Conformalized Signal Temporal Logic Inference under Covariate Shift

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

cs.LGMar 28, 2026

Conformalized Signal Temporal Logic Inference under Covariate Shift

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

Organizations: Mingyu Cai and Yixuan Wang are with Mechanical Engineering, University of California, Riverside, CA, 92521, USA. · Danyang Li and Roberto Tron are with Mechanical Engineering, Boston University. · Matthew Cleaveland is with MIT Lincoln Laboratory, Lexington, MA, 02421, USA

Abstract

Signal Temporal Logic (STL) inference learns interpretable logical rules for temporal behaviors in dynamical systems. To ensure the correctness of learned STL formulas, recent approaches have incorporated conformal prediction as a statistical tool for uncertainty quantification. However, most existing methods rely on the assumption that calibration and testing data are identically distributed and exchangeable, an assumption that is frequently violated in real-world settings. This paper proposes a conformalized STL inference framework that explicitly addresses covariate shift between training and deployment trajectories dataset. From a technical standpoint, the approach first employs a template-free, differentiable STL inference method to learn an initial model, and subsequently refines it using a limited deployment side dataset to promote distribution alignment. To provide validity guarantees under distribution shift, the framework estimates the likelihood ratio between training and deployment distributions and integrates it into an STL-robustness-based weighted conformal prediction scheme. Experimental results on trajectory datasets demonstrate that the proposed framework preserves the interpretability of STL formulas while significantly improving symbolic learning reliability at deployment time.

Figures & tables

Fig. 1 : Pipeline of proposed shift-aware conformal STL framework under covariate shift. The framework consists of four sequential stages integrating STL formula learning, distribution alignment, and weighted conformal calibration.

Panel of Figure 2 — Fig. 2 : Schematic Diagram of the Dataset

Method	MCR for $P_{\mathrm{train}}$	MCR for $P_{\mathrm{dep}}$	Time(s)	STL formula
Naval Dataset
Ours	1.25	0.07	16	$\boldsymbol{\Diamond_{[25,26]}(x\leq 38.6)\ \wedge\ \Box_{[11,12]}(y\geq 24.1)}$
TLINet	1.25	0.05	38	$\Diamond_{[55,60]}(x<25.89)\wedge\Box_{[0,16]}(y>23.77)$
LSTM/RNN	0.01	0.025	19	N/A
BCDT	0.0100	N/R	1996	$\Diamond_{[28,53]}(x<30.85)\wedge\Box_{[2,26]}(y>21.31)\wedge(x>11.10)$
DT	0.0195	N/R	140	$\neg(\Diamond_{[38,53]}(x>20.1)\wedge\Diamond_{[12,37]}(x>43.2))\vee(\Diamond_{[38,53]}(x>20.1)\wedge\neg\Diamond_{[20,59]}(y>32.2))\vee(\Diamond_{[38,53]}(x>20.1)\wedge\Diamond_{[20,59]}(y>32.2)\wedge\Box_{[14,60]}(y>30.1))$

TABLE I : Classification results of our method and baseline methods on trajectory datasets.

Fig. 3 : Weighted CP with Covariate Shift

Panel of Figure 5 — Fig. 4 : Comparison across datasets with weighted CP.

Fig. 5 : Comparison between weighted CP and Standard CP using our method in different dataset.

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Class-Conditional Conformal Prediction for Signal Temporal Logic Inference with Many Classes

Yixuan Wang, Danyang Li, Matthew Cleaveland, Roberto Tron+1

Mechanical Engineering, University of California, Riverside, CA 92521, USA · Mechanical Engineering, Boston University · MIT Lincoln Laboratory, Lexington, MA 02421, USA

Signal Temporal Logic Multivariate Time Series Classification Group-Conditional Conformal Prediction

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Signal Temporal Logic (STL) inference provides interpretable formulas for time series classification. For multiclass classification, however, accurate predictions do not necessarily ensure that the inferred formulas are satisfied consistently with the true class. We introduce the positive satisfaction rate and codeword satisfaction rate to quantify satisfaction of required formulas and consistency with the complete class codeword, respectively. We develop a two-stage procedure that first learns discriminative formulas using a margin-based objective and then refines predicate thresholds toward a small positive robustness target while fixing the formula structure and temporal intervals. Satisfaction and ranking penalties discourage violations of required formulas and loss of class separation. The decoding loss is used as a nonconformity score and calibrated separately for each class, providing finite-sample class-conditional coverage under intra-class exchangeability. Experiments on naval, synthetic, and robot manipulation trajectories demonstrate discrepancies between classification accuracy and formula satisfaction. The results further reveal a data-dependent tradeoff between robustness tightness and satisfaction, while class-conditional calibration mitigates undercoverage that can be hidden by pooled calibration.

Abstract

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