functions of kind x−a where

a > 0.

In the period from 1833 to 1849, the field of fractional calculus was under

many controversies. Actually, there were different definitions suited for different

domains. Many mathematicians of that time commented on these definitions. In

1833 Peacock [22] supported Lacriox definition and criticized Liouville’s definition,

even he made many mistakes while supporting Lacroix. Later on, another mathe-

matician Kelland supported Liouville on two different occasions 1839 and 1846. In

8

1840, Demorgan [23] and in 1843, William Center [24] also commented on Lacriox’s

and Liouville’s definitions for fractional derivatives.

In late 1800’s G. F. Bernhard Riemann worked on fractional calculus in his

student life. But he never published his work and it was published posthumously in

Gesammelte werke in 1892 [25]. He used Taylor’s series and complentary function

Ψ(x) in his work and obtained the general solution as follows:

D−vf(x) =1

Γ(x)

∫ x

c

(x− t)v−1f(t)dt+ Ψ(x) (14)

He was not sure about the lower limit c in the above equation. In order to get rid

from this ambiguity, he introduced complementary function Ψ(x).

The other great mathematicians and physicists who touched the field of frac-

tional calculus are Laurent(1884), Heaviside (1892), Al-Bassam, Davis Erdelyi,

Hardy, Kobler, Littlewood, Love, Riesz, Samko, Sneddon, Weyl, Zygmund and

Thomas J. Osler[2]. We would like to finish the history of fractional calculus with

a quote from Miller and Ross [2]. They stated:

“. . . The fractional calculus finds use in many fields of science and en-

gineering, including fluid flow, rheology, diffusive transport akin to

diffusion, electrical networks, electromagnetic theory, and probabil-

ity. . . . It seems that hardly a field of science or engineering has

remained untouched by this topic. Yet even though the subject is

old, it is rarely included in today’s curricula. Possibly, this is be-

cause many mathematicians are unfamiliar with its uses . . . ”

9

1.2.2 Mathematical Framework

The choice for using appropriate notation of fractional derivative and integration

cannot be minimized. There are different notations available for fractional deriva-

tives and integrals. We will use Jvaf(x) and Dvf(x) for fractional integral and

fractional derivative, respectively. In these notations, v is the order of integration

or differentiation and a is the lower limit of integration.

Now, we discuss the mathematical problem of defining fractional differen-

tiation and integration. It is clear from our previous discussion that the work

presented by mathematicians so far does not provide complete formalization of

fractional calculus. But they defined fractional operators for only particular class

of functions.

For every function f ; and for every number v ∈ R+, Jva and Dv should be

related to f by the following criteria [1].

1. If f(x) is an analytic function, then Jvaf(x) and Dvf(x) must also be an

analytic function of the variable x and of the order v.

2. The operations Jvaf(x) and Dvf(x) must produce the same result as ordinary

integration/differentiation when v is a positive integer n.

3. The fractional operators must be linear.

Jva [af(x) + bg(x)] = aJvaf(x) + bJvag(x) (15)

Dv[af(x) + bg(x)] = aDvf(x) + bDvg(x) (16)

4. The operation of order zero must leave the function unchanged.

J0af = f and D0f = f (17)

10

5. The law of exponents must hold for integration and differentiation of arbi-

trary order under sufficient conditions on function f ..

Jua (Jvaf) = Ju+va f and Du(Dvf) = Du+vf (18)

Fractional integrals and fractional derivatives are also referred to as Differintegrals

[26] and there are more than ten well known definitions for Differintegrals [27].

We consider two of them, which are most widely used in analyzing real-world

problems. These are the Riemann-Liouville and Grunwald-Letnikov definitions,

which are also equivalent for a wide class of functions [28].

Riemann-Liouville (RL) Definition:

Jvaf(x) =1

Γ(v)

∫ x

a

(x− t)v−1f(t)dt (19)

Where Jvaf(x) represents fractional integration with order v and lower integration

limit a. a = 0 gives the Riemann definition and a = −∞ gives the Liouville defi-

nition of fractional integration [29]. Γ in the above definition denotes the Gamma

function which is defined using the well-known improper integral as follows:

Γ(z) =

∫ ∞0

tz−1e−tdt (20)

for z > 0.

The fractional differentiation is given as follows:

Dvf(x) = (d

dx)mJm−va f(x) (21)

where m represents the ceiling of v, i.e., dve.

11

Grunwald-Letnikov (GL) Definition:

cDvxf(x) = lim

h→0h−v

[x−ch

]∑k=0

(−1)k(v

k

)f(x− kh) (22)

Grunwald-Letnikov definition caters for both fractional differentiation and integra-

tion, as positive values of v give fractional differentiation and negative values of v

give fractional integration. Here,(vk

)represents the binomial coefficient, which is

described in terms of the Gamma function.

The next section presents a brief review of the application of fractional calcu-

lus.

1.2.3 Applications of Fractional Calculus

The concept of fractional calculus has great potential to change the way we see,

model and analyze the systems. We can say that ignoring fractional calculus

is just like ignoring fractional, irrational or complex numbers. It provides good

opportunity to scientists and engineers for revisiting the origins. The theoretical

and practical interests of using fractional order operators are increasing. The

application domain of fractional calculus is ranging from accurate modeling of

the microbiological processes to the analysis of astronomical images. Next, we

will present a brief survey of applications of fractional calculus in science and

engineering.

Control Engineering

The accuracy and robustness of control systems are becoming imperative these

days. The dynamic nature of control systems requires them to be modeled using the

12

fractional calculus. In control engineering the concept of the fractional operations

is mostly used in fractional system identification [3], biomimetic (bionics) control

[4], feedback control systems [30], trajectory control of redundant manipulators

[31], temperature control [32], Model Reference based adaptive control [33], passive

vibrational control [34], fractional PIα[5] controller and fractional PDµcontrollers

[6].

Signal Processing

In the last decade, the use of fractional calculus in signal processing has tremen-

dously increased. In signal processing, the fractional operators are used in the

design of differentiator and integrator of fractional order [7], fractional order FIR

differentiator [35], IIR type digital fractional order differentiator [36] and for mod-

eling the speech signal [8]. A brief survey of application of fractional calculus in

signal processing is presented in [37].

Image Processing

In image processing, fractional calculus (fractional differentiation) is used for en-

hancing image quality, image restoration and edge detection[9]. In particularly

fractional calculus is used in satellite image classification [38] and astronomical

image processing [39].

Electromagnetic Theory

The use of fractional calculus in electromagnetic theory has emerged in the last

two decades. In 1998, Engheta [40] introduced the concept of fractional curl oper-

ators and this concept is extended by Naqvi [41]. Engheta’s [42] work gave birth

13

to the new field of research in Electromagnetics, namely, ‘Fractional Paradigms in

Electromagnetic Theory’. Nowadays fractional calculus is widely used in Electro-

magnetics to explore new results; for example, Faryad [10] used fractional calculus

for the analysis of a Rectangular Waveguide.

Communication

Chaotic Communication and Chaos synchronization are becoming very popular

nowadays. The concept of fractional calculus is recently introduced for secure

chaotic communication and very satisfactory results have been achieved [11]. In

[43], authors have used fractional calculus for informational network traffic mod-

eling.

Probability Theory

Cottone used fractional operators for the probabilistic characterization of random

variable and some remarkable results are discussed in [44].

Biology

In Biology, fractional calculus is used in neuron modeling [45], biophysical processes

[46], modeling of complex dynamics of tissues [47], modeling of infectious diseases

[48] etc.

1.3 Related Work

Fractional order systems are natural extensions of integer order systems and thus

heavily rely upon the classical concepts of integer order calculus, such as, fractional

integration and differentiation involves integer order integration. Even though,

there is a significant amount of research going on in the area of fractional order

14

systems, but to the best of our knowledge, there is no formal methods based

technique available for fractional order system analysis. In this section, we will

present existing state-of-the-art techniques for fractional order system analysis.

Traditionally, the analysis of fractional calculus based models has been done

using paper-and-pencil proof methods. However, considering the complexity of

present age engineering and scientific systems, such kind of analysis is notoriously

difficult, if not impossible, and is quite error prone. Many examples of erroneous

paper-and-pencil based proofs are available in the open literature, a recent one

can be found in [49] and its identification and correction is reported in [50]. One

of the most commonly used computer based analysis technique for fractional or-

der systems is numerical computation of fractional integration and differentiation.

Some examples include, chaos in fractional order volta systems [51], fractional PIα

controllers [5] and motion planning of redundant and hyper-redundant manipula-

tors [52]. Fractional order systems are continuous in nature and thus the first step

in their simulation based analysis is to construct a discretized system model with

minimal error. Most of the numerical algorithms are based either on Grunwald-

Letnikov definition [53] or on power series expansion (PSE) method [51]. Both of

them cannot provide valuable results due to involvement of infinite summations

in case of Grunwald-Letnikov definition and huge memory requirements in case

of the power series expansion method. Similarly, the computation of the Gamma

function Γ(x) for large values of x is not possible in such numerical computation

software packages. For example, MATLAB [54] returns 7.26e306 as the approxi-

mated value computed for x = 171 and returns Inf for all values beyond x = 171.

15

Another alternative to deal with fractional calculus proofs is computer alge-

bra systems [55], which are very efficient for computing mathematical solutions

symbolically, but they are not reliable [56] due to their limitations of dealing with

side conditions. Another limitation of computer algebra systems is the uncertain

simplification of singular expressions particularly in case of the Gamma function

[57], which are frequently used in fractional calculus due to usage of improper in-

tegrals. Another source of inaccuracy in computer algebra systems is the presence

of unverified huge symbolic manipulation algorithms in their core, which are quite

likely to contain bugs.

The early formalization [56] of main concepts of calculus in higher-order logic,

such as, limits, derivatives and integrals were laid down by Harrison . Butler [58]

reported the formalization of integral calculus in the PVS theorem prover. L. Cruz-

Filipe reported a constructive theory of analysis in the Coq theorem prover in his

Ph.D dissertation [59]. Mhamdi [60] presented the higher-order logic formalization

of Lebesgue integration theory in the HOL theorem prover, which is a fundamental

concept in many mathematical theories and allows a wider class of functions than

the Riemann and Gauge integration theory. In this thesis, we are providing a

framework that can be used to formalize improper integrals in the HOL theorem

prover. Our work uses and extends the work done by Harrison [61]. The main

reasons behind this choice include the richness of Harrison’s real analysis related

theories, which are fundamental to our work, and the ability to use Harrison’s

Gauge integral to formalize improper integrals and thus the Gamma function.

