Probabilistic Barrier Certificates

Updated 30 June 2026

Probabilistic barrier certificates are constructs that bound the safety probability of stochastic systems by ensuring state trajectories remain in designated safe sets using martingale and SOS optimization techniques.
They mitigate the curse of dimensionality by avoiding full state discretization, enabling scalable safety verification for high-dimensional, uncertain, or black-box systems.
Recent advances extend these certificates to hybrid, switched, and temporal logic systems, though challenges remain in conservatism, template selection, and handling complex uncertainties.

A probabilistic barrier certificate is an analytic, data-driven, or computational construct that certifies lower or upper bounds on the probability that a stochastic, uncertain, or partially observable dynamical system satisfies a safety property—most typically the probability that the state trajectory remains in a designated safe set over a finite time horizon—by leveraging martingale or supermartingale inequalities, chance-constrained control barrier function relaxations, or sum-of-squares (SOS) polynomial optimization. These certificates provide formal, a priori–provable stochastic safety or reach-avoid probability bounds, typically without requiring full state discretization, and are foundational for modern formal verification and safe control under uncertainty.

1. Mathematical Formulation and Invariants

To formalize probabilistic barrier certificates, consider a generic discrete-time stochastic system

$x_{k+1} = f(x_k, w_k), \qquad w_k \sim \mu,$

with safe set $S \subset X$ and unsafe set $X_u = X \setminus S$ . The key probabilistic invariant is to guarantee $\Pr_{w}\bigl\{x_k \in S,~ 0 \leq k \leq N\bigr\} \geq 1-\delta$ for a desired $\delta \in (0,1)$ .

A typical barrier certificate is a nonnegative function $B: X \to \mathbb{R}_{\geq 0}$ such that:

(Initial) $B(x) \leq \eta$ for $x \in X_0$
(Unsafe) $B(x) \geq \beta$ for $x \in X_u$
(Drift/supermartingale) $S \subset X$ 0 for all $S \subset X$ 1

Classic results (e.g., (Jagtap et al., 2018)) show that if these hold, then

$S \subset X$ 2

with the constants tuned for tightness. In continuous-time, this extends to using an infinitesimal generator or conditional expectation over SDE dynamics (Lavaei et al., 2022, Anand et al., 2021).

Probabilistic barrier certificates thus generalize the notion of forward-invariant sets via supermartingale (or c-martingale) properties, leading to analytic tail bounds for stochastic reach, finite-time invariance, and reach-avoid tasks.

2. Practical Synthesis under Uncertainty

Barrier certificates achieve tractable stochastic safety verification for high-dimensional, uncertain, or black-box systems by avoiding the curse of dimensionality intrinsic to full discretization.

Key instantiations include:

Chance-constrained control barrier functions (CBFs) (e.g., PrSBC): For control-affine systems with bounded uncertainty, chance constraints (e.g., $S \subset X$ 3) are converted to deterministic inequalities via worst-case analysis, yielding a set of linear (or quadratic) constraints suitable for quadratic programming over the control input (Luo et al., 2019, Zhang et al., 2023). This enables high-confidence collision avoidance or safety in multi-agent systems without assuming specific noise distributions.
Piecewise-constant s-CBFs: For general nonlinear systems with bounded additive noise, a piecewise-constant safe-set partition and minimax LP encoding (via dualization) synthesizes both the certificate $S \subset X$ 4 and controller $S \subset X$ 5, providing a guaranteed safety probability lower bound without requiring global polynomials (Mazouz et al., 23 Jul 2025, Mazouz et al., 2024).
Data-driven and black-box systems: Scenario-based or Bayesian-inference approaches generate barrier certificates using only sampled trajectories, learning-based GP error bounds, or input-output marginalization (Rickard et al., 17 Mar 2025, Salamati et al., 2021, Lefringhausen et al., 2 Apr 2025). PAC-style sample-complexity bounds yield high-probability stochastic safety guarantees in the absence of analytic models.

These frameworks are robust to model uncertainty and admit scalability to large-scale or compositional systems (Nejati et al., 2020, Anand et al., 2021).

3. Sum-of-Squares and Semidefinite Program Synthesis

For polynomial dynamics and semi-algebraic safe/unsafe sets, the synthesis of probabilistic barrier certificates can be encoded as Sum-of-Squares (SOS) programs and solved as semidefinite programs (SDPs) (Jagtap et al., 2018, Chen et al., 23 Jul 2025, Xue et al., 23 Sep 2025). The standard approach is:

Formulating all invariance, drift, and boundary (safe/unsafe) conditions as polynomial inequalities.
Introducing SOS multipliers (dual certificates) to enforce validity over the state constraints.
Optimizing over the polynomial coefficients (and, if required, initial values $S \subset X$ 6) to minimize the safety bound (e.g., $S \subset X$ 7) or verify feasibility.

