Formal Reliability Analysis Using Theorem Proving

Abstract

Abstract-Reliability analysis has become a tool of fundamental importance to virtually all electrical and computer engineers because of the extensive usage of hardware systems in safety and mission critical domains, such as medicine, military, and transportation. Due to the strong relationship between reliability theory and probabilistic notions, computer simulation techniques have been traditionally used to perform reliability analysis. However, simulation provides less accurate results and cannot handle large-scale systems due to its enormous CPU time requirements. To ensure accurate and complete reliability analysis and thus more reliable hardware designs, we propose to conduct a formal reliability analysis of systems within the sound core of a higher order logic theorem prover (HOL). In this paper, we present the higher order logic formalization of some fundamental reliability theory concepts, which can be built upon to precisely analyze the reliability of various engineering systems. The proposed approach and formalization is then utilized to analyze the repairability conditions for a reconfigurable memory array in the presence of stuck-at and coupling faults.

Formal Reasoning about Expectation Properties for Continuous Random Variables

“…Random variables are basically probabilistic algorithms and thus can be formalized and verified, based on their probability distribution properties, using the methodology proposed in [15]. In fact, building upon Hurd's formalization, most of the commonly used discrete [15] and continuous [11] random variables have been formalized. The above mentioned formalization of probability theory has also been used to formally reason about statistical properties, such as expectation and variance, of discrete random variables [11].…”

“…In fact, building upon Hurd's formalization, most of the commonly used discrete [15] and continuous [11] random variables have been formalized. The above mentioned formalization of probability theory has also been used to formally reason about statistical properties, such as expectation and variance, of discrete random variables [11]. Due to the fact that the discrete random variables can only attain a countable number of values, the expectation in this case has been formally defined using a summation rather than integration.…”

“…To overcome the above mentioned limitations, it has been recently proposed to conduct probabilistic analysis of systems in a higher-order-logic theorem prover [11]. The main idea behind this approach is to formally specify the behavior of systems, with random or unpredictable components, in higher-order logic, while representing the random components as formalized random variables.…”

“…The analysis carried out in this way is free from any approximation issues or flaws due to the mathematical nature of the models and the inherent soundness of the theorem proving approach. The milestones achieved so far, in this endeavor of developing a complete theorem proving based probabilistic analysis framework that is capable of analyzing any hardware or software system, include the formalization of probability theory [15], the ability to formalize discrete and continuous random variables and verify their probabilistic properties [15,11] and the ability to verify statistical properties of discrete random variables [11]. Whereas, to the best of our knowledge, the formal reasoning about statistical properties regarding continuous random variables has not been tackled in the open literature so far.…”

“…The main motivation behind this choice is the fact that most of the work that we build upon is developed in HOL, such as the formalization of the real number theory [10], probability theory [15], continuous random variables [11] and Lebesgue integration [4]. Though, it is important to note here that the ideas presented in this paper are not specific to the HOL theorem prover and can be adapted to any other higher-order-logic theorem prover as well, such as Isabelle, Coq or PVS.…”