Then the availability of the Gamma function in HOL theorem prover facilitates

16

the formal reasoning about fractional calculus which in turn paves the way to

formally analyze fractional order systems.

1.4 Proposed Framework

The objective of this thesis is mainly targeted towards the development of a theo-

rem proving based fractional order system analysis framework that can handle the

analysis of real-world systems exhibiting fractional order dynamics. In particular,

we have developed a framework characterizing:

1. The ability to formally express improper integrals in higher-order logic. Then

using the concept of improper integrals to formalize the Gamma function,

which in turn, helps to formalize Differintegrals.

2. The ability to formally verify the properties of the Gamma function and Dif-

ferintegrals in higher-order logic theorem prover. These formalized properties

can be used to develop theorems representing the characteristics of interest

for the formal model of the given system.

3. The ability to formally reason about the theorems, formalized in Step 2,

using a theorem prover.

4. The ability to utilize the above mentioned capabilities to formally model and

reason about real-world fractional order system analysis problems.

The proposed framework, given in Figure 1, outlines the main idea behind

the theorem proving based fractional order system analysis. The grey shaded

boxes in this figure represent the key contributions of the thesis that serves as

17

the fundamental requirements of conducting fractional order system analysis in a

theorem prover. Like all the system analysis tools, the input to this framework,

depicted by two rectangles with curved bottoms, is the description of the fractional

order system that needs to be analyzed and a set of properties that are required

to be checked for the given system.

Fractional order System

Properties of System

Higher -order Logic Description

(Theorem)

HOL Theorem Prover

Formal Proof of System Properties

Higher-order Logic

Real Analysis

&Integer order

calculus

Gamma function

Differintegrals

Formally Verified Properties

Formal Model

TheoremsTheorems

Theorems

Figure 1. Proposed Framework

The first step in conducting fractional order system analysis using a theorem

prover is to construct a formal model of the given system in higher-order logic.

For this purpose, the foremost requirement is the ability to formalize fractional

derivatives and integrals (Differintegrals) as higher-order logic functions. The for-

malization of Differintegrals, given in Equations (19) and (21), requires the math-

18

ematical theories of real numbers, integer order calculus and the Gamma function.

Harrison’s work on the formalization of real numbers [56] provides the first two

requirements and we built upon Harrison’s work to formalize the Gamma function

in this thesis to fulfil the third requirement. Using these fundamentals, this thesis

also presents the formalization of Differintegrals, given in Equations (19) and (21),

which in turn can be used to represent the dynamics of fractional order systems in

higher-order logic. The second step in the theorem proving based fractional order

system analysis is to utilize the formal model of fractional order system, developed

in the first step, to express system properties as higher-order logic theorems.

The third step for conducting fractional order system analysis in a theorem

prover is to formally verify the higher-order-logic theorems developed in the pre-

vious step using a theorem prover. For this verification, it would be quite handy

to have access to a library of some pre-verified theorems corresponding to some

commonly used properties of Gamma function and Differintegrals. To fulfil this

requirement, this thesis presents formal verification of the classical properties of

Gamma function, such as, Pseudo-Recurrence Relation, Factorial Generalization

and Functional Equation, and Differintegrals, such as, Identity and Linearity, using

the HOL theorem prover. Building on such a library of theorems would minimize

the interactive verification efforts and thus speed up the verification process. Fi-

nally, the output of the theorem proving based fractional order system analysis

framework, depicted by the rectangle with dashed edges, is the formal proofs of

system properties that certify that the given system properties are valid for the

given fractional order system.

19

1.5 Thesis Contributions

In summary, the main focus of this thesis is on the formal analysis of fractional

order systems using higher-order logic theorem proving. This framework allows us

to conduct precise and accurate fractional order system analysis and thus proves

to be very useful for the verification of safety critical fractional order systems. In

this endeavor, this thesis makes the following contributions.

1. It presents an infrastructure that allows us to formally specify and verify

higher-order logic theorems related to the improper integrals and it provides

the complete formalization details of one of the most famous improper inte-

gral, i.e., the Gamma function along with the verification of its key properties.

2. It presents a framework to formalize fractional order derivatives and integrals

in higher-order logic, which paves the way to model the fractional order

dynamics of real-world systems.

3. It presents the formal modeling and verification of a few real-world frac-

tional order systems, i.e., Resistoductance, a fractional differentiator and a

fractional integrator circuit in a higher-order-logic theorem prover. We be-

lieve that this is the first time that a computer based formal method has

been used to tackle these problems in a unified framework with 100% precise

results. The analysis approach for these case studies is quite general and can

be utilized to conduct the analysis of other real-world systems as well.

20

1.6 Organization of the Thesis

The rest of the thesis is organized as follows. In Chapter 2, we provide a brief in-

troduction to the HOL theorem prover and an overview of Harrison’s formalization

of the Gauge integral and derivatives to equip the reader with some notation and

concepts that are going to be used in the rest of this thesis. Chapter 3 describes

the formalization infrastructure that enables us to formally specify and verify the

properties of the Gamma function and Incomplete Gamma functions along with

the formal verification of their corresponding properties in HOL. To demonstrate

this infrastructure, the chapter 4 presents the formalization of Differintegrals along

with the verification of their useful properties, such as, identity, linearity and re-

lation to integer order calculus. Next, in Chapter 5, we illustrate the practical

effectiveness of the proposed approach (and the above mentioned formalization)

by successfully applying it for the formal analysis of Resistoductance, fractional

differentiator and integrator circuits. Finally, Chapter 5 concludes the thesis and

outlines some future research directions.

21

CHAPTER 2

Preliminaries

In this chapter, we provide a brief introduction to the HOL theorem prover and

present an overview of Harrison’s [56] formalization of derivatives and Gauge inte-

gral. The intent is to introduce the basic theories along with some notations that

are going to be used in the rest of the thesis.

2.1 Theorem Proving

Theorem proving or automatic deduction, which is one of the most developed re-

search area in automated reasoning, is concerned with the construction of mathe-

matical theorems using a computer program. These mathematical theorems can

be built on different types of logic, such as, propositional logic, first-order logic or

higher-order logic, depending upon the expressibility requirement. For example,

the use of higher-order logic is advantageous over first order logic in terms of the

availability of additional quantifiers and highly expressive nature of higher-order

logic. The main idea behind the theorem proving based formal analysis is to math-

ematically model the given system in an appropriate logic and then the properties

of interest are verified using computer based formal reasoning. Using higher-order

logic theorem proving for modeling the system behaviors makes the analysis very

flexible as it allows the formal verification of any system that can be expressed

mathematically. The core of theorem provers usually consists of some well-known

axioms and primitive inference rules. The theorem proving based verification as-

22

sures the soundness as every new theorem must be created from these basic axioms

and primitive inference rules or any other already proved theorems.

A theorem prover or proof assistant is a tool which facilitates the formal

description of a given system in terms of mathematical expressions. There are

two types of provers, i.e., automatic and interactive. In an interactive theorem

prover, significant user-computer interaction is required while automatic theorem

prover can perform different proof tasks automatically. There is a large family of

theorem provers but only few of them are in continuous development and have large

user community. Some commonly used automated provers include SATURATE,

LeanTAP, Gandalf, METEOR, SETHEO and Otter [62] and MetiTarski [63]. The

family of interactive higher-order logic based theorem provers includes Isabelle,

Coq, HOL, HOL Light, ProofPower, ACL2 and MIZAR [62].

This thesis uses the HOL theorem prover to conduct all the fractional or-

der system analysis. The main reasons behind this choice include the richness

of Harrison’s real analysis related theories, which are fundamental to our work,

and the ability to use Harrison’s Gauge integral to formalize the Gamma function

and thus the fractional calculus. Moreover, some earlier work related to formal

analysis of real-world applications, such as, Harrison’s floating point verification

of trigonometric functions [64] and Hasan’s formal analysis of optical waveguides

[65], inspired this thesis to be done in HOL theorem prover.

2.2 HOL Theorem Prover

HOL is an interactive theorem proving environment for the construction of math-

ematical proofs in higher-order logic. First version of HOL was developed by Mike

23

Gordon at Cambridge University, in 1980′s. The core of HOL is interfaced to

the functional programming language ML-Meta Language [66]. HOL utilizes the

simple type theory of Church [67] along with Hindley-Milner polymorphism [68]

to implement higher-order logic. The first version of HOL is called HOL88 and

other versions of HOL are HOL90 and HOL98 and HOL4. HOL4, the recent ver-

sion of HOL family, uses Moscow ML which is an implementation of Standard ML

(SML). The HOL core consists of only 5 basic axioms and 8 primitive inference

rules, which are implemented as ML functions. HOL has been widely used for the

formal verification of software and hardware systems along with the formalization

of mathematical theories.

2.2.1 Terms

HOL has four types of terms: constants, variables, function applications, and

lambda-terms. Variables are sequences of digits or letters beginning with a letter,

e.g., y, b, Gamma hol. The syntax of the constants is similar to the variables, but

they cannot be bounded by quantifiers. The type of an identifier, i.e., variable or a

constant, is determined by a theory; e.g., F, T. Applications in HOL represent the

evaluation of a function g at an argument y, different terms can be used instead of g

and y, e.g., f and x. In HOL, we can use λ-terms, also called lambda abstractions

for denoting functions. λ-terms has the form λx.f(x) and represent a function

which takes x and returns f(x).

2.2.2 Types

According to the lambda calculus implemented in HOL, every HOL term has a

unique type which is either one of the basic types or the result of applying a type

24

constructor to other types. In HOL, each variable and constant must be assigned

a type and variables with the same name but different types are considered as

different. When a term is entered into HOL, the type is inferred using the type

checking algorithm implemented in HOL, e.g., when (∼ y) is entered into HOL ,

the HOL type checker deduces that the variable y must have type bool because

negation (∼) has a type bool → bool. If the type of a term cannot be deduced

automatically then it is possible to explicitly mention the type of that term, e.g.,

(x : real) or (x : bool).

2.2.3 Inference Rules

Inference rules are procedures for deriving new theorems and they are represented

as ML functions. There are eight primitive inference rules in HOL and all other

rules are derived from these inference rules and axioms. The rules are Assumption

introduction, Reflexivity, Beta-conversion, Substitution, Abstraction, Type instan-

tiation, Discharging an assumption and Modus Ponens [69].

2.2.4 Theorems

A theorem is a formalized statement that may be an axiom or follows from theorems

by a inference rule. A theorem consists of a finite set of boolean terms Ω called the

assumptions and a boolean term S called the conclusion. For example, if (Ω, S) is

a theorem in HOL then it is written as Ω ` S.

