13 research outputs found
Simplified Coalgebraic Trace Equivalence
The analysis of concurrent and reactive systems is based to a large degree on
various notions of process equivalence, ranging, on the so-called
linear-time/branching-time spectrum, from fine-grained equivalences such as
strong bisimilarity to coarse-grained ones such as trace equivalence. The
theory of concurrent systems at large has benefited from developments in
coalgebra, which has enabled uniform definitions and results that provide a
common umbrella for seemingly disparate system types including
non-deterministic, weighted, probabilistic, and game-based systems. In
particular, there has been some success in identifying a generic coalgebraic
theory of bisimulation that matches known definitions in many concrete cases.
The situation is currently somewhat less settled regarding trace equivalence. A
number of coalgebraic approaches to trace equivalence have been proposed, none
of which however cover all cases of interest; notably, all these approaches
depend on explicit termination, which is not always imposed in standard
systems, e.g. LTS. Here, we discuss a joint generalization of these approaches
based on embedding functors modelling various aspects of the system, such as
transition and braching, into a global monad; this approach appears to cover
all cases considered previously and some additional ones, notably standard LTS
and probabilistic labelled transition systems
Towards Trace Metrics via Functor Lifting
We investigate the possibility of deriving metric trace semantics in a coalgebraic framework. First, we generalize a technique for systematically lifting functors from the category Set of sets to the category PMet of pseudometric spaces, by identifying conditions under which also natural transformations, monads and distributive laws can be lifted. By exploiting some recent work on an abstract determinization, these results enable the derivation of trace metrics starting from coalgebras in Set. More precisely, for a coalgebra in Set we determinize it, thus obtaining a coalgebra in the Eilenberg-Moore category of a monad. When the monad can be lifted to PMet, we can equip the final coalgebra with a behavioral distance. The trace distance between two states of the original coalgebra is the distance between their images in the determinized coalgebra through the unit of the monad. We show how our framework applies to nondeterministic automata and probabilistic automata
Towards Trace Metrics via Functor Lifting
We investigate the possibility of deriving metric trace semantics in a
coalgebraic framework. First, we generalize a technique for systematically
lifting functors from the category Set of sets to the category PMet of
pseudometric spaces, showing under which conditions also natural
transformations, monads and distributive laws can be lifted. By exploiting some
recent work on an abstract determinization, these results enable the derivation
of trace metrics starting from coalgebras in Set. More precisely, for a
coalgebra on Set we determinize it, thus obtaining a coalgebra in the
Eilenberg-Moore category of a monad. When the monad can be lifted to PMet, we
can equip the final coalgebra with a behavioral distance. The trace distance
between two states of the original coalgebra is the distance between their
images in the determinized coalgebra through the unit of the monad. We show how
our framework applies to nondeterministic automata and probabilistic automata
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
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
Stream processors and comodels
In 2009, Hancock, Pattinson and Ghani gave a coalgebraic characterisation of
stream processors drawing on ideas of
Brouwerian constructivism. Their stream processors have an intensional
character; in this paper, we give a corresponding coalgebraic characterisation
of extensional stream processors, i.e., the set of continuous functions
. Our account sites both our result and that of
op. cit. within the apparatus of comodels for algebraic effects originating
with Power-Shkaravska. Within this apparatus, the distinction between
intensional and extensional equivalence for stream processors arises in the
same way as the the distinction between bisimulation and trace equivalence for
labelled transition systems and probabilistic generative systems.Comment: 24 pages; v4: final accepted versio
A coalgebraic treatment of conditional transition systems with upgrades
We consider conditional transition systems, that model software product lines with upgrades, in a coalgebraic setting. By using Birkhoff's duality for distributive lattices, we derive two equivalent Kleisli categories in which these coalgebras live: Kleisli categories based on the reader and on the so-called lattice monad over Poset. We study two different functors describing the branching type of the coalgebra and investigate the resulting behavioural equivalence. Furthermore we show how an existing algorithm for coalgebra minimisation can be instantiated to derive behavioural equivalences in this setting
Stream processors and comodels
In 2009, Hancock, Pattinson and Ghani gave a coalgebraic characterisation of
stream processors drawing on ideas of
Brouwerian constructivism. Their stream processors have an intensional
character; in this paper, we give a corresponding coalgebraic characterisation
of extensional stream processors, i.e., the set of continuous functions
. Our account sites both our result and that of
op. cit. within the apparatus of comodels for algebraic effects originating
with Power-Shkaravska. Within this apparatus, the distinction between
intensional and extensional equivalence for stream processors arises in the
same way as the the distinction between bisimulation and trace equivalence for
labelled transition systems and probabilistic generative systems