arxiv: 2605.06087 · v1 · submitted 2026-05-07 · 💻 cs.AI · cs.SY· eess.SY

Recognition: unknown

Safety Certification is Classification

Oliver Sch\"on , Licio Romao , Sadegh Soudjani

Authors on Pith no claims yet

Pith reviewed 2026-05-08 10:35 UTC · model grok-4.3

classification 💻 cs.AI cs.SYeess.SY

keywords safety certificationkernel embeddingtrajectory classificationnon-Markovian dynamicsbarrier certificatesdynamic programminguncertain dynamical systems

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The pith

Treating safety certification as classification on trajectory data enables direct estimation of T-step safety probabilities without recursive error buildup.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that safety certification for dynamical systems under uncertainty can be cast as a classification task on sampled trajectories. Instead of estimating transition probabilities and then applying dynamic programming recursion to compute safety over T steps, the method uses kernel embeddings to classify full trajectories as safe or unsafe in one step. This avoids the accumulation of estimation errors that makes recursive bounds vacuous for longer horizons. The same framework recovers barrier certificates and robust Markov models as special cases while extending naturally to non-Markovian dynamics where history matters. Experiments on a neural-controlled quadrotor illustrate that the direct classifier stays sound and stable where recursive methods silently lose soundness.

Core claim

Safety certification is reframed as a kernel-based classification problem on trajectory data that directly estimates the T-step safety probability. The approach subsumes barrier certificates and robust Markov models, bypasses compounding errors across the horizon, and certifies systems with non-Markovian dynamics.

What carries the argument

Kernel embedding of finite trajectory samples for direct binary classification of complete safe versus unsafe trajectories, yielding a non-recursive lower bound on the T-step safety probability.

If this is right

The direct estimator's accuracy remains independent of the certification horizon T, unlike DP recursion whose error compounds.
Certification applies to non-Markovian systems where future evolution depends on trajectory history.
Existing methods such as barrier certificates arise as special cases of the classification view.
Simulation results on a quadrotor confirm that recursive certificates become unsound while the direct method stays stable.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The classification perspective could incorporate modern supervised learning models beyond kernels for systems with high-dimensional state spaces.
Data collection policies that focus on near-boundary trajectories might tighten the certified lower bound more efficiently than uniform sampling.
For deployed controllers, periodic re-certification on fresh trajectory batches could maintain soundness over changing environments.

Load-bearing premise

Finite trajectory samples plus a suitable kernel embedding produce a sound lower bound on the true T-step safety probability without the classification step introducing bias or variance that invalidates the certificate.

What would settle it

Generate ground-truth safety probabilities for a known non-Markovian system via exhaustive simulation or exact analysis, then check whether the kernel classifier's lower bound ever exceeds that ground truth as more trajectories are added.

Figures

Figures reproduced from arXiv: 2605.06087 by Licio Romao, Oliver Sch\"on, Sadegh Soudjani.

**Figure 1.** Figure 1: Indirect (top) vs. direct (bottom) safety certification on a quadrotor obstacle-avoidance task. In the learning-based setting, the DP recursion compounds estimation error at every step (UQ: uncert. quantification); for realistic horizons, direct certification is the only viable route. 2017]. Popular certification methods based on finite-state abstractions approximate dynamics by gridworld models (e.g., ro… view at source ↗

**Figure 2.** Figure 2: Overarching embedding framework recovering existing approaches and going beyond view at source ↗

**Figure 3.** Figure 3: The effect of increasingly non-Markovian dynamics ( view at source ↗

**Figure 4.** Figure 4: Synthetic neural-controlled quadrotor (Sec. 6.2, view at source ↗

**Figure 5.** Figure 5: For uniform DP data and Nˆ = NT: Results comparing DP and the direct approach for various α ∈ [0, 1) and T = 5, 10, 15 in terms of different metrics. Swift and M. Schaepper, and predict whether altitude pz > 0.2 m is maintained for 1 s after each gate passage. The initial state x0 is the 15-step (≈0.23 s) position history (px, py, pz) immediately preceding each gate crossing; each gate passage constitutes … view at source ↗