2.2.5 Theories

A HOL theory consists of a set of types, type operators, constants, definitions,

axioms and theorems. It contains list of theorems that have already been proved

25

from the axioms and definitions. The HOL theories can be loaded by user to

utilize the available definitions and theorems in that theory. The availability of

HOL theories allows the user to utilize and extend the existing results without

duplicating the efforts and have already been used in building such theories. HOL

theories are organized in a hierarchical fashion and theories can have other theories

as parents and all of the types, constants, definitions, axioms and theorems of

parent theory can be used in the child theory. For example, one of the basic

theory in HOL is bool and this is also parent theory of ind. We utilized the

HOL theories of Booleans, positive integers, real numbers, sequences, limits and

transcendental functions in our work. In fact, one of the primary motivations of

selecting the HOL theorem prover for our work was to benefit from these built-in

mathematical theories.

2.2.6 Proofs in HOL

In HOL, there are two types of interactive proof methods : forward and backward.

In a forward proof, the user starts from the primitive inference rules and tries to

prove the goals on top of these rules and already proved theorems. The forward

proof method is not an easy approach as it requires all the low level details of the

proof in advance. A backward or a goal directed proof method is the reverse of

the forward proof method. It is based on the concept of a tactic; which is an ML

function that breaks goals into simple subgoals. In the goal directed proof method,

the user starts with the desired theorem or the main goal that is further reduced

to simpler subgoals using the tactics which is a ML-function that breaks a goal

into subgoals. There are many automatic proof procedures and proof assistants

26

[70] available in HOL which helps the user in directing the proof to the end. In

interactive theorem verification user interacts with HOL proof editor and guides

the prover using the necessary tactics until the last step of the proof. In HOL,

some of the proof steps are solved automatically while others require signification

user interaction.

2.2.7 HOL Notations

Table 1 provides the mathematical interpretations of some frequently used HOL

symbols and functions in this thesis.

Table 1. HOL Symbols and Functions

HOL Symbol Standard Symbol Meaning/\ and Logical and\/ or Logical or∼ not Logical negation

==> −→ Implication<==> = Equality

!x.t ∀x.t for all x : t?x.t ∃x.t for some x : t@x.t εx.t an x such that : tλx.t λx.t Function that maps x to t(x)num 0, 1, 2, . . . Positive Integers data typereal All Real numbers Real data typesuc n (n+ 1) Successor of natural numberln x loge(x) Natural logarithm functionexp x ex Exponential functionsqrt x

√x Square root function

abs x |x| Absolute functionlim(λn.f(n)) lim

n→∞f(n) Limit of a real sequence f

convergent(λn.f(n)) ∃x. limn→∞

f(n) = x f is convergent

suminf(λn.f(n)) limk→∞

∑kn=0 f(n) Infinite summation of f

summable(λn.f(n)) ∃x. limk→∞

∑kn=0 f(n) = x Summation of f is convergent

27

2.3 Harrison’s Formalization of Integer order Calculus

In this section, we present a brief introduction to the existing higher-order-logic

formalization of the integer order derivative and Gauge integral. These are useful

for the formalization of the Gamma function and fractional calculus. The notations

and definitions of this section will be useful for the reader in the next section in

which we present our formalization of the Gamma function and fractional calculus.

2.3.1 Formalization of Derivative

Harrison [56] formalized real number theory in HOL and building on it, he formal-

ized the classical concepts of series and sequences and then verified their classical

properties. Some of the important definitions related to derivative are give as

follows:

Definition 2.1: Continuity of a Function

` ∀ f x. f contl x = ((λ h. f (x + h)) → f x) (0)

Definition 2.2: Derivative of a Function

` ∀ f l x.(f diffl l) x <=> ((λ h.(f (x + h) - f x) / h) → l) (0)

Definition 2.3: Differentiability of a Function

` ∀ f x. f differentiable x = ?l. (f diffl l) (x)

Where Definition 2.1 gives the continuity of a function f on point x, i.e., f(x +

h) −→ f(x) as h −→ 0. Definition 2.2 gives the derivative of a function f at point

x, i.e., f(x+h)−f(x)h

as h −→ 0. At the end, Definition 2.3 formalize the condition of

the differentiability of a function.

28

2.3.2 Formalization of Gauge Integral

Various integrals have been proposed in literature, namely, the Newton integral,

the Riemann integral and the Lebesgue integral, and each one of them has its

own advantages and disadvantages. For example, the Newton integral may not

exist for certain functions, e.g., the step function, the Riemann integral does not

have convenient convergence properties and the Lebesgue integral, which is better

than Riemann integral, shares one problem with the Riemann integral that the

Fundamental Theorem of Calculus is not always true. Different integrals have

been proposed to extend the Lebesgue integral for which Fundamental Theorem

of Calculus holds, but they are not constructive and requires more complicated

theorems.

In the 1960’s, Kurzweil and Henstock proposed a new integral that is simple

and powerful and it incorporates every function the others integrals can integrate.

It is normally known as the Generalized Riemann Integral, the Kurzweil-Henstock

Integral or the Gauge Integral. The main features of the Gauge integral are that

it has all the convergence properties of Lebesgue integral and it generalizes the

Riemann integral. The Gauge integral is formalized in HOL [56] as follows:

Definition 2.4: Definite Integral

` ∀ a b f k. Dint (a,b) f k = ∀ e. 0 < e ⇒

∃ g. gauge (λ x. a <= x ∧ x <= b) g ∧ ∀ D p. tdiv (a,b) (D,p)

∧ fine g (D,p) =⇒ abs (rsum (D,p) f - k) < e

Where Dint (a, b) f k is a equivalent to∫ baf = k. The definition of an integral

and integrable function is given as follows:

29

Definition 2.5: Integral

` ∀ a b f . integral (a,b) f = @k. Dint (a,b) f k

Definition 2.6: Integrable Function

` ∀ a b f . integrable (a,b) f = ∃ k. Dint (a,b) f k

Where @ is the Hilbert choice operator in HOL. Some of the important properties

of the Gauge integral are given in Table 2.

Table 2. Properties of Gauge integral

Property HOL Formalization

DINT INTEGRAL ` ∀ f g a b. (a <= b) ∧ (Dint (a,b) f g)=⇒(integral (a,b) f = g)

INTEGRAL LINEAR

` ∀ f a b c.(a <= b) ∧(integrable(a,b) f)∧ (integrable(a,b) g)

=⇒ integral(a,b) (m*f + n*g) =

m*integrable(a,b) f + n*integrable(a,b) g)

INTEGRAL BY PARTS

` ∀ f g f′ g′ a b.(a <= b)∧(∀ x. a <= x ∧ x <= b =⇒ (f diffl f′(x))(x))∧(∀ x. a <= x ∧ x <= b =⇒ (g diffl g′((x))(x))∧integrable(a,b) (λx. f′( x * g x)∧integrable(a,b) (λx. f x * g′(x)

=⇒ (integral(a,b) (λx. f x * g′ x) =

(f b * g b - f a * g a) -

integral(a,b) (λx. f′( x * g x))

INTEGRABLE CONT` ∀ f a b. (∀ x. a <= x ∧ x <= b ⇒ f contl x)

=⇒ integrable(a,b) f

INTEGRAL LE

` ∀ f g a b.(a <= b)∧(integrable(a,b) f)∧ (integrable(a,b) g)∧

(∀ x. a <= x ∧ x <= b ⇒ f(x) <= g(x))

=⇒ integral(a,b) f <= integral(a,b) g

INTEGRAL COMBINE

` ∀ f a b c.(a <= b) ∧(integrable(a,b) f)∧ (integrable(b,c) f)

=⇒ integral(a,c) f =

integral(a,b) f + integral(b,c) f)

INTEGRABLE SUBINT` ∀ f a b c d. (a <= c) ∧ (c <= d) ∧ (d <= b) ∧

integrable(a,b) f =⇒ integrable(c,d) f

30

2.4 Concluding Remarks

In this chapter, we started with a brief introduction of theorem proving and dis-

cussed different state-of-the-art theorem provers. Then we provided an overview

of the HOL theorem prover that we have used for our formalization related to

fractional order systems. Then we summarized the Harrison’s pioneering work on

the formalization of integer order calculus. We also discussed the formal definition

of derivative, continuity and differentiability of a function along with the formal-

ization details related to gauge integral. There were two main motivations of this

chapter: firstly, we intended to make the reader familiar with the HOL notations.

Secondly, Harrison’s formalization of integer order calculus is fundamental for the

formal analysis of fractional order systems. The next chapter presents the formal-

ization of the Gamma function building on top of the existing formalization of the

integer order calculus.

31

CHAPTER 3

HOL Formalization of Gamma Function

This chapter presents the higher-order logic formalization of the Gamma function

which is one of the main requirements of the proposed framework, depicted by the

gray shaded boxes in Figure 1. We have arranged the information in two sections.

The first section presents the formalization of the Gamma function and incomplete

Gamma functions in higher-order logic. The second section presents the formal

verification of important properties of the Gamma function and incomplete Gamma

functions.

3.1 Formalization of Gamma Function

The most common method for computing the factorial of an integer number m,

i.e., m!, is by recursively multiplying all the integers from 1 to m. However, this

method becomes inefficient as the value of m increases. To address this problem,

Euler proposed an integral based formula for m! in 1729, which was later upgraded

by Legendre into the widely used Gamma function [71]. One of the most important

characteristic of the Gamma function is its ability to generalize the factorial over

non-integer numbers. The most commonly used expression of the Gamma function,

for a real number z > 0, is given by an improper integral as follows:

Γ(z) =

∫ ∞0

tz−1e−tdt (23)

Improper integral is used in the above definition as it allows us to deal with the sit-

uations when the integrand becomes unbounded in the given interval or the interval

32

itself becomes unbounded. Let f(x) be defined on an interval [a,∞). For every

number t > a, the function is integrable on [a,t] and if the limit limt→∞∫ taf(x)dx,

exists, then the theory of improper integrals allows us to write:

∫ ∞a

f(x)dx = limt→∞

∫ t

a

f(x)dx (24)

Similarly, in the second case, let f(x) be a function which is bounded on [t,b] for

every a < t < b, but is unbounded on a. Though, the limit limt→a+∫ btf(x)dx,

exists, then the theory of improper integrals allows us to write:

∫ b

a

f(x)dx = limt→a+

∫ b

t

f(x)dx (25)

Similarly, many of other interesting cases can be deduced from Equations (24) and

(25).

