pdSTL: Probabilistic Differentiable Signal Temporal Logic for Stochastic Systems

Bennett Dogbey; Hemanth Manjunatha

REVIEW 2 major objections 1 minor 20 references

pdSTL: Probabilistic Differentiable Signal Temporal Logic for Stochastic Systems

T0 review · 2 major / 1 minor · reviewed 2026-06-26 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read pdSTL unifies interval-valued probabilistic semantics with differentiable robustness to optimize robot trajectories under stochastic uncertainty.

desk verdict pdSTL puts interval probabilities into differentiable STL with a recurrent formulation and shows it on real quadcopter flights, but the tightness of the propagated bounds is the part that still needs numbers. read the letter →

arxiv 2606.19561 v1 pith:ZIP5F43F submitted 2026-06-17 cs.RO cs.SYeess.SY

Bennett Dogbey , Hemanth Manjunatha This is my paper

classification cs.ROcs.SYeess.SY

keywords probabilisticsignaltemporallogicdifferentiablerobustnessstochasticsystemstrajectoryoptimizationbelief-spaceplanningautonomousrobotssafetyspecificationsintervalsemantics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Autonomous robots must satisfy temporal safety specifications despite noisy dynamics and sensing. pdSTL extends signal temporal logic by attaching interval probabilities to robustness values, then propagates those intervals through the logic operators to obtain conservative bounds on the chance a specification holds. These bounds stay differentiable because the monitoring is recast as a recurrent unfolding similar to an LSTM, allowing direct use inside gradient-based trajectory optimizers. Experiments on simulated obstacle avoidance, lane changes, and real Crazyflie flights under wind show the method keeps larger safety margins than ordinary differentiable STL. A reader cares because the approach supplies formal probabilistic certificates while remaining practical for end-to-end learning and control.

What carries the argument

Interval-valued probabilistic semantics for STL formulas, propagated compositionally through the syntax tree via recurrent operator unfolding.

What would settle it

A controlled simulation in which the actual probability that an optimized trajectory violates its STL specification exceeds the upper bound computed by pdSTL.

Watch

Extended reading notes

Core claim

pdSTL employs interval-valued probabilistic semantics to compute conservative satisfaction bounds, propagated compositionally through the STL syntax tree. The temporal robustness evaluation is formulated as a recurrent, LSTM-style unfolding of STL operators, enabling linear-time, differentiable monitoring suitable for end-to-end trajectory optimization. This supplies formal probabilistic guarantees for stochastic systems while supporting applications such as obstacle avoidance and quadcopter flight under aerodynamic disturbances.

Load-bearing premise

Interval-valued probabilistic semantics can be propagated compositionally through the STL syntax tree to produce conservative yet differentiable satisfaction bounds that remain useful for trajectory optimization.

Editorial extensions

If this is right

Robot trajectories can be optimized end-to-end while carrying formal probabilistic guarantees on temporal specifications.
The method produces larger safety margins than deterministic differentiable STL when dynamics or sensing contain real uncertainty.
Linear-time monitoring supports online replanning on platforms such as quadcopters under wind disturbances.
The same machinery applies to lane-change maneuvers and obstacle avoidance in belief space.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The recurrent unfolding could be replaced by a parallel implementation for longer horizons without changing the core bounds.
Tighter uncertainty models would narrow the interval bounds and reduce conservatism in the resulting plans.
The same interval propagation might transfer to other temporal logics once their robustness semantics are made differentiable.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main thing to know is that this paper gives a concrete way to run STL-style specs over belief trajectories while keeping everything differentiable for gradient-based planning. It does this by using interval-valued probabilities that compose through the syntax tree and by recasting the monitoring as an LSTM-style recurrence so it stays linear time.

What stands out is the hardware validation. They run obstacle avoidance and lane-change tasks in simulation, then fly a Crazyflie under aerodynamic disturbances and report that the probabilistic version keeps larger safety margins than the deterministic differentiable baseline. That kind of real-system check is still uncommon and gives the claim some weight.

The soft spot is the interval propagation itself. For nested temporal operators the over-approximation can grow, and if the resulting intervals become wide the formal guarantees turn weak and the robustness signal for optimization loses bite. The abstract does not include tables on bound width or how much tightness is lost versus exact probabilistic semantics, so it is not yet clear whether the outperformance holds once the intervals are measured. The stress-test note on this point is worth checking against the actual derivations.

The work is aimed at people already using STL for robot motion planning who need to handle sensing noise and stochastic dynamics without dropping the optimization path. A reader in that niche will find the framework and the Crazyflie results directly usable.

It is worth sending to peer review. The core construction addresses a real gap, the experiments go beyond simulation, and the open question on bound quality is the sort of thing referees can pressure-test with the full proofs and data.

