An Axiomatic Approach to Liveness for Differential Equations

This paper presents an approach for deductive liveness verification for ordinary differential equations (ODEs) with differential dynamic logic. Numerous subtleties complicate the generalization of well-known discrete liveness verification techniques, such as loop variants, to the continuous setting. For example, ODE solutions may blow up in finite time or their progress towards the goal may converge to zero. Our approach handles these subtleties by successively refining ODE liveness properties using ODE invariance properties which have a well-understood deductive proof theory. This approach is widely applicable: we survey several liveness arguments in the literature and derive them all as special instances of our axiomatic refinement approach. We also correct several soundness errors in the surveyed arguments, which further highlights the subtlety of ODE liveness reasoning and the utility of our deductive approach. The library of common refinement steps identified through our approach enables both the sound development and justification of new ODE liveness proof rules from our axioms.Comment: FM 2019: 23rd International Symposium on Formal Methods, Porto, Portugal, October 9-11, 201

Learning safe neural network controllers with barrier certificates

We provide a new approach to synthesize controllers for nonlinear continuous dynamical systems with control against safety properties. The controllers are based on neural networks (NNs). To certify the safety property we utilize barrier functions, which are represented by NNs as well. We train the controller-NN and barrier-NN simultaneously, achieving a verification-in-the-loop synthesis. We provide a prototype tool nncontroller with a number of case studies. The experiment results confirm the feasibility and efficacy of our approach

LNCS

We address the problem of analyzing the reachable set of a polynomial nonlinear continuous system by over-approximating the flowpipe of its dynamics. The common approach to tackle this problem is to perform a numerical integration over a given time horizon based on Taylor expansion and interval arithmetic. However, this method results to be very conservative when there is a large difference in speed between trajectories as time progresses. In this paper, we propose to use combinations of barrier functions, which we call piecewise barrier tube (PBT), to over-approximate flowpipe. The basic idea of PBT is that for each segment of a flowpipe, a coarse box which is big enough to contain the segment is constructed using sampled simulation and then in the box we compute by linear programming a set of barrier functions (called barrier tube or BT for short) which work together to form a tube surrounding the flowpipe. The benefit of using PBT is that (1) BT is independent of time and hence can avoid being stretched and deformed by time; and (2) a small number of BTs can form a tight over-approximation for the flowpipe, which means that the computation required to decide whether the BTs intersect the unsafe set can be reduced significantly. We implemented a prototype called PBTS in C++. Experiments on some benchmark systems show that our approach is effective

Функциональное состояние сердечно- сосудистой и пищеварительной систем организма у студентов во время занятий фитнесом

