Lecture Notes on Winning Strategies & Regions

This lecture continues the study of hybrid games and their logic, differential game logic [Pla15], that Lecture 20 on Hybrid Systems & Games started. Lecture 20 saw the introduction of differential game logic with a focus on identifying and highlighting the new dynamical aspect of adversarial dynamics. The meaning of hybrid games in differential game logic had been left informal, based on the intuition one relates to interactive gameplay and decisions in trees. While it is possible to turn such a tree-type semantics into an operational semantics for hybrid games [Pla15], the resulting development is technically rather involved. Even if such an operational semantics is interesting and touches on interesting concepts from descriptive set theory, it is unnecessarily complicated compared. This lecture will, thus, be devoted to developing a much simpler yet rigorous semantics, a denotational semantics of hybrid games. Lecture 20 already highlighted subtleties how never-ending game play ruins determinacy, simply because there never is a state in which the winner would be declared. Especially the aspect of repetition and its interplay with differential equations will need careful attention. The denotational semantics will make this subtle aspect crystal clear. 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: