We establish a generic upper bound ExpTime for reasoning with global assumptions (also known as TBoxes) in coalgebraic modal logics. Unlike earlier results of this kind, our bound does not require a tractable set of tableau rules for the instance logics, so that the result applies to wider classes of logics. Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities. We establish the theoretical upper bound using a type elimination algorithm. We also provide a global caching algorithm that potentially avoids building the entire exponential-sized space of candidate states, and thus offers a basis for practical reasoning. This algorithm still involves frequent fixpoint computations; we show how these can be handled efficiently in a concrete algorithm modelled on Liu and Smolka's linear time fixpoint algorithm. Finally, we show that the upper complexity bound is preserved under adding nominals to the logic, i.e. in coalgebraic hybrid logic.

中文翻译：

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算术模态逻辑中具有全局假设的代数推理

我们建立了一个通用上限 ExpTime，用于在代数模态逻辑中使用全局假设（也称为 TBoxes）进行推理。与早期的此类结果不同，我们的边界不需要一组易于处理的实例逻辑表规则，因此结果适用于更广泛的逻辑类别。例子是 Presburger 模态逻辑，它扩展了在后继数量上具有线性不等式的分级模态逻辑，以及在概率上具有多项式不等式的概率模态逻辑。我们使用类型消除算法建立理论上限。我们还提供了一种全局缓存算法，该算法可能避免构建候选状态的整个指数大小的空间，从而为实际推理提供基础。该算法仍然涉及频繁的固定点计算；我们展示了如何在以 Liu 和 Smolka 的线性时间定点算法为模型的具体算法中有效地处理这些问题。最后，我们表明，在向逻辑添加标称值的情况下，即在代数混合逻辑中，保留了复杂性上限。