It is not possible to formalize the definition of the Gamma function given in

Equation (23) as it does not cover the case when the integrand (tz−1e−t) becomes

unbounded at lower limit of integration when the argument of the Gamma function

z is less than one. So, it is convenient to write Equation (23) using the theory of

improper integrals as follows:

Γ(z) = lima→0+,b→∞

∫ b

a

tz−1e−tdt (26)

From the above definition, it is clear that the Gamma function involves both of

the cases of improper integrals discussed above, i.e., the interval is unbounded due

to the upper limit of integration and integrand becomes unbounded at the lower

limit of integration. We formalized Equation (26) as follows:

33

Definition 3.1: Gamma Function

` ∀ z. gamma z = lim(λn.

(lim(λb. integral( 12n, b) (λt. (t rpow (z-1))*exp(-t)))

The function rpow1 is a power function with real exponent. It takes two real

numbers x and y, and returns xy. We used limn→∞( 12n

) to model 0+ as ( 12n

) becomes

very close to 0 as n becomes very very large.

The lower and upper incomplete Gamma functions play a vital role in ob-

taining fractional integration and differentiation of periodic functions, such as,

sinusoidal response study of fractional operators [1], and can be formalized as

follows:

Definition 3.2: Upper Incomplete Gamma Function

` ∀ x z. gamma lower x z = (lim(λb.

integral(x,b) (λt. (t rpow (z-1))*exp(-t)))

Definition 3.3: Lower Incomplete Gamma Function

` ∀ s z. gamma upper s z = (lim(λn.

integral( 12n,s) (λt. (t rpow (z-1))*exp(-t)))

The availability of Definitions 3.1, 3.2 and 3.3 allows us to formally verify the most

widely used properties of the Gamma function in the next section.

3.2 Formal Verification of the Properties of Gamma Function

In this section, we define and prove some of the key properties of the Gamma

function, such as, pseudo recurrence relation, generalization of factorial, functional

1 This formalization, developed by the author, is available on hol-checkins as revision 8844, http://hol.svn.sourceforge.net/hol/?rev=8844&view=rev.

34

equation. The formal verification of these properties not only ensures the correct-

ness of our formal definitions but also paves the way to the formal reasoning about

physical systems involving the Gamma function in their analysis.

Theorem 3.1: Recurrence Relation

` ∀ z . (0 < z) =⇒ (gamma (z + 1)= z gamma (z))

Proof Sketch: We start the proof process by rewriting the left and right hand

sides using Definition 3.1. We used INTEGRAL BY PART theorem given in Table 2

(Chapter 2) along with some arithmetic reasoning to simplify the left-hand-side of

the above goal as follows:

lim(λn. lim(λb. (b + 1) rpow z ∗ (−1 ∗ exp(−1 ∗ (b + 1)))

− (1

2n) rpow z ∗ (−1 ∗ exp(−1 ∗ (

1

2n)))

+ x ∗ integral(1

2n, b)(λt.t rpow (z− 1) ∗ exp(−1 ∗ t))))

(27)

This step requires the proof of integrability of∫ b

12ntz−1e−tdt and

∫ b12ntze−tdt ,

which can be verified using the facts that integrand in each case is continuous on

limits of integration and continuity implies the integrability. Now, we first formally

verify the following two lemmas:

Lemma 3.1:

` ∀ z. (o < z) =⇒

(λb. ((b + 1) rpow z) ∗ (−1 ∗ exp(−1 ∗ (b + 1))) −→ 0)∧

(λn. (( 12n

) rpow z) ∗ (−1 ∗ exp(−1 ∗ ( 12n

))) −→ 0)

The verification of first part of Lemma 3.1 is based on the fact that both the terms

approach to zero as b becomes very large. Similarly, in case of second part of

35

Lemma 3.1, the first terms approaches to zero while the second term approaches

to 1 as n becomes very large. The verification of these two subgoals mainly depends

on the properties of limit of real sequences and availability of this Lemma allows

us to verify that the first two terms of Equation (27) approach 0 as n and b become

very large. This way we are left with the following subgoal:

lim(λn. lim(λb. z ∗ integral(1

2n, b)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t)))) =

z ∗ lim(λn. lim(λb. integral(1

2n, b)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t))))

(28)

The proof of this goal is a straightforward limit theory proof, given that we verify

the existence of the limits , i.e.,

Lemma 3.2: Existence with respect to Upper Limit

` ∀ z. (o < z) =⇒ ∃k.

(λb.integral( 12n, b)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t)))→ k)

Lemma 3.3: Existence with respect to Lower Limit

` ∀ z. (o < z) =⇒ ∃p.

(λn.integral( 12n, b)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t)))→ p)

We split the main integral of above Lemmas into sum of two integrals as follows:

integral(1

2n, b)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t)) =

integral(1

2n, 1)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t)) +

integral(1, b)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t))

(29)

36

We then rewrite the Lemmas 3.2 and 3.2 with the above result and arrive at the

following two subgoals:

∀z.(0 < z)⇒ ((∃k.(λb.integral(1

2n, 1)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t)))→ k)

∧ (∃m.(λb.integral(1, b)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t))) −→ m))

(30)

∀z.(0 < z)⇒ ((∃p.(λn.integral(1

2n, 1)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t)))→ p)

∧ (∃q.(λn.integral(1, b)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t))) −→ q))

(31)

The first part of Equation (30) and the second part of Equation (31) can be

verified based on the fact that ∀k. (λx. k)→ k. Both of the above real sequences

are monotonic and thus their convergence can be verified if we can prove their

boundedness. Lastly, we verified the upper and lower bounds of each integral as

follows:

(0 ≤∫ 1

12n

t rpow (z− 1) ∗ e−tdt ≤∫ 1

12n

t rpow (z− 1)dt)

(0 ≤∫ ∞1

t rpow (z− 1) ∗ e−tdt ≤∫ ∞1

n!

t rpow n + 1− zdt)

(32)

The verification process of upper and lower bounds requires two interesting lemmas

which are given as follows:

Lemma 3.4: Suminf lt Sum

` ∀ f n. (summable f) ∧ (∀m. n <= m =⇒ (0 < f(m)))

=⇒ sum (0, n + 1)f < suminf f

Lemma 3.5: Lower Bound of Exponential

` ∀ (t:real) (n:num). (0 < t) =⇒ (((t pow n)/(FACT n)) < exp(t))

37

Lemma 3.4 shows the lower bound of infinite summation suminf of a positive

function and its verification requires the properties of summations. Lemma 3.5

presents the lower bound of exponential function and we start its verification using

the definition of exponential function i.e.,∑∞

n=0xn

n!. The main verification steps

requires Lemma 3.5 and properties of summations along with some arithmetic

reasoning. This completes our verification of Theorem 3.1, Overall, it requires a lot

of rewriting efforts along with the formal reasoning related to integrals, derivatives

and limits.

Theorem 3.2: Functional Equation

` gamma 1 = 1

We start the proof process by rewriting the left-hand-side using the definition of

the Gamma function (Definition 3.1).

lim (λn. lim (λb.integral(1

2n, b + 1)(λt.t rpow (z− 1) ∗ exp(−1 ∗ t)))) = 1

(33)

Using Modus Ponens rule with Fundamental Theorem of calculus (FTC1) and

properties of differentiation, we can simplify the the proof goal of Equation (33)

as follows:

lim (λn. lim (λb. − exp (−(b + 1)) + exp (−(1

2n)) = 1 (34)

In the above goal first term approaches to 0 with respect to the inner limit and the

second term approaches to 1 with respect to the outer limit. We verified the above

goal based on the linearity properties of limits and some arithmetic reasoning.

38

Theorem 3.3: Generalization of Factorial

` ∀ n ∈ N . gamma (n + 1) = n!

The proof of Theorem 3.1.3 involves induction on the variable n. The base case

can be discharged by rewriting with the definition of Factorial and Theorem 3.1.2.

In the step case, we need to prove the following subgoal:

gamma (n + 1 + 1) = (n + 1)!

This can be simplified to (n + 1) gamma (n + 1) = (n + 1)n! using Theorem 3.1,

which in turn requires simple arithmetic reasoning to prove the goal.

Theorem 3.4: Reconstruction of the Gamma Function

` ∀ a z.(0 < z)∧ (0 < a) =⇒

(gamma z = gamma upper a z + gamma lower a z)

We start the verification by rewriting the left-hand-side with Definition 3.1.

lim(λn. lim(λb. integral(1

2n, b)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t))))

= gamma upper a z + gamma lower a z

(35)

Now the main step is to split the integral on the left-hand-side into the sum of two

integrals as follows:

integral(1

2n, b)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t)) =

integral(1

2n, a)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t)) +

integral(a, b)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t))

(36)

This subgoal can be verified using the Modus ponens rule with INTEGRAL COMBINE

property of the Gauge integral (Table 1). But we need to prove that 12n<= a, for

39

which we verify the property of a sequence as follows:

[∀m.(λn.fn)→ p⇔ (λn.f(n + m))→ p] (37)

Then by rewriting the left-hand-side of Equation (35) with the above property,

we need to verify [(0 < a)⇒ ( 12n+clg(ln a/ln(1/2)) <= a)], which can be proved using

the properties of logarithm and simple arithmetic reasoning. After simplifying the

subgoal of Equation (35) using Equation (36), we arrive at the following subgoal:

lim (λn. lim (λb. integral(1

2n, a)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t)) +

integral(1

2n, a)(λt. t rpow (z− 1) ∗ exp(−1 ∗ t)) =

gamma upper a z + gamma lower a z

The verification of this subgoal is based on the linearity properties of limits along

with the definitions of incomplete Gamma functions (Definition 3.2 and 3.3).

Theorem 3.5: Recurrence Relation of Upper Incomplete Gamma Function

(` ∀ s x. (0 < z)∧ (0 < s )=⇒ (gamma upper s z =

(z - 1)gamma upper s (z-1) + s rpow (z-1)exp(-s))

Theorem 3.6: Recurrence Relation of Lower Incomplete Gamma Function

` ∀ z x. (0 < z)∧ (0 < x)=⇒ gamma lower x (z + 1) =

(z)gamma lower x (z) - x rpow (z)exp(-x))

The verification steps for Theorem 3.1.5 and 3.1.6 are very similar to the ones

for Theorem 3.1.1. The major steps are to simplify the integral on the left hand

sides using integral by parts (Similar to Equation (27)) and the verification of

convergence of integrals.

40

3.3 Concluding Remarks

In this chapter, we presented the framework for the formalization of improper

integral in higher-order logic and building on the top of that we formally defined

the Gamma function and incomplete (Upper and Lower) Gamma functions in HOL

theorem prover. Next, we formally verified the most commonly used properties of

the Gamma function which are summarized in Table 3. The main challenges

in the reasoning process is to deal with improper integrals in higher-order logic.

