Barrier Certificates for Safety Verification

• Barrier certificates are scalar-valued functions defined over the state-space that separate safe states from unsafe regions, ensuring invariance under system dynamics.
• They are synthesized using techniques such as sum-of-squares programming, interval analysis, CEGIS, and neural network approaches to satisfy convex feasibility and invariance conditions.
• Their applications span autonomous systems, power grids, quantum systems, and hybrid settings while extending to probabilistic, temporal logic, and safe learning frameworks.

Barrier certificates (BCs) are scalar-valued functions over the state-space of dynamical or hybrid systems that provide functional, inductively-verifiable invariants separating reachable (safe) states from unsafe regions. The existence of such a certificate guarantees, under clearly stated analytic conditions, that no trajectory starting in the specified initial set $\mathcal{X}_0$ can enter the unsafe set $\mathcal{X}_u$. Barrier certificates are central primitives in formal methods for safety verification, controller synthesis, data-efficient learning, and probabilistic reachability analysis in continuous, discrete, hybrid, and stochastic systems.

1. Mathematical Foundations and Definitions

The canonical setting is a continuous-time system $\dot x = f(x)$, $x \in \mathcal{X} \subseteq \mathbb{R}^n$, with initial set $\mathcal{X}_0 \subset \mathcal{X}$ and unsafe set $\mathcal{X}_u \subset \mathcal{X}$. A barrier certificate is a function $B:\mathcal{X} \to \mathbb{R}$ fulfilling—typically for all $x$ in their domains—the following:

• $B(x) \leq 0$ for all $x \in \mathcal{X}_0$ (initial condition)
• $B(x) > 0$ for all $x \in \mathcal{X}_u$ (unsafe set separation)
• $\nabla B(x) \cdot f(x) \leq 0$ on the critical set, typically either $\{x: B(x)=0\}$ or everywhere outside the unsafe set (forward-invariance)

For discrete-time systems, the infinitesimal condition is replaced by $B(f(x)) - B(x) \leq 0$.

The central soundness theorem (Prajna & Jadbabaie [2004]) asserts that if such a $B$ exists, then no trajectory—deterministic, stochastic, or input-driven, according to context—starting from $x_0 \in \mathcal{X}_0$ can reach $\mathcal{X}_u$ (Wang et al., 2017, Peruffo et al., 2020, Ratschan, 2017).

Extensions are formulated for:

• Control systems: Barrier functions $h$ and control barrier certificates (CBCs) provide invariance via restricted control sets $S(x)$, ensuring forward-invariance under admissible controls (Wang et al., 2017).
• Hybrid/switched/stochastic systems: Barrier conditions are imposed mode-wise or via a family of functions tied to the infinitesimal generator or transition kernel (Anand et al., 2021, Nejati et al., 2020).
• Finite- and infinite-time, probabilistic, and temporal logic settings: Variants, including martingale- or supermartingale-type drift conditions, strengthen probabilistic safety claims (Anand et al., 2021, Jagtap et al., 2018).

A converse theorem states that for every robustly safe system, a smooth barrier certificate exists and can be constructed from finite-time reachable sets under modest regularity assumptions (Ratschan, 2017).

2. Synthesis Methodologies: Algebraic and Data-Driven Approaches

Synthesis of BCs is the major computational challenge, addressed by several methodologies:

• Sum-of-Squares Programming (SOS): Given polynomial system data and semi-algebraic domains, search for a polynomial $B$ and SOS multipliers $s_j(x)$ such that the BC conditions become convex feasibility conditions in a semidefinite program (Peruffo et al., 2020, Wongpiromsarn et al., 2014, Wooding et al., 23 Apr 2024, Wang et al., 2018). This is the most prevalent approach for moderate dimensions and polynomial dynamics.
• Interval Analysis: For arbitrary nonlinearities and general parametric barrier templates (not only polynomials), interval contractors and branch-and-bound prune the parameter space in a sound, non-relaxed but potentially inefficient manner (Djaballah et al., 2015).
• Counterexample-Guided Inductive Synthesis (CEGIS): Fix a functional template ($B$), and alternately optimize the coefficients (often by neural network optimization (Peruffo et al., 2020)) and use SMT solvers (e.g., Z3, dReal) to find counterexamples to the BC property, adding them iteratively until all constraints are satisfied or the template/family is exhausted (Peruffo et al., 2020, Abate et al., 29 Apr 2024).
• Neural and Neurosymbolic Approaches: Feedforward or specialized monotone neural networks serve as expressive, tractable templates for $B$, with learning guided by gradient losses encoding the barrier conditions, possibly under monotonicity or other architectural constraints (Peruffo et al., 2020, Nadali et al., 16 Aug 2025, Abate et al., 29 Apr 2024).
• Scenario-Based and Bayesian Methods: For systems with uncertainty or latent dynamics, candidate BCs are trained to certify safety over posterior samples of system parameters (e.g., via marginal Metropolis-Hastings sampling), with scenario theory providing statistical guarantees (Lefringhausen et al., 2 Apr 2025).
• SOS and SMT for Quantum/Complex Systems: In quantum dynamics, BCs are extended to real polynomials in $(z, \bar{z})$ on $\mathbb{C}^n$, with conditions of the form $B(z)\leq0$ on $Z_0$, $B(z)>0$ on $Z_u$, and $\partial_t B(z)\leq0$ (Lewis et al., 2023, Hu et al., 9 Jun 2025).

3. BCs for Advanced System Classes and Logic Specifications

Barrier certificates have been extended far beyond basic ODEs:

• Switched, Stochastic, and Hybrid Systems: BCs are constructed for each mode and composed under appropriate (max-type small-gain) conditions so that the maximum or suitably weighted combination yields a global BC (Anand et al., 2021, Nejati et al., 2020, Anand et al., 2021). Martingale and supermartingale arguments connect BC drift conditions to probabilistic reachability bounds (Jagtap et al., 2018).
• Temporal Logic and ω-Regular Properties: LTL or safe-LTL$_F$ specifications are automatically decomposed into finite collections of reachability subproblems via automata-theoretic constructions. BCs are synthesized for each subproblem, yielding probabilistic satisfaction bounds over complex logic properties (Wongpiromsarn et al., 2014, Anand et al., 2021, Jagtap et al., 2018, Anand et al., 2021).
• Closure and Co-Büchi Certificates: For ω-regular and recurrence properties, standard BCs are insufficient. Closure certificates generalize BCs to work over pairs $(x, y)$ and enforce inductive invariance over relational/transitive reachability, thus resolving persistence/refinement properties (Murali et al., 2023). Co-Büchi barrier certificates track finite visits to specified sets via an augmented state, certifying “no more than $k$” visits to a predicate region (Murali et al., 2023).
• Quantum Systems and Circuits: BC theory extends to the complex domain, with polynomial forms incorporating both amplitude and phase constraints. Scenario-based synthesis and SMT validation are used for efficiency and universal correctness (Lewis et al., 2023, Hu et al., 9 Jun 2025).
• Machine Teaching Dynamics: BCs certify that, under all admissible learning trajectories induced by the teaching process (modeled as a partially observable Markov process), performance thresholds (e.g., minimum belief in the target) are met within a prescribed number of steps (Ahmadi et al., 2018).

