Viewpoint Development of Stochastic Hybrid Systems

Nowadays, due to the explosive spreading of networked and highly distributed systems, mastering system complexity becomes a critical issue. Two development and verification paradigms have become more popular: viewpoints and randomisation. The viewpoints offer large freedom and introduce concurrency and compositionality in the development process. Randomisation is now a traditional method for reducing complexity (comparing with deterministic models) and it offers finer analytical analysis tools (quantification over non-determinism, multi-valued logics, etc). In this paper, we propose a combination of these two paradigms introducing a viewpoint methodology for systems with stochastic behaviours

Bisimulation, Logic and Reachability Analysis for Markovian Systems

In the recent years, there have been a large amount of investigations on safety verification of uncertain continuous systems. In engineering and applied mathematics, this verification is called stochastic reachability analysis, while in computer science this is called probabilistic model checking (PMC). In the context of this work, we consider the two terms interchangeable. It is worthy to note that PMC has been mostly considered for discrete systems. Therefore, there is an issue of improving the application of computer science techniques in the formal verification of continuous stochastic systems. We present a new probabilistic logic of model theoretic nature. The terms of this logic express reachability properties and the logic formulas express statistical properties of terms. Moreover, we show that this logic characterizes a bisimulation relation for continuous time continuous space Markov processes. For this logic we define a new semantics using state space symmetries. This is a recent concept that was successfully used in model checking. Using this semantics, we prove a full abstraction result. Furthermore, we prove a result that can be used in model checking, namely that the bisimulation preserves the probabilities of the reachable sets

Towards a General Theory of Stochastic Hybrid Systems

In this paper we set up a mathematical structure, called Markov string, to obtaining a very general class of models for stochastic hybrid systems. Markov Strings are, in fact, a class of Markov processes, obtained by a mixing mechanism of stochastic processes, introduced by Meyer. We prove that Markov strings are strong Markov processes with the cadlag property. We then show how a very general class of stochastic hybrid processes can be embedded in the framework of Markov strings. This class, which is referred to as the General Stochastic Hybrid Systems (GSHS), includes as special cases all the classes of stochastic hybrid processes, proposed in the literature

Abstractions of Stochastic Hybrid Systems

In this paper we define a stochastic bisimulation concept for a very general class of stochastic hybrid systems, which subsumes most classes of stochastic hybrid systems. The definition of this bisimulation builds on the concept of zigzag morphism defined for strong Markov processes. The main result is that this stochastic bisimulation is indeed an equivalence relation. The secondary result is that this bisimulation relation for the stochastic hybrid system models used in this paper implies the same kind of bisimulation for their continuous parts and respectively for their jumping structures

Uncertainty and Reconfigurability in Hilbertean Formal Methods

Hilbertian Formal Methods is a recently introduced paradigm for embedded systems operating in harsh physical environments. This paradigm has been more developed for the deterministic case. However, it is very rare that a physical environment follows precisely a deterministic rule and then it is more realistic to consider stochastic models. A major problem in dealing with stochastic differential equations, the ubiquitous mathematical for phenomena arising from biology, medicine, meteorology and other domains, is that they can be solved only for very particular classes (linear and quasi linear). The Hilbertian Formal Methods are designed for situations when the solutions are not known (like for non-linear stochastic equations), but enough mathematical information about them can be derived helping in solving problems like stability, controllability, convergence, system design and verification. In this paper, we present an integrated formal model for embedded systems operating in uncertain and nonlinear environments that can reconfigure their communication structure. This is achieved by introducing the observability logic, which is a formal notation for the observations of environment evolutions. This logic is integrated with a probabilistic version of the Pi-calculus that makes possible the real time communication of the measurements of the continuous evolutions, concurrency and reconfiguration of the embedded system. For example, these characteristics are necessary for mobile robot brigades, storm surge barrier systems, sensor networks or cardiac stimulators

Towards a Formal Framework for Multidimensional Codesign

Multidimensional codesign is a recently proposed paradigm for integrating different system dimensions in sensor networks. Examples of such dimensions are logical and physical mobility, continuous and discrete transitions, deterministic and random evolutions and features resulting from their interaction, like deterministic and stochastic hybrid behaviours. In this paper, we propose a unifying computational model that considers multiple dimensions and an integration framework based on domain theory. In this framework new dimensions can be incrementally added, and we illustrate this technique by adding logical mobility to the computational model. The new model has a very promising modelling power, offering all formal ingredients of a neural network. We further investigate bisimulation for systems mixing physical and logical mobility. We identify and solve a compatibility problem between bisimulation relations arising from mobility and continuous behaviours

Formal Engineering Hybrid Systems: Semantic Underpinnings

In this work we investigate some issues in applying formal methods to hybrid system development and develop a categorical framework. We study the themes of stochastic reasoning, heterogeneous formal specification and retrenchment. Hybrid systems raise a rich pallets of aspects that need to be investigated, but never the issue of how the multitude of logics, methodologies and tools can be used altogether. We attack this very difficult issue using categorical logic. When applying formal methods hybrid systems, new (formal methods) mathematics can be created. In this sense, we present new developments in categorical logic, inspired by the control engineering way of treating different aspects of systems. As stochastic reasoning has recently seized its importance in modelling and analysing hybrid systems, we present a uniform categorical formalisation of discrete, continuous and stochastic hybrid systems. A categorical, semantic framework is developed in order to help relating different aspects of hybrid system development

State constrained reachability for stochastic hybrid systems

The stochastic hybrid systems constitute well established classes of realistic models of hybrid discrete/continuous dynamics subject to random perturbations, autonomous uncontrollable transitions, nondeterminism or uncertainty. Stochastic reachability analysis is a key factor in the verification and deployment of stochastic hybrid systems. The encouraging recent progress prompts us to rene the problem to cover more realistic situations. We extend the so called constrained reachability problem from the probabilistic discrete case to stochastic hybrid systems. Then we dene mathematically this problem, and we obtain the reach probabilities as solutions of a boundary value problem. The last problem is well studied and numerical, even symbolic solutions exist. This characterization is useful in stochastic control, in probabilistic path planning and for nano-systems

Topological superposition of abstractions of stochastic processes

In this paper, we present a sound integration mechanism for Markov processes that are abstractions of stochastic hybrid systems (SHS). In a previous work, we have defined a very general model of SHS and we proved that the realization of an SHS is a Markov process. Moreover, we have developed a verification strategy for the reachability analysis problem. We develop further this line of research by making verification modularly. To achieve this, the state space is decomposed into regions that might share a common border. An abstraction can be constructed on each region and the abstraction method can vary from one region to another. We show how these abstractions can be integrated to provide an abstraction for the entire system. We illustrate this technique for the reachability analysis problem

Styles in Heterogeneous Modelling With UML

Software development is becoming increasingly heterogeneous, and therefore the formal approaches to heterogeneity are geting very important but, unacceptable extremely complex. We propose a type theoretic approach based on the concept of style, as an attempt to simplify the complex interaction of different formal aspects of system specification. We use category theory to investigate three major semantic styles - algebraic, coalgebraic and relational - and specification methodologies like viewpoints and precise metamodeling