Lecture Notes on Winning & Proving Hybrid Games

This lecture continues the study of hybrid games and their logic, differential game logic [Pla15], whose syntax was introduced in Lecture 20 on Hybrid Systems & Games and whose semantics was developed in Lecture 21 on Winning Strategies & Regions. Today’s lecture furthers the development of differential game logic to the third leg of the logical trinity: its axiomatics. This lecture will focus on the development of rigorous reasoning techniques for hybrid games as models for CPS with adversarial dynamics. Without such analysis and reasoning techniques, a logic that only comes with syntax and semantics can be used as a specification language with a precise meaning, but would not be very helpful for actually analyzing and verifying hybrid games. It is the logical trinity of syntax, semantics, and axiomatics that gives logics the power of serving as well-founded specification and verification languages with a (preferably concise) syntax, an unambiguous semantics, and actionable analytic reasoning principles. Thus, today’s lecture is the hybrid games analogue of Lecture 5 on Dynamical Systems and Dynamic Axioms. Indeed, after the logical sophistication we reached throughout the semester, this lecture will settle for a Hilbert-type calculus as in Lecture 5 on Dynamical Systems and Dynamic Axioms as opposed to the more refined and more automatable sequent calculus from Lecture 6 on Truth and Proof and subsequent lectures. Before submerging completely into the development of rigorous reasoning techniques for hybrid games as models for CPS with adversarial dynamics, however, it will be wise to take a short detour by investigating a semantical simplification of the meaning of repetition by an implicit characterization of its winning region rather than the explicit construction by iteration from Lecture 21. These lecture notes are based on [Pla15], where more information can be found on logic and hybrid games. The most important learning goals of this lecture are: