Maximal traces and path-based coalgebraic temporal logics

This paper gives a general coalgebraic account of temporal logics whose semantics involves a notion of computation path. Examples of such logics include the logic CTL* for transition systems and the logic PCTL for probabilistic transition systems. Our path-based temporal logics are interpreted over coalgebras of endofunctors obtained as the composition of a computation type (e.g. nondeterministic or stochastic) with a general transition type. The semantics of such logics relies on the existence of execution maps similar to the trace maps introduced by Jacobs and co-authors as part of the coalgebraic theory of finite traces [1]. We consider both finite execution maps derived from the theory of finite traces, and a new notion of maximal execution map that accounts for maximal, possibly infinite executions. The latter is needed to recover the logics CTL* and PCTL as specific path-based logics.