Vector Barrier Certificates and Comparison Systems

Sogokon, Andrew; Ghorbal, Khalil; Tan, Yong Kiam; Platzer, André

doi:10.1007/978-3-319-95582-7_25

Cited by 23 publications

(29 citation statements)

References 44 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…S. Prajna et al [20] had first put the idea forward. A. Sogokon et al [34] employed the comparison principle associated with the convex verification conditions, to generate vector barrier certificates in safety verification.…”

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

A Novel Approach for Solving the BMI Problem in Barrier Certificates Generation

Chen

Peng

Wang

et al. 2020

Computer Aided Verification

View full text Add to dashboard Cite

Barrier certificates generation is widely used in verifying safety properties of hybrid systems because of the relatively low computational complexity it costs. Under sum of squares (SOS) relaxation, the problem of barrier certificate generation is equivalent to that of solving a bilinear matrix inequality (BMI) with a particular type. The paper reveals the special feature of the problem, and adopts it to build a novel computational method. The proposed method introduces a sequential iterative scheme that is able to find analytical solutions, rather than the nonlinear solving procedure to produce numerical solutions used by general BMI solvers and thus is more efficient than them. In addition, different from popular LMI solving based methods, it does not make the verification conditions more conservative, and thus reduces the risk of missing feasible solutions. Benefitting from these two appealing features, it can produce barrier certificates not amenable to existing methods, which is supported by a complexity analysis as well as the experiment on some benchmarks.

show abstract

Section: Related Workmentioning

confidence: 99%

“…Compared with reachable set computation [31], barrier certificate generation requires much less computation, since the unsafe region leads to seeking a barrier certificate. Especially, it behaves very well when a safety property concerns infinite time horizon [21,34].…”

Section: Introductionmentioning

confidence: 99%

A Novel Approach for Solving the BMI Problem in Barrier Certificates Generation

Chen

Peng

Wang

et al. 2020

Computer Aided Verification

View full text Add to dashboard Cite

show abstract

“…Finally, we mention a relation between the ideas in this paper and previously proposed ideas for (non-stochastic) ODEs due to Sogokon et al [34]. The key similarity lies in the use of a non-negative matrix through which a vector of functions whose derivatives are related to their current value.…”

Section: Contributionsmentioning

confidence: 87%

Unbounded-Time Safety Verification of Stochastic Differential Dynamics

Feng

Chen

Bai

et al. 2020

Computer Aided Verification

View full text Add to dashboard Cite

In this paper, we propose a method for bounding the probability that a stochastic differential equation (SDE) system violates a safety specification over the infinite time horizon. SDEs are mathematical models of stochastic processes that capture how states evolve continuously in time. They are widely used in numerous applications such as engineered systems (e.g., modeling how pedestrians move in an intersection), computational finance (e.g., modeling stock option prices), and ecological processes (e.g., population change over time). Previously the safety verification problem has been tackled over finite and infinite time horizons using a diverse set of approaches. The approach in this paper attempts to connect the two views by first identifying a finite time bound, beyond which the probability of a safety violation can be bounded by a negligibly small number. This is achieved by discovering an exponential barrier certificate that proves exponentially converging bounds on the probability of safety violations over time. Once the finite time interval is found, a finite-time verification approach is used to bound the probability of violation over this interval. We demonstrate our approach over a collection of interesting examples from the literature, wherein our approach can be used to find tight bounds on the violation probability of safety properties over the infinite time horizon.

show abstract

“…9, this is intuitively true because the arrows always point "inwards" on the boundary of cK. To prove this, we formalized a comparison principle [33, §9.IX] and, as corollaries, variations of barrier certificate principles [31] that can be used to establish (positive) invariance. Technical details of this formalization are omitted as it is not the focus of this paper.…”

Section: Circle Examplementioning

confidence: 99%

The Poincaré-Bendixson theorem in Isabelle/HOL

Immler

Tan

2020

Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs

Self Cite

View full text Add to dashboard Cite

The Poincaré-Bendixson theorem is a classical result in the study of (continuous) dynamical systems. Colloquially, it restricts the possible behaviors of planar dynamical systems: such systems cannot be chaotic. In practice, it is a useful tool for proving the existence of (limiting) periodic behavior in planar systems. The theorem is an interesting and challenging benchmark for formalized mathematics because proofs in the literature rely on geometric sketches and only hint at symmetric cases. It also requires a substantial background of mathematical theories, e.g., the Jordan curve theorem, real analysis, ordinary differential equations, and limiting (long-term) behavior of dynamical systems. We present a proof of the theorem in Isabelle/HOL and highlight the main challenges, which include: i) combining large and independently developed mathematical libraries, namely the Jordan curve theorem and ordinary differential equations, ii) formalizing fundamental concepts for the study of dynamical systems, namely the α, ω-limit sets, and periodic orbits, iii) providing formally rigorous arguments for the geometric sketches paramount in the literature, and iv) managing the complexity of our formalization throughout the proof, e.g., appropriately handling symmetric cases. CCS Concepts • Mathematics of computing → Ordinary differential equations; • Theory of computation → Logic and verification.

show abstract

Vector Barrier Certificates and Comparison Systems

Cited by 23 publications

References 44 publications

A Novel Approach for Solving the BMI Problem in Barrier Certificates Generation

A Novel Approach for Solving the BMI Problem in Barrier Certificates Generation

Unbounded-Time Safety Verification of Stochastic Differential Dynamics

The Poincaré-Bendixson theorem in Isabelle/HOL

Contact Info

Product

Resources

About