arXiv subjects

Bai Xue

Publications and source records attributed to Bai Xue.

54 records · Page 3Linked to original sources

arXiv2022

Encoding inductive invariants as barrier certificates: synthesis via difference-of-convex programming

A barrier certificate often serves as an inductive invariant that isolates an unsafe region from the reachable set of states, and hence is widely used in proving safety of hybrid systems possibly over an infinite time horizon. We present a novel condition on barrier certificates, termed the invariant barrier-certificate condition, that witnesses unbounded-time safety of differential dynamical systems. The proposed condition is the weakest possible one to attain inductive invariance. We show that discharging the invariant barrier-certificate condition -- thereby synthesizing invariant barrier certificates -- can be encoded as solving an optimization problem subject to bilinear matrix inequalities (BMIs). We further propose a synthesis algorithm based on difference-of-convex programming, which approaches a local optimum of the BMI problem via solving a series of convex optimization problems. This algorithm is incorporated in a branch-and-bound framework that searches for the global optimum in a divide-and-conquer fashion. We present a weak completeness result of our method, namely, a barrier certificate is guaranteed to be found (under some mild assumptions) whenever there exists an inductive invariant (in the form of a given template) that suffices to certify safety of the system. Experimental results on benchmarks demonstrate the effectiveness and efficiency of our approach.

cs.LO↗

arXiv2022

Reach-avoid Verification Based on Convex Optimization

In this paper we propose novel optimization-based methods for verifying reach-avoid (or, eventuality) properties of continuous-time systems modelled by ordinary differential equations. Given a system, an initial set, a safe set and a target set of states, we say that the reach-avoid property holds if for all initial conditions in the initial set, any trajectory of the system starting at them will eventually, i.e.\ in unbounded yet finite time, enter the target set while remaining inside the safe set until that first target hit. Based on a discount value function, two sets of quantified constraints are derived for verifying the reach-avoid property via the computation of exponential/asymptotic guidance-barrier functions (they form a barrier escorting the system to the target set safely at an exponential or asymptotic rate). It is interesting to find that one set of constraints whose solution is termed exponential guidance-barrier functions is just a simplified version of the existing one derived from the moment based method, while the other one whose solution is termed asymptotic guidance-barrier functions is completely new. Furthermore, built upon this new set of constraints, we derive a set of more expressive constraints, which includes the aforementioned two sets of constraints as special instances, providing more chances for verifying the reach-avoid properties successfully. When the involved datum are polynomials, i.e., the initial set, safe set and target set are semi-algebraic, and the system has polynomial dynamics, the problem of solving these sets of constraints can be framed as a semi-definite optimization problem using sum-of-squares decomposition techniques and thus can be efficiently solved in polynomial time via interior point methods. Finally, several examples demonstrate the theoretical developments and performance of proposed methods.

math.OC↗

arXiv2022

Towards Practical Robustness Analysis for DNNs based on PAC-Model Learning

To analyse local robustness properties of deep neural networks (DNNs), we present a practical framework from a model learning perspective. Based on black-box model learning with scenario optimisation, we abstract the local behaviour of a DNN via an affine model with the probably approximately correct (PAC) guarantee. From the learned model, we can infer the corresponding PAC-model robustness property. The innovation of our work is the integration of model learning into PAC robustness analysis: that is, we construct a PAC guarantee on the model level instead of sample distribution, which induces a more faithful and accurate robustness evaluation. This is in contrast to existing statistical methods without model learning. We implement our method in a prototypical tool named DeepPAC. As a black-box method, DeepPAC is scalable and efficient, especially when DNNs have complex structures or high-dimensional inputs. We extensively evaluate DeepPAC, with 4 baselines (using formal verification, statistical methods, testing and adversarial attack) and 20 DNN models across 3 datasets, including MNIST, CIFAR-10, and ImageNet. It is shown that DeepPAC outperforms the state-of-the-art statistical method PROVERO, and it achieves more practical robustness analysis than the formal verification tool ERAN. Also, its results are consistent with existing DNN testing work like DeepGini.

cs.LG↗

arXiv2022

Synthesizing Invariant Clusters for Polynomial Programs by Semidefinite Programming

