PaperlayerView exact revision on arXiv

Source-linked AI summary

Knot So Simple: A Minimalistic Environment for Spatial Reasoning

Zizhao Chen, Yoav Artzi

arXiv:2505.18028v3cs.LGcs.AIcs.CVcs.RO

TL;DR

KnotGym addresses the challenge of evaluating visual agents that must reason spatially and manipulate ropes from image observations. It defines goal-oriented knot tasks whose crossing counts control complexity and generalization, then shows that existing methods face substantial difficulty, especially on harder tasks. The environment therefore serves as a controllable testbed for grounded visual reasoning and manipulation.

  • Problem

    Visual agents need a controlled environment for integrating perception, spatial reasoning, planning, and grounded manipulation under complex, image-only observations.

    Source: §1 · p. 2, §3.1 · p. 5

  • Method

    KnotGym provides three image-observation rope-manipulation tasks with Gauss-code goals and tunable knot-crossing complexity.

    Source: §1 · p. 2, §3.1 · p. 5, §3.2 · p. 5

  • Results

    Unknot generalizes across held-out configurations and higher crossing counts, whereas tie achieves over 80% training success but only narrowly beats random performance on new same-complexity test knots.

    Source: §4.1 · p. 8

  • Takeaways & Limitations

    KnotGym supplies a structured visual generalization ladder and exposes limitations of current methods on abstract, continuous rope manipulation.

    Source: §1 · p. 2, §3.4 · p. 6, §4.1 · p. 7

    The environment uses continuous states and actions with a large search space, while its project scope restricts configurations to those reducible to the unknot.

    Source: §3.4 · p. 6

Abstract

from arXiv · show

We propose KnotGym, an interactive environment for complex, spatial reasoning and manipulation. KnotGym includes goal-oriented rope manipulation tasks with varying levels of complexity, all requiring acting from pure image observations. Tasks are defined along a clear and quantifiable axis of complexity based on the number of knot crossings, creating a natural generalization test. KnotGym has a simple observation space, allowing for scalable development, yet it highlights core challenges in integrating acute perception, spatial reasoning, and grounded manipulation. We evaluate methods of different classes, including model-based RL, model-predictive control, and chain-of-thought reasoning, and illustrate the challenges KnotGym presents. KnotGym is available at https://github.com/lil-lab/knotgym.

1 Introduction

KNOTGYM is an accessible image-based environment for visual reasoning and rope manipulation, using knot crossings to control complexity and test generalization. Its goals are topological rather than exact visual reconstructions, while its chosen crossing measure is tailored to manipulation and observation complexity.

KNOTGYM

  • KNOTGYM trains and evaluates visual reasoning agents by having them transform an observed rope knot into a goal topology through applied forces.The goal is communicated by a secondary knot exemplar, and the environment defines three tasks with distinct starts and goals.Source: §1 · p. 2

Research Challenges

  • KNOTGYM combines spatial reasoning, action prediction, planning, and abstraction because agents must map image observations and exemplar goals to continuous action sequences.Goals form an equivalence class of valid states defined by Gauss code rather than one prescribed configuration.Source: §1 · p. 2

Generalization Ladder

  • The number of knot crossings provides a controllable complexity measure and a train-test generalization ladder beyond the complexities seen during training.It also supports curricula that gradually increase crossing counts.Source: §1 · p. 2

Research Accessibility

  • KNOTGYM is designed for research accessibility through Gymnasium-compatible interfaces, vectorized multi-CPU environments, and publicly released MIT-licensed code and baselines.The environment is positioned as a laboratory testbed for extended visual reasoning.Source: §1 · p. 2

Scope and Assumption

  • The crossing count is an appropriate KNOTGYM complexity measure for manipulation and observation even though knot mathematics more commonly emphasizes irreducible-knot complexity.Under the mathematical measure discussed, the illustrated configurations can all reduce to the unknot.Source: §1 · p. 2

2 Related Work

Related work studies spatial reasoning, generalization, deformable-object manipulation, and knot theory from several perspectives. KNOTGYM narrows the task domain while emphasizing long-horizon visual reasoning, complexity generalization, and abstract topological goals from raw images.

Spatial Reasoning in Vision-language Models

  • Unlike many spatial benchmarks and robotics tasks centered on static relations or limited reasoning, KNOTGYM demands long, complex spatial reasoning in a narrow domain.This focus creates opportunities for test-time-scaling reasoning in vision.Source: §2 · p. 3

Axes of Generalization

  • KNOTGYM proposes knot crossings as an established visual generalization axis, enabling structured evaluation beyond sequence length, graph size, recombination, arithmetic, and symbolic-operation settings.The knot-theoretic formulation provides a mathematical structure for discussing generalization.Source: §2 · p. 3

Deformable Object Manipulation

  • KNOTGYM simplifies rope manipulation to emphasize complexity generalization and evaluates abstract Gauss-code goals rather than Euclidean coordinate distance.This goal formulation relies more on abstract reasoning or searching than direct optimization.Source: §2 · p. 3

Machine Learning and Knots

  • Where prior knot-learning work uses symbolic braid words, KNOTGYM uses raw image observations to assess reasoning about intuitive physics.Its knot-manipulation setting instantiates the unknotting problem.Source: §2 · p. 3

3 KnotGym

KNOTGYM is a visual rope-manipulation environment in which agents transform knots toward exemplar-specified topological goals using continuous actions. Its tasks vary by knot crossings, enabling evaluation of visual reasoning, manipulation, and generalization across increasing complexity.

