Stochastic hybrid system : modelling and verification

Hybrid systems now form a classical computational paradigm unifying discrete and continuous system aspects. The modelling, analysis and verification of these systems are very difficult. One way to reduce the complexity of hybrid system models is to consider randomization. The need for stochastic models has actually multiple motivations. Usually, when building models complete information is not available and we have to consider stochastic versions. Moreover, non-determinism and uncertainty are inherent to complex systems. The stochastic approach can be thought of as a way of quantifying non-determinism (by assigning a probability to each possible execution branch) and managing uncertainty. This is built upon to the - now classical - approach in algorithmics that provides polynomial complexity algorithms via randomization. In this thesis we investigate the stochastic hybrid systems, focused on modelling and analysis. We propose a powerful unifying paradigm that combines analytical and formal methods. Its applications vary from air traffic control to communication networks and healthcare systems. The stochastic hybrid system paradigm has an explosive development. This is because of its very powerful expressivity and the great variety of possible applications. Each hybrid system model can be randomized in different ways, giving rise to many classes of stochastic hybrid systems. Moreover, randomization can change profoundly the mathematical properties of discrete and continuous aspects and also can influence their interaction. Beyond the profound foundational and semantics issues, there is the possibility to combine and cross-fertilize techniques from analytic mathematics (like optimization, control, adaptivity, stability, existence and uniqueness of trajectories, sensitivity analysis) and formal methods (like bisimulation, specification, reachability analysis, model checking). These constitute the major motivations of our research. We investigate new models of stochastic hybrid systems and their associated problems. The main difference from the existing approaches is that we do not follow one way (based only on continuous or discrete mathematics), but their cross-fertilization. For stochastic hybrid systems we introduce concepts that have been defined only for discrete transition systems. Then, techniques that have been used in discrete automata now come in a new analytical fashion. This is partly explained by the fact that popular verification methods (like theorem proving) can hardly work even on probabilistic extensions of discrete systems. When the continuous dimension is added, the idea to use continuous mathematics methods for verification purposes comes in a natural way. The concrete contribution of this thesis has four major milestones: 1. A new and a very general model for stochastic hybrid systems; 2. Stochastic reachability for stochastic hybrid systems is introduced together with an approximating method to compute reach set probabilities; 3. Bisimulation for stochastic hybrid systems is introduced and relationship with reachability analysis is investigated. 4. Considering the communication issue, we extend the modelling paradigm

Measures on probabilistic automata

In questa tesi consideriamo i processi probabilistici non-deterministici modellati attraverso automi. Il nostro obiettivo \`e l'analisi dei problemi di bisimulazioni approssimate. Queste relazioni sono usate, generalmente, per semplificare i modelli di alcuni sistemi e per modellare agenti e attaccanti nei protocolli di sicurezza. In questo ultimo campo ci sono diversi proposte di utilizzo di metriche, le quali sono l'analogo quantitativo della bisimulazione probabilistica e permettono una miglior precisione. Una metrica \`e grossomodo un grado di similarit\`a tra stati. Iniziando dalla formalizzazione di (bi)simulazione approssimata data nel lavoro di Turrini, definiamo due metriche su stati e su distribuzioni. Queste metriche sono basate sul concetto di errore ammesso durante la simulazione di uno stato rispetto un altro stato. Investigheremo la relazione tra queste metriche con una metrica largamente utilizzata, la metrica di Kantorovich, e scopriremo che esse sono equivalenti. Poi riadatteremo per gli automi probabilistici il trasformatore di misure proposto da De Alfaro e al., ottenendo un nuovo funzionale F che \`e una estensione conservativa dei trasformatori proposti in letteratura. Mostreremo che il minimo punto fisso di F coincide con la sua sovra-approssimazione dalle misure derivate dal lavoro di Turrini, attraverso la dimostrazione dell'esistenza di una stretta relazione tra le bisimulazioni approssimate di Turrini con le metriche in letteratura.In this thesis we consider nondeterministic probabilistic processes modeled by automata. Our purpose is the analysis of the problem of approximated bisimulations. These relations are used, generally, to simplify the models of some systems and to model agents and attackers in security protocols. For the latter field there are several proposals to use metrics, which are the quantitative analogue of probabilistic bisimilarity and allow a greater precision. A metric is about a degree of similarity between states. Starting from the formalisation of approximate (bi)simulation given in Turrini's work, we define two metrics on states and on distributions. These metrics are based on the concept of error allowed during the simulation of a state with respect to another one. We investigate the relation between these metrics with a largely used one, the Kantorovich metric, and discover that they are equivalent. Then we recast for probabilistic automata the transformer of measures proposed by De Alfaro et al., obtaining a new functional F that is a conservative extension of the transformers proposed in the literature. We show that the minimum fix point of F coincides with its over-aproximated by the measures derived from Turrini's work thus showing the existence of a strict relation between the Turrini\u2019s approximate bisimulations with the literature on metrics

