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

ACL ARR 2026 January Submission6040 Authors

Keywords: Multimodal Reasoning, Geometric Problem Solving, Reinforcement Learning, Visual-Text Interleaved

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 ($A^2PO$), a reinforcement learning paradigm for mastering strategic construction. A 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.

Paper Type: Long

Research Area: Question Answering

Research Area Keywords: math QA, multimodal QA, reasoning, logical reasoning

Contribution Types: NLP engineering experiment, Data resources

Languages Studied: English

Submission Number: 6040