From generic partition refinement to weighted tree automata minimization

Partition refinement is a method for minimizing automata and transition systems of various types. Recently, we have developed a partition refinement algorithm that is generic in the transition type of the given system and matches the run time of the best known algorithms for many concrete types of systems, e.g. deterministic automata as well as ordinary, weighted, and probabilistic (labelled) transition systems. Genericity is achieved by modelling transition types as functors on sets, and systems as coalgebras. In the present work, we refine the run time analysis of our algorithm to cover additional instances, notably weighted automata and, more generally, weighted tree automata. For weights in a cancellative monoid we match, and for non-cancellative monoids such as (the additive monoid of) the tropical semiring even substantially improve, the asymptotic run time of the best known algorithms. We have implemented our algorithm in a generic tool that is easily instantiated to concrete system types by implementing a simple refinement interface. Moreover, the algorithm and the tool are modular, and partition refiners for new types of systems are obtained easily by composing pre-implemented basic functors. Experiments show that even for complex system types, the tool is able to handle systems with millions of transitions.