Gamma function is useful in many domains, such as, probability theory (Gamma

Distribution), and our formalization can be directly utilized in such applications.

Our formalization of the Gamma function can be generalized to formalize other

improper integrals, such as, the Beta function.

Table 3. Properties of Gamma FunctionProperty HOL Formalization

Pseudo-Recurrence Relation` ∀ z.(0 < z) =⇒(gamma (z + 1)= z gamma (z))

Functional Equation` gamma 1 = 1

Factorial Generalization` ∀ n ∈ N. gamma(n + 1) = n!

Reconstruction of Gamma` ∀ x z.(0 < z)∧(0 < x) =⇒gamma z =

gamma upper x z + gamma lower x z)

Recurrence Lower Gamma` ∀ z x.(0 < z)∧(0 < x) =⇒gamma lower x (z + 1)=

(z)gamma lower x z -

x rpow (z)exp(-x)

Recurrence Upper Gamma` ∀ z x.(0 < z)∧(0 < s) =⇒gamma upper s z =

(z - 1)gamma upper s (z-1)+

s rpow (z-1)exp(-s)

Due to inherent soundness of higher-order logic theorem proving, our verifica-

tion results are exactly the same as produced by paper-and-pencil proof methods.

41

It is interesting to note that we have been able to identify a couple of critical as-

sumptions that are missed by almost all the paper-and-pencil based proof analysis,

that we came across. For example, the assumption 0 < x in the last three prop-

erties of the Gamma function (Theorem 3.4, 3.5 and 3.6) have not been specified

in anyone of the paper-and-pencil proof based analysis (e.g., [1]). Obviously the

results do not hold without this assumption and this discrepancy in the paper-and-

pencil based proofs may lead to disastrous consequences if these properties are used

without considering 0 < x for designing safety-critical fractional order systems.

It is clear from Chapter 1 that all the definitions of fractional calculus heavily

depends on infamous Gamma function. The availability of formal definition of the

Gamma function along with formally verified properties provides the basis for the

formalization and verification of the properties of fractional calculus in the next

chapter.

42

CHAPTER 4

HOL Formalization of Fractional Calculus

The second major requirement of formal reasoning about fractional order sys-

tems, is the formalization of Differintegrals, as depicted in Figure 1. This chapter

presents the higher-order logic formalization of the fractional calculus based on the

Riemann-Liouville definition. The first section presents the formal definitions of

fractional integration and fractional differentiation. The second section presents

the formal verification of the classical properties of fractional calculus.

4.1 Formalization of Differintegrals

There are more than ten well-known definitions of Differintegrals as mentioned

in Chapter 1 and two most commonly used definitions are Riemann-Liouville and

Grunwald-Letnikov . The Riemann-Liouville definition provides a way to find ana-

lytical solutions while Grunwald-Letnikov definition facilitates the numerical com-

putation of solutions. There are two motivations of using the Riemann-Liouville

definition for our formalization: Firstly, it is widely used in the modeling and anal-

ysis of engineering fractional order systems [1], Secondly, the analysis carried out

in this way is purely analytical and hence free from any kind of approximations.

On the other hand, Grunwald-Letnikov definition is more suitable for numerical

analysis based methods and thus provides approximate solutions.

According to Riemann-Liouville approach the definitions of fractional integra-

tion and fractional differentiation are given as follows:

43

Jvaf(x) =1

Γ(v)

∫ x

a

(x− t)v−1f(t)dt (38)

Dvf(x) = (d

dx)mJm−va f(x) (39)

We utilize Equations (38) and (39) to formally define fractional integration and

differentiation, respectively.

Definition 4.1: Fractional Integration

` ∀ f v a x.frac int f v a x =

if (v = 0) then f else

lim(λn. 1gamma v

∗

(integral(a,x - 12n) ((x - t) rpow (v-1)) f(t) dt))

Definition 4.2: Fractional Differentiation

` ∀ f v a x. frac diff f v a x =

n order deriv

(clg v) (frac int f (clg v - v) a x)

Where f is a function of type (real→ real), v is a real number that indicates the

order of integration/differentitiation, and a and x represent the lower and upper

limits of integration, respectively. The function n order deriv returns the nth

integer order derivative of its argument f as dnfdxn

. The function clg is the ceiling

function, which returns the least greater integer of its real number argument. It

is important to note that we have explicitly defined the case for v = 0, which

is justified based on integer order calculus and proves to be very convenient for

further manipulations [72].

44

4.2 Formal Verification of the Properties of Differintegrals

In this section, we define and prove some of the key properties of the fractional

integration and fractional differentiation in higher-order logic using Definitions 4.1

and 4.2. The formal verification of these properties not only ensures the correctness

of our formal definitions but also helps in formal reasoning about physical systems

exhibiting fractional order dynamics.

Theorem 4.1: Identity

` ∀ f a x. (a < x) =⇒

frac int f v a x = (frac int f 0 a x = f) ∧

(frac diff f 0 a x = f)

The assumption (0 < x) ensures that the upper limit of integration must be greater

than the lower limit. The verification of this theorem requires rewriting with the

definitions of fractional integration and fractional differentiation and using the facts

that ceiling of 0, i.e., clg 0 is always 0 and the identity of nth order derivative.

Theorem 4.2: Generalized Integral

` ∀ f a x v ∈ N.

(a < x)∧ (1 < v) =⇒

frac int f v a x = lim( λ n.

1(v−1)! integral (a,x - 1

2n) ( (x - t) rpow (v-1)f(t) dt))

Theorem 4.2 shows that fractional integration generalizes the integer order inte-

gration. The first assumption ensures that the upper limit of integration must be

greater than the lower limit. The second assumption ensures that the order of

45

fractional integration must be greater than 1. The verification of above theorem

heavily relies on the availability of the properties of the Gamma function which

we have presented in the previous chapter. We start the verification by rewriting

left-hand-side (RHS) and right-hand-side (LHS) with the definition of fractional

integration (Definition 4.1) which results in the following goal:

(1

gamma(v)∗ lim(λn.integral (a, x− 1

2n)(λt.(x− t) rpow (v− 1) ∗ f t )) =

(1

(v− 1)!∗ lim(λn.integral (a, x− 1

2n)(λt.(x− t) rpow (v− 1) ∗ f t ))

(40)

The verification of above goal require that gamma(v) = (v − 1)! which is an al-

ternative form of the property Factorial Generalization which we have already

verified in the previous chapter. This completes the verification of Theorem 4.2.

Theorem 4.3: frac int Linearity

` ∀ f v x a b.

(frac exists f x v)∧ (frac exists g x v) =⇒

frac int (a* f + b* g) v 0 x =

a*(frac int f v 0 x)+ b*(frac int g v 0 x)

Theorem 4.3 shows the linearity of fractional integration which is considered as one

of the most important property as it facilitates the formal reasoning about frac-

tional order systems with more than one impute sources. The first two assumptions

ensure the existence of fractional integration of functions f and g.

Proof Sketch: We start the proof process by rewriting the left and right hand

sides using the definition of fractional integration (Definition 4.1). We used dis-

46

tributivity of addition to reach the following subgoal:

1

gamma(v)∗ lim(λn.integral (a, x− 1

2n)(λt.(x− t) rpow (v− 1) ∗ (a ∗ f t)

+ (x− t)rpow(v− 1) ∗ (b ∗ g t)) =

a ∗ 1

gamma(v)∗ lim(λn.integral (a, x− 1

2n)(λt.(x− t) rpow (v− 1) ∗ f t)+

b ∗ 1

gamma(v)∗ lim(λn.integral (a, x− 1

2n)(λt.(x− t) rpow (v− 1) ∗ g t)+

(41)

Next, we split the integral on the left-hand-side of Equation (41) into the sum of

two integrals using integral add property of the Gauge integral listed in Table 2.

It requires the proof of the integrability of fractional integral of functions f and g

along with the arithmetic reasoning. We verified the integrability using the facts

that the functions f and g are continuous in the limits of integration which leads to

the continuity of the integrand and then continuity implies the integrability. Now,

we have to verify the following subgoal which conclude the proof of Theorem 4.3:

1

gamma(v)∗ lim(λn.integral (a, x− 1

2n)(λt.(x− t)rpow(v− 1) ∗ (a ∗ f t)

+ integral (a, x− 1

2n)(λt.(x− t) rpow (v− 1) ∗ (b ∗ g t))) =

a ∗ 1

gamma(v)∗ lim(λn.integral (a, x− 1

2n)(λt.(x− t) rpow (v− 1) ∗ f t)+

b ∗ 1

gamma(v)∗ lim(λn.integral (a, x− 1

2n)(λt.(x− t) rpow (v− 1) ∗ g t)

(42)

It is a straight forward limit theory proof and verification process involves prop-

erties of limit of real sequences, along with the linearity property of the Gauge

integral (Table 2).

47

Theorem 4.4: frac diff Linearity

` ∀ f v x a b.

(frac exists f x v)∧ (frac exists g x v)∧

(∀ m. (m <= clg v) ⇒

(n order deriv m (frac int f v 0 x)) differentiable x)∧

(∀ m. (m <= clg v) ⇒

(n order deriv m (frac int g v 0 x)) differentiable x)

=⇒ frac diff (a* f + b* g) v 0 x =

a*(frac diff f v 0 x)+ b*(frac diff g v 0 x))

Theorem 4.3 shows the linearity of fractional differentiation. The first two assump-

tions ensure the existence of fractional integration of functions f and g. From

Definition 4.2, it is clear that fractional differentiation involves fractional integra-

tion followed by the nth order ordinary differentiation. So, the third and fourth

assumptions of this property ensures the differentiability of (clg(v) − v)th order

fractional integral of the functions f(t) and g(t), respectively.

