ARCH-COMP19 Category Report: Stochastic Modelling

This report presents results of a friendly competition for formal verification of continuous and hybrid systems with piecewise constant dynamics. The friendly competition took place as part of the workshop Applied Verification for Continuous and Hybrid Systems (ARCH) in 2018. In this second edition, five tools have been applied to solve five different benchmark problems in the category for piecewise constant dynamics: BACH, Lyse, PHAVer, PHAVer-lite, and VeriSiMPL. Compared to last year, a new tool has participated (PHAVer-lite) and a benchmark has been made more complex (Dutch Railway Network). The result is a snapshot of the current landscape of tools and the types of benchmarks they are particularly suited for. Due to the diversity of problems, we are not ranking tools, yet the presented results probably provide the most complete assessment of tools for the safety verification of continuous and hybrid systems with piecewise constant dynamics up to this date.

This report presents results of a friendly competition for formal verification of continuous and hybrid systems with linear continuous dynamics. The friendly competition took place as part of the workshop Applied Verification for Continuous and Hybrid Systems (ARCH) in 2017. In its first edition, three tools have been applied to solve three different benchmark problems in the category ofbounded model checking of hybrid systems with piecewise constant dynamics (in alphabetical order): BACH, HyDRA, and XSpeed. The result is a snapshot of the current landscape of tools and the types of benchmarks they are particularly suited for. Due to the diversity of problems, we are not ranking tools and we also welcome more tools to join in this friendly competition in the future event.

Verifying properties of hybrid systems can be highly complex. To reduce the effort required to produce a correct proof, the use of mechanical verification techniques is promising. Recently, we extended a mechanical verification system, originally developed to reason about deterministic real-time automata, to verify properties of hybrid systems. To evaluate our approach, we applied our extended proof system to a solution, based on the Lynch-Vaandrager timed automata model, of the Steam Boiler Controller problem, a hybrid systems benchmark. This paper reviews our mechanical verification system, which builds on SRI's Prototype Verification System (PVS), and describes the features we added to handle hybrid systems. It also discusses some errors we detected in applying our system to the benchmark problem. We conclude with a summary of insights we acquired in using our system to specify and verify hybrid systems.

A methodology for safety verification of continuous and hybrid systems using barrier certificates has been proposed recently. Conditions that must be satisfied by a barrier certificate can be formulated as a convex program, and the feasibility of the program implies system safety in the sense that there is no trajectory starting from a given set of initial states that reaches a given unsafe region. The dual of this problem, i.e., the reachability problem, concerns proving the existence of a trajectory starting from the initial set that reaches another given set. Using insights from the linear programming duality appearing in the discrete shortest path problem, we show in this paper that reachability of continuous systems can also be verified through convex programming. Several convex programs for verifying safety and reachability, as well as other temporal properties such as eventuality, avoidance, and their combinations, are formulated. Some examples are provided to illustrate the application of the proposed methods. Finally, we exploit the convexity of our methods to derive a converse theorem for safety verification using barrier certificates.

Hybrid systems combining discrete and continuous dynamics arise as mathematical models of various artificial and natural systems, and as an approximation to complex continuous systems. Reachability analysis has been the principal research question in the verification of hybrid systems, even though it is a wellknown result that most non-trivial subclasses of hybrid systems reachability and most verification problems are undecidable [1]. Nonetheless, various decidable subclasses have been identified, including polygonal hybrid systems (SPDIs) [2]. SPDIs can be used, for instance, in the analysis of approximations of non-linear differential equations in two-dimensions.KeywordsModel CheckHybrid SystemPhase PortraitHybrid AutomatonReachability AnalysisThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Hybrid systems are characterized by combining discrete and continuous behaviors. Verification of hybrid systems is, in general, a diffcult task due to the potential complexity of the continuous dynamics. Currently, there are different formalisms and tools which are able to analyze specific types of hybrid systems, model checking being one of the most used approaches. In this paper, we describe an extension of Java PathFinder in order to analyze hybrid systems. We apply a general methodology which has been successfully used to extend Spin. This methodology is non-intrusive, and uses external libraries, such as the Parma Polyhedra Library, to abstract the continuous behavior of the hybrid system.

