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：我们提出了一种通用方法，用于为各类基于状态的量化系统推导线性时间逻辑，其核心在于将这些系统建模为同时包含分支行为与线性行为类型的余代数。具体而言，我们定义了语法由线性行为类型决定、真值域由分支行为类型决定的逻辑，并为它们提供了两种语义：一种类似于标准余代数逻辑的逐步语义，另一种类似于标准线性时间逻辑的路径语义。前者适用于模型检测，而后者作为更自然的语义，用于度量从给定状态出发的计算路径上满足定性性质的程度。我们的主要结果是证明这两种语义的等价性。同时，我们还对这些逻辑所诱导的逻辑距离概念进行了语义刻画。所提出逻辑的实例支持对展现特定线性时间性质的可能性、概率或最小代价进行推理。