Modal separation logics are formalisms that combine modal operators to reason locally, with separating connectives that allow to perform global updates on the models. In this work, we design Hilbert-style proof systems for the modal separation logics $\text {MSL}(\ast ,\langle \neq \rangle )$ and $\text {MSL}(\ast ,\Diamond )$, where $\ast $ is the separating conjunction, $\Diamond $ is the standard modal operator and $\langle \neq \rangle $ is the difference modality. The calculi only use the logical languages at hand (no external features such as labels) and can be divided in two main parts. First, normal forms for formulae are designed and the calculi allow to transform every formula into a formula in normal form. Second, another part of the calculi is dedicated to the axiomatization for formulae in normal form, which may still require non-trivial developments but is more manageable.

中文翻译：

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具有分离合取的模态逻辑的内部证明演算

模态分离逻辑是结合模态运算符以进行局部推理的形式，分离连接词允许对模型执行全局更新。在这项工作中，我们为模态分离逻辑 $\text {MSL}(\ast ,\langle \neq \rangle )$ 和 $\text {MSL}(\ast ,\Diamond )$ 设计了希尔伯特式证明系统，其中 $\ast $ 是分离连词，$\Diamond $ 是标准模态运算符，$\langle \neq \rangle $ 是差分模态。演算仅使用手头的逻辑语言（没有标签等外部特征），可以分为两个主要部分。首先，设计公式的范式，微积分允许将每个公式转换为范式。其次，微积分的另一部分专门用于范式公式的公理化，