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Formally Verified Approximations of Definite Integrals

Assia Mahboubi, Guillaume Melquiond, Thomas Sibut-Pinote
2018 Journal of automated reasoning  
This paper presents an efficient method for automatically computing and proving bounds on some definite integrals inside the Coq formal system.  ...  Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy.  ...  Formally Verified Approximations of Definite Integrals  ... 
doi:10.1007/s10817-018-9463-7 fatcat:g3jumo65prehlovv7mwnoqjsse

Formally Verified Approximations of Definite Integrals [chapter]

Assia Mahboubi, Guillaume Melquiond, Thomas Sibut-Pinote
2016 Lecture Notes in Computer Science  
Some of the numerical integration methods can even be made rigorous: not only do they compute an approximation of the integral value but they also bound its inaccuracy.  ...  This paper presents an efficient method for automatically computing and proving bounds on some definite integrals inside the Coq formal system.  ...  Conclusion We have presented a method for computing and formally verifying numerical enclosures of univariate definite integrals using the Coq proof assistant.  ... 
doi:10.1007/978-3-319-43144-4_17 fatcat:gbqhtd2kvvbxzlbd6i7g7wqoqa

On the formal analysis of Gaussian optical systems in HOL

Umair Siddique, Sofiène Tahar
2016 Formal Aspects of Computing  
To demonstrate the usefulness of the reported formalization, we utilize it to formally analyze some fractional order systems, namely, a fractional electrical component Resistoductance, a fractional integrator  ...  Fractional order systems, which involve integration and differentiation of non integer order, are increasingly being used in the fields of control systems, robotics, signal processing and circuit theory  ...  We verified the continuity by differentiability using the classical definition of derivative formalized in HOL [15] .  ... 
doi:10.1007/s00165-016-0367-1 fatcat:env7aa6ivzbojly4pr4rwcgj2e

Formalization of Transform Methods Using HOL Light [chapter]

Adnan Rashid, Osman Hasan
2017 Lecture Notes in Computer Science  
In particular, we present the motivation of the formalization, which is followed by the related work.  ...  Next, we present the task completed so far while highlighting some of the challenges faced during the formalization.  ...  Acknowledgements This work was supported by the National Research Program for Universities grant (number 1543) of Higher Education Commission (HEC), Pakistan.  ... 
doi:10.1007/978-3-319-62075-6_22 fatcat:nh6h44hfkbc5dh6kk7aaz4cf4y

Formalization of Transform Methods using HOL Light [article]

Adnan Rashid, Osman Hasan
2017 arXiv   pre-print
In particular, we present the motivation of the formalization, which is followed by the related work.  ...  Next, we present the task completed so far while highlighting some of the challenges faced during the formalization.  ...  Acknowledgements This work was supported by the National Research Program for Universities grant (number 1543) of Higher Education Commission (HEC), Pakistan.  ... 
arXiv:1705.10050v1 fatcat:wpigu4et6bgp5m3m7xmtqicuxm

On the Formalization of Gamma Function in HOL

Umair Siddique, Osman Hasan
2014 Journal of automated reasoning  
Firstly, we extend the existing integration theory of HOL4 by formalizing a variant of improper integrals using sequential limits.  ...  In order to illustrate the practical effectiveness and utilization of our work, we formally verify some properties of Euler's generalized power rule of differentiation, Mittag-Leffler functions and the  ...  The formalization, presented in this section, took around 5500 lines of HOL code and approximately 300 man-hours.  ... 
doi:10.1007/s10817-014-9311-3 fatcat:utrfokmtdzgwhk3fg7x5jnmzka

Formal verification of the heavy hitter problem

G. Helali, S. Tahar, O. Hasan
2012 2012 25th IEEE Canadian Conference on Electrical and Computer Engineering (CCECE)  
This model is then utilized to formally verify some interesting probabilistic and statistical properties associated with the heavy hitter problem in HOL.  ...  The heavy hitter problem is used to assess the frequency of occurrence of an element in a given data stream.  ...  However, Coble's formalization of Lebesgue integral can only consider finite-valued measures, functions and integrals.  ... 
doi:10.1109/ccece.2012.6335035 dblp:conf/ccece/HelaliTH12 fatcat:cl4fa2ay3fculaupr7m4pjwrou

Towards the Formalization of Fractional Calculus in Higher-Order Logic [article]

Umair Siddique, Osman Hasan, Sofiène Tahar
2015 arXiv   pre-print
Fractional calculus is a generalization of classical theories of integration and differentiation to arbitrary order (i.e., real or complex numbers).  ...  Mainly, we present the motivation and application of such formalization efforts, a roadmap to achieve our goals, current status of the project and future milestones.  ...  But now the current formalization seems quite stable as most of the classical properties have been formally verified for our definitions.  ... 
arXiv:1505.02140v1 fatcat:hvyf3xhf2nbh7bj4gijiav7rai

