357 research outputs found
Semantic Factorization and Descent
Let be a -category with suitable opcomma objects and
pushouts. We give a direct proof that, provided that the codensity monad of a
morphism exists and is preserved by a suitable morphism, the factorization
given by the lax descent object of the higher cokernel of is up to
isomorphism the same as the semantic factorization of , either one existing
if the other does. The result can be seen as a counterpart account to the
celebrated B\'{e}nabou-Roubaud theorem. This leads in particular to a
monadicity theorem, since it characterizes monadicity via descent. It should be
noted that all the conditions on the codensity monad of trivially hold
whenever has a left adjoint and, hence, in this case, we find monadicity to
be a -dimensional exact condition on , namely, to be an effective
faithful morphism of the -category .Comment: Full revision, new diagrams, 48 page
Notions of Monad Strength
Over the past two decades the notion of a strong monad has found wide
applicability in computing. Arising out of a need to interpret products in
computational and semantic settings, different approaches to this concept have
arisen. In this paper we introduce and investigate the connections between
these approaches and also relate the results to monad composition. We also
introduce new methods for checking and using the required laws associated with
such compositions, as well as provide examples illustrating problems and issues
that arise.Comment: In Proceedings Festschrift for Dave Schmidt, arXiv:1309.455
Coalgebraic Trace Semantics for Continuous Probabilistic Transition Systems
Coalgebras in a Kleisli category yield a generic definition of trace
semantics for various types of labelled transition systems. In this paper we
apply this generic theory to generative probabilistic transition systems, short
PTS, with arbitrary (possibly uncountable) state spaces. We consider the
sub-probability monad and the probability monad (Giry monad) on the category of
measurable spaces and measurable functions. Our main contribution is that the
existence of a final coalgebra in the Kleisli category of these monads is
closely connected to the measure-theoretic extension theorem for sigma-finite
pre-measures. In fact, we obtain a practical definition of the trace measure
for both finite and infinite traces of PTS that subsumes a well-known result
for discrete probabilistic transition systems. Finally we consider two example
systems with uncountable state spaces and apply our theory to calculate their
trace measures
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
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
Unguarded Recursion on Coinductive Resumptions
We study a model of side-effecting processes obtained by starting from a
monad modelling base effects and adjoining free operations using a cofree
coalgebra construction; one thus arrives at what one may think of as types of
non-wellfounded side-effecting trees, generalizing the infinite resumption
monad. Correspondingly, the arising monad transformer has been termed the
coinductive generalized resumption transformer. Monads of this kind have
received some attention in the recent literature; in particular, it has been
shown that they admit guarded iteration. Here, we show that they also admit
unguarded iteration, i.e. form complete Elgot monads, provided that the
underlying base effect supports unguarded iteration. Moreover, we provide a
universal characterization of the coinductive resumption monad transformer in
terms of coproducts of complete Elgot monads.Comment: 47 pages, extended version of
http://www.sciencedirect.com/science/article/pii/S157106611500079
Coalgebraic Infinite Traces and Kleisli Simulations
Kleisli simulation is a categorical notion introduced by Hasuo to verify
finite trace inclusion. They allow us to give definitions of forward and
backward simulation for various types of systems. A generic categorical theory
behind Kleisli simulation has been developed and it guarantees the soundness of
those simulations with respect to finite trace semantics. Moreover, those
simulations can be aided by forward partial execution (FPE)---a categorical
transformation of systems previously introduced by the authors.
In this paper, we give Kleisli simulation a theoretical foundation that
assures its soundness also with respect to infinitary traces. There, following
Jacobs' work, infinitary trace semantics is characterized as the "largest
homomorphism." It turns out that soundness of forward simulations is rather
straightforward; that of backward simulation holds too, although it requires
certain additional conditions and its proof is more involved. We also show that
FPE can be successfully employed in the infinitary trace setting to enhance the
applicability of Kleisli simulations as witnesses of trace inclusion. Our
framework is parameterized in the monad for branching as well as in the functor
for linear-time behaviors; for the former we mainly use the powerset monad (for
nondeterminism), the sub-Giry monad (for probability), and the lift monad (for
exception).Comment: 39 pages, 1 figur
Automata Minimization: a Functorial Approach
In this paper we regard languages and their acceptors - such as deterministic
or weighted automata, transducers, or monoids - as functors from input
categories that specify the type of the languages and of the machines to
categories that specify the type of outputs. Our results are as follows:
A) We provide sufficient conditions on the output category so that
minimization of the corresponding automata is guaranteed.
B) We show how to lift adjunctions between the categories for output values
to adjunctions between categories of automata.
C) We show how this framework can be instantiated to unify several phenomena
in automata theory, starting with determinization, minimization and syntactic
algebras. We provide explanations of Choffrut's minimization algorithm for
subsequential transducers and of Brzozowski's minimization algorithm in this
setting.Comment: journal version of the CALCO 2017 paper arXiv:1711.0306
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