arXiv subjects

Bai Xue

Publications and source records attributed to Bai Xue.

At least 19 recordsLinked to original sources

arXiv2026

Sufficient and Necessary Smooth Barrier-Like Conditions for Continuous-Time Finite-Horizon Safety Verification

We study finite-horizon safety verification for continuous-time stochastic systems modeled by stochastic differential equations. Given an open and bounded safe set, a compact initial set, a finite time horizon, and a prescribed violation-probability threshold, the objective is to certify that the exit probability from the safe set before the horizon remains below the threshold. Under uniform ellipticity of the diffusion matrix and regularity of the safe-set boundary, we develop a time-dependent barrier-like condition expressed as a set of functional inequalities, obtained by relaxing a parabolic initial-boundary value problem associated with the exit probability. If a function satisfies the condition, then the probability of leaving the safe set before the horizon is guaranteed to remain below the threshold. Conversely, when the exit probability is strictly smaller than the prescribed threshold uniformly over the initial set, the barrier-like condition is also necessary: there exists a smooth function satisfying it. Furthermore, when the dynamics are polynomial and the set descriptions are semialgebraic, the problem of verifying the condition admits a sum-of-squares formulation that is both sound and complete, yielding a complete degree hierarchy for finite-horizon stochastic safety verification.

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arXiv2026

LLM2Cert: LLM Synthesis of Certificates with SMT/Lean Verification

Lyapunov functions and barrier certificates give state-space guarantees of stability and safety, but synthesizing and verifying them---especially for nonlinear, high-dimensional systems---remains an open challenge. Existing pipelines verify candidates with numerical solvers (sum-of-squares programs), interval methods, or satisfiability-modulo-theories (SMT) solvers, which often require relaxations, are conservative, or scale poorly with model dimension and nonlinearity. These limitations have also shaped benchmark design: standard test suites are dominated by low-dimensional polynomial systems chosen to suit what solvers can handle rather than what engineering applications require. We address both sides of this challenge. First, we introduce DynaCertBench, a benchmark of 105 instances: 80 non-redundant systems curated from the literature, extended with high-dimensional and control-constrained variants and new nonlinearly coupled systems, comprehensively spanning stability, region-of-attraction, forward-invariance, and safety tasks. Second, we present LLM2Cert, an LLM-driven framework that synthesizes candidate certificates and verifies them against their proof obligations: each obligation is routed to an SMT solver or directly to the Lean Proof-Repair agent. On DynaCertBench, LLM2Cert verifies 81 of 105 instances ($77.1\%$), a $17.1$ percentage point improvement over one-shot proposal without feedback. On the 81 fixed certificates, LLM2Cert verifies 22 that All-SMT leaves unresolved. Code and data are available at https://github.com/FormalCert-AI/llm2cert.

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arXiv2026

Anytime-Valid Runtime Monitoring of Learned Barrier Certificates

Learned barrier certificates provide a principled mechanism for establishing safety of stochastic systems, but their validity may deteriorate during deployment when the underlying dynamics differ from those used for training. We propose an anytime-valid runtime monitor that audits a learned barrier certificate using observed closed-loop trajectories. The monitor is intended to detect loss of statistical support for the deployed barrier condition; it does not by itself guarantee detection before unsafe entry or safety of a subsequent intervention. Specifically, the conditional barrier-drift condition is formulated as an episode-wise sequential hypothesis test, and observed barrier increments are transformed into a nonnegative test supermartingale, yielding finite-sample false-alarm control under arbitrary stopping times. Inspired by finite-horizon safety verification, we further introduce a deterministic temporal shaping of the monitoring boundary, making the monitor more conservative early in a prescribed deployment horizon while preserving anytime-validity. The monitor is designed to detect statistical evidence that the safety certificate is no longer consistent with the observed dynamics, rather than to predict unsafe entry. To assess the monitor empirically, we evaluate empirical alarm behavior under valid and invalid certificates. Code and data are available at https://github.com/FormalCert-AI/runtime-barrier-monitor.

