Emergent Neural Automaton Policies: Learning Symbolic Structure from Visuomotor Trajectories

Yiyuan Pan; Xusheng Luo; Hanjiang Hu; Peiqi Yu; Changliu Liu

cs.ROMar 26, 2026

Emergent Neural Automaton Policies: Learning Symbolic Structure from Visuomotor Trajectories

Authors: Yiyuan Pan, Xusheng Luo, Hanjiang Hu, Peiqi Yu, Changliu Liu

Organizations: Robotics Institute, Carnegie Mellon University, Pittsburgh, PA 15213, USA

Abstract

Scaling robot learning to long-horizon tasks remains a formidable challenge. While end-to-end policies often lack the structural priors needed for effective long-term reasoning, traditional neuro-symbolic methods rely heavily on hand-crafted symbolic priors. To address the issue, we introduce ENAP (Emergent Neural Automaton Policy), a framework that allows a bi-level neuro-symbolic policy adaptively emerge from visuomotor demonstrations. Specifically, we first employ adaptive clustering and an extension of the L* algorithm to infer a Mealy state machine from visuomotor data, which serves as an interpretable high-level planner capturing latent task modes. Then, this discrete structure guides a low-level reactive residual network to learn precise continuous control via behavior cloning (BC). By explicitly modeling the task structure with discrete transitions and continuous residuals, ENAP achieves high sample efficiency and interpretability without requiring task-specific labels. Extensive experiments on complex manipulation and long-horizon tasks demonstrate that ENAP outperforms state-of-the-art (SoTA) end-to-end VLA policies by up to 27% in low-data regimes, while offering a structured representation of robotic intent (Fig. 1).

Figures & tables

- PDDL / Rule-Based
Method	Continuous	Probabilistic	Label-Free
Bilevel Learning [6]	$\checkmark$
Neuro-Symbolic IL [10]	$\checkmark$
- Automata Extraction (NLP/Theory)
RNN Extraction [2,3]			$\checkmark$
WFA Extraction [5]		$\checkmark$	$\checkmark$

TABLE I: Comparison of Symbolic Structure Discovery Approaches.

Fig. 2: Overview of the ENAP Framework. Our approach unifies structure discovery and hierarchical control through a three-stage pipeline: (i) Symbol Abstraction: Continuous trajectories from demonstrations are encoded and clustered via HDBSCAN [ 37 ] to discover a discrete alphabet $\Sigma$ , mapping sensorimotor streams to symbolic sequences. (ii) Structure Extraction via Extended L ∗ : We introduce an extended $L^{*}$ algorithm that iteratively constructs a PMM by querying the dataset. The process maintains an observation table and enforces closedness ( expanding nodes ) and consistency ( refining edge ) constraints using an RNN-based history encoder. (iii) Bi-level Control Pipeline: The learned PMM serves as a high-level planner, providing a coarse action prior $a_{\text{base}}^{t}$ . This is combined with a learned residual term $\Delta a_{t}\sim\pi_{\psi}(\cdot|q_{t},o_{t})$ to synthesize the final precise action $\hat{a}_{t}=a_{\text{base}}^{t}+\Delta a_{t}$ .

Fig. 3: Iterative Structure-Policy Co-Evolution. The training process alternates between augmenting dataset via clustering with learned encoder ( Left ) and policy learning under the given dataset ( Right ). A residual network refines the action prior from PMM and generate updated encoder for subsequent iterations.

Fig. 4: Experiments in Both Simulation and Real-World. ( Left ) Complex manipulation; ( Middle ) TAMP; ( Right ) Real-world scenarios.

- Oracle
Method	Param	DualStack	Peg
Method	(M)	Cube (%)	Insert (%)
Oracle	$2.98$	$98.3\,{\scriptstyle\pm 0.4}$	$86.7\,{\scriptstyle\pm 0.8}$
- Traditional Imitation Learning
Transformer [ 59 ]	$63.81$	$38.7\,{\scriptstyle\pm 6.0}$	$51.8\,{\scriptstyle\pm 5.5}$
GMM [ 38 ]	$46.11$	$73.6\,{\scriptstyle\pm 2.3}$	$53.1\,{\scriptstyle\pm 2.6}$

TABLE II: Comparison on Complex Manipulation Tasks.

Fig. 5: Data-scaling comparison on DualStackCube . ENAP remains robust in low-data regimes, while other methods degrade significantly.

Fig. 6: Mechanism of Logical Branching at Decision Points. We visualize how ENAP handles multi-modal StackLego tasks. At the critical decision node ( $q_{2}$ ), the extracted PMM identifies a branching structure based on distinct future outcomes ( $q_{3}$ vs. $q_{6}$ ). By conditioning on the current observation, the residual policy guides the transition into the correct logical mode. Here $I_{t}^{i}$ and $p_{t}^{i}$ denote the image observation and pose at timestep $t$ of trajectory $i$ . The predicted action $\hat{a}_{t}$ follows Eq. 3c . The feature vector $f_{t}$ is obtained by encoding the observation $o_{t}=[I_{t};p_{t}]$ via the encoder $\phi_{\theta}$ , i.e., $f_{t}=\phi_{\theta}(o_{t})$ . For the state machine involving more Lego start poses, please refer to Section D-B .

