Abstract. The tool FLOW* performs Taylor model-based flowpipe construction for non-linear (polynomial) hybrid systems. FLOW* combines well-known Taylor model arithmetic techniques for guaranteed approximations of the continuous dynamics in each mode with a combination of approaches for handling mode invariants and discrete transitions. FLOW* supports a wide variety of optimizations including adaptive step sizes, adaptive selection of approximation orders and the heuristic selection of template directions for aggregating flowpipes. This paper describes FLOW* and demonstrates its performance on a series of nonlinear continuous and hybrid system benchmarks. Our comparisons show that FLOW* is competitive with other tools. Overview of FLOW*In this paper, we present the FLOW* tool to generate flowpipes for non-linear hybrid systems using Taylor Models (TMs). TMs were originally proposed by Berz and Makino [1] to represent functions by means of higher-order Taylor polynomial expansions, bloated by an interval to represent the approximation error. TMs support functional operations such as addition, multiplication, division, derivation and antiderivation. Guaranteed integration techniques can utilize TMs to provide tight flowpipe over-approximations to non-linear ODEs, with each flowpipe segment represented by a TM [2]. However, these techniques do not naturally extend to non-linear hybrid systems consisting of multiple modes and discrete transitions (jumps). Figure 1 presents a schematic diagram of the major components of FLOW*. FLOW* accepts (i) A hybrid system model file which describes the modes, the polynomial dynamics associated with each mode and the transitions between modes; (ii) A specification file includes TM flowpipes with the state space and unsafe set specifications. For a model file, FLOW* performs a flowpipe construction for a specified time horizon [0, T ] and a maximum jump depth J such that the flowpipe set is an over-approximation of the states which can be reached in [0, T ] with at most J jumps. FLOW* also checks whether the flowpipe intersects the unsafe set and outputs a visualization of the set of reachable states using polyhedral over-approximations of the computed TM flowpipes. FLOW* is extensible in quite simple ways. Our TM output can be parsed in by other tools, including FLOW* itself to check multiple properties incrementally.

SUMMARYScalasca is a performance toolset that has been specifically designed to analyze parallel application execution behavior on large-scale systems with many thousands of processors. It offers an incremental performance-analysis procedure that integrates runtime summaries with in-depth studies of concurrent behavior via event tracing, adopting a strategy of successively refined measurement configurations. Distinctive features are its ability to identify wait states in applications with very large numbers of processes and to combine these with efficiently summarized local measurements. In this article, we review the current toolset architecture, emphasizing its scalable design and the role of the different components in transforming raw measurement data into knowledge of application execution behavior. The scalability and effectiveness of Scalasca are then surveyed from experience measuring and analyzing real-world applications on a range of computer systems.

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Abstract-We propose an approach for verifying non-linear hybrid systems using higher-order Taylor models that are a combination of bounded degree polynomials over the initial conditions and time, bloated by an interval. Taylor models are an effective means for computing rigorous bounds on the complex time trajectories of non-linear differential equations. As a result, Taylor models have been successfully used to verify properties of non-linear continuous systems. However, the handling of discrete (controller) transitions remains a challenging problem.In this paper, we provide techniques for handling the effect of discrete transitions on Taylor model flowpipe construction. We explore various solutions based on two ideas: domain contraction and range over-approximation. Instead of explicitly computing the intersection of a Taylor model with a guard set, domain contraction makes the domain of a Taylor model smaller by cutting away parts for which the intersection is empty. It is complemented by range over-approximation that translates Taylor models into commonly used representations such as template polyhedra or zonotopes, on which intersections with guard sets have been previously studied. We provide an implementation of the techniques described in the paper and evaluate the various design choices over a set of challenging benchmarks.

We present a novel method for computing reachability probabilities of parametric discrete-time Markov chains whose transition probabilities are fractions of polynomials over a set of parameters. Our algorithm is based on two key ingredients: a graph decomposition into strongly connected subgraphs combined with a novel factorization strategy for polynomials. Experimental evaluations show that these approaches can lead to a speed-up of up to several orders of magnitude in comparison to existing approaches. PreliminariesDefinition 1 (Discrete-time Markov chain). A discrete-time Markov chain (DTMC) is a tuple D = (S, I, P ) with a non-empty finite set S of states, an initial

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We investigate linear programming relaxations to synthesize Lyapunov functions that establish the stability of a given system over a bounded region. In particular, we attempt to discover functions that are more readily useful inside symbolic verification tools for proving the correctness of control systems. Our approach searches for a Lyapunov function, given a parametric form with unknown coefficients, by constructing a system of linear inequality constraints over the unknown parameters. We examine two complementary ideas for the linear programming relaxation, including interval evaluation of the polynomial form and "Handelman representations" for positive polynomials over polyhedral sets. Our approach is implemented as part of a branch-and-relax scheme for discovering Lyapunov functions. We evaluate our approach using a prototype implementation, comparing it with techniques based on Sum-of-Squares (SOS) programming. A comparison with SOSTOOLS is carried out over a set of benchmarks gathered from the related work. The evaluation suggests that our approach using Simplex is generally fast, and discovers Lyapunov functions that are simpler and easy to check. They are suitable for use inside symbolic formal verification tools for reasoning about continuous systems.

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