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arXiv2026

Sufficient and Necessary Continuous Barrier-like Conditions for Discrete-Time Stochastic Reach-Avoid Verification

This paper develops necessary and sufficient barrier-like characterizations using continuous barrier functions for infinite-horizon reach-avoid verification of discrete-time stochastic systems. Existing results establish necessary and sufficient conditions in terms of functional inequalities involving measurable or lower semicontinuous barrier functions. However, the limited regularity of such functions may hinder their numerical approximation and computational synthesis. Building on our previous barrier-like condition for finite-horizon reach-avoid verification, we show that this condition can also be used for infinite-horizon reach-avoid verification and, under a uniform absolute continuity condition on the transition kernels, admits a continuous barrier function whenever the exact reach-avoid probability is strictly larger than the prescribed threshold for every state in the initial set. We further show that the resulting continuous barrier function can be uniformly approximated by a polynomial one while preserving the required barrier-like conditions. For polynomial systems, we formulate these conditions as polynomial positivity constraints over compact basic semialgebraic sets. Putinar's Positivstellensatz then converts the positivity conditions into sum-of-squares (SOS) certificates, yielding semidefinite programming (SDP) formulations for synthesizing polynomial barrier functions. We establish both soundness and completeness of the resulting SOS-based procedure. Finally, two numerical examples illustrate the theoretical results and demonstrate the resulting SDP approach.

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arXiv2026

Sufficient and Necessary Smooth Barrier-like Conditions for Continuous-Time Stochastic Reach-Avoid Verification

In this paper, we study infinite-horizon reach-avoid verification for continuous-time stochastic systems modeled by stochastic differential equations (SDEs). We formulate this problem within a barrier-function-based framework, which transforms the verification problem into an existence problem for barrier functions satisfying barrier-like conditions expressed as functional inequalities. We provide sufficient and necessary barrier-like conditions in terms of polynomial barrier functions for infinite-horizon reach-avoid verification under suitable regularity and uniform ellipticity assumptions. We first construct a discounted value function that characterizes lower bounds on the reach-avoid probability. We then show that it is the unique classical solution of an associated elliptic Dirichlet problem. Based on this characterization, we further show that the barrier-like condition proposed in our previous work on finite-horizon reach-avoid verification is not only sufficient for infinite-horizon reach-avoid verification but also necessary whenever the reach-avoid probability is strictly larger than the specified threshold. In particular, whenever the reach-avoid probability is strictly larger than the specified threshold fro every state in the initial set, polynomial barrier functions satisfying this barrier-like condition exist. Furthermore, when the system dynamics are polynomial, we formulate the problem of finding polynomial barrier functions satisfying this barrier-like condition as sum-of-squares (SOS) programs, which are shown to be sound and complete. Finally, we demonstrate the theoretical results on two numerical examples.

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arXiv2026

Converse Barrier Certificates for Set-Based Stochastic Reach-Avoid Verification

Recent work established sufficient and necessary barrier-like conditions for infinite-horizon reach-avoid verification of stochastic discrete-time systems from a single initial state. Whether such a converse characterization extends to a set of initial states, however, remains open. In this paper, we answer this question affirmatively for compact initial sets. We consider a uniform reach-avoid specification requiring the reach-avoid probability to exceed a prescribed threshold for every initial state in a compact set. Under appropriate assumptions, including continuous system transitions, together with a strict uniform probability margin, we extend the pointwise converse characterization to the uniform setting.

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arXiv2026

Cost-Aware Adaptive Conformal Inference for Runtime Assurance in Dynamic Environments

This paper addresses the problem of providing runtime assurance for systems operating online under unknown and potentially time-varying data distributions. We propose Cost-Aware Adaptive Conformal Inference (ACI), a novel framework that incorporates constraint violation costs directly into the conformal adaptation mechanism. Our key insight is that uncertainty margins should adapt not only to the frequency of constraint violations but also to their severity. We formalize this through a cost-aware loss function that couples the miscoverage indicator with violation costs. Unlike existing methods that regulate a single controlled metric, our approach provides a dual statistical guarantee: simultaneously bounding the long-run average violation frequencies (reliability) and cumulative violation cost (harm). By weighting prediction failures according to their severity, the algorithm enables the controller to respond proportionally to violation severity, expanding prediction sets aggressively when necessary while maintaining efficiency during nominal operation. We integrate Cost-Aware ACI into a robust control synthesis framework, creating a closed-loop system that balances task performance with runtime risk control without requiring explicit model knowledge. Experiments validate its effectiveness for online risk-aware controller synthesis.

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arXiv2026

Stochastic Minimum-Cost Reach-Avoid Reinforcement Learning

We study stochastic minimum-cost reach-avoid reinforcement learning, where an agent must satisfy a reach-avoid specification with probability at least $p$ while minimizing expected cumulative costs in stochastic environments. Existing safe and constrained reinforcement learning methods typically fail to jointly enforce probabilistic reach-avoid constraints and optimize cost in the learning setting in stochastic environments. To address this challenge, we introduce reach-avoid probability certificates (RAPCs), which identify states from which stochastic reach-avoid constraints are satisfiable. Building on RAPCs, we develop a contraction-based Bellman formulation that serves as a principled surrogate for integrating reach-avoid considerations into reinforcement learning, enabling cost optimization under probabilistic constraints. We establish almost sure convergence of the proposed algorithms to locally optimal policies with respect to the resulting objective. Experiments in the MuJoCo simulator demonstrate improved cost performance and consistently higher reach-avoid satisfaction rates.

