Taolue Chen

Quantitative Model Checking of Continuous-Time Markov Chains Against Timed Automata Specifications

We study the following problem: given a continuous-time Markov chain (CTMC) C, and a linear real-time property provided as a deterministic timed automaton (DTA) A, what is the probability of the set of paths of C that are accepted by A (C satisfies A)? It is shown that this set of paths is measurable and computing its probability can be reduced to computing the reachability probability in a piecewise deterministic Markov process (PDP). The reachability probability is characterized as the least solution of a system of integral equations and is shown to be approximated by solving a system of partial differential equations. For the special case of single-clock \DTA, the system of integral equations can be transformed into a system of linear equations where the coefficients are solutions of ordinary differential equations.

Automatic verification of competitive stochastic systems

We present automatic verification techniques for the modelling and analysis of probabilistic systems that incorporate competitive behaviour. These systems are modelled as turn-based stochastic multi-player games, in which the players can either collaborate or compete in order to achieve a particular goal. We define a temporal logic called rPATL for expressing quantitative properties of stochastic multi-player games. This logic allows us to reason about the collective ability of a set of players to achieve a goal relating to the probability of an event’s occurrence or the expected amount of cost/reward accumulated. We give an algorithm for verifying properties expressed in this logic and implement the techniques in a probabilistic model checker, as an extension of the PRISM tool. We demonstrate the applicability and efficiency of our methods by deploying them to analyse and detect potential weaknesses in a variety of large case studies, including algorithms for energy management in Microgrids and collective decision making for autonomous systems.

PRISM-games: a model checker for stochastic multi-player games

We present PRISM-games, a model checker for stochastic multi-player games, which supports modelling, automated verification and strategy synthesis for probabilistic systems with competitive or cooperative behaviour. Models are described in a probabilistic extension of the Reactive Modules language and properties are expressed using rPATL, which extends the well-known logic ATL with operators to reason about probabilities, various reward-based measures, quantitative properties and precise bounds. The tool is based on the probabilistic model checker PRISM, benefiting from its existing user interface and simulator, whilst adding novel model checking algorithms for stochastic games, as well as functionality to synthesise optimal player strategies, explore or export them, and verify other properties under the specified strategy.

Model Repair for Markov Decision Processes

Markov decision processes (MDPs) are often used for modelling distributed systems with probabilistic failure or randomisation. We consider the problem of model repair for MDPs defined as follows: if the MDP fails to satisfy a property, we aim to find new values for the transition probabilities so that the property is guaranteed to hold, while at the same time the cost of repair is minimised. Because solving the MDP repair problem exactly is infeasible, in this paper we focus on approximate solution methods. We first formulate a region-based approach, which yields an interval in which the minimal repair cost is contained. As an alternative, we also consider sampling based approaches, which are faster but unable to provide lower bounds on the repair cost. We have integrated both methods into the probabilistic model checker PRISM and demonstrated their usefulness in practice using a computer virus case study.

A compositional specification theory for component behaviours

We propose a compositional specification theory for reasoning about components that interact by synchronisation of input and output (I/O) actions, in which the specification of a component constrains the temporal ordering of interactions with the environment. Such a theory is motivated by the need to support composability of components, in addition to modelling environmental assumptions, and reasoning about run-time behaviour. Models can be specified operationally by means of I/O labelled transition systems augmented by an inconsistency predicate on states, or in a purely declarative manner by means of traces. We introduce a refinement preorder that supports safe-substitutivity of components. Our specification theory includes the operations of parallel composition for composing components at run-time, logical conjunction for independent development, and quotient for incremental development. We prove congruence properties of the operations and show correspondence between the operational and declarative frameworks.

Efficient CTMC model checking of linear real-time objectives

This paper makes verifying continuous-time Markov chains (CTMCs) against deterministic timed automata (DTA) objectives practical. We show that verifying 1-clock DTA can be done by analyzing subgraphs of the product of CTMC C and the region graph of DTA A. This improves upon earlier results and allows to only use standard analysis algorithms. Our graph decomposition approach naturally enables bisimulation minimization as well as parallelization. Experiments with various examples confirm that these optimizations lead to significant speed-ups. We also report on experiments with multiple-clock DTA objectives. The objectives and the size of the problem instances that can be checked with our prototypical tool go (far) beyond what could be checked so far.

MODEL CHECKING OF CONTINUOUS-TIME MARKOV CHAINS AGAINST TIMED AUTOMATA SPECIFICATIONS

We study the following problem: given a continuous-time Markov chain (CTMC) C, and a linear real-time property provided as a deterministic timed automaton (DTA) A, what is the probability of the set of paths of C that are accepted by A (C satisfies A)? It is shown that this set of paths is measurable and computing its probability can be reduced to computing the reachability probability in a piecewise deterministic Markov process (PDP). The reachability probability is characterized as the least solution of a system of integral equations and is shown to be approximated by solving a system of partial differential equations. For the special case of single-clock \DTA, the system of integral equations can be transformed into a system of linear equations where the coefficients are solutions of ordinary differential equations.

On Stochastic Games with Multiple Objectives

We study two-player stochastic games, where the goal of one player is to satisfy a formula given as a positive boolean combination of expected total reward objectives and the behaviour of the second player is adversarial. Such games are important for modelling, synthesis and verification of open systems with stochastic behaviour. We show that finding a winning strategy is PSPACE-hard in general and undecidable for deterministic strategies. We also prove that optimal strategies, if they exists, may require infinite memory and randomisation. However, when restricted to disjunctions of objectives only, memoryless deterministic strategies suffice, and the problem of deciding whether a winning strategy exists is NP-complete. We also present algorithms to approximate the Pareto sets of achievable objectives for the class of stopping games.

Synthesis for multi-objective stochastic games: an application to autonomous urban driving

We study strategy synthesis for stochastic two-player games with multiple objectives expressed as a conjunction of LTL and expected total reward goals. For stopping games, the strategies are constructed from the Pareto frontiers that we compute via value iteration. Since, in general, infinite memory is required for deterministic winning strategies in such games, our construction takes advantage of randomised memory updates in order to provide compact strategies. We implement our methods in PRISM-games, a model checker for stochastic multi-player games, and present a case study motivated by the DARPA Urban Challenge, illustrating how our methods can be used to synthesise strategies for high-level control of autonomous vehicles.

Quantitative Verification of Implantable Cardiac Pacemakers

Implantable medical devices, such as cardiac pacemakers, must be designed and programmed to the highest levels of safety and reliability. Recently, errors in embedded software have led to a substantial increase in safety alerts, costly device recalls or even patient death. To address such issues, we propose a model-based framework for quantitative, automated verification of pacemaker software. We adapt the electrocardiogram model of Clifford et al, which generates realistic normal and abnormal heart beat behaviours, with probabilistic transitions between them, to produce a timed sequence of action potential signals that serve as pacemaker input. Working with the timed automata model of the pacemaker by Jiang et al, we develop a methodology for deriving the composition of the heart and the pacemaker, based on discretisation. The main correctness properties we consider include checking that the pacemaker corrects Bradycardia (slow heart beat) and does not induce Tachycardia (fast heart beat), for a range of realistic heart behaviours. We also analyse under sensing, through considering noise on sensor readings, and energy usage. We implement the framework using the probabilistic model checker PRISM and MATLAB and demonstrate encouraging experimental results. Our approach can be adapted to individual patients and is applicable to other pacemaker models.