Iterative computation of polyhedral invariants sets for polynomial dynamical systems

Sassi, Mohamed Amin Ben; Girard, Antoine; Sankaranarayanan, Sriram

doi:10.1109/cdc.2014.7040384

Cited by 17 publications

(22 citation statements)

References 16 publications

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“…Pegasus is a continuous invariant generator implemented in the Wolfram Language with an interface accessible through both Mathematica and KeYmaera X. 5 When KeYmaera X is faced with a continuous safety verification problem that it is unable to prove directly, it automatically invokes Pegasus to help find an appropriate invariant (if possible). KeYmaera X checks all the invariants it is supplied with-including those provided by Pegasus (see Fig.…”

Section: Invariant Generation Methods In Pegasusmentioning

confidence: 99%

Pegasus: A Framework for Sound Continuous Invariant Generation

Sogokon

Mitsch

Tan

et al. 2019

Lecture Notes in Computer Science

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Continuous invariants are an important component in deductive verification of hybrid and continuous systems. Just like discrete invariants are used to reason about correctness in discrete systems without unrolling their loops forever, continuous invariants are used to reason about differential equations without having to solve them. Automatic generation of continuous invariants remains one of the biggest practical challenges to the automation of formal proofs of safety for hybrid systems. There are at present many disparate methods available for generating continuous invariants; however, this wealth of diverse techniques presents a number of challenges, with different methods having different strengths and weaknesses. To address some of these challenges, we develop Pegasus: an automatic continuous invariant generator which allows for combinations of various methods, and integrate it with the KeYmaera X theorem prover for hybrid systems. We describe some of the architectural aspects of this integration, comment on its methods and challenges, and present an experimental evaluation on a suite of benchmarks.1 An etymological note on naming conventions. The KeY [3] prover provided the foundation for developing KeYmaera [58], an interactive theorem prover for hybrid systems. The name KeYmaera was a pun on the Chimera, a hybrid monster from Classical Greek mythology. The tactic language of the new (aXiomatic) KeYmaera X prover [24] is called Bellerophon [23], after the hero who defeats the Chimera in the myth. In keeping with an established tradition, the invariant generation framework is called Pegasus because the aid of this winged horse was crucial to the hero Bellerophon in his feat.

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Section: Invariant Generation Methods In Pegasusmentioning

confidence: 99%

Pegasus: A Framework for Sound Continuous Invariant Generation

Sogokon

Mitsch

Tan

et al. 2019

Lecture Notes in Computer Science

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“…In order to solve the bilinear sum-of-squares programs, a commonly used method is to employ some form of alteration (e.g., [19,50,30]) with a feasible initial solution to the bilinear sum-of-squares program. Recently, [43,44] proposed linear programming based methods to synthesize maximal (robustly) positive polyhedral invariants. Contrasting with aforementioned methods, in this paper we propose a semi-definite programming based method to compute semialgebraic invariant.…”

Section: Introductionmentioning

confidence: 99%

Robust Non-termination Analysis of Numerical Software

Bai

Zhan

et al. 2018

Dependable Software Engineering. Theories, Tools, and Applications

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Numerical software is widely used in safety-critical systems such as aircrafts, satellites, car engines and many other fields, facilitating dynamics control of such systems in real time. It is therefore absolutely necessary to verify their correctness. Most of these verifications are conducted under ideal mathematical models, but their real executions may not follow the models exactly. Factors that are abstracted away in models such as rounding errors can change behaviors of systems essentially. As a result, guarantees of verified properties despite the present of disturbances are needed. In this paper, we attempt to address this issue of nontermination analysis of numerical software. Nontermination is often an unexpected behaviour of computer programs and may be problematic for applications such as real-time systems with hard deadlines. We propose a method for robust conditional nontermination analysis that can be used to under-approximate the maximal robust nontermination input set for a given program. Here robust nontermination input set is a set from which the program never terminates regardless of the aforementioned disturbances. Finally, several examples are given to illustrate our approach.

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“…[35,38,6,29,26,40,17,37]), as well as tools based on verified integration of ODEs (e.g [7,27,20,21]). We are hopeful that maintaining and further populating the set of verification benchmarks will result in improvements to the existing capabilities offered by the tools for both bounded and unbounded-time safety verification.…”

Section: Discussionmentioning

confidence: 99%

“…[6]). The problems we have collected all share the property of having proofs of safety that were obtained using the methods presented in the pertinent papers (or having proofs that are immediate from the results described therein).…”

Section: Benchmarksmentioning

confidence: 99%

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Non-linear Continuous Systems for Safety Verification

Sogokon¹,

Ghorbal²,

Johnson³

EPiC Series in Computing

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Safety verification of hybrid dynamical systems relies crucially on the ability to reason about reachable sets of continuous systems whose evolution is governed by a system of ordinary differential equations (ODEs). Verification tools are often restricted to handling a particular class of continuous systems, such as e.g. differential equations with constant right-hand sides, or systems of affine ODEs. More recently, verification tools capable of working with non-linear differential equations have been developed. The behavior of non-linear systems is known to be in general extremely difficult to analyze because solutions are rarely available in closed-form. In order to assess the practical utility of the various verification tools working with non-linear ODEs it is very useful to maintain a set of verification problems. Similar efforts have been successful in other communities, such as automated theorem proving, SAT solving and numerical analysis, and have accelerated improvements in the tools and their underlying algorithms. We present a set of 65 safety verification problems featuring non-linear polynomial ODEs and for which we have proofs of safety. We discuss the various issues associated with benchmarking the currently available verification tools using these problems.

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Iterative computation of polyhedral invariants sets for polynomial dynamical systems

Cited by 17 publications

References 16 publications

Pegasus: A Framework for Sound Continuous Invariant Generation

Pegasus: A Framework for Sound Continuous Invariant Generation

Robust Non-termination Analysis of Numerical Software

Non-linear Continuous Systems for Safety Verification

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