Proof Sketch: We start the verification by rewriting the left and right hand sides

using the definition of fractional differentiation (Definition 4.2). Then, we used the

linearity of fractional integration, verified in Theorem 4.3, to reach the following

48

subgoal:

n order deriv (clgv) (λx.

a ∗ (λx. frac int f (clgv− v) 0 x) x +

b ∗ (λx. frac intg (clgv− v) 0 x) x) x =

a ∗ n order deriv (clgv) (λx. frac int f (clgv− v) 0 x) x +

b ∗ n order deriv (clgv) (λx. frac int g (clgv− v) 0 x) x

(43)

The verification of this subgoal requires the proof of the linearity of integer order

derivatives given in the following lemmas:

Lemma 4.1: Deriv Linearity

∀ m f g x. (f differentiable x) ∧ (g differentiable x) =⇒

(deriv (λx.a ∗ f x + b ∗ g x) x =

a ∗ deriv (λx.f x)x + b ∗ deriv(λx.g x) x)

Lemma 4.2: n Order Deriv Linearity

∀ m f g x.(∀m x. m <= n ⇒

(((λ x. n order deriv m f x) differentiable x) ∧

((λ x. n order deriv m f x) differentiable x)) =⇒

(n order deriv n (λx. a ∗ f x + b ∗ g x) x =

a ∗ n order deriv n(λx. f x) x + b ∗ n order deriv n (λx. g x)x)

We verified Lemma 4.1 using the properties of integer order derivative along with

arithmetic reasoning. Next, the verification of Lemma 4.2 has been done using

induction on m along with the properties of derivatives [56] and Lemma 4.1. Now,

rewriting the left-hand-side of Equation (43) with Lemma 4.2 concludes the proof

49

of Theorem 4.4. This completes our formalization of the properties of fractional

Differintegrals which to the best of our knowledge is first one in higher-order logic.

4.3 Concluding Remarks

In this chapter, we provided the details of the formalization of Differintegrals us-

ing the well-known Riemann-Liouville definition along with the formal verification

of their classical properties (listed in Table 4). The formalization, presented in

previous chapter and this chapter had to be done in an interactive way due to the

undecidable nature of higher-order logic.

Table 4. Properties of DifferintegralsProperty HOL Formalization

Identity` ∀ f a x.

(a < x) =⇒ (frac int f 0 a x = f)∧(frac diff f 0 a x = f)

Generalized Integral

` ∀ f a x v ∈ N.

(a < x)∧ (1 < v) =⇒frac int f v a x = lim(λn.

1(v-1)!

∫ x− 12n

a(x - t) rpow (v-1)f(t) dt)

frac int Linearity

` ∀ f v x a b.

(frac exists f x v)∧(frac exists g x v) =⇒frac int (a f + b g) v 0 x =

a(frac int f v 0 x)+

b(frac int g v 0 x)

frac diff Linearity

` ∀ f v x a b.

(frac exists f x v)∧(frac exists g x v)∧(∀ m. (m <= clg v) ⇒(n order deriv m (frac int f v 0 x))

differentiable x)∧(∀ m. (m <= clg v) ⇒(n order deriv m (frac int g v 0 x))

differentiable x)=⇒( frac diff (a f + b g) v 0 x =

a(frac diff f v 0 x)+

b(frac diff g v 0 x))

50

The complete formalization took around 7500 lines of HOL code and approximately

600 man hours. However, the main advantage of this rigorous exercise is that

our results can be built upon to facilitate formal reasoning about fractional order

systems. Our proof script is available for download [73] and thus can be utilized by

other researchers to conduct the formal analysis of their fractional order systems.

It is to note that the formalization of the Gamma function, presented in Chapter 3,

provided the basis for the formalization of Differintegrals, similarly, formalization

presented in this chapter facilitates the formal analysis of real-world fractional

order systems in the next chapter.

51

CHAPTER 5

Applications

In order to illustrate the utilization and effectiveness of the proposed framework,

we apply it to analyze some real-world fractional order systems. First application

presents the formalization of Euler’s generalized rule of fractional differentiation

which is very useful for computing fractional order derivative of a function in

purely analytical manners. Next, we prsent the formal analysis of three real-world

fractional order systems, i.e., a fractional electrical component Resistoductance, a

fractional integrator and a differentiator circuit. Resistoductance is used to extend

the current-voltage relationship to non-integer order and this kind of fractional

order model is usually used for modeling bio-electrodes for cardiac tissue interfacing

[27]. Fractional integrators and differentiators are the most basic components in

fractional order PID (proportional integrator differentiator) controllers and can

achieve more robustness than integer order control [74]. These systems have been

chosen as case studies in our work because of their wide usability in the field of

circuit theory and control systems. To the best of our knowledge, currently, there

is no formal technique available for the formal verification of such systems.

5.1 Euler’s Generalized Rule

Different formalisms have been proposed for computing non integer order deriva-

tives and integrals, namely, Riemann-Liouville and Grunwald-Letnikov [1]. One

of the most early definition of fractional derivative of a function, which can be

52

expressed as (f(x) = axy), was proposed by Euler in 1730 [27].

dnxm

dxn=

Γ(m+ 1)

Γ(m− n+ 1)xm−n n ∈ R+ (44)

Euler’s definition is widely used for calculating fractional derivatives because

it is simpler than other existing definitions of fractional derivatives. For example,

it can be directly utilized to reason about the fractional derivative of parabolic

step-type transition (used in edge detection [9]).

Our formalization of the Gamma function, given in Chapter 3, allows us to

formalize Euler’s derivative as follows:

Definition 3.2.1: Euler’s Derivative

` ∀ v x m. Euler deriv v x m= gamma (m + 1)

gamma (m - v + 1)x rpow (m - v)

Theorem 3.2.1: Relation of Euler’s Derivative with Integer Order Derivative

` ∀ v∈ N x m. (v < m) =⇒

Euler deriv v x m = m!(m - v)!

x rpow (m - v)

Theorem 3.2.2: Identity of Euler’s Derivative

` ∀ x m. Euler deriv 0 x m = x rpow m

We start the verification of Theorem 3.2.2 by rewriting the left-hand-side with the

Definition 3.2.1 (Euler’s Derivative) which results in the following subgoal:

gamma(m + 1)

gamma(m− v + 1)x rpow (m− v) =

m!

(m - v)!x rpow (m− v) (45)

The above goal can be verified using the fact that gamma(z + 1) = z! (Theorem

3.1.2). The verification of Theorem 3.2.2 utilizes Theorem 3.1.1, 3.1.2 and 3.1.3,

53

verified in the last section. Overall, the verification of these theorems took a few

lines of HOL code, which demonstrates the effectiveness of our formalization of the

Gamma function.

5.2 Resistoductance

Electrical components, such as, resistors, inductors and capacitors are largely used

to perform integer order calculus operations for different engineering and scientific

applications. However, actual electrical components do not posses ideal behavior

and exhibit some fractional order characteristics. Ignoring these characteristics

always results in modeling inaccuracies. Therefore, fractional calculus is being

widely used to capture real world dynamics of electrical components these days

[75]. Resistoductance is a linear electrical circuit element that posses the char-

acteristics between an ohmic resistor and an inductor. Being a fractional order

electrical component, it exhibits fractional order dynamics, which can be modeled

by Differintegrals. The model of a single Resistoductance is shown in Figure 2,

and its governing voltage and current relationship is given as follows:

i(t) =1

KJαv(t) (46)

where v(t) is the voltage and i(t) is the current through the circuit element at

time t. The range of the α is between 0 and 1. If α = 0 the circuit will be purely

resistive with K = R ohms and if α = 1 the circuit will be purely inductive with

K = L henrys.

The two important characteristics of Resistoductance are the output current

through the circuit element when constant input voltage V0 is applied and the

54

behavior of the output current for the cases when α = 0 and α = 1. These

two properties are widely used in designing Resistoductance based fractional order

systems for signal processing and control engineering applications [75].

v(t)= K D i(t) α i(t)= J v(t) α1Κ

+

−

Figure 2. Resistoductance

Now, we will present the formal verification of the above mentioned two prop-

erties of Resistoductance using our proposed framework given in Figure 1 (Chap-

ter 1). The first step in conducting the formal analysis of Resistoductance is to

construct its formal model in higher-order logic. Due to the availability formal

definition of the Gamma function (Definition 3.1, Chapter 3), the formalization of

Resistoductance can be simply done as follows:

Definition 5.1: Current through Resistoductance

` ∀ K v i alpha x. i t v 0 K v i alpha x =

(1/K)frac int v i(t) alpha 0 x

where v i is input voltage, i t is current through the circuit element, alpha is the

order of integration, and the variable x represents the upper limit of integration.

In the above definition the lower limit of integration is taken as 0 [1]. The next

step, in the proposed framework, is to utilize the formal model of Resistoductance

55

(Definition 5.1) to express the properties of interest as higher-order logic theorems

as follows:

Theorem 5.1: i t for constant voltage V 0

` ∀ K v 0 alpha x.

(0 < x) ∧ (0 < alpha) =⇒ (i t K V 0 alpha x =

(1/(K Gamma (alpha + 1))) (V 0(x rpow alpha)))

Theorem 5.2: Special Cases for i t

` ∀ x. (0 < x) =⇒

if (alpha = 0) then i t K V 0 alpha x = V 0 / K

else if (alpha = 1) then i t K V 0 alpha x = (V 0 / K ) x

Theorem 5.1 shows the relationship of the output current of Resistoductance when

constant input voltage V 0 is applied at t = 0. The formal verification of this

theorem is based on the properties of Gamma function (e.g., Pseudo-Recurrence

relation) and the definition of fractional integration. Since, these required prop-

erties have already been verified in HOL library, the interactive formal reasoning

process only consists of verifying the continuity of fractional integral. Theorem 5.2

shows an interesting feature of Resistoductance, i.e., for (alpha = 0) it behaves

as a pure resistor and for (alpha = 1) it exhibits the behavior of a pure inductor.

The verification of Theorem 2 requires Theorem 1, the properties of the real power

(rpow) function and some arithmetic reasoning.

This verification of Theorems 5.1 and 5.2 consumed approximately 350 lines

of HOL code and about two man hours and thus was very short compared to

56

the challenging verification of the theorems presented in the Chapters 3 and 4.

The verification process, besides being compact, was also very straightforward and

involved reasoning based on real analysis theories only and thus can be done with

some basic know how of higher-order-logic theorem proving. The main reason

for the above mentioned benefits is clearly the availability of formalized Gamma

function and the Differintegrals.

5.3 Fractional Differentiator and Integrator Circuits

Proportional integrator (PI) and proportional integrator differentiator (PID) con-

trollers are widely used in the industry. Therefore, numerous reliable and high

performance controllers have been designed and deployed. In recent years, it has

been observed that Fractional order (FO) controllers offer more flexibility in the

adjustment of gain and phase characteristics than integer order controllers. Due

to these flexibilities, there is a growing interest in using fractional order controllers

in industry and academia [74]. The most fundamental components of PI and PID

controllers are integrator and differentiator circuits, respectively. In this section,

we will present the formal analysis of a fractional integrator and a differentiator

circuit [1], shown in Figure 3. The output voltage-current equations for a fractional

integrator and a differentiator circuits are given as follows:

vo(t) = − 1

RCJµvi(t) Integrator (47)

v0(t) = −RCDµvi(t) Differentiator (48)

57

where R and C denotes resistance and capacitance, respectively, and their values

are used to define the reset rate of PID controllers. The variables, vo(t) and vi(t),

in the above Equations represent output and input voltages at time t, respectively.

