Embodied Turing Machines: Stateful Code for Robot Recursive Self-Improvement

Kairui Hu; Siyuan Hu; Fangzhou Hong; Zhaoxi Chen; Ziwei Liu

cs.ROOct 8, 2026

Embodied Turing Machines: Stateful Code for Robot Recursive Self-Improvement

Authors: Kairui Hu, Siyuan Hu, Fangzhou Hong, Zhaoxi Chen, Ziwei Liu

Organizations: Nanyang Technological University · Ropedia

Abstract

Most robot policies keep a model in the control loop: a VLA maps observations to actions, and an Agent Harness, such as Agent-as-Policy or Harness VLA queries a VLM for decision making at run time. We propose a different view: the embodied world is an Embodied Turing Machine, whose tape is the robot and environment state and rules are the policy. If this state can be represented accurately, the decision making can be written entirely in code. We therefore propose Code-Only-as-Policy (COAP): code measures and tracks the robot, environment, and task state from camera images and proprioception, and makes every decision from it. The same code applies across episodes, and different tasks share one library without a VLM or VLA in the loop. Compared with VLAs and Agent Harnesses, we analyze three advantages of COAP: (i) Explicit State: the state can be stored in code; (ii) Execution: code makes decision making controllable, recovers from failures flexibly, and runs fast and cheaply online; (iii) Extensibility: new tasks reuse, inherit, or extend the shared library, so capabilities can accumulate over tasks. These advantages make COAP a suitable medium for recursive self-improvement (RSI): coding agents develop the library in a closed loop, and each change is explicit and controllable. On RoboDojo's 42 bimanual tasks, the resulting library reaches a success rate of 70.24% without a model at test time. The upper bound of COAP lies in how accurately the state is represented for decision making and how robust the code logic is. We thus propose COAP as a new paradigm for embodied tasks; since it applies across episodes, it can also serve as an efficient data engine for VLAs and Agent Harnesses.

Figures & tables

Figure 1: Code-Only-as-Policy (COAP). We view the embodied world as an Embodied Turing Machine whose tape is the state of the robot and its environment (top middle). If this state is represented accurately and the code is robust enough, no agent is needed in the control loop (top right). COAP is a shared library: different tasks use the same library, and within a task the code runs across episodes. The policy is therefore code only and reaches a 70.24% success rate on RoboDojo.

Figure 2: Comparison of code-based robot policies. In prior work, a model writes code-as-policy online, turn by turn, as the robot runs (left), or one program takes its state from a learned model or the simulator oracle (middle). Our COAP reuses code across episodes and measures the state by code (right).

	Agent-as-Policy	Harness VLA	VLA / WAM	COAP (ours)
Explicit State
Inspectable	$\triangle$ reasoning text	$\triangle$ memory text	✗ black box	✓ state and rules
Persistent	$\triangle$ model context	$\triangle$ text memory	✗ recent frames	✓ code variables
Accurate	$\triangle$ pixels, some tools	$\triangle$ pixels, some tools	✗ image pixels	✓ measured values
Execution
Controllable	$\triangle$ steered by prompt	$\triangle$ steered by prompt	✗ weights decide	✓ code decides

Table 1: Four ways to build a robot policy with foundation models , compared on the properties of the three advantages in Section 4.3 . ✓ yes,

\triangle

in part, ✗ no. In the rest of the paper, we refer to Agent-as-Policy and Harness VLA together as an Agent Harness.

Figure 3: Code-Only-as-Policy. Development (left): coding agents improve the code offline in a loop. They run the current code $\theta_{k}$ on development layouts, read the state, and run alternative diffs on branches in parallel. Shared library (middle): task programs call the library from the shared code. Execution (right): at runtime the robot runs one fixed task program across episode, and an agent and a VLA are not required.

Method	Memory	Precision	Open	Generalization	Long-Horizon	Overall
	(6)	(8)	(8)	(12)	(8)	SR	Score
Agent Harness
PhysicalRSI [ 15 ] (agent + VLA)	46.6	32.5	24.9	15.6	37.3	31.38	36.27
GPT-6 Astra [ 56 ]	38.7	4.0	31.0	30.5	8.2	22.48	28.97
World Action Model (WAM)
Awomo-0.5 [ 3 ]	41.2	33.5	22.9	23.8	26.8	29.64	35.34

Table 2: Results on RoboDojo. SR is the success rate (%), the share of episodes in which the whole task succeeds; every column except Score reports SR. Score is the benchmark’s progress score.

Figure 4: Three advantages of COAP , each with three properties.

Figure 5: Explicit State. COAP keeps five categories of state as variables (left), remembers states and values (middle), and succeeds more often with more accurate measurements (right).

Figure 6: Execution. Backtracking keeps failures from compounding over long tasks (left, middle), and code runs on a CPU with a small fraction of a model’s compute (right; COAP’s compute as a share of each model’s, from reported model sizes).

Figure 7: Extensibility. New tasks run mostly library code (left), are developed from scratch within hours without data or training (middle; baseline lines are schematic), and add to what the policy can do (right, schematic).

