Stratifying Reinforcement Learning with Signal Temporal Logic

Alberto Speranzon; Justin Curry

arxiv: 2604.04923 · v1 · submitted 2026-04-06 · 💻 cs.LG · cs.LO· cs.SY· eess.SY· math.AT

Stratifying Reinforcement Learning with Signal Temporal Logic

Justin Curry , Alberto Speranzon This is my paper

Pith reviewed 2026-05-10 19:23 UTC · model grok-4.3

classification 💻 cs.LG cs.LOcs.SYeess.SYmath.AT

keywords signal temporal logicstratificationdeep reinforcement learninglatent embeddingsrobustness semanticsminigridspace-timetemporal logic

0 comments

The pith

Reinterpreting STL atomic predicates as membership tests shows that most formulas induce stratifications of space-time.

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

The paper develops a stratification-based semantics for Signal Temporal Logic in which each atomic predicate is treated as a membership test in a layered space. This perspective establishes a direct correspondence showing that the majority of STL formulas define partitions of space-time into ordered layers. The resulting framework links this structure to the latent embeddings produced by deep reinforcement learning agents, where the robustness value of an STL formula serves as a reward signal during training. Concrete illustrations use Minigrid environments to show how the induced layers appear in both the game dynamics and the agent's internal representations. The approach also supplies efficient signatures for detecting these layers computationally.

Core claim

By reinterpreting the atomic predicates of STL as membership tests within a stratified space, most STL formulas induce a stratification of space-time. This correspondence supplies a geometric lens on the embedding spaces generated by DRL agents and justifies the use of STL robustness as a reward that shapes those embeddings toward logically meaningful layers.

What carries the argument

Stratification-based semantics for STL, in which formulas are built from membership tests that together partition space-time into ordered layers.

If this is right

High-dimensional analysis tools developed for stratified spaces can be applied directly to DRL embedding spaces.
STL robustness can be used as a reward to guide agents toward embeddings that respect the induced stratifications.
Numerically computable signatures exist for uncovering stratification structure in agent embeddings.
The same stratification view applies to both the external environment dynamics and the agent's internal representations.

Where Pith is reading between the lines

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

The stratification lens could support transferring logical structure across related but non-identical RL tasks by matching layer patterns.
It suggests a route to more interpretable RL by reading off decision boundaries from the induced space-time layers.
New reward design methods might explicitly target desired stratification depths or layer transitions.

Load-bearing premise

Reinterpreting atomic predicates as membership tests in a stratified space preserves the original semantics and robustness semantics of STL sufficiently to induce a valid stratification for the full formula.

What would settle it

Train a DRL agent in a Minigrid environment using an STL robustness reward derived from a specific formula, then inspect the latent embeddings to check whether they exhibit the predicted ordered layers matching the formula's stratification.

Figures

Figures reproduced from arXiv: 2604.04923 by Alberto Speranzon, Justin Curry.

**Figure 2.** Figure 2: Empty (left) and Door-Key (right) Minigrid Environments [10] [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 5.** Figure 5: Example of an environment where an agent can move within a [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗

**Figure 8.** Figure 8: Stratification detection via HADES [18] (top) and stratification [PITH_FULL_IMAGE:figures/full_fig_p005_8.png] view at source ↗

**Figure 9.** Figure 9: Tubular Neighborhood around a closed interval. [PITH_FULL_IMAGE:figures/full_fig_p005_9.png] view at source ↗

**Figure 10.** Figure 10: Screenshots of the environment and the observation used as an [PITH_FULL_IMAGE:figures/full_fig_p006_10.png] view at source ↗

**Figure 11.** Figure 11: Main panel: UMAP embedding of the token vectors, colored by the [PITH_FULL_IMAGE:figures/full_fig_p007_11.png] view at source ↗

**Figure 13.** Figure 13: Left: UMAP embedding of the token space. Right: ISOMAP em [PITH_FULL_IMAGE:figures/full_fig_p008_13.png] view at source ↗

read the original abstract

In this paper, we develop a stratification-based semantics for Signal Temporal Logic (STL) in which each atomic predicate is interpreted as a membership test in a stratified space. This perspective reveals a novel correspondence principle between stratification theory and STL, showing that most STL formulas can be viewed as inducing a stratification of space-time. The significance of this interpretation is twofold. First, it offers a fresh theoretical framework for analyzing the structure of the embedding space generated by deep reinforcement learning (DRL) and relates it to the geometry of the ambient decision space. Second, it provides a principled framework that both enables the reuse of existing high-dimensional analysis tools and motivates the creation of novel computational techniques. To ground the theory, we (1) illustrate the role of stratification theory in Minigrid games and (2) apply numerical techniques to the latent embeddings of a DRL agent playing such a game where the robustness of STL formulas is used as the reward. In the process, we propose computationally efficient signatures that, based on preliminary evidence, appear promising for uncovering the stratification structure of such embedding spaces.

