Causality by Abstraction: Symbolic Rule Learning in Multivariate Timeseries with Large Language Models

Preetom Biswas; Giulia Pedrielli; K. Selçuk Candan

cs.LGFeb 19, 2026

Causality by Abstraction: Symbolic Rule Learning in Multivariate Timeseries with Large Language Models

Authors: Preetom Biswas, Giulia Pedrielli, K. Selçuk Candan

Organizations: Arizona State University, Tempe, AZ, USA

Abstract

Inferring causal relations in timeseries data with delayed effects is a fundamental challenge, especially when the underlying system exhibits complex dynamics that cannot be captured by simple functional mappings. Traditional approaches often fail to produce generalized and interpretable explanations, as multiple distinct input trajectories may yield nearly indistinguishable outputs. In this work, we present ruleXplain, a framework that leverages Large Language Models (LLMs) to extract formal explanations for input-output relations in simulation-driven dynamical systems. Our method introduces a constrained symbolic rule language with temporal operators and delay semantics, enabling LLMs to generate verifiable causal rules through structured prompting. ruleXplain relies on the availability of a principled model (e.g., a simulator) that maps multivariate input time series to output time series. Within ruleXplain, the simulator is used to generate diverse counterfactual input trajectories that yield similar target output, serving as candidate explanations. Such counterfactual inputs are clustered and provided as context to the LLM, which is tasked with the generation of symbolic rules encoding the joint temporal trends responsible for the patterns observable in the output times series. A closed-loop refinement process ensures rule consistency and semantic validity. We validate the framework using the PySIRTEM epidemic simulator, mapping testing rate inputs to daily infection counts; and the EnergyPlus building energy simulator, observing temperature and solar irradiance inputs to electricity needs. For validation, we perform three classes of experiments: (1) the efficacy of the ruleset through input reconstruction; (2) ablation studies evaluating the causal encoding of the ruleset; and (3) generalization tests of the extracted rules across unseen output trends with varying phase dynamics.

Figures & tables

Figure 1: ruleXplain method overview key modules and interactions.

Procedure: Explanation Generation and Refinement
Step 1: Initialize. Extract an initial ruleset $R_{0}$ derived from Cluster-derived context $[\left(\mathbf{X}^{c}_{t}\right)^{T}_{t=1},Y_{base}]_{c=1}^{C}$ pairs.
Step 2: Generate Inputs. Use $R_{i}$ and $[\left(\mathbf{X}^{c}_{t}\right)^{T}_{t=1},Y_{base}]_{c=1}^{C}$ to prompt novel input sequences $\left(\mathbf{X}^{(i)}_{t}\right)^{T}_{t=1})$ that approximately reproduce the baseline trajectory $Y_{\text{base}}$ .
Step 3: Simulate & Evaluate. Run the simulator and compute deviation $\Delta^{(i)}$ using the loss $L_{\mbox{\tiny{out}}}$ calculated from $Y^{(i)}$ and $Y_{base}$ .
Step 4: Update Rules. If $\Delta^{(i)}>\epsilon$ , update $R_{i}$ based on observed mismatches and added constraints to get $R_{i+1}$ .
Step 5: Repeat. Continue Steps 2–4 until $\Delta^{(i)}\leq\epsilon$ or the maximum number of iterations is reached, yielding the refined ruleset $R_{\mbox{\tiny{final}}}$ . Please note that $[\left(\mathbf{X}^{c}_{t}\right)^{T}_{t=1},Y_{base}]_{c=1}^{C}$ remain unchanged across iterations.

Table 1: Stepwise process for Causal Rules Inference through LLM.

Panel of Figure 2 — Figure 2: Timeseries $X_{s}$ , $X_{a}$ , and $Y$ generated using $\mathcal{R}_{final}$ for PySIRTEM .

Panel of Figure 3 — Figure 3: Timeseries $X_{d}$ , $X_{g}$ , and $Y$ generated using $\mathcal{R}_{final}$ for EnergyPlus.