In this paper, we present a novel approach to synthesize invariant clusters for polynomial programs. An invariant cluster is a set of program invariants that share a common structure, which could, for example, be used to save the needs for repeatedly synthesizing new invariants when the specifications and programs are evolving. To that end, we search for sets of parameters $R_k$ w.r.t. a parameterized multivariate polynomial $I(a, x)$ (i.e. a template) such that $I(a, x) \leq 0$ is a valid program invariant for all $a \in R_k$. Instead of using time-consuming symbolic routines such as quantifier eliminations, we show that such sets of parameters can be synthesized using a hierarchy of semidefinite programming (SDP). Moreover, we show that, under some standard non-degenerate assumptions, almost all possible valid parameters can be included in the synthesized sets. Such kind of completeness result has previously only been provided by symbolic approaches. Further extensions such as using semialgebraic and general algebraic templates (instead of polynomial ones) and allowing non-polynomial continuous functions in programs are also discussed.

eess.SY↗

arXiv2021

Ensemble Defense with Data Diversity: Weak Correlation Implies Strong Robustness

In this paper, we propose a framework of filter-based ensemble of deep neuralnetworks (DNNs) to defend against adversarial attacks. The framework builds an ensemble of sub-models -- DNNs with differentiated preprocessing filters. From the theoretical perspective of DNN robustness, we argue that under the assumption of high quality of the filters, the weaker the correlations of the sensitivity of the filters are, the more robust the ensemble model tends to be, and this is corroborated by the experiments of transfer-based attacks. Correspondingly, we propose a principle that chooses the specific filters with smaller Pearson correlation coefficients, which ensures the diversity of the inputs received by DNNs, as well as the effectiveness of the entire framework against attacks. Our ensemble models are more robust than those constructed by previous defense methods like adversarial training, and even competitive with the classical ensemble of adversarial trained DNNs under adversarial attacks when the attacking radius is large.

cs.LG↗

arXiv2021

Synthesizing Invariant Barrier Certificates via Difference-of-Convex Programming

A barrier certificate often serves as an inductive invariant that isolates an unsafe region from the reachable set of states, and hence is widely used in proving safety of hybrid systems possibly over the infinite time horizon. We present a novel condition on barrier certificates, termed the invariant barrier-certificate condition, that witnesses unbounded-time safety of differential dynamical systems. The proposed condition is by far the least conservative one on barrier certificates, and can be shown as the weakest possible one to attain inductive invariance. We show that discharging the invariant barrier-certificate condition -- thereby synthesizing invariant barrier certificates -- can be encoded as solving an optimization problem subject to bilinear matrix inequalities (BMIs). We further propose a synthesis algorithm based on difference-of-convex programming, which approaches a local optimum of the BMI problem via solving a series of convex optimization problems. This algorithm is incorporated in a branch-and-bound framework that searches for the global optimum in a divide-and-conquer fashion. We present a weak completeness result of our method, in the sense that a barrier certificate is guaranteed to be found (under some mild assumptions) whenever there exists an inductive invariant (in the form of a given template) that suffices to certify safety of the system. Experimental results on benchmark examples demonstrate the effectiveness and efficiency of our approach.

cs.LO↗

arXiv2021

Switching Controller Synthesis for Delay Hybrid Systems under Perturbations

Delays are ubiquitous in modern hybrid systems, which exhibit both continuous and discrete dynamical behaviors. Induced by signal transmission, conversion, the nature of plants, and so on, delays may appear either in the continuous evolution of a hybrid system such that the evolution depends not only on the present state but also on its execution history, or in the discrete switching between its different control modes. In this paper we come up with a new model of hybrid systems, called \emph{delay hybrid automata}, to capture the dynamics of systems with the aforementioned two kinds of delays. Furthermore, based upon this model we study the robust switching controller synthesis problem such that the controlled delay system is able to satisfy the specified safety properties regardless of perturbations. To the end, a novel method is proposed to synthesize switching controllers based on the computation of differential invariants for continuous evolution and backward reachable sets of discrete jumps with delays. Finally, we implement a prototypical tool of our approach and demonstrate it on some case studies.

eess.SY↗

arXiv2021

Robust Invariant Sets Computation for Switched Discrete-Time Polynomial Systems

In this paper we study the robust invariant sets generation problem for discrete-time switched polynomial systems subject to disturbance inputs within the optimal control framework. A robust invariant set of interest is a set of states such that every possible trajectory starting from it never leaves a specified safe set, regardless of actual disturbances. The maximal robust invariant set is shown to be the zero level set of the unique bounded solution to a Bellman type equation, which is a functional equation being widely used in discrete-time optimal control. This is the main contribution of this work. The uniqueness of bounded solutions enables us to solve the derived Bellman type equation using numerical methods such as the value iteration, which provides an approximation of the maximal robust invariant set. In order to increase the scalability of the Bellman equation based method, a semi-definite program, which is constructed based on the derived Bellman type equation, is also implemented to synthesize robust invariant sets. Finally, three examples demonstrate the performance of our methods.

