Leveraging mathematical Large Language Models (LLMs) for proof generation is a fundamental topic in LLMs research. We argue that the ability of current LLMs to prove statements largely depends on whether they have encountered the relevant proof process during training. This reliance limits their deeper understanding of mathematical theorems and related concepts. Inspired by the pedagogical method of "proof by counterexamples" commonly used in human mathematics education, our work aims to enhance LLMs' ability to conduct mathematical reasoning and proof through counterexamples. Specifically, we manually create a high-quality, university-level mathematical benchmark, CounterMATH, which requires LLMs to prove mathematical statements by providing counterexamples, thereby assessing their grasp of mathematical concepts. Additionally, we develop a data engineering framework to automatically obtain training data for further model improvement. Extensive experiments and detailed analyses demonstrate that CounterMATH is challenging, indicating that LLMs, such as OpenAI o1, have insufficient counterexample-driven proof capabilities. Moreover, our exploration into model training reveals that strengthening LLMs' counterexample-driven conceptual reasoning abilities is crucial for improving their overall mathematical capabilities. We believe that our work offers new perspectives on the community of mathematical LLMs.

We introduce thoughts of words (ToW), a novel training-time data-augmentation method for next-word prediction. ToW views next-word prediction as a core reasoning task and injects fine-grained thoughts explaining what the next word should be and how it is related to the previous contexts in pre-training texts. Our formulation addresses two fundamental drawbacks of existing next-word prediction learning schemes: they induce factual hallucination and are inefficient for models to learn the implicit reasoning processes in raw texts. While there are many ways to acquire such thoughts of words, we explore the first step of acquiring ToW annotations through distilling from larger models. After continual pre-training with only 70K ToW annotations, we effectively improve models' reasoning performances by 7% to 9% on average and reduce model hallucination by up to 10%. At the same time, ToW is entirely agnostic to tasks and applications, introducing no additional biases on labels or semantics.

How to better evaluate the capabilities of Large Language Models (LLMs) is the focal point and hot topic in current LLMs research. Previous work has noted that due to the extremely high cost of iterative updates of LLMs, they are often unable to answer the latest dynamic questions well. To promote the improvement of Chinese LLMs' ability to answer dynamic questions, in this paper, we introduce CDQA, a Chinese Dynamic QA benchmark containing question-answer pairs related to the latest news on the Chinese Internet. We obtain high-quality data through a pipeline that combines humans and models, and carefully classify the samples according to the frequency of answer changes to facilitate a more fine-grained observation of LLMs' capabilities. We have also evaluated and analyzed mainstream and advanced Chinese LLMs on CDQA. Extensive experiments and valuable insights suggest that our proposed CDQA is challenging and worthy of more further study. We believe that the benchmark we provide will become one of the key data resources for improving LLMs' Chinese question-answering ability in the future.

This study focuses on the evaluation of the Open Question Answering (Open-QA) task, which can directly estimate the factuality of large language models (LLMs). Current automatic evaluation methods have shown limitations, indicating that human evaluation still remains the most reliable approach. We introduce a new task, Evaluating QA Evaluation (QA-Eval) and the corresponding dataset EVOUNA, designed to assess the accuracy of AI-generated answers in relation to standard answers within Open-QA. Our evaluation of these methods utilizes human-annotated results to measure their performance. Specifically, the work investigates methods that show high correlation with human evaluations, deeming them more reliable. We also discuss the pitfalls of current methods and methods to improve LLM-based evaluators. We believe this new QA-Eval task and corresponding dataset EVOUNA will facilitate the development of more effective automatic evaluation tools and prove valuable for future research in this area. All resources are available at \url{https://github.com/wangcunxiang/QA-Eval} and it is under the Apache-2.0 License.

The Open-Domain Question Answering (ODQA) task involves retrieving and subsequently generating answers from fine-grained relevant passages within a database. Current systems leverage Pretrained Language Models (PLMs) to model the relationship between questions and passages. However, the diversity in surface form expressions can hinder the model's ability to capture accurate correlations, especially within complex contexts. Therefore, we utilize Abstract Meaning Representation (AMR) graphs to assist the model in understanding complex semantic information. We introduce a method known as Graph-as-Token (GST) to incorporate AMRs into PLMs. Results from Natural Questions (NQ) and TriviaQA (TQ) demonstrate that our GST method can significantly improve performance, resulting in up to 2.44/3.17 Exact Match score improvements on NQ/TQ respectively. Furthermore, our method enhances robustness and outperforms alternative Graph Neural Network (GNN) methods for integrating AMRs. To the best of our knowledge, we are the first to employ semantic graphs in ODQA.