Erika Ábrahám

Flow*: An Analyzer for Non-linear Hybrid Systems

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 non-linear continuous and hybrid system benchmarks. Our comparisons show that Flow* is competitive with other tools.

Taylor Model Flowpipe Construction for Non-linear Hybrid Systems

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 flow pipe 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.

PROPhESY: A PRObabilistic ParamEter SYnthesis Tool

We present PROPhESY, a tool for analyzing parametric Markov chains (MCs). It can compute a rational function (i.e., a fraction of two polynomials in the model parameters) for reachability and expected reward objectives. Our tool outperforms state-of-the-art tools and supports the novel feature of conditional probabilities. PROPhESY supports incremental automatic parameter synthesis (using SMT techniques) to determine “safe” and “unsafe” regions of the parameter space. All values in these regions give rise to instantiated MCs satisfying or violating the (conditional) probability or expected reward objective. PROPhESY features a web front-end supporting visualization and user-guided parameter synthesis. Experimental results show that PROPhESY scales to MCs with millions of states and several parameters. Open image in new window

Behavioral interface description of an object-oriented language with futures and promises ⋆

This paper formalizes the observable interface behavior of a concurrent, object-oriented language with futures and promises. The calculus captures the core of Creol, a language, featuring in particular asynchronous method calls and, since recently, first-class futures. The focus of the paper are open systems and we formally characterize their behavior in terms of interactions at the interface between the program and its environment. The behavior is given by transitions between typing judgments, where the absent environment is represented abstractly by an assumption context. A particular challenge is the safe treatment of promises: The erroneous situation that a promise is fulfilled twice, i.e., bound to code twice, is prevented by a resource aware type system, enforcing linear use of the writepermission to a promise. We show subject reduction and the soundness of the abstract interface description.

DTMC Model Checking by SCC Reduction

Discrete-Time Markov Chains (DTMCs) are a widely-used formalism to model probabilistic systems. On the one hand, available tools like PRISM or MRMC offer efficient model checking algorithms and thus support the verification of DTMCs. However, these algorithms do not provide any diagnostic information in the form of counterexamples, which are highly important for the correction of erroneous systems. On the other hand, there exist several approaches to generate counterexamples for DTMCs, but all these approaches require the model checking result for completeness. In this paper we introduce a model checking algorithm for DTMCs that also supports the generation of counterexamples. Our algorithm, based on the detection and abstraction of strongly connected components, offers abstract counterexamples, which can be interactively refined by the user.

Minimal critical subsystems for discrete-time markov models

We propose a new approach to compute counterexamples for violated ω-regular properties of discrete-time Markov chains and Markov decision processes. Whereas most approaches compute a set of system paths as a counterexample, we determine a critical subsystem that already violates the given property. In earlier work we introduced methods to compute such subsystems based on a search for shortest paths. In this paper we use SMT solvers and mixed integer linear programming to determine minimal critical subsystems.

Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

In this paper, we examine linear programming (LP) based relaxations for synthesizing polynomial Lyapunov functions to prove the stability of polynomial ODEs. A common approach to Lyapunov function synthesis starts from a desired parametric polynomial form of the polynomial Lyapunov function. Subsequently, we encode the positive-definiteness of the function, and the negative-definiteness of its derivative over the domain of interest. Therefore, the key primitives for this encoding include: (a) proving that a given polynomial is positive definite over a domain of interest, and (b) encoding the positive definiteness of a given parametric polynomial, as a constraint on the unknown parameters. We first examine two classes of relaxations for proving polynomial positivity: relaxations by sum-of-squares (SOS) programs, against relaxations that produce linear programs. We compare both types of relaxations by examining the class of polynomials that can be shown to be positive in each case. Next, we present a progression of increasingly more powerful LP relaxations based on expressing the given polynomial in its Bernstein form, as a linear combination of Bernstein polynomials. The wellknown bounds on Bernstein polynomials over the unit box help us formulate increasingly precise LP relaxations that help us establish the positive definiteness of a polynomial over a bounded domain. Subsequently, we show how these LP relaxations can be used to search for Lyapunov functions for polynomial ODEs by formulating LP instances. We compare our approaches to synthesizing Lyapunov functions with approaches based on SOS programming relaxations. The approaches are evaluated on a suite of benchmark examples drawn from the literature and automatically synthesized benchmarks. Our evaluation clearly demonstrates the promise of LP relaxations, especially for finding polynomial local Lyapunov functions that prove that the system is asymptotically stable over a given bounded region containing the equilibrium. In particular, the LP approach is shown to be as fast as the SOS programming approach, but less prone to numerical problems.

A Greedy Approach for the Efficient Repair of Stochastic Models

For discrete-time probabilistic models there are efficient methods to check whether they satisfy certain properties. If a property is refuted, available techniques can be used to explain the failure in form of a counterexample. However, there are no scalable approaches to repair a model, i.e., to modify it with respect to certain side conditions such that the property is satisfied. In this paper we propose such a method, which avoids expensive computations and is therefore applicable to large models. A prototype implementation is used to demonstrate the applicability and scalability of our technique.

SMT-RAT: An Open Source C++ Toolbox for Strategic and Parallel SMT Solving

During the last decade, popular SMT solvers have been extended step-by-step with a wide range of decision procedures for different theories. Some SMT solvers also support the user-defined tuning and combination of such procedures, typically via command-line options. However, configuring solvers this way is a tedious task with restricted options.

Accelerating Parametric Probabilistic Verification

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.