Higher-order Logic Formalization Research Articles

Research on the higher-order logic formalization of fractance element

Fractance element reflects the fractional order behavior of circuits, which can show the characteristics of the actual circuits. Higher-order logic theorem proving is based on the rigorous and correct mathematical theories. It becomes more and more important in the verifications of high-reliability systems. Fractance element is formalized using the proof of higher-order logic theorem in this paper. Firstly, the formalized model of fractional calculus which is based on Caputo definition is established in higher-order logic theorem proof tool. Then some properties of fractional calculus are proved, including the zero order property, the fractional differential of a constant and the consistency of fractional calculus and integer order calculus. Finally, fractance element and fractional differential circuit constituted by fractance element are formally analyzed. These formalizations demonstrate the effectiveness of the formal method in the analysis of fractance element.

Towards the formal performance analysis of multistate coherent systems using HOL theorem proving

Many practical engineering systems and their components have multiple performance levels and failure modes. If these systems form a monotonically increasing structure function (system model) with respect to the performance of their components and also if all of their components affect the overall system performance, then they are said to be multistate coherent systems. Traditionally, the reliability analysis of these multistate coherent systems has been carried out using paper-and-pencil or simulation based methods. The former method is often prone to human errors, while the latter requires high computational resources for large and complex systems having components with multiple operational states. As a complimentary approach, we propose to use Higher-order-logic (HOL) theorem proving to develop a sound reasoning framework to analyze the reliability of multistate coherent systems in this paper. This framework allows us to formally verify generic mathematical properties about multistate coherent systems with an arbitrary number of components and their states. Particularly, we present the HOL formalization of series and parallel multistate coherent systems and formally verify their deterministic and probabilistic properties using the HOL4 theorem prover. For illustration purposes, we present the formal reliability analysis of the multistate oil and gas pipeline to demonstrate the effectiveness of our proposed framework.

Formalization of Reliability Block Diagrams in Higher-order Logic

Reliability Block Diagrams (RBDs) allow us to model the failure relationships of complex systems and their sub-components and are extensively used for system reliability, availability and maintainability analyses. Traditionally, these RBD-based analyses are done using paper-and-pencil proofs or computer simulations, which cannot ascertain absolute correctness due to their inaccuracy limitations. As a complementary approach, we propose to use the higher-order logic theorem prover HOL to conduct RBD-based analysis. For this purpose, we present a higher-order logic formalization of commonly used RBD configurations, such as series, parallel, parallel-series and series-parallel, and the formal verification of their equivalent mathematical expressions. A distinguishing feature of the proposed RBD formalization is the ability to model nested RBD configurations, which are RBDs having blocks that also represent RBD configurations. This generality allows us to formally analyze the reliability of many real-world systems. For illustration purposes, we formally analyze the reliability of a generic Virtual Data Center (VDC) in a cloud computing infrastructure exhibiting the nested series-parallel RBD configuration.

Higher-Order Logic Formalization of Conformal Geometric Algebra and its Application in Verifying a Robotic Manipulation Algorithm

Conformal geometric algebra (CGA) is an advanced geometric language used in solving three-dimensional Euclidean geometric problems due to its simple, compact and coordinate-free formulations. It promises to stimulate new methods and algorithms in all areas of science dealing with geometric properties, especially for engineering applications. This paper presents a higher-order logic formalization of CGA theories in the HOL-Light theorem prover. First, we formally define the classical algebraic operations and representations of geometric entities in the new framework. Second, we use these results to reason about the correctness of operation properties and geometric features such as the distance between the geometric entities and their rigid transformations in higher-order logic. Finally, in order to demonstrate the practical effectiveness and utilization of this formalization, we use it to formally model the grasping algorithm of a robot based on the conformal geometric control technique and verify the property that whether the robot can grasp firmly or not.

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.

Formalization of Function Matrix Theory in HOL

Function matrices, in which elements are functions rather than numbers, are widely used in model analysis of dynamic systems such as control systems and robotics. In safety-critical applications, the dynamic systems are required to be analyzed formally and accurately to ensure their correctness and safeness. Higher-order logic (HOL) theorem proving is a promise technique to match the requirement. This paper proposes a higher-order logic formalization of the function vector and the function matrix theories using the HOL theorem prover, including data types, operations, and their properties, and further presents formalization of the differential and integral of function vectors and function matrices. The formalization is implemented as a library in the HOL system. A case study, a formal analysis of differential of quadratic functions, is presented to show the usefulness of the proposed formalization.

Formalization of Bernstein Polynomials and Applications to Global Optimization

This paper presents a formalization in higher-order logic of a practical representation of multivariate Bernstein polynomials. Using this representation, an algorithm for finding lower and upper bounds of the minimum and maximum values of a polynomial has been formalized and verified correct in the Prototype Verification System (PVS). The algorithm is used in the definition of proof strategies for formally and automatically solving polynomial global optimization problems.

Formalization of the Standard Uniform random variable

Continuous random variables are widely used to mathematically describe random phenomena in engineering and the physical sciences. In this paper, we present a higher-order logic formalization of the Standard Uniform random variable as the limit value of the sequence of its discrete approximations. We then show the correctness of this specification by proving the corresponding probability distribution properties within the HOL theorem prover, summarizing the proof steps. The formalized Standard Uniform random variable can be transformed to formalize other continuous random variables, such as Uniform, Exponential, Normal, etc., by using various non-uniform random number generation techniques. The formalization of these continuous random variables will enable us to perform an error free probabilistic analysis of systems within the framework of a higher-order-logic (HOL) theorem prover. For illustration purposes, we present the formalization of the Continuous Uniform random variable based on the formalized Standard Uniform random variable, and then utilize it to perform a simple probabilistic analysis of roundoff error in HOL.

Formalization of Fixed-Point Arithmetic in HOL

This paper addresses the formalization in higher order logic of fixed-point arithmetic. We encoded the fixed-point number system and specified the different quantization modes in fixed-point arithmetic such as the directed and even quantization modes. We also considered the formalization of exceptions detection and their handling like overflow and invalid operation. An error analysis is then performed to check the correctness of the quantized result after carrying out basic arithmetic operations, such as addition, subtraction, multiplication and division against their mathematical counterparts. Finally, we showed by an example how this formalization can be used to enable the verification of the transition from floating-point to fixed-point algorithmic level in the signal processing design flow.