Abstract

Engineering systems often have components that exhibit random behavior. This randomness in many cases is normally distributed. To verify such systems, proba- bilistic analysis is used. Such engineering systems have applications in domains like transportation, medicine and military. Despite the safety-critical nature of these ap- plications, most of the analysis is done using informal techniques like simulation and paper-and-pencil analysis, and thus cannot be completely relied upon. The unreliable results produced by such methods may result in heavy financial loss or even the loss of a human life. To overcome the limitation of traditional methods, we propose to conduct the analysis of such systems within the trusted kernel of a higher-order-logic theorem prover HOL4. The soundness and the deduction style of the theorem prover guarantee the validity of the analysis and the results of this type of analysis are generic and valid for any instance of the system. For this purpose, we provide HOL4 formalization of Lebesgue measure and normal random variables along with the proof of their classical properties. We also ported the theory of Gauge integral and other required foundational concepts from HOL Light and Isabelle/HOL theorem provers. To illustrate the usefulness of our formalization, we conducted the formal analysis of two applications, i.e., error probability of binary transmission in the presence of Gaussian noise and probabilistic clock synchronization in wireless sensor networks.