We present Llemma, a large language model for mathematics. We continue pretraining Code Llama on the Proof-Pile-2, a mixture of scientific papers, web data containing mathematics, and mathematical code, yielding Llemma. On the MATH benchmark Llemma outperforms all known openly released models, as well as the unreleased Minerva model suite on an equi-parameter basis. Moreover, Llemma is capable of tool use and formal theorem proving without any finetuning. We openly release all artifacts, including 7 billion and 34 billion parameter models, the Proof-Pile-2, and code to replicate our experiments.

This paper presents a novel approach to premise selection, a crucial reasoning task in automated theorem proving. Traditionally, symbolic methods that rely on extensive domain knowledge and engineering effort are applied to this task. In contrast, this work demonstrates that contrastive training with the transformer architecture can achieve higher-quality retrieval of relevant premises, without the knowledge or feature engineering overhead. Our method, Magnushammer, outperforms the most advanced and widely used automation tool in interactive theorem proving called Sledgehammer. On the PISA and miniF2f benchmarks Magnushammer achieves $59.5\%$ (against $38.3\%$) and $34.0\%$ (against $20.9\%$) success rates, respectively. By combining Magnushammer with a language-model-based automated theorem prover, we further improve the state-of-the-art proof success rate from $57.0\%$ to $71.0\%$ on the PISA benchmark using $4$x fewer parameters. Moreover, we develop and open source a novel dataset for premise selection, containing textual representations of (proof state, relevant premise) pairs. To the best of our knowledge, this is the largest available premise selection dataset, and the first dataset of this kind for the Isabelle proof assistant.

Autoformalization is the task of translating natural language materials into machine-verifiable formalisations. Progress in autoformalization research is hindered by the lack of a sizeable dataset consisting of informal-formal pairs expressing the same essence. Existing methods tend to circumvent this challenge by manually curating small corpora or using few-shot learning with large language models. But these methods suffer from data scarcity and formal language acquisition difficulty. In this work, we create mma, a large, flexible, multi-language, and multi-domain dataset of informal-formal pairs, by using a language model to translate in the reverse direction, that is, from formal mathematical statements into corresponding informal ones. Experiments show that language models fine-tuned on mma can produce up to $29-31$\% of statements acceptable with minimal corrections on the miniF2F and ProofNet benchmarks, up from $0$\% with the base model. We demonstrate that fine-tuning on multi-language formal data results in more capable autoformalization models even on single-language tasks.

Text embeddings are essential for tasks such as document retrieval, clustering, and semantic similarity assessment. In this paper, we study how to contrastively train text embedding models in a compute-optimal fashion, given a suite of pretrained decoder-only language models. Our innovation is an algorithm that produces optimal configurations of model sizes, data quantities, and fine-tuning methods for text-embedding models at different computational budget levels. The resulting recipe, which we obtain through extensive experiments, can be used by practitioners to make informed design choices for their embedding models. Specifically, our findings suggest that full fine-tuning and Low-Rank Adaptation fine-tuning produce optimal models at lower and higher computational budgets respectively.

Ontologies are useful for automatic machine processing of domain knowledge as they represent it in a structured format. Yet, constructing ontologies requires substantial manual effort. To automate part of this process, large language models (LLMs) have been applied to solve various subtasks of ontology learning. However, this partial ontology learning does not capture the interactions between subtasks. We address this gap by introducing OLLM, a general and scalable method for building the taxonomic backbone of an ontology from scratch. Rather than focusing on subtasks, like individual relations between entities, we model entire subcomponents of the target ontology by finetuning an LLM with a custom regulariser that reduces overfitting on high-frequency concepts. We introduce a novel suite of metrics for evaluating the quality of the generated ontology by measuring its semantic and structural similarity to the ground truth. In contrast to standard metrics, our metrics use deep learning techniques to define more robust distance measures between graphs. Both our quantitative and qualitative results on Wikipedia show that OLLM outperforms subtask composition methods, producing more semantically accurate ontologies while maintaining structural integrity. We further demonstrate that our model can be effectively adapted to new domains, like arXiv, needing only a small number of training examples. Our source code and datasets are available at https://github.com/andylolu2/ollm.

In theorem proving, the task of selecting useful premises from a large library to unlock the proof of a given conjecture is crucially important. This presents a challenge for all theorem provers, especially the ones based on language models, due to their relative inability to reason over huge volumes of premises in text form. This paper introduces Thor, a framework integrating language models and automated theorem provers to overcome this difficulty. In Thor, a class of methods called hammers that leverage the power of automated theorem provers are used for premise selection, while all other tasks are designated to language models. Thor increases a language model's success rate on the PISA dataset from $39\%$ to $57\%$, while solving $8.2\%$ of problems neither language models nor automated theorem provers are able to solve on their own. Furthermore, with a significantly smaller computational budget, Thor can achieve a success rate on the MiniF2F dataset that is on par with the best existing methods. Thor can be instantiated for the majority of popular interactive theorem provers via a straightforward protocol we provide.

Autoformalization is the process of automatically translating from natural language mathematics to formal specifications and proofs. A successful autoformalization system could advance the fields of formal verification, program synthesis, and artificial intelligence.While the long-term goal of autoformalization seemed elusive for a long time, we show large language models provide new prospects towards this goal. We make the surprising observation that LLMs can correctly translate a significant portion ($25.3\%$) of mathematical competition problems perfectly to formal specifications in Isabelle/HOL. We demonstrate the usefulness of this process by improving a previously introduced neural theorem prover via training on these autoformalized theorems. Our methodology results in a new state-of-the-art result on the MiniF2F theorem proving benchmark, improving the proof rate from~$29.6\%$ to~$35.2\%$.