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Conformal Prediction for Signal Temporal Logic Inference

Danyang Li, Yixuan Wang, Matthew Cleaveland, Mingyu Cai, Roberto Tron

TL;DR

This work addresses the lack of formal confidence guarantees in STL inference by introducing an end-to-end differentiable conformal-prediction (CP) framework, TLICP, for learning STL formulas from time-series data. It defines a robustness-based, unit-invariant nonconformity score and embeds a smooth CP layer into training, paired with a single-term loss that optimizes both accuracy and CP efficiency; after training, exact CP provides formal guarantees on unseen data. The approach yields tighter, more reliable STL specifications and reduced prediction-set sizes without requiring extensive hyperparameter tuning, outperforming state-of-the-art baselines on benchmark tasks. The method has practical significance for safety-critical domains where interpretable temporal rules with quantifiable uncertainty are essential.

Abstract

Signal Temporal Logic (STL) inference seeks to extract human-interpretable rules from time-series data, but existing methods lack formal confidence guarantees for the inferred rules. Conformal prediction (CP) is a technique that can provide statistical correctness guarantees, but is typically applied as a post-training wrapper without improving model learning. Instead, we introduce an end-to-end differentiable CP framework for STL inference that enhances both reliability and interpretability of the resulting formulas. We introduce a robustness-based nonconformity score, embed a smooth CP layer directly into training, and employ a new loss function that simultaneously optimizes inference accuracy and CP prediction sets with a single term. Following training, an exact CP procedure delivers statistical guarantees for the learned STL formulas. Experiments on benchmark time-series tasks show that our approach reduces uncertainty in predictions (i.e., it achieves high coverage while reducing prediction set size), and improves accuracy (i.e., the number of misclassifications when using a fixed threshold) over state-of-the-art baselines.

Conformal Prediction for Signal Temporal Logic Inference

TL;DR

This work addresses the lack of formal confidence guarantees in STL inference by introducing an end-to-end differentiable conformal-prediction (CP) framework, TLICP, for learning STL formulas from time-series data. It defines a robustness-based, unit-invariant nonconformity score and embeds a smooth CP layer into training, paired with a single-term loss that optimizes both accuracy and CP efficiency; after training, exact CP provides formal guarantees on unseen data. The approach yields tighter, more reliable STL specifications and reduced prediction-set sizes without requiring extensive hyperparameter tuning, outperforming state-of-the-art baselines on benchmark tasks. The method has practical significance for safety-critical domains where interpretable temporal rules with quantifiable uncertainty are essential.

Abstract

Signal Temporal Logic (STL) inference seeks to extract human-interpretable rules from time-series data, but existing methods lack formal confidence guarantees for the inferred rules. Conformal prediction (CP) is a technique that can provide statistical correctness guarantees, but is typically applied as a post-training wrapper without improving model learning. Instead, we introduce an end-to-end differentiable CP framework for STL inference that enhances both reliability and interpretability of the resulting formulas. We introduce a robustness-based nonconformity score, embed a smooth CP layer directly into training, and employ a new loss function that simultaneously optimizes inference accuracy and CP prediction sets with a single term. Following training, an exact CP procedure delivers statistical guarantees for the learned STL formulas. Experiments on benchmark time-series tasks show that our approach reduces uncertainty in predictions (i.e., it achieves high coverage while reducing prediction set size), and improves accuracy (i.e., the number of misclassifications when using a fixed threshold) over state-of-the-art baselines.

Paper Structure

This paper contains 20 sections, 1 theorem, 18 equations, 6 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Signal Temporal Logic
  4. Conformal Prediction
  5. Problem Statement
  6. Nonconformity Score for STL Inference
  7. STL margin
  8. Nonconformity Score
  9. Differentiable Conformal Predictor for STL Inference
  10. Differentiable Nonconformity Score
  11. ConfTr using differentiable thresholds
  12. TLICP using $p$-values
  13. Learning Conformal Prediction for STL Inference
  14. ConfTr with Inefficiency Regularization
  15. TLICP with $p$-Value Optimization
  16. ...and 5 more sections

Key Result

Lemma 1

The proposed nonconformity scores are invariant to changes of measurement units (equivalently, to positive rescalings of the robustness).

Figures (6)

  • Figure 1: An example of binary-class STL margin.
  • Figure 2: The approximated nonconformity score $\tilde{E}_{\theta}(X,Y)$ with $m=5$, $M=5$, $T_1 = T_2 = T_3 = 0.5$.
  • Figure 3: VIMA dataset. (a) Task environment setup. (b) Representative trajectories demonstrating whether blocks are placed into the basket.
  • Figure 4: Confidence Level ($\alpha$) vs Average Inefficiency (Set Size) for Task 1 and Task 2.
  • Figure 5: Specification regions inferred via STL from each method. Colored/linestyled boxes denote the learned axis-aligned predicates; points show test trajectory endpoints (positive and negative).
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition 1: Binary-class STL margin
  • Lemma 1
  • proof
  • Remark 1