cs.DM↗

arXiv2020

Synthesizing Robust Domains of Attraction for State-Constrained Perturbed Polynomial Systems

In this paper we propose a novel semi-definite programming based method to compute robust domains of attraction for state-constrained perturbed polynomial systems. A robust domain of attraction is a set of states such that every trajectory starting from it will approach an equilibrium while never violating a specified state constraint, regardless of the actual perturbation. The semi-definite program is constructed by relaxing a generalized Zubov's equation. The existence of solutions to the constructed semi-definite program is guaranteed and there exists a sequence of solutions such that their strict one sub-level sets inner-approximate the interior of the maximal robust domain of attraction in measure under appropriate assumptions. Some illustrative examples demonstrate the performance of our method.

eess.SY↗

arXiv2020

Improving Neural Network Verification through Spurious Region Guided Refinement

We propose a spurious region guided refinement approach for robustness verification of deep neural networks. Our method starts with applying the DeepPoly abstract domain to analyze the network. If the robustness property cannot be verified, the result is inconclusive. Due to the over-approximation, the computed region in the abstraction may be spurious in the sense that it does not contain any true counterexample. Our goal is to identify such spurious regions and use them to guide the abstraction refinement. The core idea is to make use of the obtained constraints of the abstraction to infer new bounds for the neurons. This is achieved by linear programming techniques. With the new bounds, we iteratively apply DeepPoly, aiming to eliminate spurious regions. We have implemented our approach in a prototypical tool DeepSRGR. Experimental results show that a large amount of regions can be identified as spurious, and as a result, the precision of DeepPoly can be significantly improved. As a side contribution, we show that our approach can be applied to verify quantitative robustness properties.

cs.AI↗

arXiv2020

PAC Model Checking of Black-Box Continuous-Time Dynamical Systems

In this paper we present a novel model checking approach to finite-time safety verification of black-box continuous-time dynamical systems within the framework of probably approximately correct (PAC) learning. The black-box dynamical systems are the ones, for which no model is given but whose states changing continuously through time within a finite time interval can be observed at some discrete time instants for a given input. The new model checking approach is termed as PAC model checking due to incorporation of learned models with correctness guarantees expressed using the terms error probability and confidence. Based on the error probability and confidence level, our approach provides statistically formal guarantees that the time-evolving trajectories of the black-box dynamical system over finite time horizons fall within the range of the learned model plus a bounded interval, contributing to insights on the reachability of the black-box system and thus on the satisfiability of its safety requirements. The learned model together with the bounded interval is obtained by scenario optimization, which boils down to a linear programming problem. Three examples demonstrate the performance of our approach.

eess.SY↗

arXiv2020

Unbounded-Time Safety Verification of Stochastic Differential Dynamics

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.

math.DS↗

arXiv2020

Nonlinear Craig Interpolant Generation

Interpolation-based techniques have become popularized in recent years because of their inherently modular and local reasoning, which can scale up existing formal verification techniques like theorem proving, model-checking, abstraction interpretation, and so on, while the scalability is the bottleneck of these techniques. Craig interpolant generation plays a central role in interpolation-based techniques, and therefore has drawn increasing attentions. In the literature, there are various works done on how to automatically synthesize interpolants for decidable fragments of first-order logic, linear arithmetic, array logic, equality logic with uninterpreted functions (EUF), etc., and their combinations. But Craig interpolant generation for non-linear theory and its combination with the aforementioned theories are still in infancy, although some attempts have been done. In this paper, we first prove that a polynomial interpolant of the form $h(\mathbf{x})>0$ exists for two mutually contradictory polynomial formulas $ϕ(\mathbf{x},\mathbf{y})$ and $ψ(\mathbf{x},\mathbf{z})$, with the form $f_1\ge0\wedge\cdots\wedge f_n\ge0$, where $f_i$ are polynomials in $\mathbf{x},\mathbf{y}$ or $\mathbf{x},\mathbf{z}$, and the quadratic module generated by $f_i$ is Archimedean. Then, we show that synthesizing such interpolant can be reduced to solving a semi-definite programming problem (${\rm SDP}$). In addition, we propose a verification approach to assure the validity of the synthesized interpolant and consequently avoid the unsoundness caused by numerical error in ${\rm SDP}$ solving. Finally, we discuss how to generalize our approach to general semi-algebraic formulas.

cs.FL↗

arXiv2020

Robust Regions of Attraction Generation for State-Constrained Perturbed Discrete-Time Polynomial Systems

