In traditional justification logic, evidence terms have the syntactic form of polynomials, but they are not equipped with the corresponding algebraic structure. We present a novel semantic approach to justification logic that models evidence by a semiring. Hence justification terms can be interpreted as polynomial functions on that semiring. This provides an adequate semantics for evidence terms and clarifies the role of variables in justification logic. Moreover, the algebraic structure makes it possible to compute with evidence. Depending on the chosen semiring this can be used to model trust, probabilities, cost, etc. Last but not least the semiring approach seems promising for obtaining a realization procedure for modal fixed point logics.

中文翻译：

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证据的半环

在传统的论证逻辑中，证据项具有多项式的句法形式，但没有相应的代数结构。我们提出了一种新颖的证明逻辑语义方法，该方法通过半环对证据进行建模。因此，对齐项可以解释为该半环上的多项式函数。这为证据术语提供了足够的语义，并阐明了变量在证明逻辑中的作用。此外，代数结构使得有证据计算成为可能。根据选择的半环，这可用于对信任、概率、成本等进行建模。最后但并非最不重要的一点是，半环方法似乎有望获得模态不动点逻辑的实现过程。