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 sketches a stratification lens on STL but the key step of preserving standard robustness for temporal operators is missing.

read the letter

The main contribution is a new semantics that treats STL atomic predicates as membership tests in a stratified space, then claims most formulas induce a stratification on space-time. This is presented as novel and is used to analyze the geometry of deep RL embeddings. The Minigrid illustrations and the DRL experiment (where STL robustness serves as reward) are the concrete parts that work best; they show how the idea might let people apply existing high-dimensional tools to latent spaces and they introduce some candidate signatures for detecting the structure. Those pieces are straightforward and give a reader something to look at. The theoretical core is thinner. The abstract and examples assert the correspondence but do not supply the inductive clauses that would show the stratified interpretation reproduces the usual min/max robustness semantics for operators such as until, always, and eventually. Without that step the claim stays at the level of a rephrasing rather than a proven equivalence. There is also the practical worry that training on robustness and then analyzing the resulting embeddings can make the stratification look stronger than it is. The numerical results are labeled preliminary and lack error bars or controls that would let a reader judge how reliable the signatures are. This work is aimed at people who already care about formal methods in reinforcement learning or about geometric views of embedding spaces. A reader who wants a finished theorem or a fully validated method will find it short on both. A serious editor could send it to review if the authors add the missing inductive argument and tighten the experimental section, but on the current draft the theoretical gap is large enough that many referees would ask for substantial revision before acceptance.

Referee Report

3 major / 2 minor

Summary. The paper develops a stratification-based semantics for Signal Temporal Logic (STL) by reinterpreting atomic predicates as membership tests within a stratified space. This yields a claimed correspondence principle under which most STL formulas induce a stratification of space-time. The framework is illustrated in Minigrid environments and applied to latent embeddings of a DRL agent trained with STL robustness as the reward signal; computationally efficient signatures are proposed to recover the stratification structure in these embeddings.

Significance. If the central correspondence is placed on a rigorous footing, the work would supply a novel theoretical bridge between stratification theory, STL semantics, and the geometry of DRL embedding spaces, potentially allowing reuse of existing high-dimensional analysis tools. The Minigrid illustrations and preliminary DRL experiments provide concrete grounding, but the absence of formal derivations, inductive proofs, or error bounds currently limits the result to a promising conceptual proposal rather than a fully substantiated framework.

major comments (3)

[Abstract and stratified-semantics section] The correspondence principle (abstract and the section developing the stratified semantics) asserts that reinterpreting atoms as stratified membership tests extends inductively to Boolean connectives and temporal operators while preserving both truth values and quantitative robustness. No inductive clauses or theorem are supplied showing that the stratified definition of 'phi U_[a,b] psi' reproduces the standard STL robustness (the specific min/max over the interval). This equivalence is load-bearing for the claim that STL formulas induce the original space-time stratification rather than a new, non-equivalent notion.
[DRL experiments section] § on DRL experiments: STL robustness is used both as the training reward and as the quantity whose stratification structure is subsequently recovered from the learned embeddings. This creates a circularity risk; the signatures may simply rediscover the reward signal rather than independently uncovering an intrinsic stratification of the decision space. An ablation or control experiment separating the reward from the analysis is needed to substantiate the second claimed significance.
[Numerical results and signatures section] The manuscript provides only preliminary numerical illustrations with no error analysis, convergence guarantees, or comparison against standard STL robustness computation. Without these, it is impossible to assess whether the proposed signatures reliably recover the stratification for formulas beyond the simple Minigrid cases shown.

minor comments (2)

[Preliminaries] Notation for the stratified membership test and the induced robustness measure should be introduced with explicit comparison to the classical STL robustness definition to aid readability.
[Abstract] The abstract claims 'most STL formulas' induce a stratification; the precise class of formulas for which the correspondence holds should be stated explicitly (e.g., negation-free, bounded-horizon, etc.).

