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This tutorial volume presents a coherent and well-balanced introduction to the validation of stochastic systems; it is based on a GI/Dagstuhl research seminar. Supervised by the seminar organizers and volume editors, established researchers in the area as well as graduate students put together a collection of articles competently covering all relevant issues in the area. The lectures are organized in topical sections on: modeling stochastic systems, model checking of stochastic systems, representing large state spaces, deductive verification of stochastic systems.
It is common practice in structural dynamics to develop mathematical models for system behavior, and the authors are now capable of developing stochastic models, i.e., models whose parameters are random variables. Such models have random characteristics that are meant to simulate the randomness in characteristics of experimentally observed systems. This paper suggests a formal statistical procedure for the validation of mathematical models of stochastic systems when data taken during operation of the stochastic system are available. The statistical characteristics of the experimental system are obtained using the bootstrap, a technique for the statistical analysis of non-Gaussian data. The authors propose a procedure to determine whether or not a mathematical model is an acceptable model of a stochastic system with regard to user-specified measures of system behavior. A numerical example is presented to demonstrate the application of the technique.
Abstract: This thesis establishes theoretical and computational frameworks for formal verification and control synthesis for discrete-time stochastic systems. Given a temporal logic specification, the system is analyzed to determine the probability that the specification is achieved, and an input law is automatically generated to maximize this probability. The approach consists of three main steps: constructing an abstraction of the stochastic system as a finite Markov model, mapping the given specification onto this abstraction, and finding a control policy to maximize the probability of satisfying the specification. The framework uses Probabilistic Computation Tree Logic (PCTL) as the specification language. The verification and synthesis algorithms are inspired by the field of probabilistic model checking. In abstraction, a method for the computation of the exact transition probability bounds between the regions of interest in the domain of the stochastic system is first developed. These bounds are then used to construct an Interval-valued Markov Chain (IMC) or a Bounded-parameter Markov Decision Process (BMDP) abstraction for the system. Then, a representative transition probability is used to construct an approximating Markov chain (MC) for the stochastic system. The exact bound of the approximation error and an explicit expression for its growth over time are derived. To achieve a desired error value, an adaptive refinement algorithm that takes advantage of the linear dynamics of the system is employed.To verify the properties of the continuous domain stochastic system against a finite-time PCTL specification, IMC and BMDP verification algorithms are designed. These algorithms have low computational complexity and are inspired by the MC model checking algorithms. The low computational complexity is achieved by over approximating the probabilities of satisfaction. To increase the precision of the method, two adaptive refinement procedures are proposed. Furthermore, a method of generating the control strategy that maximizes the probability of satisfaction of a PCTL specification for Markov Decision Processes (MDPs) is developed. Through a similar method, a formal synthesis framework is constructed for continuous domain stochastic systems by utilizing their BMDP abstractions. These methodologies are then applied in robotics applications as a means of automatically deploying a mobile robot subject to noisy sensors and actuators from PCTL specifications. This technique is demonstrated through simulation and experimental case studies of deployment of a robot in an indoor environment. The contributions of the thesis include verification and synthesis frameworks for discrete time stochastic linear systems, abstraction schemes for stochastic systems to MCs, IMCs, and BMDPs, model checking algorithms with low computational complexity for IMCs and BMDPs against finite-time PCTL formulas, synthesis algorithms for Markov Decision Processes (MDPs) from PCTL formulas, and a computational framework for automatic deployment of a mobile robot from PCTL specifications. The approaches were validated by simulations and experiments. The algorithms and techniques in this thesis help to make discrete-time stochastic systems a more useful and effective class of models for analysis and control of real world systems.
