64 research outputs found
Relating coalgebraic notions of bisimulation
The theory of coalgebras, for an endofunctor on a category, has been proposed
as a general theory of transition systems. We investigate and relate four
generalizations of bisimulation to this setting, providing conditions under
which the four different generalizations coincide. We study transfinite
sequences whose limits are the greatest bisimulations
A Definition Scheme for Quantitative Bisimulation
FuTS, state-to-function transition systems are generalizations of labeled
transition systems and of familiar notions of quantitative semantical models as
continuous-time Markov chains, interactive Markov chains, and Markov automata.
A general scheme for the definition of a notion of strong bisimulation
associated with a FuTS is proposed. It is shown that this notion of
bisimulation for a FuTS coincides with the coalgebraic notion of behavioral
equivalence associated to the functor on Set given by the type of the FuTS. For
a series of concrete quantitative semantical models the notion of bisimulation
as reported in the literature is proven to coincide with the notion of
quantitative bisimulation obtained from the scheme. The comparison includes
models with orthogonal behaviour, like interactive Markov chains, and with
multiple levels of behavior, like Markov automata. As a consequence of the
general result relating FuTS bisimulation and behavioral equivalence we obtain,
in a systematic way, a coalgebraic underpinning of all quantitative
bisimulations discussed.Comment: In Proceedings QAPL 2015, arXiv:1509.0816
Interaction and observation: categorical semantics of reactive systems trough dialgebras
We use dialgebras, generalising both algebras and coalgebras, as a complement
of the standard coalgebraic framework, aimed at describing the semantics of an
interactive system by the means of reaction rules. In this model, interaction
is built-in, and semantic equivalence arises from it, instead of being
determined by a (possibly difficult) understanding of the side effects of a
component in isolation. Behavioural equivalence in dialgebras is determined by
how a given process interacts with the others, and the obtained observations.
We develop a technique to inter-define categories of dialgebras of different
functors, that in particular permits us to compare a standard coalgebraic
semantics and its dialgebraic counterpart. We exemplify the framework using the
CCS and the pi-calculus. Remarkably, the dialgebra giving semantics to the
pi-calculus does not require the use of presheaf categories
Powerset-Like Monads Weakly Distribute over Themselves in Toposes and Compact Hausdorff Spaces
The powerset monad on the category of sets does not distribute over itself. Nevertheless a weaker form of distributive law of the powerset monad over itself exists and it essentially stems from the canonical Egli-Milner extension of the powerset to the category of relations. On the other hand, any regular category yields a category of relations, and some regular categories also possess a powerset-like monad, as is the Vietoris monad on compact Hausdorff spaces. We derive the Egli-Milner extension in three different frameworks : sets, toposes, and compact Hausdorff spaces. We prove that it corresponds to a monotone weak distributive law in each case by showing that the multiplication extends to relations but the unit does not. We provide an application to coalgebraic determinization of alternating automata
Universal Constructions for (Co)Relations: categories, monoidal categories, and props
Calculi of string diagrams are increasingly used to present the syntax and
algebraic structure of various families of circuits, including signal flow
graphs, electrical circuits and quantum processes. In many such approaches, the
semantic interpretation for diagrams is given in terms of relations or
corelations (generalised equivalence relations) of some kind. In this paper we
show how semantic categories of both relations and corelations can be
characterised as colimits of simpler categories. This modular perspective is
important as it simplifies the task of giving a complete axiomatisation for
semantic equivalence of string diagrams. Moreover, our general result unifies
various theorems that are independently found in literature and are relevant
for program semantics, quantum computation and control theory.Comment: 22 pages + 3 page appendix, extended version of arXiv:1703.0824
Coalgebraic Weak Bisimulation from Recursive Equations over Monads
Strong bisimulation for labelled transition systems is one of the most
fundamental equivalences in process algebra, and has been generalised to
numerous classes of systems that exhibit richer transition behaviour. Nearly
all of the ensuing notions are instances of the more general notion of
coalgebraic bisimulation. Weak bisimulation, however, has so far been much less
amenable to a coalgebraic treatment. Here we attempt to close this gap by
giving a coalgebraic treatment of (parametrized) weak equivalences, including
weak bisimulation. Our analysis requires that the functor defining the
transition type of the system is based on a suitable order-enriched monad,
which allows us to capture weak equivalences by least fixpoints of recursive
equations. Our notion is in agreement with existing notions of weak
bisimulations for labelled transition systems, probabilistic and weighted
systems, and simple Segala systems.Comment: final versio
A Uniform Framework for Timed Automata
Timed automata, and machines alike, currently lack a general mathematical characterisation. In this paper we provide a uniform coalgebraic understanding of these devices. This framework encompasses known behavioural equivalences for timed automata and paves the way for the extension of these notions to new timed behaviours and for the instantiation of established results from the coalgebraic theory as well. Key to this work is the use of lax functors for they allow us to model time flow as a context property and hence offer a general and expressive setting where to study timed systems: the index category encodes "how step sequences form executions" (e.g. whether steps duration get accumulated or kept distinct) whereas the base category encodes "step nature and composition" (e.g. non-determinism and labels). Finally, we develop the notion of general saturation for lax functors and show how equivalences of interest for timed behaviours are instances of this notion. This characterisation allows us to reason about the expressiveness of said notions within a uniform framework and organise them in a spectrum independent from the behavioural aspects encoded in the base category
On coalgebras with internal moves
In the first part of the paper we recall the coalgebraic approach to handling
the so-called invisible transitions that appear in different state-based
systems semantics. We claim that these transitions are always part of the unit
of a certain monad. Hence, coalgebras with internal moves are exactly
coalgebras over a monadic type. The rest of the paper is devoted to supporting
our claim by studying two important behavioural equivalences for state-based
systems with internal moves, namely: weak bisimulation and trace semantics.
We continue our research on weak bisimulations for coalgebras over order
enriched monads. The key notions used in this paper and proposed by us in our
previous work are the notions of an order saturation monad and a saturator. A
saturator operator can be intuitively understood as a reflexive, transitive
closure operator. There are two approaches towards defining saturators for
coalgebras with internal moves. Here, we give necessary conditions for them to
yield the same notion of weak bisimulation.
Finally, we propose a definition of trace semantics for coalgebras with
silent moves via a uniform fixed point operator. We compare strong and weak
bisimilation together with trace semantics for coalgebras with internal steps.Comment: Article: 23 pages, Appendix: 3 page
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