exercises effect the activation of internal systems’ work in the body. Analyzing the correlation of recreative fitness with students’ nutrition provides an opportunity to identify prospects for improving cardiovascular and digestive systems, improving general condition and achieving the desired student athletic performance. The study found that 92.5% of students were positive about the new content of training in modern fitness technology and only 7.5% said their attitude was uncertain. During the experiment, the dynamics of changes in the parameters of the cardiovascular and digestive systems’ functional state (experimental/control groups) were confirmed: the mean body weight decreased by 2.8 kg/1.5 kg, the pulse at rest decreased to 65.25 beats./min.+0.25/74.46 bpm+0.64; systolic pressure decreased to 112.75 mm Hg+0.35/118.21 mm Hg+0.79; diastolic pressure decreased to 72.21 mm Hg+0.39/76.57 mm Hg+0.43; the vital capacity of the lungs increased by 0.4 l/0.2 l, which is evidence of the cardiovascular system’s economic activity and confirms the feasibility of the applied methodology of health fitness technologies. Regarding the expediency of dietary change: 77% of students said they needed change, while 21% said they wanted it, and only 2% said they didn't want to change their diet.В роботі проведено дослідження студентів, що займаються сучасними фітнес-технологіями та встановлено, що заняття фізичними вправами впливають на активізацію роботи внутрішніх систем організму. Аналіз взаємозв’язку оздоровчого фітнесу з раціональним харчуванням студентів, дає можливість виявити перспективи покращення діяльності серцево-судинної та травної систем, покращення самопочуття та досягнення бажаних спортивних результатів студентів. За результатами дослідження було встановлено, що 92,5 % студентів позитивно віднеслися до нового змісту занять за сучасними фітнес-технологіями і лише 7,5 % своє ставлення назвали невизначеним. Під час проведення експериментальної роботи підтвердилась динаміка змін параметрів функціонального стану серцево-судинної та травної систем (експериментальна /контрольна групи): середній показник маси тіла зменшився на 2,8 кг / 1,5 кг, пульс в стані спокою знизився до 65,25 уд./хвл. + 0,25 / 74,46 уд./хвл. +0,64; систолічний тиск знизився до 112,75 мм рт.ст.+0,35 / 118,21 мм рт.ст.+0,79; діастолічний тиск знизився до 72,21 мм рт.ст. +0,39 / 76,57 мм рт. ст. +0,43 ; життєва ємність легень збільшилась на 0,4 л / 0,2 л, що є свідченням економічної діяльності серцево-судинної системи та підтверджує доцільність використаної методики оздоровчих фітнес-технологій. Щодо доцільності змін у харчуванні: 77% студентів висловились за необхідність змін, тоді як 21% назвали зміни бажаними і лише 2% виявили небажання змінювати свій режим харчування.В работе проведено исследование студентов, которые занимаются современными фитнес- технологиями и установлено, что занятия физическими упражнениями влияют на активизацию работы внутренних систем организма. Анализ взаимосвязи оздоровительного фитнеса с режимом питания студентов, дает возможность выявить перспективы улучшения деятельности сердечнососудистой и пищеварительной систем, улучшения самочувствия и достижения желаемых спортивных результатов. По результатам исследования установлено, что 92,5 % студентов позитивно отнеслись к новому содержанию занятий за современными фитнес -технологиями и только 7,5 % свое отношение назвали неопределенным. Что касается изменений в режиме питании: 77% высказались за необходимость изменений, тогда как 21% назвали изменения желаемыми и только 2% выявили нежелание изменять свой режим питания. При проведении эксперимента подтвердилась динамика изменений параметров физического состояния организма: средний показатель массы тела снизился на 2,8 кг, пульс в состоянии покоя снизился до 65,25 + 0,25 уд. /мин.; систолическое давление снизилось до 112,75 мм рт.ст.+0,35; диастолическое давление снизилось до 72,21 мм рт.ст. +0,39, что является показателем экономной деятельности сердечно-сосудистой системы

Pegasus: a framework for sound continuous invariant generation

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 automation of formal proofs of safety in 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.<br/

Discrete Abstraction of Multiaffine Systems

Many biological systems can be modeled as multiaffine hybrid systems. Due to the nonlinearity of multiaffine systems, it is difficult to verify their properties of interest directly. A common strategy to tackle this problem is to construct and analyze a discrete overapproximation of the original system. However, the conservativeness of a discrete abstraction significantly determines the level of confidence we can have in the properties of the original system. In this paper, in order to reduce the conservativeness of a discrete abstraction, we propose a new method based on a sufficient and necessary decision condition for computing discrete transitions between states in the abstract system. We assume the state space partition of a multiaffine system to be based on a set of multivariate polynomials. Hence, a rectangular partition defined in terms of polynomials of the form (xi − c) is just a simple case of multivariate polynomial partition, and the new decision condition applies naturally. We analyze and demonstrate the improvement of our method over the existing methods using some examples

LNCS

Piecewise Barrier Tubes (PBT) is a new technique for flowpipe overapproximation for nonlinear systems with polynomial dynamics, which leverages a combination of barrier certificates. PBT has advantages over traditional time-step based methods in dealing with those nonlinear dynamical systems in which there is a large difference in speed between trajectories, producing an overapproximation that is time independent. However, the existing approach for PBT is not efficient due to the application of interval methods for enclosure-box computation, and it can only deal with continuous dynamical systems without uncertainty. In this paper, we extend the approach with the ability to handle both continuous and hybrid dynamical systems with uncertainty that can reside in parameters and/or noise. We also improve the efficiency of the method significantly, by avoiding the use of interval-based methods for the enclosure-box computation without loosing soundness. We have developed a C++ prototype implementing the proposed approach and we evaluate it on several benchmarks. The experiments show that our approach is more efficient and precise than other methods in the literature