This paper investigates first-order game logic and first-order modal mu-calculus, which extend their propositional modal logic counterparts with first-order modalities of interpreted effects such as variable assignments. Unlike in the propositional case, both logics are shown to have the same expressive power and their proof calculi to have the same deductive power. Both calculi are also mutually relatively complete. In the presence of differential equations, corollaries obtain usable and complete translations between differential game logic, a logic for the deductive verification of hybrid games, and the differential mu-calculus, the modal mu-calculus for hybrid systems. The differential mu-calculus is complete with respect to first-order fixpoint logic and differential game logic is complete with respect to its ODE-free fragment.
翻译:本文调查了一阶游戏逻辑和一阶模式模型计算模型,这些逻辑扩展了它们的假设模式逻辑对等,具有各种解释效果的一阶模式,如可变分配。与本假设的情况不同,两种逻辑都显示具有相同的表达力和证据计算力,具有相同的推算力。两种计算法也是相对完整的。在存在差异方程式的情况下,混合游戏的分级逻辑、混合游戏的分级核查逻辑和混合系统不同的混合计算法、混合计算法的分级混合计算法、分级确定点逻辑和分级游戏逻辑之间可以得到可用和完整的翻译。