4. Permissive and Data-Efficient Variants

To minimize conservativeness and address data- or sample-efficient safe exploration in uncertain systems:

• Permissive Barrier Certificates: The PBC problem seeks to maximize the volume of the safe-and-stabilizable region, often by optimizing over the shape and extent of the BC level set. The approach uses iterative (coordinate-descent) SOS algorithms, ensuring that the computed region strictly contains the Lyapunov sublevel-set region (Wang et al., 2018).
• Safe Learning and Adaptive Certification: In model learning with safety constraints (e.g., quadrotor dynamics), BCs parameterized by Gaussian process posteriors yield high-probability safety via mean-plus-confidence interval constraints. Adaptive sampling expands the safe set efficiently, while real-time recursive updates to the GP ensure on-the-fly computation (Wang et al., 2017). Bayesian scenario-based methods similarly provide high-confidence probabilistic safety for latent-state systems (Lefringhausen et al., 2 Apr 2025).
• Monotone and Large-Scale Systems: For monotone systems, BC verification reduces to localized checks on the boundary of the state simplex. Monotone neural network barriers—architecturally constrained so that $h(x)$ is monotone-increasing—enable scalable synthesis and verification in very high dimensions with only linear sample complexity (Nadali et al., 16 Aug 2025).

5. Computational Workflows, Software, and Scalability

A variety of tools and algorithmic paradigms support BC synthesis:

• Sum-of-Squares and SDP Tools: Python-based APIs (e.g., PRoTECT (Wooding et al., 23 Apr 2024)), MATLAB SOSTOOLS, and other toolkits automate SOS-based search for BCs across four major classes (continuous/discrete, deterministic/stochastic), favoring parallelization over degree and domain parameters to maximize computational throughput.
• Learner-Verifier Loops and Meta-Networks: CEGIS frameworks coordinate gradient-based NN (or polynomial) learners with SMT verifiers, extending to meta-networks that generalize BC generation to unseen initial/unsafe regions with minimal latency (Peruffo et al., 2020, Abate et al., 29 Apr 2024).
• Scenario and Sample-Driven Methods: Probabilistic or Bayesian scenario approaches provide statistical validation of BCs synthesized on sampled instances, efficiently closing the gap between finite sample feasibility and universal properties (Lefringhausen et al., 2 Apr 2025, Hu et al., 9 Jun 2025).
• Practical Performance: Modern workflows demonstrate BC synthesis for high-order (4–13,000 dimensional) systems, with efficient numerical convergence, formal soundness guarantees, and application to real-world safety-critical domains including autonomous driving, power grids, and urban traffic (Wooding et al., 23 Apr 2024, Nadali et al., 16 Aug 2025).

6. Limitations, Extensions, and Research Directions

Current limitations and future opportunities for BC research include:

• Completeness and Expressivity: No guarantee exists for finding a BC in a fixed parametric template, especially in non-polynomial or high-dimensional settings (Peruffo et al., 2020). Research explores richer function classes (e.g., rational, spline, monomial sum networks) and automated template selection.
• Handling Recurrence and Liveness: Standard BCs are inherently safety-type tools. For infinite-horizon recurrence, persistence, and liveness, closure certificates (Murali et al., 2023) and co-Büchi barrier certificates (Murali et al., 2023) provide structurally more powerful certificates, at the expense of higher-dimensional search spaces.
• Hybrid/Uncertain/Dynamically Evolving Systems: BC theory is being extended to more general hybrid systems, data-driven and black-box settings, and systems with stochastic dynamics or complex interconnections, sometimes via compositional and small-gain frameworks (Nejati et al., 2020, Anand et al., 2021).
• Quantum and Noncommutative Extensions: Recent advances adapt BC techniques for state-spaces over $\mathbb{C}^n$, quantum circuit unitaries, and infinite-dimensional dynamics, with mixed-integer and hybrid constraint formulations for general circuits (Lewis et al., 2023, Hu et al., 9 Jun 2025).
• Software and Reproducibility: Open-source tools (e.g., PRoTECT (Wooding et al., 23 Apr 2024)) and rigorous SMT/SDP pipelines are increasingly being adopted for reproducible, verifiable safety guarantees across research and industrial sectors.

Barrier certificates and their extensions now constitute a unifying thread in the verification and design of safe autonomous, cyber-physical, and hybrid systems, bridging formal analysis, control theory, machine learning, and applied logic at scale.