Abstract. We introduce a fixedpoint algorithm for verifying safety properties of hybrid systems with differential equations whose right-hand sides are polynomials in the state variables. In order to verify nontrivial systems without solving their differential equations and without numerical errors, we use a continuous generalization of induction, for which our algorithm computes the required differential invariants. As a means for combining local differential invariants into global system invariants in a sound way, our fixedpoint algorithm works with a compositional verification logic for hybrid systems. To improve the verification power, we further introduce a saturation procedure that refines the system dynamics successively with differential invariants until safety becomes provable. By complementing our symbolic verification algorithm with a robust version of numerical falsification, we obtain a fast and sound verification procedure. We verify roundabout maneuvers in air traffic management and collision avoidance in train control.

Abstract. This paper presents a constraint-based technique for discovering a rich class of inductive invariants (disjunctions of polynomial inequalities of bounded degree) for verification of hybrid systems. The key idea is to introduce a template for the unknown invariants and then translate the verification condition of the hybrid system into an ∃ ∀ constraint over the template unknowns (which are variables over reals) by making use of the fact that vector fields must point inwards at the boundary. These constraints are then solved using Farkas lemma. We also present preliminary experimental results that demonstrate the feasibility of our approach of solving the ∃ ∀ constraints generated from models of realworld hybrid systems. 1

Abstract. We generalise dynamic logic to a logic for differential-algebraic programs, i.e., discrete programs augmented with first-order differentialalgebraic formulas as continuous evolution constraints in addition to first-order discrete jump formulas. These programs characterise interacting discrete and continuous dynamics of hybrid systems elegantly and uniformly. For our logic, we introduce a calculus over real arithmetic with discrete induction and a new differential induction with which differential-algebraic programs can be verified by exploiting their differential constraints algebraically without having to solve them. We develop the theory of differential induction and differential refinement and analyse their deductive power. As a case study, we present parametric tangential roundabout maneuvers in air traffic control and prove collision avoidance in our calculus.

Abstract. We present an algorithm for finding valid polynomial relations (i. e. invariants) among program variables for imperative loops. The algorithm is implemented in the verification environment for imperative programs (using Hoare logic) in the frame of the Theorema system (www.theorema.org). We use techniques from (polynomial) algebra and combinatorics, namely Gröbner Bases, variable elimination, algebraic dependencies and symbolic summation (the Gosper algorithm, handling geometric series, C-finite solving). These methods are demonstrated on several examples which have been treated completely automatically by our implementation.

Recently there has been a growing interest in models and methods targeted towards the (co)design of stream processing applications; e.g. those for audio/video processing. Streams processed by such applications tend to be highly bursty and exhibit a high data-dependent variability in their processing requirements. As a result, classical event and service models such as periodic, sporadic, etc. can be overly pessimistic when dealing with such applications. In this paper, we present a new model called Event Count Automata (ECA) for capturing the timing properties of such streams. Our model can be used to cleanly formulate properties relevant to stream processing on heterogeneous multiprocessor architectures, such as buffer overflow/underflow constraints. It can also provide the basis for developing analysis methods to compute delay/timing properties of the processed streams under different scheduling policies. Our ECAs, though similar in flavor to timed and hybrid automata, have a different semantics, are more light-weight, and are specifically suited for modeling stream processing applications and architectures. We present the basic aspects of this model and illustrate its modeling potential. We then apply it in a specific stream processing setting and develop an analysis technique based on the formalism of Colored Petri Nets (CPNs). Finally, we validate our modeling and analysis techniques with the help of preliminary experimental results generated using the CPN simulation tool.

Abstract. We propose techniques for the verification of hybrid systems using template polyhedra, i.e., polyhedra whose inequalities have fixed expressions but with varying constant terms. Given a hybrid system description and a set of template linear expressions as inputs, our technique constructs over-approximations of the reachable states using template polyhedra. Therefore, operations used in symbolic model checking such as intersection, union and post-condition across discrete transitions over template polyhedra can be computed efficiently using template polyhedra without requiring expensive vertex enumeration. Additionally, the verification of hybrid systems requires techniques to handle the continuous dynamics inside discrete modes. We propose a new flowpipe construction algorithm using template polyhedra. Our technique uses higher-order Taylor series expansion to approximate the time trajectories. The terms occurring in the Taylor series expansion are bounded using repeated optimization queries. The location invariant is used to enclose the remainder term of the Taylor series, and thus truncate the expansion. Finally, we have implemented our technique as a part of the tool TimePass for the analysis of affine hybrid automata. 1

Abstract — In this paper, we present a method for computing a basin of attraction to a target region for non-linear ordinary differential equations. This basin of attraction is ensured by a Lyapunov-like polynomial function that we compute using an interval based branch-and-relax algorithm. This algorithm relaxes the necessary conditions on the coefficients of the Lyapunov-like function to a system of linear interval inequalities that can then be solved exactly, and iteratively reduces the relaxation error by recursively decomposing the state space into hyper-rectangles. Tests on an implementation are promising. I.

Abstract. Box invariant sets are box-shaped positively invariant sets. We show that box invariants are computable for a large class of nonlinear and hybrid systems. The technique for computing these invariants is based on nonlinear constraint solving. This paper also shows that the class of multiaffine systems, which has been used successfully for modeling and analyzing regulatory and biochemical reaction networks, can be generalized to the class of componentwise monotone and componentwise quasi monotone systems without losing any of its nice properties.