Differential Game Logic for Hybrid Games

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Differential Game Logic

Differential game logic (dGL) is a logic for specifying and verifying properties of hybrid games, i.e. games that combine discrete, continuous, and adversarial dynamics. Unlike hybrid systems, hybrid games allow choices in the system dynamics to be resolved adversarially by different players with different objectives. The logic dGL can be used to study the existence of winning strategies for such hybrid games, i.e. ways of resolving the player's choices in some way so that he wins by achieving his objective for all choices of the opponent. Hybrid games are determined, i.e. from each state, one player has a winning strategy, yet computing their winning regions may take transfinitely many steps. The logic dGL, nevertheless, has a sound and complete axiomatization relative to any expressive logic. Separating axioms are identified that distinguish hybrid games from hybrid systems. Finally, dGL is proved to be strictly more expressive than the corresponding logic of hybrid systems by characterizing the expressiveness of both.

Keywords: differential game logic, game logic, hybrid games, axiomatization, expressiveness [2]

Differential game logic
Differential game logic for hybrid games has been implemented in the KeYmaera X theorem prover based on the uniform substitution calculus for differential game logic [4,6].

Differential Hybrid Games

The ACM TOCL 2017 article introduces differential hybrid games, which combine differential games with hybrid games. In both kinds of games, two players interact with continuous dynamics. The difference is that hybrid games also provide all the features of hybrid systems and discrete games, but only deterministic differential equations. Differential games, instead, provide differential equations with input by both players, but not the luxury of hybrid games, such as mode switches and discrete or alternating interaction. This article augments differential game logic with modalities for the combined dynamics of differential hybrid games. It shows how hybrid games subsume differential games and introduces differential game invariants and differential game variants for proving properties of differential games inductively.

Keywords: differential games, hybrid games, differential game game logic, differential game invariants, partial differential equations, viscosity solutions, real algebraic geometry [3]

Illustration for the superdifferential of a non-differentiable function Illustration for the subdifferential of a non-differentiable function used for differential game invariants Local safety zones for Zeppelin obstacle parcours with a response trajectory. The zones are proved by differential game invariants.

Differential Game Invariants

Differential game invariants [3] generalize differential invariance reasoning principles to differential games. Differential game invariants provide an inductive reasoning principle for differential games. A differential game is a differential equation x'=f(x,y,z) in which one player controls the choice of y while the opponent controls the choice of z.
Illustration of differential game invariant and control actions by the two players of a differential game
Even if the ultimate proof rule for differential game invariants is pleasantly simple, the major challenge was its soundness justification [3].

Constructive Differential Game Logic

Constructive differential game logic (CdGL) [8] is a constructive version of dGL enabling the constructive study of the existence of winning strategies in hybrid games with discrete, continuous, and adversarial dynamics. Constructive truth of CdGL formulas about hybrid games implies the constructive existence of winning strategies that are programs that win the hybrid game in question. This Curry-Howard correspondence ensures that constructive proofs for constructive hybrid games correspond to programs implementing their winning strategies.

Constructive Differential Game Logic


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    Differential Game Logic for Hybrid Games.
    School of Computer Science, Carnegie Mellon University, CMU-CS-12-105, March 2012.
    Also see new results.
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