Safety verification and reachability analysis for hybrid systems

A Tool for Modular Modelling and Verification of Hybrid Systems

Hybrid systems are networks of interacting digital devices and continuous plants reacting to a changing environment. Our multiple agent hybrid control architecture ([KN93b], [KN93c]) is based on the notion of hybrid system state. The latter incorporates evolution models using differential or difference equations, logic constraints, and geometric constraints. The set of hybrid states of a hybrid system can be construed in a variety of ways as a differentiable (or a C ∞) manifold which we have called the carrier manifold ([KNRG95]). We have suggested that for control problems the coordinates of points of the carrier manifolds should be selected to incorporate all information about system state, control state, and environment needed to choose new values of control parameters.This paper provides an outline of our ongoing work which expresses a wide class of control problems as relaxed, or convexified, calculus of variations problems on carrier manifolds. Weierstrass pointed out that many calculus of variations problems with time as an independent variable can be rewritten as parametric problems, that is, as problems where time is not an explicit variable, by moving to a larger manifold where time is an extra variable, x n+1=t, with the extra constraint x n+1=1. Then Finsler (1918) pointed out that each such parametric calculus of variations problem induces a metric ground form ds 2 associated with the manifold. We reduce a wide class of optimal control problems to calculating the Cartan affine connection associated with the Finsler metric ground form induced by the corresponding convexified calculus of variations problem. For any prescribed ε of deviation from the variational minimum, we then use chatterings of control flows on the manifold to approximate to a connection and use this to compute a finite state control program which enforces ε-optimal behavior. Thus we find that Finsler manifolds and their associated differential geometry and control flows form a computational basis for extraction and verification of hybrid systems and intelligent control systems in general. We remark that other connections, not arising from metrics, can be usefully employed in control too.Here we discuss only the simplest problem of the calculus of variations and optimal control: find an “optimal” path x(t) among admissible paths on the manifold M extending from initial point x(t 0 )=x0 to endpoint x(t 1 )=x1 with a given end direction. x(t 1 )=y 1. The theory of Finsler manifolds is not familiar to most electrical engineers and certainly not to computer scientists. The goal of this paper is to give some historical background and a brief description of how one interprets connections as optimal controls and extraction of finite state control programs as extraction of approximations to connections. One can also unwind and use connections similarly for non-parametric problems. A book length mathematical treatment is in preparation.KeywordsHybrid SystemTangent SpaceTangent VectorParallel TransportFinsler SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Intelligent environments can be described as hybrid systems, which combine continuous dynamics, modelling the behaviour of physical components, and discrete dynamics, modelling the software components that control the evolution of the physical variables. The growing boom in intelligent environments makes the construction of complex discrete components necessary, which may require the use of more sophisticated concurrent languages. Hy-tccp is a concurrent language for modelling hybrid systems with high-level notations that facilitate the description of computational systems, abstracting away from the implementation details. In this paper, we present an operational semantics based on hybrid automata for Hy-tccp which is a theoretical basis for the analysis and verification of hybrid systems like intelligent environment.

The interplay of random phenomena with discrete-continuous dynamics deserves increased attention in many systems of growing importance. Their verification needs to consider both stochastic behaviour and hybrid dynamics. In the verification of classical hybrid systems, one is often interested in deciding whether unsafe system states can be reached. In the stochastic setting, we ask instead whether the probability of reaching particular states is bounded by a given threshold. In this thesis, we consider stochastic hybrid systems and develop a general abstraction framework for deciding such problems. This gives rise to the first mechanisable technique that can, in practice, formally verify safety properties of systems which feature all the relevant aspects of nondeterminism, general continuous-time dynamics, and probabilistic behaviour. Being based on tools for classical hybrid systems, future improvements in the effectiveness of such tools directly carry over to improvements in the effectiveness of our technique. We extend the method in several directions. Firstly, we discuss how we can handle continuous probability distributions. We then consider systems which we are in partial control of. Next, we consider systems in which probabilities are parametric, to analyse entire system families at once. Afterwards, we consider systems equipped with rewards, modelling costs or bonuses. Finally, we consider all orthogonal combinations of the extensions to the core model.

Interval-based Solving of Hybrid Constraint Systems

We describe our UTP theory of hybrid relations, which extends the relational calculus with continuous variables and differential equations. This enables the use of UTP in modelling and verification of hybrid systems, supported by our mechanisation in Isabelle/UTP. The hybrid relational calculus is built upon the same foundation as the UTP’s theory of reactive processes, which is accomplished through a generalised trace algebra and a model of piecewise-continuous functions. From this foundation, we give semantics to hybrid programs, including ordinary differential equations and preemption, and show how the theory can be used to reason about sequential hybrid systems.

REACHABILITY COMPUTATION FOR UNCERTAIN PLANAR AFFINE SYSTEMS USING LINEAR ABSTRACTIONS

In control theory, models of physical processes, such as systems of differential equations, are usually checked against specifications, such as stability and set invariance. In formal methods, rich specifications, such as languages and formulae of temporal logics, are checked against models of software programs and digital circuits, such as finite transition graphs. With the development and integration of cyber physical and safety critical systems, there is an increasing need for computational tools for verification and control of complex systems from rich, temporal logic specifications. The formal verification and synthesis problems have been shown to be undecidable even for very simple classes of infinitespace continuous and hybrid systems. However, provably correct but conservative approaches, in which the satisfaction of a property by a dynamical system is implied by the satisfaction of the property by a finite over-approximation (abstraction) of the system, have received a lot of attention in recent years. Some classes of systems allowing for computationally efficient verification and control from temporal logic specifications are reviewed. For continuous and discrete-time linear systems and continuous-time multi-linear systems, it is shown that finite abstractions can be constructed through polyhedral operations only. By using techniques from model checking and automata games, this allows for verification and control from specifications given as Linear Temporal Logic (LTL) formulae over linear predicates in the state variables. A connection between the existence of Lyapunov functions and finite bisimulations is established for discrete-time linear and switched linear systems. Finally, optimality and correctness requirements are combined in a model predictive approach to generate control strategies for discrete-time linear systems. The usefulness of these computational tools is illustrated with various examples such as verification and synthesis of biological circuits in synthetic biology and motion planning and control in robotics.