**Figure 6.** Figure 6: For uniform DP data and Nˆ = NT: Safety probability estimates (in color) and unknown true dynamics (mean) vector field (white arrows; all identical) for the fully Markovian dynamics (α = 0). With ϱ 100≈0.96 the DP recursion barely contracts, yet DP is still severely miscalibrated: REL is 18× and 26× higher than Direct for Swift and Schaepper respectively ( view at source ↗

**Figure 7.** Figure 7: For uniform DP data and Nˆ = NT: Safety probability estimates (in color) and unknown true dynamics (mean) vector field (white arrows; all identical) for the fully Markovian dynamics (α = 0.95). 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5 R M S E m e a n ± 2std DP T = 5 DP T = 10 DP T = 15 Direct T = 5 Direct T = 10 Direct T = 15 (a) RMSE (lower is better) 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 0.5 E xcess… view at source ↗

**Figure 8.** Figure 8: For uniform DP data and Nˆ = N: Results comparing DP and the direct approach for various α ∈ [0, 1) and T = 5, 10, 15 in terms of different metrics. Using same hyperparameters as for Nˆ = NT. 17 view at source ↗

**Figure 9.** Figure 9: For uniform DP data and Nˆ = N: Safety probability estimates (in color) and unknown true dynamics (mean) vector field (white arrows; all identical) for the fully Markovian dynamics (α = 0). Using same hyperparameters as for Nˆ = NT. 18 view at source ↗

**Figure 10.** Figure 10: For uniform DP data and Nˆ = N: Safety probability estimates (in color) and unknown true dynamics (mean) vector field (white arrows; all identical) for the fully Markovian dynamics (α = 0.95). Using same hyperparameters as for Nˆ = NT. 0.0 0.2 0.4 0.6 0.8 0.02 0.04 0.06 0.08 0.10 0.12 0.14 R M S E m e a n ± 2std DP T = 5 DP T = 10 DP T = 15 Direct T = 5 Direct T = 10 Direct T = 15 (a) RMSE (lower is bette… view at source ↗

**Figure 11.** Figure 11: For dependent DP data and Nˆ = NT: Results comparing DP and the direct approach for various α ∈ [0, 1) and T = 5, 10, 15 in terms of different metrics. 19 view at source ↗

**Figure 12.** Figure 12: For dependent DP data and Nˆ = NT: Safety probability estimates (in color) and unknown true dynamics (mean) vector field (white arrows; all identical) for the fully Markovian dynamics (α = 0). 20 view at source ↗

**Figure 13.** Figure 13: For dependent DP data and Nˆ = NT: Safety probability estimates (in color) and unknown true dynamics (mean) vector field (white arrows; all identical) for the fully Markovian dynamics (α = 0.95). 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 R M S E m e a n ± 2std DP T = 5 DP T = 10 DP T = 15 Direct T = 5 Direct T = 10 Direct T = 15 (a) RMSE (lower is better) 0.0 0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3 0.4 E xcess R M … view at source ↗

**Figure 14.** Figure 14: For dependent DP data and Nˆ = N: Results comparing DP and the direct approach for various α ∈ [0, 1) and T = 5, 10, 15 in terms of different metrics. Using same hyperparameters as for Nˆ = NT. 21 view at source ↗

**Figure 15.** Figure 15: For dependent DP data and Nˆ = N: Safety probability estimates (in color) and unknown true dynamics (mean) vector field (white arrows; all identical) for the fully Markovian dynamics (α = 0). Using same hyperparameters as for Nˆ = NT. 22 view at source ↗

**Figure 16.** Figure 16: For dependent DP data and Nˆ = N: Safety probability estimates (in color) and unknown true dynamics (mean) vector field (white arrows; all identical) for the fully Markovian dynamics (α = 0.95). Using same hyperparameters as for Nˆ = NT. 23 view at source ↗

**Figure 17.** Figure 17: Distribution of predicted safety probabilities for safe (blue) and unsafe (orange) gate view at source ↗