Formal Reasoning about Expectation Properties for Continuous Random Variables [chapter]

Osman Hasan, Naeem Abbasi, Behzad Akbarpour, Sofiène Tahar, Reza Akbarpour
2009 Lecture Notes in Computer Science  
Starting from the relatively complex higher-order-logic definition of expectation, based on Lebesgue integration, we formally verify key expectation properties that allow us to reason about expectation  ...  This paper presents an infrastructure that can be used to formally reason about expectation properties of most of the continuous random variables in a theorem prover.  ...  Besides the formalization of the above definitions, many useful properties regarding the Lebesgue integral have also been verified in [4] as higher-orderlogic theorems.  ... 
doi:10.1007/978-3-642-05089-3_28 fatcat:4ppnedhwjbf3pp3b7jsc6moski

Towards the Formalization of Fractional Calculus in Higher-Order Logic [chapter]

Umair Siddique, Osman Hasan, Sofiène Tahar
2015 Lecture Notes in Computer Science  
Fractional calculus is a generalization of classical theories of integration and differentiation to arbitrary order (i.e., real or complex numbers).  ...  Mainly, we present the motivation and application of such formalization efforts, a roadmap to achieve our goals, current status of the project and future milestones.  ...  But now the current formalization seems quite stable as most of the classical properties have been formally verified for our definitions.  ... 
doi:10.1007/978-3-319-20615-8_21 fatcat:lgx5adcufvfenedqbkpuuo4wnu

A Certificate-Based Approach to Formally Verified Approximations

Florent Bréhard, Assia Mahboubi, Damien Pous, Michael Wagner
2019 International Conference on Interactive Theorem Proving  
Finally, we provide an implementation of verified Chebyshev approximations, and we discuss a few examples of computations.  ...  We present a library to verify rigorous approximations of univariate functions on real numbers, with the Coq proof assistant.  ...  I T P 2 0 1 9 8:16 A Certificate-Based Approach to Formally Verified Approximations Error bounding.  ... 
doi:10.4230/lipics.itp.2019.8 dblp:conf/itp/BrehardMP19 fatcat:b2twcbftqrdihpid6s3fxlw4ha

Formalization of Lerch's Theorem using HOL Light [article]

Adnan Rashid, Osman Hasan
2018 arXiv   pre-print
In this paper, we extend our higher-order logic formalization of the Laplace transform, which includes the formal definition of the Laplace transform and verification of its various classical properties  ...  For illustration, we present the formal analysis of a 4-π soft error crosstalk model, which is widely used in nanometer technologies, such as, Integrated Circuits (ICs).  ...  We formally verified these using our new definition of the Laplace transform [27] .  ... 
arXiv:1806.03049v1 fatcat:dxqnz5t2pvhvzaueddrd2hxq7a

Formalization of Laplace Transform Using the Multivariable Calculus Theory of HOL-Light [chapter]

Syeda Hira Taqdees, Osman Hasan
2013 Lecture Notes in Computer Science  
In order to demonstrate the practical effectiveness of this formalization, we use it to formally verify the transfer function of Linear Transfer Converter (LTC) circuit, which is a commonly used electrical  ...  To facilitate formal analysis of these systems, we present the formalization of Laplace transform using the multivariable calculus theories of HOL-Light.  ...  This work was supported by the National Research Program for Universities grant (number 1543) of Higher Education Commission (HEC), Pakistan.  ... 
doi:10.1007/978-3-642-45221-5_50 fatcat:fks2jihksrfalktlbopv5weapy

One-parameter extension of the Doi-Peliti formalism and its relation with orthogonal polynomials

Jun Ohkubo
2012 Physical Review E  
An extension of the Doi-Peliti formalism for stochastic chemical kinetics is proposed. Using the extension, path-integral expressions consistent with previous studies are obtained.  ...  In addition, the extended formalism is naturally connected to orthogonal polynomials.  ...  Here, we briefly explain some basic definitions for a coherent-state path-integral expression, which are especially useful in the Doi-Peliti formalism.  ... 
doi:10.1103/physreve.86.042102 pmid:23214630 fatcat:4d5ytjszkjhe3dvkzij7bsuewq

Formal Probabilistic Analysis: A Higher-Order Logic Based Approach [chapter]

Osman Hasan, Sofiène Tahar
2010 Lecture Notes in Computer Science  
Some major contributions include the formalization of both discrete and continuous random variables and the verification of some of their corresponding probabilistic and statistical properties.  ...  Recently, a significant amount of formalization has been done in higherorder logic that allows us to conduct precise probabilistic analysis using theorem proving and thus overcome the limitations of the  ...  In order to verify the correctness of the above definitions of expectation, they are utilized in [20, 21] to formally verify the following classical expectation properties.  ... 
doi:10.1007/978-3-642-11811-1_2 fatcat:hhk4xlpnevflrf2w3ijc3hweni
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