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. The comments identify key areas where the manuscript can be strengthened with additional formalization and experiments. We address each major comment below and commit to the corresponding revisions in the next version of the manuscript.

read point-by-point responses

Referee: [Abstract and stratified-semantics section] The correspondence principle (abstract and the section developing the stratified semantics) asserts that reinterpreting atoms as stratified membership tests extends inductively to Boolean connectives and temporal operators while preserving both truth values and quantitative robustness. No inductive clauses or theorem are supplied showing that the stratified definition of 'phi U_[a,b] psi' reproduces the standard STL robustness (the specific min/max over the interval). This equivalence is load-bearing for the claim that STL formulas induce the original space-time stratification rather than a new, non-equivalent notion.

Authors: We agree that a formal inductive proof is necessary to substantiate the correspondence. Although the stratified semantics is constructed to mirror the standard inductive definition of STL, the manuscript does not explicitly state or prove the equivalence. In the revised version we will add a new theorem (with full inductive proof) establishing that the stratified robustness semantics coincides with the classical quantitative STL semantics for every formula, including the until operator. The proof will explicitly verify preservation of the min/max operations over intervals via the stratification membership tests. revision: yes
Referee: [DRL experiments section] § on DRL experiments: STL robustness is used both as the training reward and as the quantity whose stratification structure is subsequently recovered from the learned embeddings. This creates a circularity risk; the signatures may simply rediscover the reward signal rather than independently uncovering an intrinsic stratification of the decision space. An ablation or control experiment separating the reward from the analysis is needed to substantiate the second claimed significance.

Authors: This is a legitimate concern. To eliminate the circularity, the revised manuscript will include a control experiment in which we train separate DRL agents on the same Minigrid tasks using only standard sparse task-completion rewards (without any STL robustness term). We will then extract embeddings from these agents and apply the proposed signatures, comparing the recovered stratification structure against the STL-rewarded case. This will demonstrate that the signatures can identify meaningful structure even when the training signal is independent of STL. revision: yes
Referee: [Numerical results and signatures section] The manuscript provides only preliminary numerical illustrations with no error analysis, convergence guarantees, or comparison against standard STL robustness computation. Without these, it is impossible to assess whether the proposed signatures reliably recover the stratification for formulas beyond the simple Minigrid cases shown.

Authors: We acknowledge that the current numerical section is preliminary. In the revision we will augment it with: (i) quantitative error analysis measuring the discrepancy between signature-derived stratifications and direct STL robustness values on the Minigrid environments, (ii) empirical convergence plots showing how signature accuracy improves with sample size and embedding dimension, and (iii) direct runtime and accuracy comparisons against standard STL robustness computation on the same formulas. Where analytic bounds are derivable we will include them; otherwise we will report empirical error statistics. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained as a proposed reinterpretation

full rationale

The paper introduces a stratification-based semantics for STL by reinterpreting atomic predicates as membership tests in a stratified space and claims this induces a correspondence with standard STL formulas. No equations or definitions in the abstract or described content show the new semantics being defined in terms of the target result (or vice versa), nor any fitted parameters from data being relabeled as predictions. The use of STL robustness as a DRL reward is an application choice for generating embeddings, not a load-bearing step that forces the theoretical correspondence by construction. The central claim is presented as a novel perspective rather than a closed derivation loop, and no self-citation chains or ansatzes reduce the result to its inputs. This is the expected honest non-finding for a paper offering a fresh theoretical framework.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents exhaustive extraction; the framework appears to rest on standard definitions of stratified spaces and STL semantics without introducing new free parameters or invented entities in the visible text.

pith-pipeline@v0.9.0 · 5490 in / 1102 out tokens · 49872 ms · 2026-05-10T19:23:38.263739+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

each atomic predicate is interpreted as a membership test in a stratified space... most STL formulas can be viewed as inducing a stratification of space-time
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

poset stratification of X is then a continuous map q:X→P... Alexandrov topology where down-sets are closed

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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J. Curry, B. Lagasse, N. B. Lam, G. Cox, D. Rosenbluth, and A. Speranzon, “Exploring the stratified space structure of an RL game with the volume growth transform,”arXiv:2507.22010, 2025

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Semi-algebraic neighborhoods of closed semi-algebraic sets,

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