Figure 8: Code structure. (a) The layers of the library. (b) Functions are abstracted at a sensible level: grasping is written by grasp method, not by object. (c) A simplified task program; each call is colored by its layer.

Figure 9: Why Code-Only-as-Policy is a suitable medium for RSI. Each row is an attribute important for RSI, which comes from an advantage in Section 4.3 (check: yes; half circle: partly; cross: no).

Figure 10: Parallel branching over the variants of one decision. (a) Development as a BFS: eight grasps of a mug run at once, and the best code logic is chosen (decisions 2 and 3 are schematic). (b) Of 36 branches, each measured against its parent, few help. (c) With more branches, an accepted change is more likely and larger. (d) With more branches, an accepted change comes sooner in wall-clock time, for more GPU time.

Figure 11: Four classes of error. Each column gives the share of failures in one class, two frames of one failed episode from the official video, and the fix.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Outcome	Share	Failed step (reattempted)	Share	Saved
Reattempted, succeeded	12%	grasp missed or slipped	49%	17%
Reattempted, failed	22%	dropped while carrying	32%	48%
Noticed and reported	42%	stuck on release	18%	23%
no further method	37%	press did not latch	8%	100%
step budget	4%	pour incomplete	1%	100%
Hit the step limit	1%

Appendix

Table 3: What COAP did when something went wrong. Left: outcomes of these episodes. Right: failed steps of the reattempted episodes.

Figure 12: Compute moves offline. COAP uses models only offline; a VLA and an Agent Harness run them online.

Step limit	Tasks	COAP	Agent Harness	VLA	GPT-as-policy
200–500	12	78.1	47.3	21.9	26.5
550–800	15	78.4	30.6	38.0	26.9
900–1900	15	52.0	13.2	22.1	14.8

Appendix

Table 4: Success (%) by the step limit of a task. GPT-as-policy: an Agent Harness without a VLA.

Figure 13: One example of each grasp method.

Method	Approach	Contact	Examples	Tasks
Top	from above	across the body	blocks, digits, a lying pen, eggs	28
Side	level, toward the axis	around the body or neck	a standing bottle, a cup	6
Handle	toward the handle	around the bar	a basket’s arch, a broom, a mallet	3
Rim	down onto the wall	one jaw inside, one outside	a bowl, a mug, a plate	3
Edge	along the thin side	on the two faces	a coin, a bread slice, a key	4
Cloth pinch	down onto the hem	fabric layers	a long-sleeved top	1

Appendix

Table 5: Grasp methods. Tasks: number of task programs using it.

	make_toast	hang_mugs	sweep_blocks	fill_pen_holder
First evaluation	3.2 h	3.2 h	3.9 h	0.2 h
First success	4.7 h	5.2 h	6.4 h	4.0 h
First merge	5.8 h	6.8 h	7.9 h	2.7 h
Success at merges	50 $\to$ 77	7 $\to$ 50 $\to$ 57	0 $\to$ 60 $\to$ 50	0 $\to$ 20 $\to$ 27

Appendix

Table 6: Timeline of the new tasks. Hours from the start, and validation success (%) at each merge.

Figure 14: Extension without breaking. (a) The code each new task runs, by layer. (b) One library function reaches many tasks. (c) sweep_blocks reuses slide, written for push_T.

Layer	Holds	Lines	Used by
Task programs	order, parameters, special cases	23.5k	2.5%
Object families	what an object is, how to hold it	28.5k	18%
Manipulation	pick, carry, place, push; grasping	5.4k	78%
Perception	look, measure, confirm	5.3k	100%
Motion	inverse kinematics, planning, collision	3.3k	100%

Appendix

Table 7: Layers of the library. Used by: share of the 40 task programs that use the layer’s most used module.

Class	Typical cases
Perception error	unrecognized randomized look; wrong final check; split or merged object
Contact error	slip or drop in the grip; push stops short; tip at release
Incomplete logic	no grasp that also reaches the goal; untried order; unchecked collision
Poor code design	fixed size filter; fixed grip-width band; fixed match threshold
Unknown	no lead in the trace

Appendix

Table 8: Typical cases of each class of error.

Task	SR	Score	Task	SR	Score
Memory			Generalization
cover_blocks	100.0	100.0	push_T	97.3	97.3
press_by_number	100.0	100.0	stack_blocks	94.7	95.2
swap_T	100.0	100.0	stack_bowls	93.3	94.0
swap_blocks	100.0	100.0	fold_clothes	90.0	91.6
imitate_sorting_sequence	74.7	78.5	pour_liquid_into_cup	89.3	89.3

Appendix

Table 9: Per-task results of COAP , by dimension. † Task without a program.

Figure 15: Case studies: memory, recovery, and one program in two scenes.

Figure 16: Execution trajectories (1/4). Successful official episodes.

Figure 17: Execution trajectories (2/4). Successful official episodes.

Figure 18: Execution trajectories (3/4). Successful official episodes.

Figure 19: Execution trajectories (4/4). Successful official episodes.

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