Umair Siddique

Formal Modeling and Analysis of the MAL-Associated Biological Regulatory Network: Insight into Cerebral Malaria

The discrete modeling formalism of René Thomas is a well known approach for the modeling and analysis of Biological Regulatory Networks (BRNs). This formalism uses a set of parameters which reflect the dynamics of the BRN under study. These parameters are initially unknown but may be deduced from the appropriately chosen observed dynamics of a BRN. The discrete model can be further enriched by using the model checking tool HyTech along with delay parameters. This paves the way to accurately analyse a BRN and to make predictions about critical trajectories which lead to a normal or diseased response. In this paper, we apply the formal discrete and hybrid (discrete and continuous) modeling approaches to characterize behavior of the BRN associated with MyD88-adapter-like (MAL) – a key protein involved with innate immune response to infections. In order to demonstrate the practical effectiveness of our current work, different trajectories and corresponding conditions that may lead to the development of cerebral malaria (CM) are identified. Our results suggest that the system converges towards hyperinflammation if Brutons tyrosine kinase (BTK) remains constitutively active along with pre-existing high cytokine levels which may play an important role in CM pathogenesis.

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.

On the Formalization of Z-Transform in HOL

System analysis based on difference or recurrence equations is the most fundamental technique to analyze biological, electronic, control and signal processing systems. Z-transform is one of the most popular tool to solve such difference equations. In this paper, we present the formalization of Z-transform to extend the formal linear system analysis capabilities using theorem proving. In particular, we use differential, transcendental and topological theories of multivariate calculus to formally define Z-transform in higher-order logic and reason about the correctness of its properties, such as linearity, time shifting and scaling in z-domain. To illustrate the practical effectiveness of the proposed formalization, we present the formal analysis of an infinite impulse response (IIR) digital signal processing filter.

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.

Rewriting-Based Runtime Verification for Alternation-Free HyperLTL

Analysis of complex security and privacy policies e.g., information flow involves reasoning about multiple execution traces. This stems from the fact that an external observer may gain knowledge about the system through observing and comparing several executions. Monitoring of such policies is in particular challenging because most existing monitoring techniques are limited to the analysis of a single trace at run time. In this paper, we present a rewriting-based technique for runtime verification of the full alternation-free fragment of HyperLTL, a temporal logic for specification of hyperproperties. The distinguishing feature of our proposed technique is its space complexity, which is independent of the number of trace quantifiers in a given HyperLTL formula.

Formal modeling and verification of integrated photonic systems

The prominent advantages of photonics are high bandwidth, low power and the possibility of better electromagnetic interference immunity. As a result, photonics technology is increasingly used in ubiquitous applications such as telecommunication, medicine, avionics and robotics. One of the main critical requirements is to verify the corresponding functional properties of these systems. In this perspective, we identify the most widely used modeling techniques (e.g., transfer matrices, difference equations and block diagrams) for the modeling and analysis of photonic components. Considering the safety and cost critical nature of the application domain, we discuss the potential of using formal methods as a complementary analysis approach. In particular, we propose a framework to formally specify and verify the critical properties of complex photonic systems within the sound core of a higher-order-logic theorem prover. For illustration purposes, we present the formal specification of a microring resonator based photonic filter along with the verification of some important design properties such as spectral power and filtering rejection ratio.

Towards the formal analysis of microresonators based photonic systems

Recent developments in the fabrication technology attracted the attention of optical engineers and physicists in the area of VLSI photonics. Due to the physical nature of light-wave systems and their usage in safety critical domains such as human surgeries and high budget space missions, it is indispensable to build high assurance systems. 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 higher-order logic theorem proving to improve the analysis in the domain of integrated optics or VLSI photonics. In particular, this paper provides a higher-order logic formalization of optical microresonators which are the most fundamental building blocks of many photonic devices. In order to illustrate the practical utilization of our work, we present the formal analysis of 2-D microresonator lattice optical filters.

On the Formalization of Gamma Function in HOL

The Gamma function is a special transcendental function that is widely used in probability theory, fractional calculus and analytical number theory. This paper presents a higher-order logic formalization of the Gamma function using the HOL4 theorem prover. The contribution of this paper can be mainly divided into two parts. Firstly, we extend the existing integration theory of HOL4 by formalizing a variant of improper integrals using sequential limits. Secondly, we build upon these results to formalize the Gamma function and verify some of its main properties, such as pseudo-recurrence relation, functional equation and factorial generalization. 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 relationship between the Exponential and Gamma random variables.

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.