This tutorial provides an introduction to probabilistic model checking, a technique for automatically verifying quantitative properties of probabilistic systems. We focus on Markov decision processes (MDPs), which model both stochastic and nondeterministic behaviour. We describe methods to analyse a wide range of their properties, including specifications in the temporal logics PCTL and LTL, probabilistic safety properties and cost- or reward-based measures. We also discuss multi-objective probabilistic model checking, used to analyse trade-offs between several different quantitative properties. Applications of the techniques in this tutorial include performance and dependability analysis of networked systems, communication protocols and randomised distributed algorithms. Since such systems often comprise several components operating in parallel, we also cover techniques for compositional modelling and verification of multi-component probabilistic systems. Finally, we describe three large case studies which illustrate practical applications of the various methods discussed in the tutorial.

/pdf/automated-verification-techniques-for-probabilistic-systems-21hj56l7q5.pdf

Automated Verification Techniques for Probabilistic Systems

Randomization is a key element in sequential and distributed computing. Reasoning about randomized algorithms is highly non-trivial. In the 1980s, this initiated first proof methods, logics, and model-checking algorithms. The field of probabilistic verification has developed considerably since then. This paper surveys the algorithmic verification of probabilistic models, in particular probabilistic model checking. We provide an informal account of the main models, the underlying algorithms, applications from reliability and dependability analysis---and beyond---and describe recent developments towards automated parameter synthesis.

The Probabilistic Model Checking Landscape

We present a verification framework for analysing multiple quantitative objectives of systems that exhibit both nondeterministic and stochastic behaviour. These systems are modelled as probabilistic automata, enriched with cost or reward structures that capture, for example, energy usage or performance metrics. Quantitative properties of these models are expressed in a specification language that incorporates probabilistic safety and liveness properties, expected total cost or reward, and supports multiple objectives of these types. We propose and implement an efficient verification framework for such properties and then present two distinct applications of it: firstly, controller synthesis subject to multiple quantitative objectives; and, secondly, quantitative compositional verification. The practical applicability of both approaches is illustrated with experimental results from several large case studies.

/pdf/quantitative-multi-objective-verification-for-probabilistic-224cau1ppv.pdf

Quantitative multi-objective verification for probabilistic systems

Statistical model checking has become a promising technique to circumvent the state space explosion problem in model-based verification. It trades time for memory, via a probabilistic simulation and exploration of the model behaviour--often combined with effective a posteriori hypothesis testing. However, as a simulation-based approach, it can only provide sound verification results if the underlying model is a stochastic process. This drastically limits its applicability in verification, where most models are indeed variations of nondeterministic transition systems. In this paper, we describe a sound extension of statistical model checking to scenarios where nondeterminism is present. We focus on probabilistic automata, and discuss how partial order reduction can be twisted such as to apply statistical model checking to models with spurious nondeterminism. We report on an implementation of this technique and on promising results in the context of verification and dependability analysis of distributed systems.

/pdf/partial-order-methods-for-statistical-model-checking-and-5dyqwqqm21.pdf

Partial order methods for statistical model checking and simulation

Probabilistic model checking is a powerful technique for formally verifying quantitative properties of systems that exhibit stochastic behaviour. Such systems are found in many application domains: for example, probabilistic behaviour may arise due to the presence of failures in unreliable hardware, message loss in wireless communication channels, or the use of randomisation in distributed protocols. This chapter starts with an introduction to the technique of probabilistic model checking. We then survey some recent advances in the area, including controller synthesis, compositional verification, probabilistic real-time systems and parametric model checking. We illustrate the application of the various techniques with a combination of toy examples and descriptions of larger case studies. The chapter concludes with a discussion of some of the key challenges in the field.

Probabilistic model checking: advances and applications

A stochastic timed automaton is a purely stochastic process defined on a timed automaton, in which both delays and discrete choices are made randomly. We study the almost-sure model-checking problem for this model, that is, given a stochastic timed automaton A and a property $\Phi$, we want to decide whether A satisfies $\Phi$ with probability 1. In this paper, we identify several classes of automata and of properties for which this can be decided. The proof relies on the construction of a finite abstraction, called the thick graph, that we interpret as a finite Markov chain, and for which we can decide the almost-sure model-checking problem. Correctness of the abstraction holds when automata are almost-surely fair, which we show, is the case for two large classes of systems, single- clock automata and so-called weak-reactive automata. Techniques employed in this article gather tools from real-time verification and probabilistic verification, as well as topological games played on timed automata.

Stochastic Timed Automata

A stochastic timed automaton is a purely stochastic process defined on a timed automaton, in which both delays and discrete choices are made randomly. We study the almost-sure model-checking problem for this model, that is, given a stochastic timed automaton A and a property ϕ, we want to decide whether A satisfies ϕ with probability 1. In this paper, we identify several classes of automata and of properties for which this can be decided. The proof relies on the construction of a finite abstraction, called the thick graph, that we interpret as a finite Markov chain, and for which we can decide the almost-sure model-checking problem. Correctness of the abstraction holds when automata are almost-surely fair, which we show, is the case for two large classes of systems, single-clock automata and so-called weak-reactive automata. Techniques employed in this article gather tools from real-time verification and probabilistic verification, as well as topological games played on timed automata.

https://hal.inria.fr/hal-01102368/document

Stochastic timed automata

It is well-known that fairness assumptions can be crucial for verifying progress, reactivity or other liveness properties for interleaving models. This also applies to Markov decision processes as an operational model for concurrent probabilistic systems and the task to establish tight lower or upper probability bounds for events that are specified by liveness properties. In this paper, we study general notions of strong and weak fairness constraints for Markov decision processes, formalized in an action- or state-based setting. We present a polynomially time-bounded algorithm for the quantitative analysis of an MDP against *** -automata specifications under fair worst- or best-case scenarios. Furthermore, we discuss the treatment of strong and weak fairness and process fairness constraints in the context of partial order reduction techniques for Markov decision processes that have been realized in the model checker LiQuor and rely on a variant of Peled's ample set method.

Quantitative Analysis under Fairness Constraints

In the past, several model checking algorithms have been proposed to verify probabilistic reactive systems. In contrast to the non-probabilistic setting where various techniques have been suggested and successfully applied to combat the state space-explosion problem in the context of model checking the techniques used for probabilistic systems have mainly concentrated on symbolic methods with variants of decision diagrams or abstraction methods. Only recently results have been published that give criteria on applying partial order reduction for verifying quantitative linear time properties as well as branching time properties for probabilistic systems. This paper summarizes the results that have been established so far about partial order reduction for Markov decision processes. We present the different reduction conditions and provide a comparison of the corresponding results.

Partial order reduction for markov decision processes: a survey

Partial order reduction for markov decision processes : A survey

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