330 research outputs found
Semipullbacks of labelled Markov processes
A labelled Markov process (LMP) consists of a measurable space together
with an indexed family of Markov kernels from to itself. This structure has
been used to model probabilistic computations in Computer Science, and one of
the main problems in the area is to define and decide whether two LMP and
"behave the same". There are two natural categorical definitions of
sameness of behavior: and are bisimilar if there exist an LMP and
measure preserving maps forming a diagram of the shape ; and they are behaviorally equivalent if there exist some
and maps forming a dual diagram .
These two notions differ for general measurable spaces but Doberkat
(extending a result by Edalat) proved that they coincide for analytic Borel
spaces, showing that from every diagram one
can obtain a bisimilarity diagram as above. Moreover, the resulting square of
measure preserving maps is commutative (a "semipullback").
In this paper, we extend the previous result to measurable spaces
isomorphic to a universally measurable subset of a Polish space with the trace
of the Borel -algebra, using a version of Strassen's theorem on common
extensions of finitely additive measures.Comment: 10 pages; v2: missing attribution to Doberka
Distribution-based bisimulation for labelled Markov processes
In this paper we propose a (sub)distribution-based bisimulation for labelled
Markov processes and compare it with earlier definitions of state and event
bisimulation, which both only compare states. In contrast to those state-based
bisimulations, our distribution bisimulation is weaker, but corresponds more
closely to linear properties. We construct a logic and a metric to describe our
distribution bisimulation and discuss linearity, continuity and compositional
properties.Comment: Accepted by FORMATS 201
Bisimulation for Labelled Markov Processes
AbstractIn this paper we introduce a new class of labelled transition systems—labelled Markov processes— and define bisimulation for them. Labelled Markov processes are probabilistic labelled transition systems where the state space is not necessarily discrete. We assume that the state space is a certain type of common metric space called an analytic space. We show that our definition of probabilistic bisimulation generalizes the Larsen–Skou definition given for discrete systems. The formalism and mathematics is substantially different from the usual treatment of probabilistic process algebra. The main technical contribution of the paper is a logical characterization of probabilistic bisimulation. This study revealed some unexpected results, even for discrete probabilistic systems. •Bisimulation can be characterized by a very weak modal logic. The most striking feature is that one has no negation or any kind of negative proposition.•We do not need any finite branching assumption, yet there is no need of infinitary conjunction.
We also show how to construct the maximal autobisimulation on a system. In the finite state case, this is just a state minimization construction. The proofs that we give are of an entirely different character than the typical proofs of these results. They use quite subtle facts about analytic spaces and appear, at first sight, to be entirely nonconstructive. Yet one can give an algorithm for deciding bisimilarity of finite state systems which constructs a formula that witnesses the failure of bisimulation
Generalized labelled Markov processes, coalgebraically
Coalgebras of measurable spaces are of interest in probability theory as a formalization of Labelled Markov Processes (LMPs). We discuss some general facts related to the notions of bisimulation and cocongruence on these systems, providing a faithful characterization of bisimulation on LMPs on generic measurable
spaces. This has been used to prove that bisimilarity on single LMPs is an equivalence, without assuming the state space to be analytic. As the second main contribution, we introduce the first specification rule format to define well-behaved composition operators for LMPs. This allows one to define process description languages on LMPs which are always guaranteed to have a fully-abstract semantics
Unprovability of the Logical Characterization of Bisimulation
We quickly review labelled Markov processes (LMP) and provide a
counterexample showing that in general measurable spaces, event bisimilarity
and state bisimilarity differ in LMP. This shows that the logic in Desharnais
[*] does not characterize state bisimulation in non-analytic measurable spaces.
Furthermore we show that, under current foundations of Mathematics, such
logical characterization is unprovable for spaces that are projections of a
coanalytic set. Underlying this construction there is a proof that stationary
Markov processes over general measurable spaces do not have semi-pullbacks.
([*] J. Desharnais, Labelled Markov Processes. School of Computer Science.
McGill University, Montr\'eal (1999))Comment: Extended introduction and comments; extra section on semi-pullbacks;
11 pages Some background details added; extra example on the non-locality of
state bisimilarity; 14 page
Bisimulations for non-deterministic labelled Markov processes
We extend the theory of labelled Markov processes to include internal non-determinism, which is a fundamental concept for the further development of a process theory with abstraction on non-deterministic continuous probabilistic systems. We define non-deterministic labelled Markov processes (NLMP) and provide three definitions of bisimulations: a bisimulation following a traditional characterisation; a state-based bisimulation tailored to our 'measurable' non-determinism; and an event-based bisimulation. We show the relations between them, including the fact that the largest state bisimulation is also an event bisimulation. We also introduce a variation of the Hennessy-Milner logic that characterises event bisimulation and is sound with respect to the other bisimulations for an arbitrary NLMP. This logic, however, is infinitary as it contains a denumerable. We then introduce a finitary sublogic that characterises all bisimulations for an image finite NLMP whose underlying measure space is also analytic. Hence, in this setting, all the notions of bisimulation we consider turn out to be equal. Finally, we show that all these bisimulation notions are different in the general case. The counterexamples that separate them turn out to be non-probabilistic NLMPs.Fil: D'argenio, Pedro Ruben. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física. Sección Ciencias de la Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; ArgentinaFil: Sanchez Terraf, Pedro Octavio. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física. Sección Ciencias de la Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; ArgentinaFil: Wolovick, Nicolás. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física. Sección Ciencias de la Computación; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba; Argentin
Semipullbacks of labelled Markov processes
A labelled Markov process (LMP) consists of a measurable space together
with an indexed family of Markov kernels from to itself. This structure has
been used to model probabilistic computations in Computer Science, and one of
the main problems in the area is to define and decide whether two LMP and
"behave the same". There are two natural categorical definitions of
sameness of behavior: and are bisimilar if there exist an LMP and
measure preserving maps forming a diagram of the shape ; and they are behaviorally equivalent if there exist some
and maps forming a dual diagram .
These two notions differ for general measurable spaces but Doberkat
(extending a result by Edalat) proved that they coincide for analytic Borel
spaces, showing that from every diagram one can
obtain a bisimilarity diagram as above. Moreover, the resulting square of
measure preserving maps is commutative (a semipullback).
In this paper, we extend the previous result to measurable spaces
isomorphic to a universally measurable subset of a Polish space with the trace
of the Borel -algebra, using a version of Strassen's theorem on common
extensions of finitely additive measures
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