Referee Report

2 major / 1 minor

Summary. The paper introduces pdSTL, a framework unifying probabilistic semantics with differentiable robustness for Signal Temporal Logic (STL) over belief trajectories in stochastic systems. It employs interval-valued probabilistic semantics propagated compositionally through the STL syntax tree and formulates temporal robustness evaluation as an LSTM-style recurrent unfolding of operators to enable linear-time differentiable monitoring for end-to-end trajectory optimization. Validation is presented on simulated obstacle avoidance and lane-change maneuvers plus real-world Crazyflie quadcopter experiments under aerodynamic disturbances, with claims of efficient optimization and superior safety-margin maintenance relative to deterministic differentiable STL.

Significance. If the central claims on sound yet sufficiently tight interval propagation hold, pdSTL would address an important gap by enabling gradient-based optimization of complex temporal specifications under belief-space uncertainty with formal probabilistic guarantees. The real-world quadcopter validation provides concrete evidence of practical utility beyond simulation.

major comments (2)

[Abstract (validation paragraph)] The central claim that pdSTL 'significantly outperform[s] deterministic differentiable STL in maintaining safety margins' depends on the interval-valued semantics remaining both conservative and tight enough to produce useful robustness signals. The manuscript should provide explicit quantitative evidence (e.g., interval-width ratios or bound tightness metrics) for nested temporal operators in the lane-change and obstacle-avoidance cases; without it, the outperformance may reduce to the deterministic baseline when intervals become vacuous.
[Abstract (framework description)] The assumption that compositional interval propagation through the STL syntax tree yields 'conservative satisfaction bounds' suitable for optimization is load-bearing. The paper must demonstrate (via theorem or empirical counter-example) that the LSTM-style unfolding does not accumulate over-approximation that nullifies the probabilistic guarantees for specifications with multiple nested 'always' or 'eventually' operators.

minor comments (1)

[Abstract] The abstract states 'formal probabilistic guarantees' but does not name the precise probability measure or the exact form of the interval bounds; a short clarifying sentence would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our submission. Below we respond point-by-point to the two major comments. We agree that additional quantitative evidence on bound tightness will strengthen the manuscript and will incorporate it in the revision.

read point-by-point responses

Referee: [Abstract (validation paragraph)] The central claim that pdSTL 'significantly outperform[s] deterministic differentiable STL in maintaining safety margins' depends on the interval-valued semantics remaining both conservative and tight enough to produce useful robustness signals. The manuscript should provide explicit quantitative evidence (e.g., interval-width ratios or bound tightness metrics) for nested temporal operators in the lane-change and obstacle-avoidance cases; without it, the outperformance may reduce to the deterministic baseline when intervals become vacuous.

Authors: We agree that explicit quantitative metrics would better substantiate the claim. The current experiments already compare safety margins under uncertainty, but we will add in the revised manuscript a dedicated table reporting interval-width ratios and tightness metrics (e.g., ratio of interval width to deterministic robustness value) for the nested operators appearing in the lane-change and obstacle-avoidance specifications. This will demonstrate that the intervals remain sufficiently tight to produce useful gradient signals. revision: yes
Referee: [Abstract (framework description)] The assumption that compositional interval propagation through the STL syntax tree yields 'conservative satisfaction bounds' suitable for optimization is load-bearing. The paper must demonstrate (via theorem or empirical counter-example) that the LSTM-style unfolding does not accumulate over-approximation that nullifies the probabilistic guarantees for specifications with multiple nested 'always' or 'eventually' operators.

Authors: The semantics section establishes soundness of the interval propagation by induction over the STL syntax tree, ensuring conservativeness at each operator. The LSTM-style unfolding is a direct, finite-horizon unrolling of the same recursive definitions and therefore inherits the same soundness. Our evaluated specifications already contain nested 'always' and 'eventually' operators; the reported results show non-vacuous bounds that enable successful optimization. To address the accumulation concern explicitly, we will add an empirical subsection in the revision that plots bound width growth versus nesting depth for the tested formulas. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity detected; framework presented as novel construction

full rationale

The abstract and provided description introduce pdSTL as a new unification of probabilistic interval semantics with differentiable STL robustness, using compositional propagation and LSTM-style unfolding. No load-bearing claim reduces by definition or self-citation to its own inputs; the central semantics and monitoring procedure are constructed rather than fitted or renamed from prior results. The derivation chain is self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Based on abstract only; the framework rests on standard STL syntax and the novel interval-valued probabilistic semantics plus recurrent unfolding, but no explicit free parameters, axioms, or invented entities are detailed.

