Source-linked AI summary
Figure It Out: Improve the Frontier of Reasoning with Executable Visual States
Meiqi Chen, Fandong Meng, Jie Zhou
TL;DR
Complex spatial and geometric constraints can be difficult to maintain through text-only reasoning. FIGR integrates executable visual construction into multi-turn reasoning and adaptively regulates its use; the paper reports improved reasoning stability and accuracy on complex mathematical problems.
Problem
Text-only reasoning struggles to capture complex spatial and geometric constraints that are not explicitly encoded in text.
Source: §1 · p. 1, §2.1 · p. 2Method
FIGR interleaves textual reasoning with executable code that constructs diagrams, using an adaptive reward mechanism to regulate visual construction.
Source: §1 · p. 2Results
Takeaways & Limitations
Controllable, revisable visual states support reasoning over global structural constraints that are difficult to maintain through text alone.
Source: §5 · p. 8Multi-turn reasoning and code execution add computational cost, and visual construction may benefit local or purely symbolic tasks less.
Source: Limitations · p. 9
Abstract
from arXiv · showhide
Complex reasoning problems often involve implicit spatial and geometric relationships that are not explicitly encoded in text. While recent reasoning models perform well across many domains, purely text-based reasoning struggles to capture structural constraints in complex settings. In this paper, we introduce FIGR, which integrates executable visual construction into multi-turn reasoning via end-to-end reinforcement learning. Rather than relying solely on textual chains of thought, FIGR externalizes intermediate hypotheses by generating executable code that constructs diagrams within the reasoning loop. An adaptive reward mechanism selectively regulates when visual construction is invoked, enabling more consistent reasoning over latent global properties that are difficult to infer from text alone. Experiments on eight challenging mathematical benchmarks demonstrate that FIGR outperforms strong text-only chain-of-thought baselines, improving the base model by 13.12% on AIME 2025 and 11.00% on BeyondAIME. These results highlight the effectiveness of precise, controllable figure construction of FIGR in enhancing complex reasoning ability.
1 Introduction
Text-only reasoning can struggle to maintain complex spatial constraints, while existing visual approaches trade precision for control or cannot construct task-specific diagrams. FIGR integrates executable visual construction into multi-turn reasoning and learns when to use it without supervised visual-reasoning cold-start.
1 Introduction
- Text-only reasoning must maintain spatial relations implicitly, while generated images can lack precision and fixed visual tools cannot autonomously build task-specific diagrams.These limitations are especially relevant to problems with subtle geometric constraints.Source: §1 · pp. 1–2
- FIGR integrates executable code and revisable diagrams into multi-turn reasoning, using rendered figures as explicit feedback for geometric consistency.The model can construct and iteratively refine diagrams during inference rather than relying on unconstrained image generation or fixed toolsets.Source: §1 · p. 2
- FIGR uses an adaptive reward mechanism to encourage beneficial visual construction and discourage unnecessary rendering without supervised visual-reasoning cold-start.It learns when to invoke visual reasoning and how to integrate execution feedback, starting from an instruction-tuned model and using reinforcement learning.Source: §1 · p. 2
- The paper proposes active visual thinking during multi-turn reasoning and reports improvements over strong text-only baselines across diverse mathematical benchmarks.Its stated contributions also include regulating selective visual reasoning without supervised cold-start.Source: §1 · p. 2
2 Related Work
Prior approaches leave intermediate structure implicit, produce visuals without reliable geometric constraints, or restrict visual operations to given images and predefined transformations. The related work frames FIGR against these distinct limitations.
