Lecture Notes on Differential & Temporal Logics

This course is devoted to the study of the Foundations of Cyber-Physical Systems [Pla12c, Pla12b]. Lecture 3 on Choice & Control explained hybrid programs, a program notation for hybrid systems [Pla08, Pla10, Pla12c, Pla12a]. Lecture 4 on Safety & Contracts defined differential dynamic logic [Pla08, Pla10, Pla12c, Pla12a] as a specification and verification logic for hybrid programs. Lecture 5 on Dynamical Systems & Dynamic Axioms and subsequent lectures studied proof principles for differential dynamic logic with which we can prove correctness properties of hybrid systems. In your labs, you have demonstrated aptly how you can model, specify, and verify quite sophisticated and challenging robots. Yet, there was one rather puzzling phenomenon that we noticed in Lecture 4 only then did not have a chance to consider any further. For a hybrid program α and differential dynamic logic formula φ, the modal formula