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Formal methods refer to rigorous, mathematical approaches to system
development and have played a key role in establishing the correctness of
safety-critical systems. The main building blocks of formal methods are models
and specifications, which are analogous to behaviors and requirements in system
design and give us the means to verify and synthesize system behaviors with
formal guarantees.
This monograph provides a survey of the current state of the art on
applications of formal methods in the autonomous systems domain. We consider
correct-by-construction synthesis under various formulations, including closed
systems, reactive, and probabilistic settings. Beyond synthesizing systems in
known environments, we address the concept of uncertainty and bound the
behavior of systems that employ learning using formal methods. Further, we
examine the synthesis of systems with monitoring, a mitigation technique for
ensuring that once a system deviates from expected behavior, it knows a way of
returning to normalcy. We also show how to overcome some limitations of formal
methods themselves with learning. We conclude with future directions for formal
methods in reinforcement learning, uncertainty, privacy, explainability of
formal methods, and regulation and certification.
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