We present a generic partition refinement algorithm that quotients coalgebraic systems by be-havioural equivalence, an important task in reactive verification; coalgebraic generality implies in particular that we cover not only classical relational systems but also various forms of weighted systems. Under assumptions on the type functor that allow representing its finite coalgebras in terms of nodes and edges, our algorithm runs in time O(m⋅log n) where n and m are the numbers of nodes and

more »

... , respectively. Instances of our generic algorithm thus match the runtime of the best known algorithms for unlabelled transition systems, Markov chains, and deterministic automata (with fixed alphabets), and improve the best known algorithms for Segala systems. 1 Introduction Minimization under bisimilarity is the task of identifying all states in a reactive system that exhibit the same behaviour. Minimization appears as a subtask in state space reduction (e.g. [5]) or non-interference checking [34]. The notion of bisimulation was first defined for relational systems [33, 24, 26]; it was later extended to other system types including probabilistic systems [23, 9] and weighted automata [6]. In fact, the importance of minimization under bisimilarity appears to increase with the complexity of the underlying system type. E.g., while in LTL model checking, minimization drastically reduces the state space but, depending on the application, does not necessarily lead to a speedup in the overall balance [11], in probabilistic model checking, minimization under strong bisimilarity does lead to substantial efficiency gains [19]. The algorithmics of minimization, often referred to as partition refinement or lumping, has received a fair amount of attention. Since bisimilarity is a greatest fixpoint, it is more or less immediate that it can be calculated in polynomial time by approximating this fixpoint from above following Kleene's fixpoint theorem. In the relational setting, Kanellakis and * Full version with all proof details available at http://arxiv.org/abs/1705.08362. † This work forms part of the DFG-funded project COAX (MI 717/5-1 and SCHR 1118/12-1).