# Learning the Boundary of Solvability: Aligning LLMs to Detect Unsolvable Problems

arXiv: https://arxiv.org/abs/2512.01661 · HTML: https://mlsanity.com/paper/2512.01661/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Dec 2025.

## Built on by (most cited first, 4 of 4)

- [OmniSIFT: Modality-Asymmetric Token Compression for Efficient Omni-modal Large Language Models](https://mlsanity.com/paper/2602.04804/) (2026)
- [Code2Math: Can Your Code Agent Effectively Evolve Math Problems Through Exploration?](https://mlsanity.com/paper/2603.03202/) (2026)
- [Bridging the Detection-to-Abstention Gap in Reasoning Models under Insufficient Information](https://mlsanity.com/paper/2605.28070/) (2026)
- [Visual Abstention in Unified Multimodal Models](https://mlsanity.com/paper/2610.07887/) (2026)

## Abstract (arXiv)

Ensuring large language model (LLM) reliability requires distinguishing objective unsolvability (inherent contradictions) from subjective capability limitations (tasks exceeding model competence). Current LLMs often conflate these dimensions, leading to hallucinations in which they return confident answers to inherently unsolvable queries. To address this issue, we propose a multi-domain dataset containing both solvable and unsolvable questions, UnsolvableQA, together with an alignment framework, UnsolvableRL. First, we construct UnsolvableQA by "Reverse Construction" that systematically injects logical contradictions into otherwise valid reasoning chains. Second, we introduce UnsolvableRL, a reinforcement learning paradigm that balances objective unsolvability detection with calibrated confidence under capability limits. Empirically, our approach achieves robust unsolvability detection (>85% detection rate) and boosts solvable reasoning accuracy from 43.4% to 69.4% on Qwen3-4B-Instruct. Crucially, we identify a data-training interaction: strict alignment constraints induce Capability Collapse without unsolvable data, but act as a regularizer for rigor when such data are included, thereby improving overall robustness. Our code and data are available at https://github.com/sfasfaffa/unsolvableQA .