What Makes a Good Symbol for Hybrid Discrete-Continuous Planning?

Yiyuan Pan; Changliu Liu; Jiayuan Mao

cs.ROOct 16, 2026

What Makes a Good Symbol for Hybrid Discrete-Continuous Planning?

Authors: Yiyuan Pan, Changliu Liu, Jiayuan Mao

Organizations: Robotics Institute, Carnegie Mellon University · University of Pennsylvania

Abstract

The physical world has continuous state and action spaces, yet robot planning often benefits from hybrid discrete-continuous formulations that explicitly expose combinatorial structure, such as contact modes or discrete primitive actions. Existing methods typically rely on human-defined symbol sets or symbols derived from language supervision, without asking what makes a symbol computationally useful. In this paper, we take an inference-algorithm--centric view to define usefulness: a good symbol should expose the discrete commitments that simplify solving the original planning problem. We first introduce the concept of solver-sufficient symbolization by studying finite-horizon piecewise-affine systems, for which solver sufficiency can be precisely characterized, and develop a counterexample-guided algorithm for symbol invention. We then extend this principle to invent symbols for optimization objectives parameterized by general neural networks as energy functions, showing that solver-sufficient symbols can nevertheless be induced from their continuous optimization landscapes. Experiments on simulated and real-robot tasks show that the resulting symbols reduce solving cost while preserving solution quality, while also improving interpretability and steerability. Furthermore, our theoretical and algorithmic insights help explain why certain human-defined symbols are computationally useful.

Figures & tables

Figure 1: Solver-Sufficient Symbolization for PWA Systems. Relative affine-TU groups optimization regions with a compact symbol set. Each conditional optimization subproblem $\mathcal{P}_{\mathrm{joint}}^{\psi}$ , given by an assignment of discrete symbols $\psi$ , reduces to an LP-relaxable optimization problem.

Panel of Figure 2 — Figure 2: One CEGAR Skeleton, Two Instantiations. Both algorithms share three stages: Verify , Sparsify ( eliminate false obstacles ), and Split ( pay one symbol for real obstacles ).

Figure 3: Solver-Sufficient Symbolization for NN Systems. $\tau$ -descent faithfulness partitions a multimodal state-conditioned objective into regions. A symbol-guided control algorithm performs one confined descent per region and selects the lowest-value candidate across all regions.

Figure 4: Exemplar Symbol Invention in One Round. Take Lunar Lander as the exemplar task.

Figure 5: Task Suites. Dashed line divides the tasks for PWA systems and for network systems.

Method

2D Manipulation

In-Hand Reorientation

MPushT

Hook

Shelf

Block

Cuboid

Pen

\uparrow

\uparrow

\uparrow

\uparrow

\uparrow

\uparrow

\uparrow

\uparrow

\uparrow

\uparrow

\uparrow

\uparrow

- Solver-Oriented Methods

CoCo

0.75

3.57\times

0.63

\mathbf{3.18\times}

0.75

1.39\times

0.69

\mathbf{2.70\times}

0.81

\mathbf{1.61\times}

0.75

1.28\times

GBD-MPC

0.94

0.12\times

0.94

0.06\times

0.81

0.09\times

\mathbf{0.94}

0.32\times

0.88

0.21\times

\mathbf{0.88}

0.06\times

- Symbol-Oriented Methods

Table 1: Solver Sufficiency on PWA Systems. For inference-time speedup IS , E2E is

1.00\times

Figure 6: Symbol Count vs. Conditioned Solving Time. The dashed line shows the fitted trend across methods.

Method	Avoiding		Pushing		Aligning		Sorting		Matching
Method	SR $\uparrow$	DC ${}_{\delta}\downarrow$	SR $\uparrow$	DC ${}_{\delta}\downarrow$	SR $\uparrow$	DC ${}_{\delta}\downarrow$	SR $\uparrow$	DC ${}_{\delta}\downarrow$	SR $\uparrow$	DC ${}_{\delta}\downarrow$
DDPM	$\mathbf{1.00}$	$7.03$	$0.90$	$\mathbf{1.00}$	$\mathbf{0.94}$	$10.31$	$\mathbf{0.75}$	$5.15$	$\mathbf{0.88}$	$1.68$
ACT	$0.91$	$833.69$	$0.91$	$5.93$	$0.91$	$9.25$	$0.69$	$173.45$	$0.63$	$259.67$
BESO	$0.97$	$329.03$	$0.63$	$198.57$	$0.75$	$338.03$	$\mathbf{0.75}$	$87.48$	$0.69$	$147.85$
IBC	$0.81$	$690.34$	$0.63$	$312.70$	$0.63$	$349.94$	$0.63$	$340.78$	$0.69$	$562.95$
BeT	$0.72$	$495.50$	$0.30$	$105.63$	$0.53$	$94.66$	$0.56$	$936.83$	$0.75$	$1325.94$

Table 2: Solver Sufficiency on Network Systems. For DC δ , we take

\delta=0.001

Figure 7: Human-Defined Concept Analysis.