This methodology enables automated certificate construction up to moderate state dimensions and polynomial degrees, and supports both finite-horizon and infinite-horizon settings. For reach-avoid or temporal logic, automata-based decomposition reduces temporal formulas to sequential reachability, each verified by a corresponding barrier certificate (Jagtap et al., 2018, Anand et al., 2021).

4. Extensions: Temporal Logic, Hybrid, and Switched Systems

Probabilistic barrier certificates support specification classes beyond basic safety:

Temporal logic properties: By decomposing the automaton of the negation of an LTL (or safe-LTL) property into sequential reachability problems, one applies barrier certificates to each subtask. The overall violation probability is upper-bounded by a sum-product structure over automaton runs, while the satisfaction bound is one minus this aggregate (Jagtap et al., 2018, Anand et al., 2021, Anand et al., 2021).
Hybrid and switched systems: For stochastic hybrid models with both continuous flows and Poisson/discrete jumps, augmented barrier certificates (ACBCs) operate on an expanded state-space and blend generator-based (diffusion) and expectation-based (jump) drift conditions (Lavaei et al., 2022). For stochastic switching, mode-dependent barrier certificates and small-gain–type compositional analysis provide global safety certification (Nejati et al., 2020).
Uncertain temporal predicates: Barrier certificates on product spaces (state × predicate parameter) yield rigorous probability bounds for specifications whose atomic propositions are themselves random sets (Mamduhi et al., 10 May 2026).

5. Variants and Theoretical Advances

Recent work further generalizes the probabilistic barrier certificate paradigm:

k-inductive and interpolation-inspired barrier certificates: Multiple-barrier-function chains or inductive relaxations allow the synthesis of certificates when a single supermartingale does not exist in a tractable template, yielding nontrivial lower safety probability bounds in otherwise intractable cases (Oumer et al., 21 Apr 2025).
Occupation time and multiplicative barriers: For cumulative properties such as guaranteed $S \subset X$ 8-visitation or persistent surveillance, multiplicative barrier functions encode occupancy via geometric amplification, enabling exponential tail bounds on visit counts (Xue et al., 20 Apr 2026).
Refined (unbounded domain) certificates: Recent conditions remove prior boundedness assumptions for polynomial certificates, expanding applicability to unbounded state spaces in both discrete and continuous time (Xue et al., 23 Sep 2025).

6. Application Domains and Empirical Results

Probabilistic barrier certificates are broadly applied for

Multi-robot and autonomous vehicle safety: Ensuring finite-time collision avoidance under localization and actuation uncertainty (Luo et al., 2019).
Learning-enabled control and model uncertainty: Safety for systems with learned dynamics estimates, with practical GP-based or scenario-theoretic generalization bounds (Kumar et al., 2021, Mazouz et al., 2024).
Building automation, energy networks, and high-dimensional ensembles: Efficient scalable guarantees for networks with hundreds to thousands of subsystems via compositionality (Anand et al., 2021, Nejati et al., 2020).
Complex temporal tasks (e.g., POMDP verification, reach-avoid constraints): Structured automation-based barrier composition for certification of rich, multi-step logic requirements (Ahmadi et al., 2018, Jagtap et al., 2018).
Black-box dynamical systems: Empirical validation that high-probability guarantees on stochastic safety can be synthesized solely from finite data without analytic models (Rickard et al., 17 Mar 2025, Salamati et al., 2021, Lefringhausen et al., 2 Apr 2025).

Empirical results consistently show that barrier-certificate–based methods, whether analytic, data-driven, or learning-augmented, maintain empirically validated upper bounds on violation probabilities and typically outperform classical deterministic barrier approaches in the presence of uncertainty.

7. Limitations and Ongoing Challenges

While the probabilistic barrier certificate methodology is powerful and extensible, key limitations and challenges remain:

Conservatism: Worst-case or union-bound–based probability aggregation across automaton runs or multi-step invariance may lead to conservative bounds, especially for complex temporal logic (Anand et al., 2021).
Scalability: While scenario-based and compositional techniques enable scaling, classical polynomial or SOS-based synthesis remains computationally intensive for very high-degree or high-dimensional systems.
Template dependence: The choice of certificate template (e.g., polynomial degree, piecewise structure) critically impacts the ability to find certificates and the tightness of bounds; advanced relaxation strategies such as k-induction or interpolation are sometimes required (Oumer et al., 21 Apr 2025).
Handling of non-polynomial and non-Gaussian uncertainty: The foundational approach presumes polynomial or bounded-support uncertainty; adapting certificates to more general distributions or non-polynomial systems is nontrivial.
Verification under unmodeled or time-varying uncertainties: Ongoing work seeks to extend guarantees to more adversarial or time-varying regimes, including real-time learning and adaptation.

Despite these challenges, probabilistic barrier certificates furnish a mathematically principled, versatile, and rapidly evolving foundation for formal verification and safety-critical control in stochastic, uncertain, and complex dynamical systems.