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Cite this review

Pith. "Pith review of pdSTL: Probabilistic Differentiable Signal Temporal Logic for Stochastic Systems." pith.science (2026). https://pith.science/paper/ZIP5F43F

@misc{pith2026260619561,
  author       = {Pith},
  title        = {Pith review of: pdSTL: Probabilistic Differentiable Signal Temporal Logic for Stochastic Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZIP5F43F}},
  note         = {Machine review of arXiv:2606.19561}
}

read the original abstract

Autonomous robots operating in uncertain environments must satisfy complex temporal and safety specifications despite stochastic dynamics and sensing noise. While Signal Temporal Logic (STL) offers robustness measures for gradient-based optimization, existing extensions either lack differentiability or ignore belief-space uncertainty. We introduce pdSTL (probabilistic differentiable Signal Temporal Logic), a framework that unifies probabilistic semantics with differentiable robustness over belief trajectories. pdSTL employs interval-valued probabilistic semantics to compute conservative satisfaction bounds, propagated compositionally through the STL syntax tree. We formulate the temporal robustness evaluation as a recurrent, LSTM-style unfolding of STL operators, enabling linear-time, differentiable monitoring suitable for end-to-end trajectory optimization. We validate pdSTL on simulated obstacle avoidance, lane-change maneuvers, and real-world Crazyflie quadcopter flight experiments under aerodynamic disturbances. Results demonstrate that pdSTL achieves efficient optimization with formal probabilistic guarantees, significantly outperforming deterministic differentiable STL in maintaining safety margins under real-world uncertainty.

Figures

Figures reproduced from arXiv: 2606.19561 by the authors.

**Figure 2.** Differentiable probabilistic STL framework. For a given belief state [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** □[1,2](x ≥ 50) valuation of the probabilistic STL monitor. Top: Piecewise signal with uncertainty bounds and threshold. Middle: Predicate satisfaction probability bounds. Bottom: Temporal operator output computed via recurrent aggregation. TABLE I: Predicate Probability Intervals and SlidingWindow Evaluation for □[1,2]φ t p↓ t p ↑ t Suffix s ∈ [t + 1, t + 2] r □[1,2]φ t 0 0.0002 0.0668 {1, 2} [0.9332, 0.9998] 1 0.9… view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: Lane-merging behavior across two scenarios. Top: [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]

**Figure 4.** Figure 4: A) Belief-space trajectory optimization under pdSTL. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 6.** Figure 6: Experimental setup for real-world validation of [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

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Reference graph

Works this paper leans on

20 extracted references · 5 canonical work pages

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Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.

[1] [1]

On signal temporal logic

Alexandre Donz ´e. “On signal temporal logic”. In: International conference on runtime verification. Springer Berlin Heidelberg Berlin, Heidelberg. 2013, pp. 382–383

2013

[2] [2]

Robust satis- faction of temporal logic over real-valued signals

Alexandre Donz ´e and Oded Maler. “Robust satis- faction of temporal logic over real-valued signals”. In:International conference on formal modeling and analysis of timed systems. Springer. 2010, pp. 92–106

2010

[3] [3]

Continuous-Time Control Synthesis for Multiple Quadrotors under Signal Temporal Logic Specifications

Yating Yuan. “Continuous-Time Control Synthesis for Multiple Quadrotors under Signal Temporal Logic Specifications”. In:arXiv preprint arXiv:2505.07240 (2025)

work page arXiv 2025

[4] [4]

Power line inspection tasks with multi-aerial robot systems via signal temporal logic specifications

Giuseppe Silano et al. “Power line inspection tasks with multi-aerial robot systems via signal temporal logic specifications”. In:IEEE Robotics and Automa- tion Letters6.2 (2021), pp. 4169–4176

2021

[5] [5]

Guiding autonomous exploration with signal temporal logic

Fernando S Barbosa et al. “Guiding autonomous exploration with signal temporal logic”. In:IEEE Robotics and Automation Letters4.4 (2019), pp. 3332–3339

2019

[6] [6]

Learning from demonstrations using signal temporal logic in stochastic and contin- uous domains

Aniruddh G Puranic, Jyotirmoy V Deshmukh, and Stefanos Nikolaidis. “Learning from demonstrations using signal temporal logic in stochastic and contin- uous domains”. In:IEEE Robotics and Automation Letters6.4 (2021), pp. 6250–6257

2021

[7] [7]

Control with Probabilistic Signal Temporal Logic

Chanyeol Yoo and Calin Belta. “Control with prob- abilistic signal temporal logic”. In:arXiv preprint arXiv:1510.08474(2015)

work page Pith review arXiv 2015

[8] [8]

Probabilistic Safety Programs

Ashish Kapoor, Debadeepta Dey, and Shital Shah. “Probabilistic safety programs”. In:arXiv preprint arXiv:1610.05376(2016)

work page Pith review arXiv 2016