**Figure 18.** Figure 18: Finite-state abstractions in the embedding framework. view at source ↗

read the original abstract

The goal of this paper is certifying safety of dynamical systems subject to uncertainty. Existing approaches use trajectory data to estimate transition probabilities, and compute safety probabilities recursively via dynamic programming (DP). This recursion may lead to compounding errors in the certified safety probability, thus collapsing to a vacuous lower bound for growing horizons $T$. We propose a kernel embedding framework that treats safety certification as a classification problem on trajectory data, directly estimating the $T$-step safety probability without recursion. We show that the framework subsumes well-established approaches from the literature (e.g., barrier certificates, robust Markov models) as special cases, and allows us to go beyond their limitations. As the main consequence, it bypasses compounding error across the horizon and enables certification for systems with non-Markovian dynamics. We demonstrate that direct estimators remain stable independent of the certification horizon and in the non-Markovian setting, whilst DP-based certificates silently go unsound -- confirmed in simulation on a neural-controlled quadrotor.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper reframes safety certification as direct kernel classification on full trajectories to skip recursive DP error buildup, with a simulation showing stability on a quadrotor, but the finite-data lower bound may not be provably conservative.

read the letter

The main takeaway is that this work turns T-step safety probability estimation into a one-shot classification problem on entire trajectories via kernel mean embeddings. That move avoids the compounding approximation errors that make standard dynamic programming collapse for long horizons, and it extends naturally to non-Markovian dynamics where recursion is awkward anyway. They also position the approach as generalizing barrier certificates and robust Markov models as special cases. The quadrotor simulation is the clearest evidence they provide: the direct estimator stays numerically stable as T grows, while the recursive baseline silently stops being a valid lower bound. That contrast is useful for anyone who has watched DP-based certificates become vacuous in practice. The framing itself is clean and the empirical point lands. The soft spot is whether the classification step actually produces a sound, non-vacuous lower bound from finite samples. Kernel embeddings yield approximations whose error depends on coverage of the trajectory space and the kernel choice. Without an explicit one-sided concentration result or a conservative surrogate that accounts for both sampling variance and embedding bias, the output can be optimistically biased precisely when the method is most needed, i.e., large T or non-Markovian cases where data coverage thins. The abstract claims the result is a certificate, but the details on how bias is controlled would need checking. This is aimed at researchers in safe control and verification for robotics and learned systems. Readers who already work with kernel methods or who need to certify long-horizon behavior will get the most out of the reduction and the simulation comparison. The paper shows clear thinking and honest engagement with the limitations of recursive methods, so it deserves a serious referee to examine the statistical guarantees and the subsumption claims. I would send it to peer review rather than desk reject.

Referee Report

3 major / 2 minor

Summary. The paper proposes reframing safety certification of uncertain dynamical systems as a kernel-based classification task on full trajectory samples. This yields a direct estimator for the T-step safety probability that avoids recursive dynamic programming, thereby sidestepping compounding approximation error. The framework is positioned as a generalization that recovers barrier certificates and robust Markov models as special cases and extends to non-Markovian dynamics. Empirical support is provided via simulation on a neural-controlled quadrotor, where the direct estimator remains numerically stable for large T while DP-based bounds become vacuous.

Significance. A sound, non-recursive method for producing non-vacuous lower bounds on finite-horizon safety probabilities would be a useful contribution to safe control and formal verification, especially for systems whose dynamics or controllers induce non-Markovian behavior. The subsumption claim and the reported stability contrast with DP are potentially valuable if backed by appropriate guarantees; the quadrotor demonstration illustrates the practical issue the authors target.

major comments (3)

[Section 3 (Kernel Embedding Framework) and Theorem 1] The central claim that the kernel classifier produces a valid (non-overestimating) lower bound on the true T-step safety probability from finite trajectories is not supported by an explicit one-sided concentration inequality or conservative surrogate (e.g., slack variables or worst-case embedding) that accounts for both sampling variance and RKHS approximation error. This is load-bearing for the certification interpretation and is especially acute for non-Markovian dynamics where the trajectory measure has high effective dimension.
[Section 3.3 (Special Cases)] The subsumption of barrier certificates and robust Markov models as special cases is asserted but not accompanied by a precise statement of the kernel and label choices that recover each prior method; without this, it is unclear whether the generalization preserves the soundness properties of the special cases or merely their functional form.
[Section 5 (Experiments)] The quadrotor simulation (Section 5) demonstrates numerical stability of the direct estimator but reports no statistical confidence intervals, bootstrap estimates, or worst-case analysis on the certified probabilities. Consequently the experiments show empirical behavior rather than verified certificates, weakening the claim that DP-based methods “silently go unsound” while the new method remains sound.

