Abstract. We present a scalable reachability algorithm for hybrid systems with piecewise affine, non-deterministic dynamics. It combines polyhedra and support function representations of continuous sets to compute an over-approximation of the reachable states. The algorithm improves over previous work by using variable time steps to guarantee a given local error bound. In addition, we propose an improved approximation model, which drastically improves the accuracy of the algorithm. The algorithm is implemented as part of SpaceEx, a new verification platform for hybrid systems, available at spaceex.imag.fr. Experimental results of full fixed-point computations with hybrid systems with more than 100 variables illustrate the scalability of the approach. 1

Abstract. We propose algorithms significantly extending the limits for maintaining exact representations in the verification of linear hybrid systems with large discrete state spaces. We use AND-Inverter Graphs (AIGs) extended with linear constraints (LinAIGs) as symbolic representation of the hybrid state space, and show how methods for maintaining compactness of AIGs can be lifted to support model-checking of linear hybrid systems with large discrete state spaces. This builds on a novel approach for eliminating sets of redundant constraints in such rich hybrid state representations by a suitable exploitation of the capabilities of SMT solvers, which is of independent value beyond the application context studied in this paper. We used a benchmark derived from an Airbus flap control system (containing 2 20 discrete states) to demonstrate the relevance of the approach. 1

In this paper, we show how to compute an over-approximation for the reachable set of uncertain nonlinear continuous dynamical systems by using guaranteed set integration. We introduce two ways to do so. The first one is a full interval method which handles whole domains for set computation and relies on state-of-the-art validated numerical integration methods. The second one relies on comparison theorems for differential inequalities in order to bracket the uncertain dynamics between two dynamical systems where there is no uncertainty. Since the derived bracketing systems are piecewise-differentiable functions, validated numerical integration methods cannot be used directly. Hence, our contribution resides in the use of hybrid automata to model the bounding systems. We give a rule for building these automata and we show how to run them and address mode switching in a guaranteed way in order to compute the over approximation for the reachable set. The computational cost of our method is also analyzed and shown to be smaller that the one of classical interval techniques. Sufficient conditions are given which ensure the-practical stability of the enclosures given by our hybrid bounding method. Two examples are also given which show that the performance of our method is very promising.

We address nonlinear reachability computation for uncertain monotone systems, those for which flows preserve a suitable partial orderings on initial conditions. In a previous work [1], we introduced a nonlinear hybridization approach to nonlinear continuous reachability computation. By analysing the signs of off-diagonal elements of system’s Jacobian matrix, a hybrid automaton can be obtained, which yields component-wise bounds for the reachable sets. One shortcoming of the method is induced by the need to use whole sets for addressing mode switching. In this paper, we improve this method and show that for the broad class of monotone dynamical systems, component-wise bounds can be obtained for the reachable set in a separate manner. As a consequence, mode switching no longer needs to use whole solution sets. We give examples which show the potentials of the new approach.

Abstract. In this paper we describe reachability computation for continuous and hybrid systems and its potential contribution to the process of building and debugging biological models. We then develop a novel algorithm for computing reachable states for nonlinear systems and report experimental results obtained using a prototype implementation. We believe these results constitute a promising contribution to the analysis of complex models of biological systems. 1

Abstract. Computing over-approximations of all possible time trajectories is an important task in the analysis of hybrid systems. Sankaranarayanan et al. [20] suggested to approximate the set of reachable states using template polyhedra. In the present paper, we use a max-strategy improvement algorithm for computing an abstract semantics for affine hybrid automata that is based on template polyhedra and safely over-approximates the concrete semantics. Based on our formulation, we show that the corresponding abstract reachability problem is in co−NP. Moreover, we obtain a polynomial-time algorithm for the time elapse operation over template polyhedra. 1

Abstract. There has been much recent progress on invariant generation techniques for continuous systems whose dynamics are described by Ordinary Differential Equations (ODE). In this paper, we present a simple abstraction scheme for hybrid systems that abstracts continuous dynamics by relating any state of the system to a state that can potentially be reached at some future time instant. Such relations are then interpreted as discrete transitions that model the continuous evolution of states over time. We adapt template-based invariant generation techniques for continuous dynamics to derive relational abstractions for continuous systems with linear as well as non-linear dynamics. Once a relational abstraction hasbeen derived,theresultingsystemis apurelydiscrete, infinite-statesystem. Therefore, techniquessuchas k-inductioncan be directly applied to this abstraction to prove properties, and bounded model-checking techniques applied to find potential falsifications. We present the basic underpinnings of our approach and demonstrate its use on many benchmark systems to derive simple and usable abstractions. 1

Abstract. This paper is concerned with the problem of computing the image of a set by a polynomial function. Such image computations constitute a crucial component in typical tools for set-based analysis of hybrid systems and embedded software with polynomial dynamics, which found applications in various engineering domains. One typical example is the computation of all states reachable from a given set in one step by a continuous dynamics described by a differential or difference equation. We propose a new algorithm for over-approximating such images based on the Bernstein representation of polynomial functions. The images are stored using template polyhedra. Using a prototype implementation, the performance of the algorithm was demonstrated on two practical systems as well as a number of randomly generated examples. 1

On using hybridization methods for the analysis of biomolecular networks

On using hybridization methods for the analysis of biomolecular networks