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arXiv2026

Convex Computations for Controlled Safety Invariant Sets of Black-box Discrete-time Dynamical Systems

Identifying controlled safety invariant sets (CSISs) is essential for safety-critical systems. This paper addresses the problem of computing CSISs for black-box discrete-time systems, where the dynamics are unknown and only limited simulation data are available. Traditionally, a CSIS requires that for every state in the set, there exists a control input that keeps the system within the set at the next step. However, enforcing such universal invariance, i.e., requiring the set to remain controlled invariant for all states, is often overly restrictive or impractical for black-box systems. To address this, we introduce the notion of a Probably Approximately Correct (PAC) CSIS, in which, with prescribed confidence, there exists a suitable control input to keep the system within the set at the next step for at least a specified fraction of the states. Our approach leverages barrier functions and scenario optimization, yielding a tractable linear programming method for estimating PAC CSISs. Several illustrative examples demonstrate the effectiveness of the proposed framework.

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arXiv2026

Refined Barrier Conditions for Finite-Time Safety and Reach-Avoid Guarantees in Stochastic Systems

Providing finite-time probabilistic safety and reach-avoid guarantees is crucial for safety-critical stochastic systems. Existing state-of-the-art barrier methods often rely on a restrictive boundedness assumption for auxiliary functions, limiting their applicability. This paper presents refined barrier conditions that remove this assumption. Specifically, we establish conditions for deriving upper bounds on finite-time safety probabilities in discrete-time systems and lower bounds on finite-time reach-avoid probabilities in continuous-time systems. This relaxation expands the class of verifiable systems, especially those with unbounded state spaces, and facilitates the use of advanced optimization techniques, such as semi-definite programming with polynomial functions. Numerical examples demonstrate the effectiveness of the approach.

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arXiv2026

Quantitative Verification of Finite-Time Constrained Occupation Measures for Continuous-time Stochastic Systems

This paper addresses the quantitative verification of finite-time constrained occupation time for stochastic continuous-time systems governed by stochastic differential equations (SDEs). Unlike classical reachability analysis, which focuses on single-event properties such as entering a target set, many autonomous tasks-including surveillance, wireless charging, and chemical mixing-require a system to accumulate a prescribed duration within a target region while strictly maintaining safety constraints. We propose a barrier-certificate framework to compute rigorous upper and lower bounds on the probability that such cumulative specifications are satisfied over a finite time horizon. By introducing a stopped process that freezes the system once it reaches the boundary of the safe set, we derive three classes of certificates: one for upper bounds and two for lower bounds. The proposed approaches are validated through numerical examples implemented using semidefinite programming.

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arXiv2026

Quantitative Verification of Constrained Occupation Time for Stochastic Discrete-time Systems

This paper addresses the quantitative verification of constrained occupation time in stochastic discrete-time systems, focusing on the probability of visiting a target set at least $k$ times while maintaining safety. Such cumulative properties are essential for certifying repeated behaviors like surveillance and periodic charging. To address this, we present the first barrier certificate framework capable of certifying these behaviors. We introduce multiplicative stochastic barrier functions that encode visitation counts implicitly within the algebraic structure of a scalar barrier. By adopting a switched-system reformulation to handle safety, we derive rigorous probabilistic bounds for both finite and infinite horizons. Specifically, we show that dissipative barriers establish upper bounds ensuring the exponential decay of frequent visits, while attractive barriers provide lower bounds via submartingale analysis. The efficacy of the proposed framework is demonstrated through numerical examples.

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arXiv2026

PAC Finite-Time Safety Guarantees for Stochastic Systems with Unknown Disturbance Distributions

We investigate the problem of establishing finite-time probabilistic safety guarantees for discrete-time stochastic dynamical systems subject to unknown disturbance distributions, using barrier certificate methods. Our approach develops a data-driven safety certification framework that relies only on a finite collection of independent and identically distributed (i.i.d.) disturbance samples. Within this framework, we propose a certification procedure such that, with confidence at least $1-δ$ over the sampled disturbances, if the output of the certification procedure is accepted, the probability that the system remains within a prescribed safe set over a finite horizon is at least $1-ε$. A key challenge lies in formally characterizing the probably approximately correct (PAC) generalization behavior induced by finite samples. To address this, we derive PAC generalization bounds using tools from VC dimension, scenario optimization, and Rademacher complexity. These results illuminate the fundamental trade-offs between sample size, model complexity, and safety tolerance, providing both theoretical insight and practical guidance for designing reliable, data-driven safety certificates in discrete-time stochastic systems.