3.1 Environment

  • KNOTGYM represents knot manipulation as a partially observable environment where agents receive images of current and goal configurations and apply 3D forces to rope points.The environment uses a Gymnasium interface and MuJoCo transitions, while the agent lacks direct access to world state or Gauss code.Source: §3 · p. 3, §3.1 · p. 4
  • Success requires matching the goal Gauss code, producing sparse reward while allowing visually different configurations with the same topology.The criterion abstracts over multiple valid termination states rather than requiring exact geometric reconstruction.Source: §3.1 · p. 5

3.2 Three Tasks

  • The three tasks—unknot, tie, and convert—share one interface but differ in initial states and goal codes, with convert requiring recognition, goal inference, and planning.Complexity is tunable through the number of crossings, and experiments use sampled training and testing configurations with a 50-step horizon.Source: §3.2 · pp. 5–6, Table 4 · p. 5

3.3 Evaluation and Generalization

  • Evaluation measures success rate on unseen configurations and on complexity generalization from trained crossing counts to knots with more crossings.The latter creates a train-test split over knot complexity rather than only over instances.Source: §3.2 · p. 5, §3.3 · p. 6

3.4 Knot So Simple

  • KNOTGYM combines fine-grained crossing perception, continuous spatial reasoning, and a large search space whose possible Gauss codes grow factorially with crossings.Even discretizing knot structure would still require grounding symbolic moves back into continuous actions, while the current scope restricts configurations to reducible knots.Source: Table 3 · p. 5, §3.4 · p. 6

4 Experiments

KNOTGYM experiments compare RL and prompting methods across tasks, crossing complexities, and generalization settings. Results show that unknot is tractable, whereas tie and convert expose substantial training, generalization, and grounded-action challenges.

4 Experiments

  • The benchmark evaluates PPO, DreamerV3, TM-MPC2, and GPT-4.1-nano prompting over nine KNOTGYM setups, with RL measured after 1M environment steps.Table 5 reports training-split success rates over rollouts, while Figure 3 summarizes RL training success with 95% confidence intervals.Source: §4 · p. 6, Table 5 · p. 7, §4.1 · p. 7

4.1 Reinforcement Learning Methods

  • At #X=4, no RL method outperforms a random policy on tie or convert, indicating sharply increased difficulty for tasks with many possible goals.At #X=3, only DreamerV3 is clearly better than random on these tasks.Source: Figure 3 · p. 7, §4.1 · p. 8
  • RL methods achieve non-trivial success on unknot, but tie and convert remain substantially harder, especially as goal complexity increases.DreamerV3 performs particularly well on unknot, while methods struggle on the more demanding tasks.Source: §4.1 · pp. 7–8, Figure 3 · p. 7
  • Qualitative rollouts reveal a consistent pulling strategy for unknot but diverse, unstructured actions for tie.The contrast highlights why tie requires more careful reasoning about the goal and action sequence.Source: §4.1 · pp. 7–8, Figure 6 · p. 8
  • Increasing goal-configuration diversity challenges RL training, motivating experiments that reduce the training pool from 20 configurations to 10, 5, and 1.The study links the variety of goal configurations to training difficulty.Source: Figure 4 · p. 7, §4.1 · p. 8
  • Unknot policies generalize across held-out configurations and, to a lesser degree, higher crossing counts, whereas tie policies show weak test and goal generalization.A tie policy exceeding 80% training success only narrowly beats random on held-out same-complexity knots.Source: §4.1 · p. 8

4.2 Prompting

  • GPT-4.1-nano can recognize the goal Gauss code and form a reasonable plan, yet its rollout actions are too weak and imprecise for effective manipulation.The example illustrates brittleness in naive VLM world modeling even when the intended plan is sensible.Source: Figure 7 · p. 9, §4.2 · p. 9
  • Prompting-based methods generally underperform RL baselines, with most falling below random except on unknot at #X=2 and #X=3.Stateful prompting appears helpful on unknot, but closed-loop feedback does not eliminate action imprecision.Source: §4.2 · pp. 8–9

5 Discussion

KNOTGYM supports multiple visual-reasoning research directions, while current methods face data-efficiency and grounded-action challenges. The appendix provides reproducible prompt instances across open, stateless, and stateful modes.

