22 research outputs found
Bisimilar States in Uncertain Structures
We provide a categorical notion called uncertain bisimilarity, which allows
to reason about bisimilarity in combination with a lack of knowledge about the
involved systems. Such uncertainty arises naturally in automata learning
algorithms, where one investigates whether two observed behaviours come from
the same internal state of a black-box system that can not be transparently
inspected. We model this uncertainty as a set functor equipped with a partial
order which describes possible future developments of the learning game. On
such a functor, we provide a lifting-based definition of uncertain bisimilarity
and verify basic properties. Beside its applications to Mealy machines, a
natural model for automata learning, our framework also instantiates to an
existing compatibility relation on suspension automata, which are used in
model-based testing. We show that uncertain bisimilarity is a necessary but not
sufficient condition for two states being implementable by the same state in
the black-box system. To remedy the failure of the one direction, we
characterize uncertain bisimilarity in terms of coalgebraic simulations
Efficient and Modular Coalgebraic Partition Refinement
We present a generic partition refinement algorithm that quotients
coalgebraic systems by behavioural equivalence, an important task in system
analysis and verification. Coalgebraic generality allows us to cover not only
classical relational systems but also, e.g. various forms of weighted systems
and furthermore to flexibly combine existing system types. Under assumptions on
the type functor that allow representing its finite coalgebras in terms of
nodes and edges, our algorithm runs in time where
and are the numbers of nodes and edges, respectively. The generic
complexity result and the possibility of combining system types yields a
toolbox for efficient partition refinement algorithms. Instances of our generic
algorithm match the run-time of the best known algorithms for unlabelled
transition systems, Markov chains, deterministic automata (with fixed
alphabets), Segala systems, and for color refinement.Comment: Extended journal version of the conference paper arXiv:1705.08362.
Beside reorganization of the material, the introductory section 3 is entirely
new and the other new section 7 contains new mathematical result
A Coalgebraic View on Reachability
Coalgebras for an endofunctor provide a category-theoretic framework for
modeling a wide range of state-based systems of various types. We provide an
iterative construction of the reachable part of a given pointed coalgebra that
is inspired by and resembles the standard breadth-first search procedure to
compute the reachable part of a graph. We also study coalgebras in Kleisli
categories: for a functor extending a functor on the base category, we show
that the reachable part of a given pointed coalgebra can be computed in that
base category