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arXiv2026

PAC One-Step Safety Certification for Black-Box Discrete-Time Stochastic Systems

This paper investigates the problem of safety certification for black-box discrete-time stochastic systems, where both the system dynamics and disturbance distributions are unknown, and only sampled data are available. Under such limited information, ensuring robust or classical quantitative safety over finite or infinite horizons is generally infeasible. To address this challenge, we propose a data-driven framework that provides theoretical one-step safety guarantees in the Probably Approximately Correct (PAC) sense. This one-step guarantee can be applied recursively at each time step, thereby yielding step-by-step safety assurances over extended horizons. Our approach formulates barrier certificate conditions based solely on sampled data and establishes PAC safety guarantees by leveraging the VC dimension, scenario approaches, Markov's inequality, and Hoeffding's inequality. Two sampling procedures are proposed, and three methods are proposed to derive PAC safety guarantees. The properties and comparative advantages of these three methods are thoroughly discussed. Finally, the effectiveness of the proposed methods are demonstrated through several numerical examples.

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arXiv2026

Converse Barrier Certificates for Finite-time Safety Verification of Continuous-time Perturbed Deterministic Systems

In this paper, we investigate the problem of verifying the finite-time safety of continuous-time perturbed deterministic systems represented by ordinary differential equations in the presence of measurable disturbances. Given a finite-time horizon, if the system is safe, it, starting from a compact initial set, will remain within an open and bounded safe region throughout the specified time horizon, regardless of the disturbances. The main contribution of this work is a converse theorem: we prove that a continuously differentiable, time-dependent barrier certificate exists if and only if the system is safe over the finite-time horizon. The existence problem is explored by finding a continuously differentiable approximation of a unique Lipschitz viscosity solution to a Hamilton-Jacobi equation.

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arXiv2026

Sufficient and Necessary Barrier-like Conditions for Safety and Reach-avoid Verification of Stochastic Discrete-time Systems

This paper investigates necessary and sufficient barrier-like conditions for infinite-horizon safety and reach-avoid verification of stochastic discrete-time systems, derived via a relaxation of the Bellman equations. Unlike prior approaches that primarily focus on sufficient conditions, our work rigorously establishes both necessity and sufficiency for infinite-horizon properties. Safety verification concerns certifying that, starting from a given initial state, the system remains within a safe set at all future time steps with probability at least equal to a specified threshold. For this purpose, we formulate a necessary and sufficient barrier-like condition that captures this infinite-time safety property. In contrast, reach-avoid verification generalizes safety verification by also incorporating reachability. Specifically, it aims to ensure that the probability of the system, starting from a given initial state, eventually reaching a target set while remaining within the safe set until the first hit of the target is no less than a prescribed bound. Under suitable assumptions, we establish two necessary and sufficient barrier-like conditions for this reach-avoid specification.

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arXiv2025

A New Framework for Bounding Reachability Probabilities of Continuous-time Stochastic Systems

This manuscript presents an innovative framework for constructing barrier functions to bound reachability probabilities for continuous-time stochastic systems described by stochastic differential equations (SDEs). The reachability probabilities considered in this paper encompass two aspects: the probability of reaching a set of specified states within a predefined finite time horizon, and the probability of reaching a set of specified states at a particular time instant. The barrier functions presented in this manuscript are developed either by relaxing a parabolic partial differential equation that characterizes the exact reachability probability or by applying the Grönwall's inequality. In comparison to the prevailing construction method, which relies on Doob's non-negative supermartingale inequality (or Ville's inequality), the proposed barrier functions provide stronger alternatives, complement existing methods, or fill gaps.

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arXiv2025

Comparative Analysis of Barrier-like Function Methods for Reach-Avoid Verification in Stochastic Discrete-Time Systems

In this paper, we compare several representative barrier-like conditions from the literature for infinite-horizon reach-avoid verification of stochastic discrete-time systems. Our comparison examines both their theoretical properties and computational tractability, highlighting each condition's strengths and limitations that affect applicability and conservativeness. Finally, we illustrate their practical performance through computational experiments using semidefinite programming (SDP) and counterexample-guided inductive synthesis (CEGIS).

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