2.1 Text-based and Programmatic Reasoning
- Textual and programmatic reasoning leave intermediate states implicit, making complex spatial relations difficult to maintain and relational errors liable to accumulate.The passage names geometry and kinematics among structurally demanding settings.Source: §2.1 · p. 2
2.2 Unconstrained Visual Generation for Reasoning
- Unconstrained image generation lacks explicit geometric-precision and relational-consistency mechanisms, so visual inaccuracies can propagate in mathematical and scientific reasoning.The passage describes such generated representations as noisy or spatially imprecise.Source: §2.2 · p. 3
2.3 Tool-Augmented Visual Reasoning
- Tool-augmented visual reasoning is controllable but limited to supplied images and predefined transformations, preventing autonomous construction of task-specific visual structures.This restricts its applicability when reasoning requires dynamically generated and refined diagrams.Source: §2.3 · p. 3
3 Methodology
FIGR interleaves textual inference with executable diagram construction, then uses trajectory-level reinforcement learning and an adaptive reward to regulate visual reasoning. The reward ties visual invocation to task suitability, execution, and answer correctness.
3.1 Executable Visual Construction within the Reasoning Loop
- FIGR alternates textual inference and code generation, executing code in a sandbox to return textual feedback and/or a rendered diagram that updates the reasoning context.The process repeats until termination or a step limit, providing executable and reproducible visual feedback.Source: §3.1 · p. 3
3.2 Reinforcement Learning with Multi-turn Executable Reasoning
- GRPO optimizes multi-turn reasoning at the trajectory level because an action's utility depends on the final outcome and resulting visual states.It samples groups of candidate trajectories and assigns rewards for answer correctness, format compliance, and appropriate visual construction; the resulting learning signal compares trajectories without step-wise supervision or an explicit value model.Source: §3.2 · pp. 3–5
3.3 Adaptive Reward Mechanism
- The adaptive reward mechanism addresses unnecessary rendering by using a suitability classifier to modulate trajectory-level visual-invocation rewards.A positive reward requires correct answers and successful code execution, with higher weight when diagrammatic reasoning is predicted to suit the task; this discourages redundant visual actions while preserving purposeful use.Source: §3.3 · p. 5
4 Experiments
Across seven mathematical benchmarks, FIGR exceeds both the base policy and a text-only RL baseline. Ablations and a case study examine how visual construction, reward guidance, and visual feedback relate to its reasoning behavior.
4.1 Experimental Setup
- FIGR is trained on DeepMath-103K and evaluated across seven benchmarks, with pass@1 reported using k=64 for AIME 2024/2025 and k=1 elsewhere.Source: §4.1 · pp. 5–6
4.2 Main Results
- 74.11% average accuracy across seven mathematical benchmarks exceeds Qwen3-VL-32B-Instruct by 6.77% and the text-only RL baseline by 4.88%.Source: Table 1 · p. 6, §4.2 · p. 6
4.3 Ablation Study
- Prompt engineering yields moderate gains but unstable drawing, while SFT lowers AIME 2025 accuracy from 66.20% to 53.33%; text-only RL improves performance consistently.Source: §4.3 · pp. 6–7
- Removing ARM or visual feedback degrades performance versus full FIGR; without ARM, code ratio and length collapse to zero after brief early use.Source: Table 2 · p. 7, §4.3 · pp. 7–8, Figure 3 · p. 7
- Passive diagrams do not consistently improve over the base model or text-only RL, while UMMs and TAVLMs perform weakly, with cascading errors or predefined-operation limits.Source: §4.3 · p. 7
- Full FIGR sustains a high code ratio, longer code blocks, and a high execution pass rate, while executable visual feedback supports validation of global structure.Source: §4.3 · p. 8
4.4 Case Study
- In the representative case, visual state lets FIGR observe global patterns and adjust later reasoning, unlike text-only feedback that leaves reasoning local and algebraic.Source: §4.4 · p. 8
5 Conclusion
FIGR embeds controllable, revisable visual states in multi-turn reasoning to support global structural constraints that are difficult to maintain through text alone. The conclusion reports improved reasoning stability and accuracy on complex mathematical problems.