minor comments (2)

[Section 3.1] The notation for the trajectory kernel and the safety label function should be introduced with an explicit definition of the feature map and the RKHS inner product before the classification objective is stated.
[Introduction] Several sentences in the introduction equate “stable numerical value” with “certificate”; this terminology should be replaced by precise statements about lower bounds once the theoretical guarantee is supplied.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for the constructive review and for highlighting these important points regarding guarantees, subsumption, and experimental validation. We address each major comment below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

Referee: [Section 3 (Kernel Embedding Framework) and Theorem 1] The central claim that the kernel classifier produces a valid (non-overestimating) lower bound on the true T-step safety probability from finite trajectories is not supported by an explicit one-sided concentration inequality or conservative surrogate (e.g., slack variables or worst-case embedding) that accounts for both sampling variance and RKHS approximation error. This is load-bearing for the certification interpretation and is especially acute for non-Markovian dynamics where the trajectory measure has high effective dimension.

Authors: Theorem 1 establishes consistency of the estimator to the true safety probability in the infinite-sample limit. The direct (non-recursive) formulation avoids compounding error by design, but we agree that an explicit finite-sample one-sided bound is needed to fully support the certification claim, particularly in high-dimensional non-Markovian settings. We will revise Section 3 to include a discussion of conservative lower bounds derived from existing concentration results on kernel mean embeddings (e.g., via Hilbert-norm deviation bounds) together with a slack-variable surrogate on the classification threshold. revision: yes
Referee: [Section 3.3 (Special Cases)] The subsumption of barrier certificates and robust Markov models as special cases is asserted but not accompanied by a precise statement of the kernel and label choices that recover each prior method; without this, it is unclear whether the generalization preserves the soundness properties of the special cases or merely their functional form.

Authors: We will expand Section 3.3 with explicit constructions: barrier certificates are recovered by an indicator kernel on the safe set with labels given by the barrier-function sign; robust Markov models are recovered by a feature map that embeds the transition kernel with labels equal to the one-step safety indicator. These choices ensure the general estimator reduces exactly to the original sound methods, thereby inheriting their guarantees. revision: yes
Referee: [Section 5 (Experiments)] The quadrotor simulation (Section 5) demonstrates numerical stability of the direct estimator but reports no statistical confidence intervals, bootstrap estimates, or worst-case analysis on the certified probabilities. Consequently the experiments show empirical behavior rather than verified certificates, weakening the claim that DP-based methods “silently go unsound” while the new method remains sound.

Authors: The simulations are designed to illustrate the practical phenomenon that DP bounds become vacuous for large T while the direct estimator remains stable. We agree that statistical quantification would better support the soundness contrast. In the revision we will add bootstrap confidence intervals computed over independent trajectory batches and a brief worst-case sensitivity analysis. revision: yes

Circularity Check

0 steps flagged

No circularity: direct kernel classification estimator is independent of recursive DP inputs

full rationale

The paper's central derivation reframes safety certification as kernel mean embedding classification over full trajectories to produce a direct T-step probability estimate. No equations or self-citations in the abstract or described framework reduce this estimate to a fitted parameter, prior result, or input quantity by construction. The subsumption of barrier certificates and Markov models is presented as a derived special case rather than a definitional equivalence, and the avoidance of compounding error follows from the non-recursive structure without circular re-use of DP quantities. The approach is positioned against external benchmarks (recursive DP) with independent data requirements, making the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard assumptions from kernel methods and statistical learning applied to safety properties; no new free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)

domain assumption Trajectory data can be embedded via kernels to enable accurate classification of multi-step safety properties.
Core modeling choice that allows direct estimation without recursion.

pith-pipeline@v0.9.0 · 5468 in / 1340 out tokens · 52823 ms · 2026-05-08T10:35:03.320611+00:00 · methodology

discussion (0)

Reference graph

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