Figure 8: Behavior Steering with Invented Symbols. Different symbol values correspond to distinct basin-level behaviors, allowing steering the controller toward preferences.

Task	Ours	PGS	pix2pred
MPushT	$\displaystyle\sum_{\tau_{c}<0}z_{c}-\sum_{\tau_{c}>0}z_{c}$ $\triangleright$ is clockwise push $z_{\mathrm{top}}-z_{\mathrm{bottom}}$ $\triangleright$ is push above	$\xi_{\theta}\leq-0.9$ $\triangleright$ T orientation $\leq$ threshold $\xi_{\mathrm{dist}}\leq 1.0$ $\triangleright$ near the goal	stem_pointing_down_right rotation_error_small
Block	$z_{\rm s\rightarrow ff}$ $\triangleright$ index finger contacts object $z_{\rm th\rightarrow palm}$ $\triangleright$ thumb contacts palm	$\xi_{\rm ff\rightarrow thu}>0.5$ $\triangleright$ index finger near thumb $\xi_{\rm 11}\leq 0.6$ $\triangleright$ control variable $\leq$ threshold	orientation_over_bound orientation_southeast

Table 3: Semantics of CPR-Invented Symbols. We only show two invented symbols per setting.

Table 12

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 13 — Algorithm 3 Detailed CPR Algorithm (PWA Systems)

Panel of Figure 14 — Algorithm 4 Detailed CPR Algorithm (Network Systems)

Figure 9: Quadratic Symbol Axes. It captures the curved decision boundary with fewer symbolic directions.

Table 16

Task	Factors	Semantics
MPushT	$z_{0:4}$	$F_{1}$ , push along $-y$ ; cumulative runs $c_{0}\!\rightarrow\!c_{4}$
	$z_{5}$	$F_{2}$ , push along $-x$ at $c_{5}$
	$z_{6:7}$	$F_{3}$ , push along $+y$ from $c_{6}$ to $c_{7}$
	$z_{8:10}$	$F_{4}$ , push along $-x$ at $c_{8}$ and along $c_{9}\!\rightarrow\!c_{10}$
	$z_{11}$	$F_{5}$ , push along $+y$ at $c_{11}$
	$z_{12:14}$	$F_{6}$ , push along $+x$ from $c_{12}$ to $c_{14}$

Appendix

Table 8: Discrete Factors for 2D Manipulation.

Task	Factors	Semantics
Block	$z_{0:4}$	active finger: ff, mf, rf, lf, or th
	$z_{5:12}$	admissible finger–face assignment
	$z_{13:18}$	contacted face: palm-facing, dorsal, thumb-side, little-side, distal, or proximal
	$z_{19}$	idle unit, allowing only two active finger contacts
Cuboid	$z_{0:19}$	same finger, finger–face, face, and idle-factor semantics as Block ; contact feasibility follows the cuboid geometry
Pen	$z_{0:2}$	active finger group: opposition, medial, or ulnar

Appendix

Table 9: Discrete Factors for In-Hand Reorientation.

Form.	Factors	Semantics
$P_{1}$	$z_{0:2}$	$F_{1}$ : push at $c_{0}$ , extend leftward to $c_{1}$ , or move to $c_{3}$
	$z_{3,4,7,10,11}$	single pushes on $F_{2},F_{3},F_{5},F_{7},F_{8}$ at $c_{5},c_{6},c_{11},c_{15},c_{17}$
	$z_{5:6}$	$F_{4}$ : push at $c_{8}$ or move to $c_{9}$
	$z_{8:9}$	$F_{6}$ : push at $c_{12}$ or extend to $c_{13}$
$P_{2}$	$z_{0:7}$	one occupancy bit per face $F_{1},\ldots,F_{8}$ , pushing at its innermost contact point
$P_{3}$	$z_{0:3}$	push from the top, bottom, right, or left side

Appendix

Table 10: Alternative Expert-Designed MPushT Factorizations.

P_{0}

is described in Table 8 .

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Intelligent agents must reason over both continuous dynamics and discrete representations to generate effective plans in complex environments. Previous studies have shown that symbolic abstractions can emerge from neural effect predictors trained with a robot's unsupervised exploration. However, these methods rely on deterministic symbolic domains, lack mechanisms to verify the generated symbolic plans, and operate only at the abstract level, often failing to capture the continuous dynamics of the environment. To overcome these limitations, we propose a bilevel neuro-symbolic framework in which learned probabilistic symbolic rules generate candidate plans rapidly at the high level, and learned continuous effect models verify these plans and perform forward search when necessary at the low level. Our experiments on multi-object manipulation tasks demonstrate that the proposed bilevel method outperforms symbolic-only approaches, reliably identifying failing plans through verification, and achieves planning performance statistically comparable to continuous forward search while resolving most problems via efficient symbolic reasoning.

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