(a)

(b)

Figure 3. (a) Integrator (b) Differentiator

The output response of integrator and differentiator circuit is usually analyzed

for benchmark input signals, such as, the unit step, which is defined as follows:

u(t) =

0 if t ≤ 0;

1 if t > 1;

The first step in the formal analysis of integrator and differentiator circuits,

when unit step signal is applied at the input, is to construct the formal model

of these circuits and unit step signal in higher-order logic. Since the governing

equations (Equations (47), (48)) of integrator and differentiator circuits involve

58

fractional integration and differentiation, thus we utilize our formalized definitions

(Definition 3.3.1, 3.3.2) of Differintegrals as follows:

Definition 5.2: Fractional Order Integrator

` ∀ R C v i mu x. v I 0 R C v i mu x =

-(1/RC)frac int v i(t) mu 0 x

Definition 5.2: Fractional Order Differentiator

` ∀ R C v i mu x. v D 0 R C v i mu x=

-(RC)frac diff v i(t) mu 0 x

Definition 5.3: Unit Step

` ∀ t. unit t = if (0 <= t) then 1 else 0

where v I 0 and v D 0 are output voltages of integrator and differentiator circuits,

respectively. v i is the input voltage, R, C, mu and x represent resistance, capaci-

tance, order of integration/differention and upper limit of integration, respectively.

Now, the next step in the formal analysis of fractional integrator and dif-

ferentiator, as mentioned in Fig 1, is to describe their properties of interest as

higher-order logic theorems. The following two theorems describe the input-output

relationship of these circuits.

Theorem 5.3: Output of Fractional Integrator Circuit

` ∀ R C mu x .

(0 < x) ∧ (0 < mu) ∧ ( mu < 1)=⇒ (v I 0 R C (unit t) mu x =

(-1/(RC Gamma (mu + 1)) ((x rpow ( mu))))

59

Theorem 5.4: Output of Fractional Differentiator Circuit

` ∀ R C mu x .

(0 < x) ∧ (0 < mu) ∧ ( mu < 1)=⇒ (v D 0 R C (unit t) mu x =

((-1/(RC (Gamma ((1 - mu)))) ((x rpow (- mu))))

The next step in the theorem proving based fractional order system analysis is the

verification of above mentioned theorems using the already verified properties and

lemmas presented in Chapters 3 and 4. Theorem 5.3 gives the output response

of a fractional integrator circuit for unit step signal, and its formal verification

certifies the output response under the given conditions. The availability of already

verified properties of Gamma function and Differintegrals (Chapter 4) led us to

the simple subgoal, i.e., the proof of continuity of the integrand, which involves

multiplication of power function and the unit step signal. We verified the continuity

by differentiability using the classical definition of derivative formalized in HOL

[56].

Theorem 5.4 describes the output response of the fractional differentiator cir-

cuit with unit step signal as an input. The second and third assumptions in

Theorem 5.4 ensure that the order of the fractional differentiation mu is between 0

and 1 which means that clg of mu will always be 1. So the verification of this the-

orem requires fractional integration of the order 1− mu followed by the fractional

differentiation of order 1. This requires Theorem 5.3 along with some arithmetic

reasoning. Just like the case of the Resistoductance, the verification of Theorem

5.3 and Theorem 5.4 was very straightforward and took about 400 lines of HOL

code and about 2.5 man hours.

60

5.4 Concluding Remarks

In this chapter, we presented the real-world applications of our proposed frame-

work. In the first application, we formalized the Euler’s generalized rule of differen-

tiation along with the formal verification of its corresponding properties. Next, we

presented the formal analysis of a fractional electrical component Resistoductance,

fractional differentiator and fractional integrator circuits. The above case studies

clearly demonstrate the effectiveness of the proposed theorem proving based frac-

tional order system analysis technique. Due to the formal nature of the model

and inherent soundness of higher-order logic theorem proving, we have been able

to verify the properties of given fractional order systems with 100% accuracy; a

feature that, to the best of our knowledge, is not available in any other com-

puter based analysis technique. This additional benefit comes at the cost of the

time and effort spent, while formalizing the Differintegrals and formally reasoning

about their properties. But, the availability of such a formalized infrastructure

significantly reduces the time required to analyze fractional order systems, as the

verification task of the properties of Resistoductance and a fractional integrator

and differentiator circuits took just a couple of man hours each.

Our formal analysis of above mentioned case studies is fairly general and can

be extended to other safety-critical fractional order systems. For example, our

analysis can be utilized to conduct formal analysis of a Resistocapcitance (having

characteristics between pure resistor and capacitor), fractional differentiator and

integrator based control systems, semi-infinite lossless transmission lines, the frac-

tional dynamics of a battery and different configurations of operational amplifier

61

with lumped elements [1]. Another, interesting application of current work can

be the formalization of viscoelastic stress-strain relationship which is known to be

best modeled with fractional calculus and its application domain is ranging from

material and mechanical modeling to the modeling of neurons [1].

62

CHAPTER 6

Conclusions and Future Work

6.1 Conclusions

In this thesis, we have presented a novel application of formal methods in the area

of analyzing fractional order systems. In particular, we have developed a framework

for accurate and reliable analysis of fractional order systems within the sound core

of the HOL theorem prover. Due to the formal nature of our models, the analysis

conducted by this framework will be free from errors, such as, discretization error,

round-off error and precision error. This approach can thus be of great benefit for

the analysis of fractional order systems used in safety-critical domains, such as,

medicine and transportation.

In the development of formal framework for fractional order system analysis,

the contributions of the thesis can be divided into two parts. Firstly, it presents

the framework for the formalization of improper integrals in higher-order logic by

formalizing of infamous Gamma function and verifying its classical properties us-

ing the HOL theorem prover. The formalization of the Gamma function is quite

general and can be applied for the formalization of other improper integrals and

special functions. Secondly, it provides the complete formalization details of Dif-

ferintegrals along with the formal verification of their important properties. The

availability of the Gamma function and Differintegrals allowed us to conduct the

formal analysis of Resistoductance, a fractional differentiator and a fractional in-

tegrator circuit. To the best of our knowledge, this is the first time that a formal

63

method technique has been used to conduct the analysis of fractional order sys-

tems. However, the verification using the proposed framework required significant

user interaction because the user needs to guide the theorem prover. Because of

above mentioned limitation of the proposed approach, it should be taken as com-

plementary technique with traditional approaches, which can be of great benefit

for the verification of safety-critical parts of fractional order systems.

6.2 Future Work

The reported formalization opens the doors to many interesting and novel direc-

tions of research. Building on the top of our results, more important features can

be added to increase the strength of the theorem proving based analysis of frac-

tional order systems. Some of the worth mentioning extensions are outlined as

follows:

• Enriching the library of the formally verified properties of Differentagrals

with law of exponents and relationship with Beta function to broaden the

scope of formal fractional order system analysis

• The reported formalization can be utilized to formalize the fractional laplace

transform theory, which in turn can be utilized for the the formal analysis of

industrial fractional order control systems

• Our formalization was done using real numbers and the same formalization

can also be extended to cover the complex numbers using the higher-order-

logic formalization of complex number theory [61], which would allow us to

formalize fractional electromagnetic systems, such as, fractional rectangular

64

waveguides [10].

• The formalization of the other definitions of Differintegrals, for example,

Caputo derivatives [27] which is best suited in the situations when we need

to include initial conditions in fractional order system analysis. This kind

of formalization can be built upon the existing HOL formalization of the

Lebesgue integration theory [60], which is fundamental to this extension.

65

BIBLIOGRAPHY

[1] S. Das, Functional Fractional Calculus for System Identification and Controls,1st ed., 2007.

[2] K. S. Miller and B. Ross, An Introduction to Fractional Calculus and Frac-tional Differential Equations. John Willey and Sons,Inc, 1993.

[3] T. T. Hartley and C. F. Lorenzo, “Fractional System Identification: An Ap-proach Using Continuous Order Distributions,” National Aeronautics andSpace Administration Glenn Research Cente NASA TM, Tech. Rep., 1999,.

[4] Y. Q. Chen, D. Xue, and H. Dou, “Fractional Calculus and Biomimetic Con-trol,” in Robotics and Biomimetics, 2004. ROBIO 2004. IEEE, 2004, pp. 901–906.

[5] G. Maione and P. Lino, “New Tuning Rules for Fractional piα Controllers,”Nonlinear Dynamics, vol. 49, 2007.

[6] F. Y. D. J. Zhuang and Y. Lin, “Evaluation of a Vehicle Directional Controlwith a Fractional Order pdλ Controller,” International Journal of AutomotiveTechnology, vol. 9, no. 6, pp. 679–685, 2008.

[7] B. T. Krishna and K. V. V. S. Reddy, “Design of Digital Differentiators andIntegrators of Order 1

2,” World Journal of Modelling and Simulation, UK,

vol. 4, pp. 182–187, 2008.

[8] W. A. K. Assaleh, “Modeling of Speech Signals Using Fractional Calculus,”9th International Symposium on Signal Processing and Its Applications,, pp.1–4, 2007.

[9] B. M. P. M. A. Oustaloup and C. Ceyral, “Fractional Differentiation for EdgeDetection,” Signal Processing, vol. 83, pp. 2421–2432, 2003.

[10] M. Faryad and Q. A. Naqvi, “Fractional Rectangular Waveguide,” ProgressIn Electromagnetics Research, PIER, vol. 75, pp. 383–396, 2007.

[11] N. P. A. Kiani-B, K. Fallahi and H. Leung, “A Chaotic Secure CommunicationScheme Using Fractional Chaotic Systems Based on an Extended FractionalKalman Filter,” Communications in Nonlinear Science and Numerical Simu-lation, vol. 14, pp. 863–879, 2009.

[12] Maplesoft, http://www.maplesoft.com/, 2011.

[13] A. Gupta, “Formal Hardware Verification Methods: A Survey.” Formal Meth-ods in System Design.

66

[14] E. M. C. Jr., O. Grumberg, and D. A. Peled, Model Checking. The MITPress, 1999.

[15] M. J. C. Gordon and T. F. Melham, Eds., Introduction to HOL: A TheoremProving Environment for Higher-Order Logic. Cambridge University Press,1993.

[16] G. W. Leibnitz, “Leibnitzens Mathematische Schriften,” SIGDA Newsl.,vol. 2, pp. 301–302, 1962.