2008 Abstracts Collection -- IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science

This volume contains the proceedings of the 28th international conference on the Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2008), organized under the auspices of the Indian Association for Research in Computing Science (IARCS)

Robust Control for Dynamical Systems With Non-Gaussian Noise via Formal Abstractions

Controllers for dynamical systems that operate in safety-critical settings must account for stochastic disturbances. Such disturbances are often modeled as process noise in a dynamical system, and common assumptions are that the underlying distributions are known and/or Gaussian. In practice, however, these assumptions may be unrealistic and can lead to poor approximations of the true noise distribution. We present a novel controller synthesis method that does not rely on any explicit representation of the noise distributions. In particular, we address the problem of computing a controller that provides probabilistic guarantees on safely reaching a target, while also avoiding unsafe regions of the state space. First, we abstract the continuous control system into a finite-state model that captures noise by probabilistic transitions between discrete states. As a key contribution, we adapt tools from the scenario approach to compute probably approximately correct (PAC) bounds on these transition probabilities, based on a finite number of samples of the noise. We capture these bounds in the transition probability intervals of a so-called interval Markov decision process (iMDP). This iMDP is, with a user-specified confidence probability, robust against uncertainty in the transition probabilities, and the tightness of the probability intervals can be controlled through the number of samples. We use state-of-the-art verification techniques to provide guarantees on the iMDP and compute a controller for which these guarantees carry over to the original control system. In addition, we develop a tailored computational scheme that reduces the complexity of the synthesis of these guarantees on the iMDP. Benchmarks on realistic control systems show the practical applicability of our method, even when the iMDP has hundreds of millions of transitions.Comment: To appear in the Journal of Artificial Intelligence Research (JAIR). arXiv admin note: text overlap with arXiv:2110.1266

Coalgebra Encoding for Efficient Minimization

Recently, we have developed an efficient generic partition refinement algorithm, which computes behavioural equivalence on a state-based system given as an encoded coalgebra, and implemented it in the tool CoPaR. Here we extend this to a fully fledged minimization algorithm and tool by integrating two new aspects: (1) the computation of the transition structure on the minimized state set, and (2) the computation of the reachable part of the given system. In our generic coalgebraic setting these two aspects turn out to be surprisingly non-trivial requiring us to extend the previous theory. In particular, we identify a sufficient condition on encodings of coalgebras, and we show how to augment the existing interface, which encapsulates computations that are specific for the coalgebraic type functor, to make the above extensions possible. Both extensions have linear run time

Computer Aided Verification

This open access two-volume set LNCS 11561 and 11562 constitutes the refereed proceedings of the 31st International Conference on Computer Aided Verification, CAV 2019, held in New York City, USA, in July 2019. The 52 full papers presented together with 13 tool papers and 2 case studies, were carefully reviewed and selected from 258 submissions. The papers were organized in the following topical sections: Part I: automata and timed systems; security and hyperproperties; synthesis; model checking; cyber-physical systems and machine learning; probabilistic systems, runtime techniques; dynamical, hybrid, and reactive systems; Part II: logics, decision procedures; and solvers; numerical programs; verification; distributed systems and networks; verification and invariants; and concurrency