Intelligent Modeling and Verification (Editorial)

published_or_final_versio

Formalization of Normal Random Variables

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

Intelligent Modeling and Verification

System modeling tends to have many complex features, and uncertainties often lead to numerous complications and influence many important aspects related to its applications. Intelligent modeling merges mathematical and computer based approaches, and it utilizes pioneering new scientific methods and cutting-edge technologies

Formalization of Real Analysis: A Survey of Proof Assistants and Libraries

International audienceIn the recent years, numerous proof systems have improved enough to be used for formally verifying non-trivial mathematical results. They, however, have different purposes and it is not always easy to choose which one is adapted to undertake a formalization effort. In this survey, we focus on properties related to real analysis: real numbers, arithmetic operators, limits, differentiability, integrability, and so on. We have chosen to look into the formalizations provided in standard by the following systems: Coq, HOL4, HOL Light, Isabelle/HOL, Mizar, ProofPower-HOL, and PVS. We have also accounted for large developments that play a similar role or extend standard libraries: ACL2(r) for ACL2, C-CoRN/MathClasses for Coq, and the NASA PVS library. This survey presents how real numbers have been defined in these various provers and how the notions of real analysis described above have been formalized. We also look at the methods of automation these systems provide for real analysis

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

Formally Verified Compositional Algorithms for Factored Transition Systems

Artificial Intelligence (AI) planning and model checking are two disciplines that found wide practical applications. It is often the case that a problem in those two fields concerns a transition system whose behaviour can be encoded in a digraph that models the system's state space. However, due to the very large size of state spaces of realistic systems, they are compactly represented as propositionally factored transition systems. These representations have the advantage of being exponentially smaller than the state space of the represented system. Many problems in AI~planning and model checking involve questions about state spaces, which correspond to graph theoretic questions on digraphs modelling the state spaces. However, existing techniques to answer those graph theoretic questions effectively require, in the worst case, constructing the digraph that models the state space, by expanding the propositionally factored representation of the syste\ m. This is not practical, if not impossible, in many cases because of the state space size compared to the factored representation. One common approach that is used to avoid constructing the state space is the compositional approach, where only smaller abstractions of the system at hand are processed and the given problem (e.g. reachability) is solved for them. Then, a solution for the problem on the concrete system is derived from the solutions of the problem on the abstract systems. The motivation of this approach is that, in the worst case, one need only construct the state spaces of the abstractions which can be exponentially smaller than the state space of the concrete system. We study the application of the compositional approach to two fundamental problems on transition systems: upper-bounding the topological properties (e.g. the largest distance between any two states, i.e. the diameter) of the state spa\ ce, and computing reachability between states. We provide new compositional algorithms to solve both problems by exploiting different structures of the given system. In addition to the use of an existing abstraction (usually referred to as projection) based on removing state space variables, we develop two new abstractions for use within our compositional algorithms. One of the new abstractions is also based on state variables, while the other is based on assignments to state variables. We theoretically and experimentally show that our new compositional algorithms improve the state-of-the-art in solving both problems, upper-bounding state space topological parameters and reachability. We designed the algorithms as well as formally verified them with the aid of an interactive theorem prover. This is the first application that we are aware of, for such a theorem prover based methodology to the design of new algorithms in either AI~planning or model checking