We describe a general approach to deriving linear-time logics for a wide variety of state-based, quantitative systems, by modelling the latter as coalgebras whose type incorporates both branching and linear behaviour. Concretely, we define logics whose syntax is determined by the type of linear behaviour, and whose domain of truth values is determined by the type of branching behaviour, and we provide two semantics for them: a step-wise semantics akin to that of standard coalgebraic logics, and a path-based semantics akin to that of standard linear-time logics. The former semantics is useful for model checking, whereas the latter is the more natural semantics, as it measures the extent with which qualitative properties hold along computation paths from a given state. Our main result is the equivalence of the two semantics. We also provide a semantic characterisation of a notion of logical distance induced by these logics. Instances of our logics support reasoning about the possibility, likelihood or minimal cost of exhibiting a given linear-time property.

翻译：本文提出一种通用方法，通过将基于状态、定量系统的广泛类别建模为兼具分支行为与线性行为类型的余代数，来推导其线性时态逻辑。具体而言，我们定义了一种逻辑：其语法由线性行为类型决定，真值域由分支行为类型决定，并为其提供两种语义——类似于标准余代数逻辑的逐步语义，以及类似于标准线性时态逻辑的基于路径语义。前者适用于模型检验，后者作为更自然的语义，可衡量从给定状态出发的计算路径上定性性质的满足程度。本文主要结论证明两种语义的等价性，同时给出该类逻辑诱导的逻辑距离概念的语义刻画。该逻辑的实例支持对系统展现特定线性时态性质的可能性、似然性或最小代价进行推理。