43,129 research outputs found
Weighted Modal Transition Systems
Specification theories as a tool in model-driven development processes of
component-based software systems have recently attracted a considerable
attention. Current specification theories are however qualitative in nature,
and therefore fragile in the sense that the inevitable approximation of systems
by models, combined with the fundamental unpredictability of hardware
platforms, makes it difficult to transfer conclusions about the behavior, based
on models, to the actual system. Hence this approach is arguably unsuited for
modern software systems. We propose here the first specification theory which
allows to capture quantitative aspects during the refinement and implementation
process, thus leveraging the problems of the qualitative setting.
Our proposed quantitative specification framework uses weighted modal
transition systems as a formal model of specifications. These are labeled
transition systems with the additional feature that they can model optional
behavior which may or may not be implemented by the system. Satisfaction and
refinement is lifted from the well-known qualitative to our quantitative
setting, by introducing a notion of distances between weighted modal transition
systems. We show that quantitative versions of parallel composition as well as
quotient (the dual to parallel composition) inherit the properties from the
Boolean setting.Comment: Submitted to Formal Methods in System Desig
Explaining Behavioural Inequivalence Generically in Quasilinear Time
We provide a generic algorithm for constructing formulae that distinguish behaviourally inequivalent states in systems of various transition types such as nondeterministic, probabilistic or weighted; genericity over the transition type is achieved by working with coalgebras for a set functor in the paradigm of universal coalgebra. For every behavioural equivalence class in a given system, we construct a formula which holds precisely at the states in that class. The algorithm instantiates to deterministic finite automata, transition systems, labelled Markov chains, and systems of many other types. The ambient logic is a modal logic featuring modalities that are generically extracted from the functor; these modalities can be systematically translated into custom sets of modalities in a postprocessing step. The new algorithm builds on an existing coalgebraic partition refinement algorithm. It runs in time ?((m+n) log n) on systems with n states and m transitions, and the same asymptotic bound applies to the dag size of the formulae it constructs. This improves the bounds on run time and formula size compared to previous algorithms even for previously known specific instances, viz. transition systems and Markov chains; in particular, the best previous bound for transition systems was ?(m n)
Quantitative Hennessy-Milner Theorems via Notions of Density
The classical Hennessy-Milner theorem is an important tool in the analysis of concurrent processes;
it guarantees that any two non-bisimilar states in finitely branching labelled transition systems can
be distinguished by a modal formula. Numerous variants of this theorem have since been established
for a wide range of logics and system types, including quantitative versions where lower bounds on
behavioural distance (e.g. in weighted, metric, or probabilistic transition systems) are witnessed
by quantitative modal formulas. Both the qualitative and the quantitative versions have been
accommodated within the framework of coalgebraic logic, with distances taking values in quantales,
subject to certain restrictions, such as being so-called value quantales. While previous quantitative
coalgebraic Hennessy-Milner theorems apply only to liftings of set functors to (pseudo)metric spaces,
in the present work we provide a quantitative coalgebraic Hennessy-Milner theorem that applies more
widely to functors native to metric spaces; notably, we thus cover, for the first time, the well-known
Hennessy-Milner theorem for continuous probabilistic transition systems, where transitions are given
by Borel measures on metric spaces, as an instance of such a general result. In the process, we also
relax the restrictions imposed on the quantale, and additionally parametrize the technical account
over notions of closure and, hence, density, providing associated variants of the Stone-WeierstraĂź
theorem; this allows us to cover, for instance, behavioural ultrametrics.publishe
Alternation-free weighted mu-calculus : decidability and completeness
In this paper we introduce WMC, a weighted version of the alternation-free modal mu-calculus for weighted transition systems. WMC subsumes previously studied weighted extensions of CTL and resembles previously proposed time-extended versions of the modal mu-calculus. We develop, in addition, a symbolic semantics for WMC and demonstrate that the notion of satisfiability coincides with that of symbolic satisfiability. This central result allows us to prove two major meta-properties of WMC. The first is decidability of satisfiability for WMC. In contrast to the classical modal mu-calculus, WMC does not possess the finite model-property. Nevertheless, the finite model property holds for the symbolic semantics and decidability readily follows; and this contrasts to resembling logics for timed transitions systems for which satisfiability has been shown undecidable. As a second main contribution, we provide a complete axiomatization, which applies to both semantics. The completeness proof is non-standard, since the logic is non-compact, and it involves the notion of symbolic models
Discrete Dynamical Systems: A Brief Survey
Dynamical system is a mathematical formalization for any fixed rule that is described in time dependent fashion. The time can be measured by either of the number systems - integers, real numbers, complex numbers. A discrete dynamical system is a dynamical system whose state evolves over a state space in discrete time steps according to a fixed rule. This brief survey paper is concerned with the part of the work done by José Sousa Ramos [2] and some of his research students. We present the general theory of discrete dynamical systems and present results from applications to geometry, graph theory and synchronization
Compositionality for Quantitative Specifications
We provide a framework for compositional and iterative design and
verification of systems with quantitative information, such as rewards, time or
energy. It is based on disjunctive modal transition systems where we allow
actions to bear various types of quantitative information. Throughout the
design process the actions can be further refined and the information made more
precise. We show how to compute the results of standard operations on the
systems, including the quotient (residual), which has not been previously
considered for quantitative non-deterministic systems. Our quantitative
framework has close connections to the modal nu-calculus and is compositional
with respect to general notions of distances between systems and the standard
operations
GSOS for non-deterministic processes with quantitative aspects
Recently, some general frameworks have been proposed as unifying theories for
processes combining non-determinism with quantitative aspects (such as
probabilistic or stochastically timed executions), aiming to provide general
results and tools. This paper provides two contributions in this respect.
First, we present a general GSOS specification format (and a corresponding
notion of bisimulation) for non-deterministic processes with quantitative
aspects. These specifications define labelled transition systems according to
the ULTraS model, an extension of the usual LTSs where the transition relation
associates any source state and transition label with state reachability weight
functions (like, e.g., probability distributions). This format, hence called
Weight Function SOS (WFSOS), covers many known systems and their bisimulations
(e.g. PEPA, TIPP, PCSP) and GSOS formats (e.g. GSOS, Weighted GSOS,
Segala-GSOS, among others).
The second contribution is a characterization of these systems as coalgebras
of a class of functors, parametric on the weight structure. This result allows
us to prove soundness of the WFSOS specification format, and that
bisimilarities induced by these specifications are always congruences.Comment: In Proceedings QAPL 2014, arXiv:1406.156
Structural Refinement for the Modal nu-Calculus
We introduce a new notion of structural refinement, a sound abstraction of
logical implication, for the modal nu-calculus. Using new translations between
the modal nu-calculus and disjunctive modal transition systems, we show that
these two specification formalisms are structurally equivalent.
Using our translations, we also transfer the structural operations of
composition and quotient from disjunctive modal transition systems to the modal
nu-calculus. This shows that the modal nu-calculus supports composition and
decomposition of specifications.Comment: Accepted at ICTAC 201
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