This course will discuss different ways of formally modeling requirements of interest for autonomous systems. Examples of such requirements include stability, invariance, reachability, regular languages, omega-regular languages, and linear temporal logic properties. In addition, it will introduce non-deterministic finite and büchi automata for recognizing, respectively, regular languages and omega-regular languages.
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What you'll learn
- Utilize formal methods to specify and verify requirements for autonomous systems.
- Model system behaviors and verify stability using various analytical methods.
- Apply reachable set computation and robustness analysis in system design.
Syllabus
Course Introduction
In this course, we delve into both low-level and high-level specifications, fundamental to the development of safe autonomous systems. This module is specifically designed to equip students with an in-depth understanding of expressing system behaviors through formal methods, including linear temporal logic and automata on both finite and infinite strings. Through a collection of detailed examples and practical applications, participants will acquire the skills needed to define and analyze key properties of autonomous systems, such as safety and reachability.
Low-Level Specifications
This module offers a concise introduction to normed vector spaces and stability concepts in autonomous systems, encompassing both asymptotic stability and global asymptotic stability. It emphasizes the application of Lyapunov's Stability Theorem for the formal verification of these properties in complex systems, including its application to various simple systems, such as linear ones. Through illustrative examples, we will demonstrate the significance of these concepts in analyzing and ensuring the stability of systems.
High-Level Specifications: Reachability, Safety, Regular and ω-Regular Properties
Delve into the topic of reachable sets and uncover their critical role in guaranteeing system safety. This module introduces frameworks for exploring computational techniques to over-approximate reachable sets across diverse system classes. You will have the chance to apply your knowledge in real-world contexts, investigate the use of zonotopes, and recognize their beneficial properties in the computation of reachable sets. Moreover, we delve into fundamental concepts of formal languages, and regular and omega-regular expressions, offering succinct and formal methods to express regular and omega-regular languages, respectively.
Nondeterministic Finite and Büchi Automata (NFA and NBA)
This module immerses you in the essential principles of regular and ω-regular properties and how they are represented via non-deterministic finite automata (NFA) and Büchi automata (NBA), respectively. You will study the notation and architecture of NFAs and NBAs, master the construction of regular and ω-regular expressions, and grasp their correlation with these automata. The course will navigate you through the conversion of NFAs to regular expressions and NBAs to ω-regular expressions and the inverse, elucidating the significance of these concepts in the verification of finite and infinite behaviors of systems.
Linear Temporal Logic Formulae
This module provides an in-depth exploration of Linear Temporal Logic (LTL) formulas, a mathematical formalism for describing languages containing infinite words. It presents a framework for articulating the temporal dimensions of system behaviors, offering a syntax that closely mirrors natural language. By melding propositional logic with temporal operators, LTL furnishes a powerful toolkit for specifying the rich behaviors of systems.
