We introduce a novel real-valued endogenous logic for expressing properties of probabilistic transition systems called Riesz modal logic. The design of the syntax and semantics of this logic is directly inspired by the theory of Riesz spaces, a mature field of mathematics at the intersection of universal algebra and functional analysis. By using powerful results from this theory, we develop the duality theory of the Riesz modal logic in the form of an algebra-to-coalgebra correspondence. This has a number of consequences including: a sound and complete axiomatization, the proof that the logic characterizes probabilistic bisimulation and other convenient results such as completion theorems. This work is intended to be the basis for subsequent research on extensions of Riesz modal logic with fixed-point operators.

中文翻译：

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基于 Riesz 空间的概率逻辑

我们引入了一种新的实值内生逻辑，用于表达称为 Riesz 模态逻辑的概率转换系统的特性。该逻辑的句法和语义设计直接受到 Riesz 空间理论的启发，这是一个成熟的数学领域，处于泛代数和泛函分析的交叉点。通过使用该理论的强大结果，我们以代数到余代数对应的形式发展了 Riesz 模态逻辑的对偶理论。这有许多结果，包括：合理且完整的公理化，证明该逻辑表征概率互模拟和其他方便的结果，例如完成定理。这项工作旨在成为后续研究使用定点算子扩展 Riesz 模态逻辑的基础。