Figure 4: Causal Encoding Comparison of $\mathcal{R}_{\mbox{\tiny{final}}}$ through scenarios S1-S3 for PySIRTEM .

Figure 5: Causal Encoding Comparison of $\mathcal{R}_{\mbox{\tiny{final}}}$ through scenarios S1-S3 for EnergyPlus.

Panel of Figure 6 — Figure 6: Input Inference for an alternate (left) $Y^{1}_{\mbox{\tiny{base}}}$ with Prominent Phase Changes (right) $Y^{2}_{\mbox{\tiny{base}}}$ with Non-Prominent Phase Changes. E1 results are in the left subplot, E2 in the right subplot for each.

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: Clustering of the initial solution pool $\left\{(X^{p}_{s},X^{p}_{a})\right\}_{p}$ from solving $P_{\mbox{\tiny{LEARN}}}(p)$ for PySIRTEM setup.

Rule ID	Input Window	$X_{s}$ Trend	$X_{a}$ Trend	Output Window	$Y$ Phase	Symbolic Rule
R1	[0, 14]	Low	SlowRise	[15, 29]	InitialGrowth	$\Box_{[0,14]}(\mathrm{Low}(X_{s})\wedge\mathrm{SlowRise}(X_{a}))\rightarrow\Diamond_{[15,29]}\mathrm{InitialGrowth}(Y)$
R2	[14, 28]	Moderate	SlowRise	[29, 43]	PeakFormation	$\Box_{[14,28]}(\mathrm{Moderate}(X_{s})\wedge\mathrm{SlowRise}(X_{a}))\rightarrow\Diamond_{[29,43]}\mathrm{PeakFormation}(Y)$
R3	[28, 42]	SharpSpike	Low	[43, 57]	PostPeak	$\Box_{[28,42]}(\mathrm{SharpSpike}(X_{s})\wedge\mathrm{Low}(X_{a}))\rightarrow\Diamond_{[43,57]}\mathrm{PostPeak}(Y)$
R4	[42, 56]	DipThenRise	Low	[57, 71]	Decline	$\Box_{[42,56]}(\mathrm{DipThenRise}(X_{s})\wedge\mathrm{Low}(X_{a}))\rightarrow\Diamond_{[57,71]}\mathrm{Decline}(Y)$
R5	[56, 70]	SlowRise	HighStable	[71, 85]	Resurgence	$\Box_{[56,70]}(\mathrm{SlowRise}(X_{s})\wedge\mathrm{HighStable}(X_{a}))\rightarrow\Diamond_{[71,85]}\mathrm{Resurgence}(Y)$
R6	[70, 100]	Moderate	SlowRise	[85, 100]	PeakFormation	$\Box_{[70,100]}(\mathrm{Moderate}(X_{s})\wedge\mathrm{SlowRise}(X_{a}))\rightarrow\Diamond_{[85,100]}\mathrm{PeakFormation}(Y)$

Appendix

Table 2: Causal rules extracted from timeseries examples and context for PySIRTEM Setup.

