Thinking with Constructions: A Benchmark and Policy Optimization for Visual-Text Interleaved Geometric Reasoning

2026-03-19

AI Summary

Existing multimodal large language models lack the ability to actively construct effective visual aids for geometric reasoning, relying instead on static diagrams for passive inference. To address this limitation, this work proposes a vision–language interleaved chain-of-thought framework that dynamically generates visual assistance aligned with each reasoning step, thereby enhancing geometric problem-solving capabilities. We introduce GeoAux-Bench, the first benchmark specifically designed for auxiliary construction in geometry, which demonstrates the superiority of vision–language synergy over unimodal approaches and offers a novel perspective framing construction as entropy reduction. Furthermore, we develop an Action Applicability Policy Optimization (A2PO) reinforcement learning algorithm, integrating counterfactual sampling and adaptive reward shaping to refine construction behaviors. Experiments show that our method outperforms strong baselines by 3.51% on GeoAux-Bench, validating the efficacy of selective visual auxiliary construction.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Visual Reasoning & Symbolic Representations
• Natural Language Processing: Language Grounding & Multi-modal NLP

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

Abstract

Geometric reasoning inherently requires "thinking with constructions" -- the dynamic manipulation of visual aids to bridge the gap between problem conditions and solutions. However, existing Multimodal Large Language Models (MLLMs) are largely confined to passive inference with static diagrams, lacking the strategic knowledge of when and how to construct effective visual aids. To address this, we present a framework for Visual-Text Interleaved Chain-of-Thought. We first introduce GeoAux-Bench, the first benchmark comprising 4,334 geometry problems that aligns textual construction steps with ground-truth visual updates. Our pilot study reveals two critical insights: (1) interleaved visual-textual aids outperform single-modality counterparts, which cannot losslessly capture geometric synergy; and (2) valid constructions act as entropy reducers, strongly correlating with reduced reasoning perplexity. Building on these findings, we propose Action Applicability Policy Optimization (A2PO), a reinforcement learning paradigm for mastering strategic construction. A2PO employs Adaptive Reward Shaping to regulate the timing and quality of visual aids via counterfactual sampling to distinguish necessary from redundant constructions. Experiments demonstrate our approach enables MLLMs to leverage selective auxiliary constructions, yielding a 3.51% gain over strong baselines. Code and data are available on GitHub.

Problem

• geometric reasoning
• visual-text interleaved reasoning
• construction-based thinking
• multimodal large language models
• visual aids

Innovation

• Visual-Text Interleaved Reasoning
• Geometric Construction
• Action Applicability Policy Optimization
• GeoAux-Bench
• Multimodal Large Language Models