[17] G. W. Leibniz, “Letter from Hanover, Germany, to Johann Bernoulli,,” Math-ematische Schriften; Reprinted 1962,, 1695.

[18] G. W. Leibniz, “Letter from Hanover, Germany, to John Wallis,,” Mathema-tische Schriften; Reprinted 1962,, 1697.

[19] S. F. Lacroix, Traite du Calcul Differentiel et du Calcul Integral. Courtier,Paris, 1819.

[20] J. B. J. Fourier, “Theorie Analytique de la Chaleur,” Oeuvres de Fourier,vol. 1, 1822.

[21] N. H. Abel, “Solution de Quelques Problemes a l’aide D’integrales Definies,Oeuvres Completes,” vol. 1, pp. 16–18, 1881.

[22] G. Peacock, “Report on the Recent Progress and Present State of Affairs ofCertain Branches of Analysis,” British Association for the Advancemnet ofScince, Tech. Rep., 1833.

[23] A. De Morgan, “The Differential and Integral Calculus Differential Equations,Differences, Summation, Calculus of Variations with Applications to Algebra,Plane abs Solid Geometry,” 1840.

[24] W. Center, “On the Value of (d/dx)θx0 When θ is a Positive Proper Fraction,”Cambridge Dublin Math., 1848.

[25] B. Riemann, “Versuch Einer Allgemeinen Auffassung der Integration und Dif-ferentiation,” Gesammelte Werke, 1892.

[26] K. B. Oldham and J. Spanier, The Fractional Calculus. New York, AcademicPress, 1974.

[27] M. Dalir and M. Bashour, “Application of Fractional Calculus,” Applicationsof Fractional Calculus in Physics, vol. 4, no. 21, p. 12, 2010.

[28] I. Podlubny, “Fractional Differential Equations, Academic Press,” 1999.

67

[29] B. Ross, “A Brief History And Exposition of The Fundamental Theory ofFractional Calculus,” in Fractional Calculus and Its Applications, ser. LectureNotes in Mathematics. Springer, 1975, vol. 457, pp. 1–36.

[30] I. Petras, “Fractional Order Feedback Control of a DC Motor,” Journal ofElectrical Engineering, vol. 60, no. 3, 2009.

[31] F. B. M. Duarte and J. A. T. Machado, “Pseudoinverse Trajectory Controlof Redundant Manipulators: A Fractional Calculus Perspective,” in ICRA,2002, pp. 2406–2411.

[32] I. Petras and B. Vinagre, “Practical Application of Digital Fractional-OrderController to Temperature Control,” Acta Montanistica Slovaca, vol. 7, no. 2,pp. 131–137, 2002.

[33] I. P. B. M. Vinagre, I. Petras and Y. Q. Chen, “Using Fractional OrderAdjustment Rules and Fractional Order Reference Models in Model-ReferenceAdaptive Control,” Nonlinear Dynamics, vol. 49, 2002.

[34] C. R.-S. X. Moreau and A. Outstlaup, “Fractional Differentiation in PassiveVibration Control,” Nonlinear Dynamics, vol. 29, 2002.

[35] C. Tseng, “Design of Fractional Order Digital FIR Differentiators,” IEEESignal Processing Letters,, vol. 8, no. 3.

[36] B. M. V. Y. Chena, “Fractional Differentiation for Edge Detection,” SignalProcessing, vol. 83, pp. 2359–2365, 2003.

[37] R. Y. Chen and A. Zhou, “An overview of Fractional Order Signal Processing(fosp)Techniques,” Proceedings of the ASME 2007 International Design Engi-neering Technical Conferences and Computers and Information in EngineeringConference IDETC/CIE 2007. ASME, September 2007.

[38] E. C. C. Quintanoa, “Improving Satellite Image Classification by Using Frac-tional Type Convolution Filtering,” International Journal of Applied EarthObservation and Geoinformation, vol. 12, pp. 298–301, 2010.

[39] P. M. A.C. Sparavigna, “Using Fractional Differentiation in Astronomy,” In-strumentation and Methods for Astrophysics.

[40] N. Engheta, “Fractional Curl Operator in Electromagnetics,” Microwave Opt.Tech. Lett., vol. 17, pp. 86–91, 1998.

[41] Q. A. Naqvi and M. Abbas, “Complex and Higher Order Fractional CurlOperator in Electromagnetics,” Optics Communications, vol. 241, pp. 349–355, 2004.

68

[42] N. Engheta, “On the Role of Fractional Calculus in Electromagnetic Theory,”Antenna and Propagation Magzine, vol. 39, no. 4, pp. 35–46, 1997.

[43] V. Zaborovsky and R. Meylanov, “Informational Network Traffic Model Basedon Fractional Calculus,” Proceedings of the International Conference Info-techand Info-net, ICII 2001, Beijing, China,, vol. 1, pp. 58–63, 2001.

[44] G. Cottone and M. D. Paola, “On the Use of Fractional Calculus for theProbabilistic Characterization of Random Variables,” Probabilistic Engineer-ing Mechanics, vol. 24, pp. 321–330, 2009.

[45] T. J. Auastasio, “The Fractional-Order Dynamics of Brainstem Vestibulo-Oculomotor Neurons,” Biological cybemetics, 1994.

[46] M. M. Darıo Manuel Domınguez and M. Camacho, “Macrophage Ion Currentsare Fit by a Fractional Model and Therefore are a Time Series with Memory,”Eur Biophys J., vol. 38, pp. 457–464, 2009.

[47] R. L. Magin, “Fractional Calculus Models of Complex Dynamics in BiologicalTissues,” Computers and Mathematics with Applications, vol. 59, pp. 1586–1593, 2010.

[48] Y. Ding and H. Ye, “A Fractional-Order Differential Equation Model of HIVInfection of cd4+ T-cells,” Computers and Mathematics with Applications,vol. 50, pp. 386–392, 2009.

[49] Q. C. . T. J. Cui and C. Zhang, “Waves in Planar Waveguide ContainingChiral Nihility Metamaterial,” Optics and Communication, vol. 274, pp. 317–321, 2007.

[50] A. Naqvi, “Comments on Waves in Planar Waveguide Containing Chiral Ni-hility Metamaterial,” Optics and Communication, vol. 284, pp. 215–216, 2011.

[51] I. Petras, “Chaos in the Fractional-Order Volta’s System: Modeling and Sim-ulation,” Nonlinear Dynamics, vol. 57, pp. 157–170, 2009.

[52] M. D. G. Marcos, J. A. Tenreiro Machado, and T. P. Azevedo-Perdicoulis,“A Fractional Approach for the Motion Planning of Redundant and Hyper-Redundant Manipulators,” Signal Process., vol. 91, pp. 562–570, March 2011.

[53] K. Diethelm, N. J. Ford, A. D. Freed, and Y. Luchko, “Algorithms for theFractional Calculus: A Selection of Numerical Methods,” Computer Methodsin Applied Mechanics and Engineering, vol. 194, no. 6-8, pp. 743 –773, 2005.

[54] MATLAB, http://www.mathworks.com/products/matlab/, 2011.

[55] G. Baumann, “Fractional Calculus and Symbolic Solution of Fractional Dif-ferential Equations,” in Fractals in Biology and Medicine, ser. Mathematicsand Biosciences in Interaction. Birkhuser Basel, 2005, pp. 287–298.

69

[56] J. Harrison, Theorem Proving with the Real Numbers. Springer-Verlag, 1998.

[57] U. S. E. in Mathematica, http://reference.wolfram.com/mathematica/ref/FullSimplify.html, 2011.

[58] R. W. Butler, “Formalization of the Integral Calculus in the PVS TheoremProver,” Journal of Formalized Reasoning, vol. 2, no. 1, pp. 1–26, 2009.

[59] L. Cruz-Filipe, “Constructive real analysis: a type-theoretical formalizationand applications,” Ph.D. dissertation, University of Nijmegen, Apr. 2004.

[60] T. Mhamdi, O. Hasan, and S. Tahar, “On the Formalization of the LebesgueIntegration Theory in HOL,” in Interactive Theorem Proving. Springer,LNCS, 2010, pp. 387–402.

[61] J. Harrison, “Formalizing Basic Complex Analysis,” in From Insight to Proof:Festschrift in Honour of Andrzej Trybulec, ser. Studies in Logic, Grammarand Rhetoric, vol. 10(23). University of Bia lystok, 2007, pp. 151–165.

[62] J. Harrison, “A List of Theorem Provers,” http://www.cl.cam.ac.uk/users/jrh/ar.html, 2011.

[63] B. Akbarpour and L. C. Paulson, “Metitarski: An Automatic Prover for theElementary Functions,” in AISC/MKM/Calculemus, 2008, pp. 217–231.

[64] J. Harrison, “Formal Verification of Floating Point Trigonometric Functions,”in FMCAD, 2000, pp. 217–233.

[65] O. Hasan, S. K. Afshar, and S. Tahar, “Formal Analysis of Optical Waveguidesin HOL,” in TPHOLs, 2009, pp. 228–243.

[66] L. C. Paulson, ML for the Working Programmer. Cambridge UniversityPress, 1996.

[67] A. Church, “A Formulation of the Simple Theory of Types,” Journal of Sym-bolic Logic, vol. 5, pp. 56–68, 1940.

[68] R. Milner, “A Theory of Type Polymorphism in Programming,” Journal ofComputer and System Sciences, vol. 17, pp. 348–375, 1978.

[69] “The HOL System Description,” http://hol.sourceforge.net/documentation.html, 2011.

[70] J. Harrison, “Formalized Mathematics,” Turku Centre for Computer Science,Tech. Rep. 36, 1996.

[71] C. G. van der Laan and N. M. Temme, Calculation of Special Functions:The Gamma Function, The Exponential Integrals and Error-like Functions.Amsterdam, The Netherlands: Centrum voor Wiskunde en Informatica, 1984.

70

[72] K. Diethelm, The Analysis of Fractional Differential Equations, 1st ed., J.-M. M. C. F. T. Groningen, and B. T. Paris, Eds. Springer-Verlag Berlin /Heidelberg, 2010.

[73] U. Siddique, “Formal Analysis of Fractional Order Systems - HOL ProofScript,” http://save.seecs.nust.edu.pk/students/umair/fc.html, 2011.

[74] Y. Q. Chen, I. Petras, and D. Xue, “Fractional Order Control - A Tutorial.”American Control Conference, 2009, pp. 1397–1411.

[75] G. W. Bohannan, S. K. Hurst, and L. Spangler, “Electrical Components withFractional Order Impedence,” Patent, US 2006/0267595 A1, Nov. 2006.

71