Rule ID	Input Window	$X_{t,1}$ Trend	$X_{t,2}$ Trend	Output Window	$Y$ Phase	Symbolic Rule
R1	[0, 9]	Low	Low	[0, 8]	Decline	$\Box_{[0,9]}(\mathrm{Low}(X_{t,1})\wedge\mathrm{Low}(X_{t,2}))\rightarrow\Diamond_{[0,8]}\mathrm{Decline}(Y)$
R2	[9, 18]	SlowRise	SharpSpike	[9, 17]	InitialGrowth	$\Box_{[9,18]}(\mathrm{SlowRise}(X_{t,1})\wedge\mathrm{SharpSpike}(X_{t,2}))\rightarrow\Diamond_{[9,17]}\mathrm{InitialGrowth}(Y)$
R3	[18, 21]	DipThenRise	LowModerate	[18, 20]	PostPeak	$\Box_{[18,21]}(\mathrm{DipThenRise}(X_{t,1})\wedge\mathrm{LowModerate}(X_{t,2}))\rightarrow\Diamond_{[18,20]}\mathrm{PostPeak}(Y)$
R4	[21, 30]	Low	Low	[21, 29]	Decline	$\Box_{[21,30]}(\mathrm{Low}(X_{t,1})\wedge\mathrm{Low}(X_{t,2}))\rightarrow\Diamond_{[21,29]}\mathrm{Decline}(Y)$
R5	[30, 42]	Moderate	HighStable	[30, 41]	PeakFormation	$\Box_{[30,42]}(\mathrm{Moderate}(X_{t,1})\wedge\mathrm{HighStable}(X_{t,2}))\rightarrow\Diamond_{[30,41]}\mathrm{PeakFormation}(Y)$
R6	[42, 48]	DipThenRise	LowModerate	[42, 47]	PostPeak	$\Box_{[42,48]}(\mathrm{DipThenRise}(X_{t,1})\wedge\mathrm{LowModerate}(X_{t,2}))\rightarrow\Diamond_{[42,47]}\mathrm{PostPeak}(Y)$

Appendix

Table 3: Causal rules extracted from timeseries examples and context for EnergyPlus Setup.

Panel of Figure 8 — Figure 8: ruleXplain input timeseries produced Mean Squared Difference across iterations for (left) PySIRTEM (right) EnergyPlus

Panel of Figure 9 — Figure 9: Prompt for generating the initial ruleset.

Panel of Figure 10 — Figure 10: Prompt for Novel Input series generation using inferred ruleset $\mathcal{R}_{i}$ .

Panel of Figure 11 — Figure 11: Prompt for Ruleset Refinement.

Panel of Figure 12 — Figure 12: Prompt for Input Generation with no rules + no clustering context

Rule ID	Input Window	$X_{t,1}$ Trend	$X_{t,2}$ Trend	Output Window	$Y$ Phase	Symbolic Rule
R1	[0, 14]	SharpSpike	Low	[15, 29]	InitialGrowth	$\Box_{[0,14]}(\mathrm{SharpSpike}(X_{s})\wedge\mathrm{Low}(X_{a}))\rightarrow\Diamond_{[15,29]}\mathrm{InitialGrowth}(Y)$
R2	[14, 28]	HighStable	HighStable	[29, 43]	PeakFormation	$\Box_{[14,28]}(\mathrm{HighStable}(X_{s})\wedge\mathrm{HighStable}(X_{a}))\rightarrow\Diamond_{[29,43]}\mathrm{PeakFormation}(Y)$
R3	[28, 42]	Low	DipThenRise	[43, 57]	PostPeak	$\Box_{[28,42]}(\mathrm{Low}(X_{s})\wedge\mathrm{DipThenRise}(X_{a}))\rightarrow\Diamond_{[43,57]}\mathrm{PostPeak}(Y)$
R4	[42, 56]	SharpSpike	HighStable	[57, 71]	Decline	$\Box_{[42,56]}(\mathrm{SharpSpike}(X_{s})\wedge\mathrm{HighStable}(X_{a}))\rightarrow\Diamond_{[57,71]}\mathrm{Decline}(Y)$
R5	[56, 70]	Moderate	Low	[71, 85]	Resurgence	$\Box_{[56,70]}(\mathrm{Moderate}(X_{s})\wedge\mathrm{Low}(X_{a}))\rightarrow\Diamond_{[71,85]}\mathrm{Resurgence}(Y)$
R6	[70, 100]	HighStable	HighStable	[85, 100]	PeakFormation	$\Box_{[70,100]}(\mathrm{HighStable}(X_{s})\wedge\mathrm{HighStable}(X_{a}))\rightarrow\Diamond_{[85,100]}\mathrm{PeakFormation}(Y)$

Appendix

Table 4: Inaccurate Causal Rules used for Input Generation for Generalizability Experiment for PySIRTEM

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