Method	Param	Speed	Stack	Pick	Hanger
Method	(M)	(ms)	Lego	Place	Task
$\pi_{0.5}$ [ 20 ]	$3403$	$6841$	$58.82$	$76.47$	$64.71$
ENAP ∗ (DINO)	$23$	$281$	$\mathbf{88.24}$	$\mathbf{94.12}$	$\mathbf{94.12}$

TABLE III: Real-World Evaluation.

Method/Param (M)	Seq. (%)		Hier. (%)
Method/Param (M)	$3/5$	$5/5$	$3/5$	$5/5$
HULC ( $47$ ) [ 39 ]	$3.0\,{\scriptstyle\pm 0.3}$	$3.0\,{\scriptstyle\pm 0.2}$	$87.0\,{\scriptstyle\pm 0.6}$	$2.0\,{\scriptstyle\pm 0.2}$
LCD ( $68$ ) [ 64 ]	$11.0\,{\scriptstyle\pm 0.5}$	$9.0\,{\scriptstyle\pm 0.4}$	$57.2\,{\scriptstyle\pm 0.7}$	$5.0\,{\scriptstyle\pm 0.3}$
MDT ( $440$ ) [ 51 ]	$78.4\,{\scriptstyle\pm 0.9}$	$53.7\,{\scriptstyle\pm 1.1}$	$93.8\,{\scriptstyle\pm 0.8}$	$21.9\,{\scriptstyle\pm 0.6}$
FLOWER ( $947$ ) [ 52 ]	$91.0\,{\scriptstyle\pm 0.6}$	$90.6\,{\scriptstyle\pm 0.5}$	$90.8\,{\scriptstyle\pm 0.7}$	$15.9\,{\scriptstyle\pm 0.4}$
ENAP (FLOWER) ( $571$ )	$\mathbf{97.0\,{\scriptstyle\pm 0.4}}$	$\mathbf{96.8\,{\scriptstyle\pm 0.3}}$	$\mathbf{95.5\,{\scriptstyle\pm 0.5}}$	$\mathbf{28.2\,{\scriptstyle\pm 0.6}}$

TABLE IV: Comparison on Long-Horizon TAMP Tasks.

Fig. 7: Comparison of Learned Policies from PegInsert . (Left) Radar chart of six structural metrics. (Right) Visualized PMMs for DINO and GMM variants. For the PMM from Oracle, see Emergent Neural Automaton Policies: Learning Symbolic Structure from Visuomotor Trajectories .

Metric (%)	DP [ 8 ]	GMM [ 38 ]	$\pi_{0}$ [ 6 ]	ENAP (Oracle)
Either	$0.41\,{\scriptstyle\pm 0.05}$	$0.50\,{\scriptstyle\pm 0.05}$	$0.25\,{\scriptstyle\pm 0.02}$	$\mathbf{0.94\,{\scriptstyle\pm 0.03}}$
Nearest	$0.38\,{\scriptstyle\pm 0.07}$	$0.38\,{\scriptstyle\pm 0.03}$	$0.13\,{\scriptstyle\pm 0.04}$	$\mathbf{0.84\,{\scriptstyle\pm 0.02}}$

TABLE V: MultiGoalPushT Performance Comparison.

Fig. 9: Ablation on Structure Discovery Parameters. Success rates across varying similarity thresholds $\tau_{\text{sim}}$ and RNN loss configurations. Black dots indicate measured data points, while the surface shows interpolated values.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Fig. 16: Iteration 3 : Discovery of the decision point.

Fig. 17: Iteration 4 : Stabilized hypothesis capturing the full topology.

Fig. 18: Impact of Phase-Aware Contrastive Loss. Without phase-aware contrastive loss, the automaton is fragmented with ambiguous transitions.

Fig. 19: Generalization of Structural Branching. Instead of overfitting to six distinct branches, the model discovers three logical pathways ( $q_{2},q_{4},q_{5}$ ), grouping visually and kinetically similar poses (e.g., along the line of sight) into shared phases.

Component	ENAP (Oracle)	ENAP ∗ (DINO)
- Encoder Architecture
Visual Backbone	NatureCNN (3-layer Conv)	DINOv2 ViT-S/14 (Frozen)
Visual Head	Linear $\to$ 256	Spatial Softmax $\to$ MLP $\to$ 256
Proprioception	Linear $\to$ 256	Fourier (7-freq) $\to$ SpecNorm $\to$ 128
- Feature Dimensions (Vis + Pos [+ Delta_Pos])
StackCube	512 (256+256)	384 (256+128)

Appendix

TABLE X: Network Architectures for Complex Manipulation.

Component	ENAP (FLOWER)
- Encoder Architecture
Visual Backbone	Florence-2 DaVit (Frozen)
Language Encoder	Florence-2 Text Encoder
Fusion	Concat $\to$ Florence-2 Encoder $\to$ Mean Pool
Feature Dim	1024
- Residual Policy Network

Appendix

TABLE XI: Network Architectures for Long-Horizon TAMP.

Component	ENAP ∗ (DINO)
- Encoder Architecture
Visual Backbone	DINOv2 ViT-S/14 (Frozen)
Visual Head	Spatial Softmax $\to$ MLP (768 $\to$ 128)
Proprioception	Fourier (7-freq) $\to$ SpecNorm $\to$ 128
Fusion	Base (128) + Wrist (128) + Prop (128) = 384
- Policy Network

Appendix

TABLE XII: Network Architectures for Real-World Manipulation.

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