5 Conclusion
- FIGR integrates executable visual construction into multi-turn reasoning through end-to-end reinforcement learning.Its controllable, revisable visual states are intended to support reasoning over global structural constraints that are difficult to maintain through text alone.Source: §5 · p. 8
- Learning when and how to externalize intermediate structure as executable visual states significantly improves reasoning stability and accuracy on complex mathematical problems.Source: §5 · p. 9
Limitations
The method adds computational cost through multi-turn reasoning and code execution, and may offer less benefit for tasks solvable through local or purely symbolic reasoning. Evaluation is limited to mathematical benchmarks emphasizing structural consistency and verifiable outcomes.
Limitations
- Multi-turn reasoning and code execution add computational cost, motivating more efficient reasoning–execution loops.The passage identifies compact visual representations or adaptive control strategies as future directions for reducing this overhead.Source: Limitations · p. 9
- Visual externalization may provide less benefit on tasks solvable through local or purely symbolic reasoning.More precise identification of the task regimes that benefit remains an open research question.Source: Limitations · p. 9
- The evaluation focuses on mathematical benchmarks emphasizing structural consistency and verifiable outcomes.Scientific reasoning, program synthesis, and long-horizon planning are proposed as future extensions.Source: Limitations · p. 9
A Dataset Details
The evaluation suite spans competition problems, standardized mathematics datasets, and advanced reasoning tasks. The supplied dataset descriptions specify the scope and size of several benchmarks.
Dataset suite
- The evaluation suite spans competition problems, standardized mathematics datasets, and advanced reasoning tasks.Source: A Dataset Details · p. 12
Competition benchmarks
- AIME 2024 and AIME 2025 each comprise 30 competition problems, while Beyond AIME contains 100 problems designed to reduce memorization and increase reasoning complexity.Source: A Dataset Details · p. 12
Mathematics benchmarks
- MATH 500 is a 500-problem high-difficulty subset of the 12,500-problem MATH dataset, and the AMC dataset contains 83 problems.Source: A Dataset Details · p. 12
Advanced reasoning tasks
- MinervaMath contains 272 undergraduate-level quantitative reasoning problems requiring logical deductions and multi-step solutions.Source: A Dataset Details · p. 12
- The evaluated OlympiadBench subset, OE_TO_maths_en_COMP, contains 674 English open-ended, text-only comprehensive math problems emphasizing rigorous reasoning.Source: A Dataset Details · p. 12
B Prompt Templates
The appendix describes templates for judging visual suitability and for prompting different reasoning settings. FIGR uses an interactive prompt, while the text-only RL baseline and other models use single-turn prompting.
B.1 Prompt Template for Suitability Classifier
- The suitability-classifier template asks whether visual sketches are necessary and useful based solely on the problem statement.Source: §B.1 · p. 12
B.2 Prompt Template for Reasoning
- The reasoning prompts comprise three categories corresponding to different reasoning settings.Source: §B.2 · p. 12
- FIGR, the variant without visual feedback, and the variant without ARM use a multi-turn prompt that lets models iteratively generate output and receive interpreter feedback.Source: §B.2 · pp. 12–13
B.1 Prompt Template for Suitability Classifier
- Figure 5 presents the prompt template used for the suitability classifier.Source: Figure 5 · p. 13
B.2 Prompt Template for Reasoning
- The text-only RL baseline and other models use a single-turn prompt requiring a direct final answer without intermediate interactions.Source: §B.2 · p. 13
- DeepEyes and Chain-of-Focus use the prompt templates recommended in their original papers.Source: §B.2 · p. 13
C Additional Experimental Results
This section points to full experimental results in Table 3 and includes prompt templates for multi-turn and single-turn chain-of-thought.
C Additional Experimental Results
- Table 3 presents full experimental results, including unified multimodal and tool-augmented vision-language models partially reported in the main context.The passage does not provide the table's individual scores.Source: C Additional Experimental Results · p. 13
- Figure 6 provides a prompt template for multi-turn chain-of-thought.Source: Figure 6 · p. 14
- Figure 7 provides a prompt template for single-turn chain-of-thought.Source: Figure 7 · p. 14