214 research outputs found
Generic Trace Semantics via Coinduction
Trace semantics has been defined for various kinds of state-based systems,
notably with different forms of branching such as non-determinism vs.
probability. In this paper we claim to identify one underlying mathematical
structure behind these "trace semantics," namely coinduction in a Kleisli
category. This claim is based on our technical result that, under a suitably
order-enriched setting, a final coalgebra in a Kleisli category is given by an
initial algebra in the category Sets. Formerly the theory of coalgebras has
been employed mostly in Sets where coinduction yields a finer process semantics
of bisimilarity. Therefore this paper extends the application field of
coalgebras, providing a new instance of the principle "process semantics via
coinduction."Comment: To appear in Logical Methods in Computer Science. 36 page
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
Behavioural equivalences for timed systems
Timed transition systems are behavioural models that include an explicit
treatment of time flow and are used to formalise the semantics of several
foundational process calculi and automata. Despite their relevance, a general
mathematical characterisation of timed transition systems and their behavioural
theory is still missing. We introduce the first uniform framework for timed
behavioural models that encompasses known behavioural equivalences such as
timed bisimulations, timed language equivalences as well as their weak and
time-abstract counterparts. All these notions of equivalences are naturally
organised by their discriminating power in a spectrum. We prove that this
result does not depend on the type of the systems under scrutiny: it holds for
any generalisation of timed transition system. We instantiate our framework to
timed transition systems and their quantitative extensions such as timed
probabilistic systems
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
A necessary and sufficient condition for induced model structures
A common technique for producing a new model category structure is to lift
the fibrations and weak equivalences of an existing model structure along a
right adjoint. Formally dual but technically much harder is to lift the
cofibrations and weak equivalences along a left adjoint. For either technique
to define a valid model category, there is a well-known necessary "acyclicity"
condition. We show that for a broad class of "accessible model structures" - a
generalization introduced here of the well-known combinatorial model structures
- this necessary condition is also sufficient in both the right-induced and
left-induced contexts, and the resulting model category is again accessible. We
develop new and old techniques for proving the acyclity condition and apply
these observations to construct several new model structures, in particular on
categories of differential graded bialgebras, of differential graded comodule
algebras, and of comodules over corings in both the differential graded and the
spectral setting. We observe moreover that (generalized) Reedy model category
structures can also be understood as model categories of "bialgebras" in the
sense considered here.Comment: 49 pages; final journal version to appear in the Journal of Topolog
Monoidal computer III: A coalgebraic view of computability and complexity
Monoidal computer is a categorical model of intensional computation, where
many different programs correspond to the same input-output behavior. The
upshot of yet another model of computation is that a categorical formalism
should provide a much needed high level language for theory of computation,
flexible enough to allow abstracting away the low level implementation details
when they are irrelevant, or taking them into account when they are genuinely
needed. A salient feature of the approach through monoidal categories is the
formal graphical language of string diagrams, which supports visual reasoning
about programs and computations.
In the present paper, we provide a coalgebraic characterization of monoidal
computer. It turns out that the availability of interpreters and specializers,
that make a monoidal category into a monoidal computer, is equivalent with the
existence of a *universal state space*, that carries a weakly final state
machine for any pair of input and output types. Being able to program state
machines in monoidal computers allows us to represent Turing machines, to
capture their execution, count their steps, as well as, e.g., the memory cells
that they use. The coalgebraic view of monoidal computer thus provides a
convenient diagrammatic language for studying computability and complexity.Comment: 34 pages, 24 figures; in this version: added the Appendi
Coalgebraic Semantics for Timed Processes
We give a coalgebraic formulation of timed processes and their operational semantics. We model time by a monoid called a “time domain”, and we model processes by “timed transition systems”, which amount to partial monoid actions of the time domain or, equivalently, coalgebras for an “evolution comonad ” generated by the time domain. All our examples of time domains satisfy a partial closure property, yielding a distributive law of a monad for total monoid actions over the evolution comonad, and hence a distributive law of the evolution comonad over a dual comonad for total monoid actions. We show that the induced coalgebras are exactly timed transition systems with delay operators. We then integrate our coalgebraic formulation of time qua timed transition systems into Turi and Plotkin’s formulation of structural operational semantics in terms of distributive laws. We combine timing with action via the more general study of the combination of two arbitrary sorts of behaviour whose operational semantics may interact. We give a modular account of the operational semantics for a combination induced by that of each of its components. Our study necessitates the investigation of products of comonads. In particular, we characterise when a monad lifts to the category of coalgebras for a product comonad, providing constructions with which one can readily calculate. Key words: time domains, timed transition systems, evolution comonads, delay operators, structural operational semantics, modularity, distributive laws
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