Formalization of Infinite Dimension Linear Spaces with Application to Quantum Theory

Mahmoud, Mohamed Yousri; Aravantinos, Vincent; Tahar, Sofiène

doi:10.1007/978-3-642-38088-4_28

Cited by 18 publications

(17 citation statements)

References 11 publications

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“…As a part of the optics formal verification project [6], Mahmoud et al [60] provided a support for the formal analysis of the quantum systems using HOL Light. In particular, the authors formalized the infinite dimension linear spaces and used it for formally verifying a quantum beam splitter.…”

Section: Verification Of Physical Componentsmentioning

confidence: 99%

Formal Verification of Cyber-Physical Systems Using Theorem Proving

Rashid

Siddique

Tahar

2020

Communications in Computer and Information Science

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Due to major breakthroughs in software and engineering technologies, embedded systems are increasingly being utilized in areas ranging from aerospace and next-generation transportation systems, to smart grid and smart cities, to health care systems, and broadly speaking to what is known as Cyber-Physical Systems (CPS). A CPS is primarily composed of several electronic, communication and controller modules and some actuators and sensors. The mix of heterogeneous underlying smart technologies poses a number of technical challenges to the design and more severely to the verification of such complex infrastructure. In fact, a CPS shall adhere to strict safety, reliability, performance and security requirements, where one needs to capture both physical and random aspects of the various CPS modules and then analyze their interrelationship across interlinked continuous and discrete dynamics. Oftentimes however, system bugs remain uncaught during the analysis and in turn cause unwanted scenarios that may have serious consequences in safety-critical applications. In this paper, we introduce some of the challenges surrounding the design and verification of contemporary CPS with the advent of smart technologies. In particular, we survey recent developments in the use of theorem proving, a formal method, for the modeling, analysis and verification of CPS, and overview some real world CPS case studies from the automotive, avionics and healthtech domains from system level to physical components.

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Section: Verification Of Physical Componentsmentioning

confidence: 99%

Formal Verification of Cyber-Physical Systems Using Theorem Proving

Rashid

Siddique

Tahar

2020

Communications in Computer and Information Science

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“…The former three focus essentially on n-dimensional Euclidean and complex spaces, whereas our work generalizes it to (possibly) infinite-dimension vector spaces of complex numbers (more precisely, complex-valuedfunction spaces). More details about this formalization are available in [52].…”

Section: Quantum Opticsmentioning

confidence: 99%

“…In a nutshell, we believe that we have enough mathematical and physical foundations that allow us to formalize quantum devices and complete quantum systems. Some of our formalization details can be found in [52], in addition to the formalization of beam splitters which are commonly used in building quantum computers [65]. Our future work is to extend our library to some new optical elements, e.g., Mach-Zehnder interferometer [62], photon detectors [28], and parametric amplifiers [49] that are used for building quantum gates and quantum networks.…”

Section: Hol Light Implementationmentioning

confidence: 99%

Formal Analysis of Optical Systems

et al. 2014

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Optical systems are becoming increasingly important by resolving many bottlenecks in today's communication, electronics, and biomedical systems. However, given the continuous nature of optics, the inability to efficiently analyze optical system models using traditional paperand-pencil and computer simulation approaches sets limits especially in safety-critical applications. In order to overcome these limitations, we propose to employ higher-order-logic theorem proving as a complement to computational and numerical approaches to improve optical model analysis in a comprehensive framework. The proposed framework allows formal analysis of optical systems at four abstraction levels, i.e., ray, wave, electromagnetic, and quantum.Mathematics Subject Classification (2010). Primary (68T15,68Q60); Secondary (78A05,78A25, 81V80).

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“…The authors reason about why formal techniques are necessary for the analysis of large-scale systems that combine quantum and classical components, e.g., quantum optics, however, their approach addresses system level designs. We also can refer to the work of Mahmoud et al 13,14 in quantum optics where formal verification of beam splitters and quantum flip gates are presented, respectively. To the best of our knowledge, the most comprehensive work on formalization of optical systems is a general framework presented in 2013, by Khan-Afshar et al, 15 to use theorem proving as a complement to computational and numerical approaches to improve analysis of optical system models, in ray, electromagnetic and quantum optics.…”

Section: Introductionmentioning

confidence: 99%

Formal analysis of electromagnetic optics

Khan-Afshar¹,

Hasan²,

Tahar³

2014

SPIE Proceedings

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Optical systems are increasingly being used in safety-critical applications. Due to the complexity and sensitivity of optical systems, their verification raises many challenges for engineers. Traditionally, the analysis of such systems has been carried out by paper-and-pencil based proofs and numerical computations. However, these techniques cannot provide accurate results due to the risk of human error and inherent approximations of numerical algorithms. In order to overcome these limitations, we propose to use theorem proving (i.e., a computer-based technique that allows to express mathematical expressions and reason about their correctness by taking into account all the details of mathematical reasoning) as a complementary approach to improve optical system analysis. This paper provides a higher-order logic (a language used to express mathematical theories) formalization of electromagnetic optics in the HOL Light theorem prover. In order to demonstrate the practical effectiveness of our approach, we present the analysis of resonant cavity enhanced photonic devices.

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Formalization of Infinite Dimension Linear Spaces with Application to Quantum Theory

Cited by 18 publications

References 11 publications

Formal Verification of Cyber-Physical Systems Using Theorem Proving

Formal Verification of Cyber-Physical Systems Using Theorem Proving

Formal Analysis of Optical Systems

Formal analysis of electromagnetic optics

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