Vincent Aravantinos

Formal Analysis of Optical Systems

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 paper-and-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.

Formal Stability Analysis of Optical Resonators

An optical resonator usually consists of mirrors or lenses which are configured in such a way that the beam of light is confined in a closed path. Resonators are fundamental components used in many safety-critical optical and laser applications such as laser surgery, aerospace industry and nuclear reactors. Due to the complexity and sensitivity of optical resonators, their verification poses many challenges to optical engineers. Traditionally, the stability analysis of such resonators, which is the most critical design requirement, has been carried out by paper-and-pencil based proof methods and numerical computations. However, these techniques cannot provide accurate results due to the risk of human error and the inherent incompleteness of numerical algorithms. In this paper, we propose to use higher-order logic theorem proving for the stability analysis of optical resonators. Based on the multivariate analysis library of HOL Light, we formalize the notion of light ray and optical system (by defining medium interfaces, mirrors, lenses, etc.). This allows us to derive general theorems about the behaviour of light in such optical systems. In order to illustrate the practical effectiveness of our work, we present the formal analysis of a Fabry-Perot resonator with fiber rod lens.

Formalization of Infinite Dimension Linear Spaces with Application to Quantum Theory

Linear algebra is considered an essential mathematical theory that has many engineering applications. While many theorem provers support linear spaces, they only consider finite dimensional spaces. In addition, available libraries only deal with real vectors, whereas complex vectors are extremely useful in many fields of engineering. In this paper, we propose a new linear space formalization which covers both finite and infinite dimensional complex vector spaces, implemented in HOL-Light. We give the definition of a linear space and prove many properties about its operations, e.g., addition and scalar multiplication. We also formalize a number of related fundamental concepts such as linearity, hermitian operation, self-adjoint, and inner product space. Using the developed linear algebra library, we were able to implement basic definitions about quantum mechanics and use them to verify a quantum beam splitter, an optical device that has many applications in quantum computing.

On the Formal Analysis of Geometrical Optics in HOL

Geometrical optics, in which light is characterized as rays, provides an efficient and scalable formalism for the modeling and analysis of optical and laser systems. The main applications of geometrical optics are in stability analysis of optical resonators, laser mode locking and micro opto-electro-mechanical systems. Traditionally, the analysis of such applications has been carried out by informal techniques like paper-and-pencil proof methods, simulation and computer algebra systems. These traditional techniques cannot provide accurate results and thus cannot be recommended for safety-critical applications, such as corneal surgery, process industry and inertial confinement fusion. On the other hand, higher-order logic theorem proving does not exhibit the above limitations, thus we propose a higher-order logic formalization of geometrical optics. Our formalization is mainly based on existing theories of multivariate analysis in the HOL Light theorem prover. In order to demonstrate the practical effectiveness of our formalization, we present the modeling and stability analysis of some optical resonators in HOL Light.

Formal reasoning about classified markov chains in HOL

Classified Markov chains have been extensively applied to model and analyze various stochastic systems in many engineering and scientific domains. Traditionally, the analysis of these systems has been conducted using computer simulations and, more recently, also probabilistic model-checking. However, these methods either cannot guarantee accurate analysis or are not scalable due to the unacceptable computation times. As an alternative approach, this paper proposes to reason about classified Markov chains using HOL theorem proving. We provide a formalization of classified discrete-time Markov chains with finite state space in higher-order logic and the formal verification of some of their widely used properties. To illustrate the usefulness of the proposed approach, we present the formal analysis of a generic LRU (least recently used) stack model.

A new approach for the verification of optical systems

Optical systems are increasingly used in microsystems, telecommunication, aerospace and laser industry. Due to the complexity and sensitivity of optical systems, their verification poses many challenges to 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 perfectly 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 them by taking into account all the details of mathematical reasoning) as an alternative to computational and numerical approaches to improve optical system analysis in a comprehensive framework. In particular, this paper provides a higher-order logic (a language used to express mathematical theories) formalization of ray optics in the HOL Light theorem prover. Based on the multivariate analysis library of HOL Light, we formalize the notion of light ray and optical system (by defining medium interfaces, mirrors, lenses, etc.), i.e., we express these notions mathematically in the software. This allows us to derive general theorems about the behavior of light in such optical systems. In order to demonstrate the practical effectiveness, we present the stability analysis of a Fabry-Perot resonator.

Linear Temporal Logic and Propositional Schemata, Back and Forth

This paper relates the well-known Linear Temporal Logic with the logic of propositional schemata introduced in elsewhere by the authors. We prove that LTL is equivalent to a class of schemata in the sense that polynomial-time reductions exist from one logic to the other. Some consequences about complexity are given. We report about first experiments and the consequences about possible improvements in existing implementations are analyzed.

Formal Verification of Optical Quantum Flip Gate

Quantum computers are promising to efficiently solve hard computational problems, especially NP problems. In this paper, we propose to tackle the formal verification of quantum circuits using theorem proving. In particular, we focus on the verification of quantum computing based on coherent light, which is typically light produced by laser sources. We formally verify the behavior of the quantum flip gate in HOL Light: we prove that it can flip a zero-quantum-bit to a one-quantum-bit and vice versa. To this aim, we model two optical devices: the beam splitter and the phase conjugating mirror and prove relevant properties about them. Then by cascading the two elements and utilizing these properties, the complete model of the flip gate is formally verified. This requires the formalization of some fundamental mathematics like exponentiation of linear transformations.

Formalization of Complex Vectors in Higher-Order Logic

Complex vector analysis is widely used to analyze continuous systems in many disciplines, including physics and engineering. In this paper, we present a higher-order-logic formalization of the complex vector space to facilitate conducting this analysis within the sound core of a theorem prover: HOL Light. Our definition of complex vector builds upon the definitions of complex numbers and real vectors. This extension allows us to extensively benefit from the already verified theorems based on complex analysis and real vector analysis. To show the practical usefulness of our library we adopt it to formalize electromagnetic fields and to prove the law of reflection for the planar waves.

Implicational Rewriting Tactics in HOL

Reducing the distance between informal and formal proofs in interactive theorem proving is a long-standing matter. An approach to this general topic is to increase automation in theorem provers: indeed, automation turns many small formal steps into one big step. In spite of the usual automation methods, there are still many situations where the user has to provide some information manually, whereas this information could be derived from the context. In this paper, we characterize some very common use cases where such situations happen, and identify some general patterns behind them. We then provide solutions to deal with these situations automatically, which we implemented as HOL Light and HOL4 tactics. We find these tactics to be extremely useful in practice, both for their automation and for the feedback they provide to the user.