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Rank-1 Modal Logics are Coalgebraic

机译：等级1模态逻辑是联合代数

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摘要

Coalgebras provide a unifying semantic framework for a wide variety of modal logics. It has previously been shown that the class of coalgebras for an endofunctor can always be axiomatized in rank 1. Here we establish the converse, i.e. every rank-1 modal logic has a sound and strongly complete coalgebraic semantics. This is achieved by constructing for a given modal logic a canonical coalgebraic semantics, consisting of a signature functor and interpretations of modal operators, which turns out to be final among all such structures. The canonical semantics may be seen as a coalgebraic reconstruction of neighbourhood semantics, broadly construed. A finitary restriction of the canonical semantics yields a canonical weakly complete semantics which moreover enjoys the Hennessy-Milner property. As a consequence, the machinery of coalgebraic modal logic, in particular generic decision procedures and upper complexity bounds, becomes applicable to arbitrary rank-1 modal logics, without regard to their semantic status; we thus obtain purely syntactic versions of such results. As an extended example, we apply our framework to recently defined deontic logics. In particular, our methods lead to the new result that these logics are strongly complete.
机译：Coalgebras为多种模态逻辑提供了统一的语义框架。先前已经证明，endofunctor的类代数总是可以在等级1上公理化。在这里，我们建立了相反的说法，即，每个等级1模态逻辑都具有健全且非常完整的类代语义。这是通过为给定的模态逻辑构造规范的联合代词语义来实现的，该规范的语义包括签名函子和模态算符的解释，而模态算符在所有此类结构中都是最终的。规范语义可以被看作是广泛解释的邻域语义的联合构造。对规范语义的最终限制会产生规范的弱完整语义，而且还具有Hennessy-Milner属性。结果，混数模态逻辑的机制，特别是通用决策程序和复杂性上限，变得适用于任意等级1模态逻辑，而与它们的语义状态无关。因此，我们获得了这种结果的纯粹句法版本。作为扩展示例，我们将我们的框架应用于最近定义的宗法逻辑。尤其是，我们的方法导致这些逻辑完全完成的新结果。

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