FaithformBench: Benchmarking Faithfulness of Mathematical Chain-of-Thought Autoformalisation

Autoformalisation (AF) systems map natural language reasoning steps into formal statements in a proof assistant such as Lean. We consider how to assess the faithfulness of these systems. Existing approaches require expensive human-annotated ground truth, or rely on LLM judges or embedding models, which come with limited guarantees of accuracy. In addition, these methods typically only consider inputs that are known to be correct, and therefore do not assess whether the AF translates incorrect inputs faithfully. To address these limitations, we propose a new benchmark for AF faithfulness that is cheap to apply, sound under weak assumptions, and assesses both positive and negative examples. Our method is based on automatically generating perturbed reasoning steps that are designed to be invalid, and then measuring validity preservation on unperturbed steps and invalidity preservation on perturbed steps. We apply our method to eight AF systems across four mathematical datasets, and observe pervasive sycophancy: many AFs "silently correct" invalid inputs into provable statements. The most validity-preserving fine-tuned AFs are also the most sycophantic, suggesting a tension between validity and invalidity preservation in current AF systems.

KA2L: A Knowledge-Aware Active Learning Framework for LLMs

Fine-tuning large language models (LLMs) with high-quality knowledge has been shown to enhance their performance effectively. However, there is a paucity of research on the depth of domain-specific knowledge comprehension by LLMs and the application of targeted active learning to improve their expertise. To address this gap, we introduce the Knowledge-Aware Active Learning (KA2L) framework. This framework assesses LLMs' mastery of specific knowledge points to aid in constructing unanswerable or unknowable questions through latent space analysis. This active learning strategy enhances training efficiency by focusing on knowledge the model has yet to master, thereby minimizing redundancy in learning already acquired information. This study innovatively employs a knowledge distribution probing technique to examine the hidden states of specific Transformer layers and identify the distribution of known and unknown knowledge within the LLM. Additionally, a hidden-state decoding method is proposed to generate numerous unknown questions in natural language from the latent knowledge space. In our experiments, we selected nine open-source LLMs to validate the effectiveness of the proposed framework. Results indicate that KA2L not only significantly reduces 50% annotation and computation costs across two open-domain and one vertical-domain dataset but also achieves better performance, offering valuable insights into active learning strategies for LLMs. The code is available at https://anonymous.4open.science/r/KA2L-F15C.