In this paper we propose a convex programming based method for computing robust regions of attraction for state-constrained perturbed discrete-time polynomial systems. The robust region of attraction of interest is a set of states such that every possible trajectory initialized in it will approach an equilibrium state while never violating the specified state constraint, regardless of the actual perturbation. Based on a Bellman equation which characterizes the interior of the maximal robust region of attraction as the strict one sub-level set of its unique bounded and continuous solution, we construct a semi-definite program for computing robust regions of attraction. Under appropriate assumptions, the existence of solutions to the constructed semi-definite program is guaranteed and there exists a sequence of solutions such that their strict one sub-level sets inner-approximate and converge to the interior of the maximal robust region of attraction in measure. Finally, we demonstrate the method by two examples.

math.DS↗

arXiv2020

Over- and Under-Approximating Reachable Sets for Perturbed Delay Differential Equations

This note explores reach set computations for perturbed delay differential equations (DDEs). The perturbed DDEs of interest in this note is a class of DDEs whose dynamics are subject to perturbations, and their solutions feature the local homeomorphism property with respect to initial states. Membership in this class of perturbed DDEs is determined by conducting sensitivity analysis of solution mappings with respect to initial states to impose a bound constraint on the time-lag term. The homeomorphism property of solutions to such class of perturbed DDEs enables us to construct over- and under-approximations of reach sets by performing reachability analysis on just the boundaries of their permitted initial sets, thereby permitting an extension of reach set computation methods for ordinary differential equations to perturbed DDEs. Three examples demonstrate the performance of our approach.

eess.SY↗

arXiv2019

Reach-Avoid Differential Games Based on Invariant Generation

Reach-avoid differential games play an important role in collision avoidance, motion planning and control of aircrafts, and related applications. The central problem is the computation of the set of initial states from which the ego player can enforce the satisfiability of safety specifications over a specified time horizon. Previous methods addressing this problem mostly focus on finite time horizons. We study this problem in the context of the infinite time horizon, where the ego player aims to perpetually force the system to satisfy certain safety specification while the mutual other player attempts to enforce a violation of this safety specification. The problem is studied within the Hamilton-Jacobi reachability framework with unique Lipschitz continuous viscosity solutions. The continuity and uniqueness property of the viscosity solution facilitates the use of contemporary numerical methods to solve this problem with an appropriate number of state variables. An example adopted from a Moore-Greitzer jet-engine model is employed to illustrate our approach.

math.OC↗

arXiv2019

Inner-Approximating Reachable Sets for Polynomial Systems with Time-Varying Uncertainties

In this paper we propose a convex programming based method to address a long-standing problem of inner-approximating backward reachable sets of state-constrained polynomial systems subject to time-varying uncertainties. The backward reachable set is a set of states, from which all trajectories starting will surely enter a target region at the end of a given time horizon without violating a set of state constraints in spite of the actions of uncertainties. It is equal to the zero sub-level set of the unique Lipschitz viscosity solution to a Hamilton-Jacobi partial differential equation (HJE). We show that inner-approximations of the backward reachable set can be formed by zero sub-level sets of its viscosity super-solutions. Consequently, we reduce the inner-approximation problem to a problem of synthesizing polynomial viscosity super-solutions to this HJE. Such a polynomial solution in our method is synthesized by solving a single semi-definite program. We also prove that polynomial solutions to the formulated semi-definite program exist and can produce a convergent sequence of inner-approximations to the interior of the backward reachable set in measure under appropriate assumptions. This is the main contribution of this work. Several illustrative examples demonstrate the merits of our approach.

math.OC↗

arXiv2018

Robust Non-termination Analysis of Numerical Software

Numerical software are widely used in safety-critical systems such as aircrafts, satellites, car engines and so on, facilitating dynamics control of such systems in real time, it is therefore absolutely necessary to verify their correctness. It is a long standing challenge to guarantee verified properties of numerical software are indeed satisfied by their real behaviours, because most of these verifications are conducted under ideal mathematical models, but their real executions could be influenced essentially by uncertain inputs accounting for round-off errors and additive perturbations from real-world phenomena to hardware, which are abstracted away in these ideal mathematical models. In this paper, we attempt to address this issue focusing on nontermination analysis of numerical software, where nontermination is often an unexpected behaviour of computer programs and may be problematic for applications such as real-time systems having hard deadlines on transaction execution time, and propose a method for robust conditional nontermination analysis, which can be used to under-approximate the maximal robust nontermination input set for a given program, from which the program never terminates, regardless of the aforementioned disturbances. Finally, several examples are employed to illustrate our approach.

math.OC↗