340 research outputs found
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
Bisimulation of Labelled State-to-Function Transition Systems Coalgebraically
Labeled state-to-function transition systems, FuTS for short, are
characterized by transitions which relate states to functions of states over
general semirings, equipped with a rich set of higher-order operators. As such,
FuTS constitute a convenient modeling instrument to deal with process languages
and their quantitative extensions in particular. In this paper, the notion of
bisimulation induced by a FuTS is addressed from a coalgebraic point of view. A
correspondence result is established stating that FuTS-bisimilarity coincides
with behavioural equivalence of the associated functor. As generic examples,
the equivalences underlying substantial fragments of major examples of
quantitative process algebras are related to the bisimilarity of specific FuTS.
The examples range from a stochastic process language, PEPA, to a language for
Interactive Markov Chains, IML, a (discrete) timed process language, TPC, and a
language for Markov Automata, MAL. The equivalences underlying these languages
are related to the bisimilarity of their specific FuTS. By the correspondence
result coalgebraic justification of the equivalences of these calculi is
obtained. The specific selection of languages, besides covering a large variety
of process interaction models and modelling choices involving quantities,
allows us to show different classes of FuTS, namely so-called simple FuTS,
combined FuTS, nested FuTS, and general FuTS
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
Bisimulation of Labeled State-to-Function Transition Systems of Stochastic Process Languages
Labeled state-to-function transition systems, FuTS for short, admit multiple
transition schemes from states to functions of finite support over general
semirings. As such they constitute a convenient modeling instrument to deal
with stochastic process languages. In this paper, the notion of bisimulation
induced by a FuTS is proposed and a correspondence result is proven stating
that FuTS-bisimulation coincides with the behavioral equivalence of the
associated functor. As generic examples, the concrete existing equivalences for
the core of the process algebras ACP, PEPA and IMC are related to the
bisimulation of specific FuTS, providing via the correspondence result
coalgebraic justification of the equivalences of these calculi.Comment: In Proceedings ACCAT 2012, arXiv:1208.430
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
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
A Coalgebraic Approach to Kleene Algebra with Tests
Kleene algebra with tests is an extension of Kleene algebra, the algebra of
regular expressions, which can be used to reason about programs. We develop a
coalgebraic theory of Kleene algebra with tests, along the lines of the
coalgebraic theory of regular expressions based on deterministic automata.
Since the known automata-theoretic presentation of Kleene algebra with tests
does not lend itself to a coalgebraic theory, we define a new interpretation of
Kleene algebra with tests expressions and a corresponding automata-theoretic
presentation. One outcome of the theory is a coinductive proof principle, that
can be used to establish equivalence of our Kleene algebra with tests
expressions.Comment: 21 pages, 1 figure; preliminary version appeared in Proc. Workshop on
Coalgebraic Methods in Computer Science (CMCS'03
Uniform Interpolation for Coalgebraic Fixpoint Logic
We use the connection between automata and logic to prove that a wide class
of coalgebraic fixpoint logics enjoys uniform interpolation. To this aim, first
we generalize one of the central results in coalgebraic automata theory, namely
closure under projection, which is known to hold for weak-pullback preserving
functors, to a more general class of functors, i.e.; functors with
quasi-functorial lax extensions. Then we will show that closure under
projection implies definability of the bisimulation quantifier in the language
of coalgebraic fixpoint logic, and finally we prove the uniform interpolation
theorem
Bisimilarity is not Borel
We prove that the relation of bisimilarity between countable labelled
transition systems is -complete (hence not Borel), by reducing the
set of non-wellorders over the natural numbers continuously to it.
This has an impact on the theory of probabilistic and nondeterministic
processes over uncountable spaces, since logical characterizations of
bisimilarity (as, for instance, those based on the unique structure theorem for
analytic spaces) require a countable logic whose formulas have measurable
semantics. Our reduction shows that such a logic does not exist in the case of
image-infinite processes.Comment: 20 pages, 1 figure; proof of Sigma_1^1 completeness added with
extended comments. I acknowledge careful reading by the referees. Major
changes in Introduction, Conclusion, and motivation for NLMP. Proof for Lemma
22 added, simpler proofs for Lemma 17 and Theorem 30. Added references. Part
of this work was presented at Dagstuhl Seminar 12411 on Coalgebraic Logic
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