5 Discussion

  • KNOTGYM offers a testbed for frontier visual reasoning agents, including verbalized action reasoning, latent or image-space rollouts, video world models, and physics-based visual coding.The discussion names several agent classes for future evaluation.Source: §5 · p. 10
  • Current methods struggle differently: reinforcement learning faces data inefficiency over the broad goal space, while prompted vision-language models cannot produce fine, grounded actions.Multi-turn interaction history improves performance only to a limited extent.Source: §5 · p. 10
  • The planned simulator improvements target fidelity for higher crossing counts and longer ropes, plus throughput beyond the current 20 environment steps per second on one CPU.The proposed direction is moving simulation from CPUs to GPUs.Source: §5 · p. 10

A Prompt Examples

B Benchmarking Details

This reproducibility section states that unspecified hyperparameters use official implementation defaults. Deviations from those defaults are highlighted in bold.

B Benchmarking Details

  • Unspecified hyperparameters use the default values from the official implementation.This establishes the fallback rule for reproducing experiments.Source: B Benchmarking Details · p. 14
  • Deviations from default hyperparameter settings are bolded.The formatting identifies departures from the defaults.Source: B Benchmarking Details · p. 14
  • The section is explicitly intended to support reproducibility.It introduces the conventions used in the detailed benchmarking appendices.Source: B Benchmarking Details · p. 14

B.1 PPO

The PPO benchmark uses Stable-Baselines3 vision PPO with a 16.5M-parameter model and 32 vectorized environments. Each run uses 32 CPU cores, 96 GB RAM, and one NVIDIA GTX 2080 Ti GPU for about four hours.

B.1 PPO

  • Stable-Baselines3 vision PPO contains 16.5 M trainable parameters and runs with 32 vectorized environments.The implementation and environment parallelism define the benchmark setup.Source: §B.1 · p. 14
  • Each PPO training run uses 32 CPU cores, 96 GB RAM, and one NVIDIA GTX 2080 Ti GPU with 11GB memory.These are the reported hardware resources for each run.Source: §B.1 · p. 14
  • A PPO training run takes around 4 hours.The runtime is reported for the stated hardware configuration.Source: §B.1 · p. 14

B.2 DreamerV3

The DreamerV3 benchmark uses the official Hafner et al. [2025a] code with a 12M-parameter model and default repository hyperparameters. Runs use 32 vectorized environments and an H100 GPU, taking about 14 hours each.

B.2 DreamerV3

  • DreamerV3 uses the official Hafner et al. [2025a] codebase with 12M trainable parameters, roughly matching the PPO model-free policy size.The released implementation and model scale are kept consistent with the reported setup.Source: §B.2 · p. 14
  • The benchmark retains the exact default hyperparameters from the released repository and uses 32 vectorized environments.These choices are documented in Table 7 according to the passage.Source: §B.2 · p. 14
  • Each DreamerV3 run uses 33 CPU cores, 125 GB RAM, and one NVIDIA H100 GPU with 80GB memory, taking around 14 hours.The passage reports the resource configuration and runtime per training run.Source: §B.2 · p. 14

B.3 TD-MPC2

The TD-MPC2 setup uses the released default 5M-parameter architecture, modified for KNOTGYM’s larger image observations and accelerated with parallel environments.

B.3 TD-MPC2

  • TD-MPC2 uses the official Hansen et al. [2024] code base with the default 5M trainable-parameter architecture.Larger policies up to 48M parameters did not significantly improve KNOTGYM performance.Source: §B.3 · p. 15
  • The convolutional encoder adds two strided convolutional layers and a fully connected layer to support [3, 128, 256] observations.Source: §B.3 · p. 15
  • Training uses 16 parallel environments on 16 CPU cores, 148 GB RAM, and one NVIDIA GTX 2080 Ti GPU, with runs taking up to 20 hours.Source: §B.3 · p. 15
  • Hyperparameter configurations are listed in Tables 7 and 8 for DreamerV3 and TD-MPC2, respectively.Source: Table 7 · p. 16, Table 8 · p. 16

C KNOTGYM Implementation Details

KNOTGYM simulates bead-chain knots in MuJoCo and renders image observations while standardizing the software stack and environment specifications.

C KNOTGYM Implementation Details

D Additional Results

Additional results show limited goal generalization and substantial learning difficulty for RL methods as knot complexity increases.

D Additional Results

  • As training success rises sharply, test success on unseen goals with the same #X increases only modestly, indicating difficulty generalizing across goal configurations.The analysis uses a DreamerV3 tie #X=2 run with 20 goal configurations; Figure 12 reports 95% confidence intervals across three seeds.Source: D Additional Results · p. 17, Figure 12 · p. 17
  • For the unknot task, all RL methods learn despite differing sample efficiency, whereas tie and convert do not surpass random baseline when #X>2.Figure 13 reports success rates over the first million steps with 95% confidence intervals across three seeds.Source: D Additional Results · p. 17, Figure 13 · p. 17
  • The RL training curves cover nine KNOTGYM tasks during the first million environment steps.Source: Figure 13 · p. 17
Loading 2505.18028v3…