Stochastic hybrid systems consist of software-controlled physical processes, where uncertainties manifest due to either disturbance in the environment in which the physical systems operate or the noise in sensors/actuators through which they interact with the software. The safety analysis of such systems is challenging due to complex dynamics, uncertainties, and infinite state space. This thesis introduces fully automated methods for bounded/unbounded safety analysis of certain subclasses of the stochastic hybrid systems against given safety specifications. Our first contribution is to compute the maximum/minimum bounded probability of reachability of polyhedral probabilistic hybrid systems, where plant dynamics are expressed as a set of linear constraints over the rate of state variables, and uncertainties are involved in discrete transitions represented as discrete probability distributions over the set of locations/modes. Our broad approach is to encode all possible bounded probabilistic behaviors into an appropriate logic along with the given safety specifications. Then, we perform optimization over all possible behaviors for the maximum/minimum probability via state-of-the-art optimization solvers. The second contribution is to present fully automatic unbounded safety analysis of the polyhedral probabilistic hybrid systems (PHS). We present a novel counterexample guided abstraction refinement (CEGAR) algorithm for polyhedral PHS. Developing a CEGAR algorithm for the polyhedral PHS is complex owing to the uncertainties in the discrete transitions, and the infinite state space due to the real-valued variables. We present a practical algorithm by choosing a succinct representation for counterexamples, an efficient validation algorithm and a constructive method for refinement that ensures progress towards the elimination of a spurious abstract counterexample. The third contribution is to extend unbounded safety analysis to the class of linear probabilistic hybrid systems (PHS). Developing an abstraction for the linear PHS is a challenge when the dynamics is linear, because the solutions are exponential and require solving exponential constraints to construct the finite state MDP. Hence, we consider a hierarchical abstraction, where we first abstract a linear PHS to a polyhedral PHS using hybridization and then apply predicate abstraction to construct the finite state MDP. Finally, we consider uncertainties in plant dynamics, and develop an abstraction based method for both bounded/unbounded safety analysis of linear stochastic systems. The bounded safety analysis is similar to the encoding in bounded model checking, where we encode bounded stochastic behaviors instead of continuous behaviors and solve the encoding using a semi-definite program solver. For the unbounded safety analysis, we abstract the linear stochastic system into a finite state system, and analyze its safety using graph search algorithms.
At the dawn of the 21st century and the information age, communication and c- puting power are becoming ever increasingly available, virtually pervading almost every aspect of modern socio-economical interactions. Consequently, the potential for realizing a signi?cantly greater number of technology-mediated activities has emerged. Indeed, many of our modern activity ?elds are heavily dependant upon various underlying systems and software-intensive platforms. Such technologies are commonly used in everyday activities such as commuting, traf?c control and m- agement, mobile computing, navigation, mobile communication. Thus, the correct function of the forenamed computing systems becomes a major concern. This is all the more important since, in spite of the numerous updates, patches and ?rmware revisions being constantly issued, newly discovered logical bugs in a wide range of modern software platforms (e. g. , operating systems) and software-intensive systems (e. g. , embedded systems) are just as frequently being reported. In addition, many of today’s products and services are presently being deployed in a highly competitive environment wherein a product or service is succeeding in most of the cases thanks to its quality to price ratio for a given set of features. Accordingly, a number of critical aspects have to be considered, such as the ab- ity to pack as many features as needed in a given product or service while c- currently maintaining high quality, reasonable price, and short time -to- market.
Stochastic Satisfiability Modulo Theories (SSMT) is a quantitative extension of Satisfiability Modulo Theories (SMT) inspired by stochastic logics. It extends SMT by randomized quantifiers, facilitating capture of stochastic game properties in the logic, like reachability analysis of hybrid-state Markov decision processes. Solving SSMT formulae with quantification over finite and thus discrete domain has been addressed by Tino Teige et al. A major limitation of the SSMT solving approach is that all quantifiers are confined to range over finite domains. As this implies that the support of probability distributions have to be finite, a large number of phenomena cannot be expressed within the SSMT framework. To overcome this limitation, this thesis relaxes the constraints on the domains of randomized variables, now also admitting dense probability distributions in SSMT solving, which yields SSMT over continuous quantifier domains (CSSMT). engl.