155 research outputs found
Weak bisimulation for coalgebras over order enriched monads
The paper introduces the notion of a weak bisimulation for coalgebras whose
type is a monad satisfying some extra properties. In the first part of the
paper we argue that systems with silent moves should be modelled
coalgebraically as coalgebras whose type is a monad. We show that the visible
and invisible part of the functor can be handled internally inside a monadic
structure. In the second part we introduce the notion of an ordered saturation
monad, study its properties, and show that it allows us to present two
approaches towards defining weak bisimulation for coalgebras and compare them.
We support the framework presented in this paper by two main examples of
models: labelled transition systems and simple Segala systems.Comment: 44 page
Generalized Vietoris Bisimulations
We introduce and study bisimulations for coalgebras on Stone spaces [14]. Our
notion of bisimulation is sound and complete for behavioural equivalence, and
generalizes Vietoris bisimulations [4]. The main result of our paper is that
bisimulation for a coalgebra is the topological closure of
bisimulation for the underlying coalgebra
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
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
Coalgebra Encoding for Efficient Minimization
Recently, we have developed an efficient generic partition refinement
algorithm, which computes behavioural equivalence on a state-based system given
as an encoded coalgebra, and implemented it in the tool CoPaR. Here we extend
this to a fully fledged minimization algorithm and tool by integrating two new
aspects: (1) the computation of the transition structure on the minimized state
set, and (2) the computation of the reachable part of the given system. In our
generic coalgebraic setting these two aspects turn out to be surprisingly
non-trivial requiring us to extend the previous theory. In particular, we
identify a sufficient condition on encodings of coalgebras, and we show how to
augment the existing interface, which encapsulates computations that are
specific for the coalgebraic type functor, to make the above extensions
possible. Both extensions have linear run time
Well-Pointed Coalgebras
For endofunctors of varieties preserving intersections, a new description of
the final coalgebra and the initial algebra is presented: the former consists
of all well-pointed coalgebras. These are the pointed coalgebras having no
proper subobject and no proper quotient. The initial algebra consists of all
well-pointed coalgebras that are well-founded in the sense of Osius and Taylor.
And initial algebras are precisely the final well-founded coalgebras. Finally,
the initial iterative algebra consists of all finite well-pointed coalgebras.
Numerous examples are discussed e.g. automata, graphs, and labeled transition
systems
Topo-Bisimulations are Coalgebraic
We show that the topological interpretation of the modal
logic S4 can be reformulated using a special kind of coalgebras for the
filter functor. Thus the topological semantics is subsumed in coalgebraic
semantics. Moreover, the relational notion of topo-bisimulation can
be characterized via spans of open and continuous maps of topological
spaces or via spans of coalgebras morphisms
- …