An Epistemic - Geometric Extension of Game Logic

The notions of uncertainty and closeness have long played an important role in games. Many games such as dart and 20Q involve uncertainty and, furthermore one can describe a notion of closeness in such games. Closeness of the dart to the aim or closeness of the golf ball to the hole can be considered as the instances of the situations that involve uncertainty. In dart, for instance, a player cannot guarantee that he will be able to hit the aim precisely. However, he can guarantee that he will be able to hit a point within a neighborhood about the aim. Such neighborhoods vary from player to player, from move to move. Players can improve their skills to make the neighborhood smaller. In some situations players can guarantee a certain “point” once the neighborhood entirely lies in a certain section of the dart board. Nevertheless, game logics which have been suggested and discussed so far do not involve such components in their semantics in order to be able to formalize the uncertainty and closeness in games. In this preliminary report, our aim is to make use of epistemic logical apparatus in the context of game logic in order to give an epistemic-geometric semantics for games that involve uncertainty. Some previous work, on the other hand, tend to converge to our current agenda where the logic of epistemic programs were discussed extensively with dynamic epistemic logical concerns and puzzles in Kripke structures (Baltag & Moss, 2004). We also have a hidden agenda. This work can immediately be considered as a first step towards a geometrical (or even topological) semantics for propositional dynamic logic (PDL, henceforth). The reason for that is the fact that game logic is an extension of PDL with an additional operator for duality. However, the additional operator in the game logic is not trivial as the completeness proof of game logic with dual has not yet been given. The organization of the paper is as follows. We first introduce the basics of game logic and subset space logic where the latter is the basis of our geometric and epistemic semantics. Then, we discuss our extension of game logic where we extend the basic language with epistemic modalities and make use of a geometrical semantics. Finally, we will conclude with some future research directions.