# NaturalProver: Grounded Mathematical Proof Generation with Language Models

@article{Welleck2022NaturalProverGM, title={NaturalProver: Grounded Mathematical Proof Generation with Language Models}, author={Sean Welleck and Jiacheng Liu and Ximing Lu and Hannaneh Hajishirzi and Yejin Choi}, journal={ArXiv}, year={2022}, volume={abs/2205.12910} }

Theorem proving in natural mathematical language – the mixture of symbolic and natural language used by humans – plays a central role in mathematical advances and education, and tests aspects of reasoning that are core to intelligence. Yet it has remained underexplored with modern generative models. We study large-scale language models on two new generation tasks: suggesting the next step in a mathematical proof, and full proof generation. We develop N ATURAL P ROVER , a language model that…

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## References

SHOWING 1-10 OF 53 REFERENCES

### GLEU: Automatic Evaluation of Sentence-Level Fluency

- Computer ScienceACL
- 2007

An automatic evaluation metric to estimate fluency alone is developed, by examining the use of parser outputs as metrics, and it is shown that they correlate with human judgements of generated text fluency.

### NaturalProofs: Mathematical Theorem Proving in Natural Language

- Computer ScienceNeurIPS Datasets and Benchmarks
- 2021

NATHURALPROOFS is developed, a multi-domain corpus of mathematical statements and their proofs, written in natural mathematical language that unifies broad coverage, deep coverage, and low-resource mathematical sources, allowing for evaluating both in-distribution and zero-shot generalization.

### WebGPT: Browser-assisted question-answering with human feedback

- Computer ScienceArXiv
- 2021

GPT-3 is tuned to answer long-form questions using a text-based web-browsing environment, which allows the model to search and navigate the web, and the best model is obtained by using behavior cloning, and then performing rejection sampling against a reward model trained to predict human preferences.

### Retrieval Augmentation Reduces Hallucination in Conversation

- Computer ScienceEMNLP
- 2021

This work explores the use of neural-retrieval-in-the-loop architectures recently shown to be effective in open-domain QA for knowledge-grounded dialogue, a task that is arguably more challenging as it requires querying based on complex multi-turn dialogue context and generating conversationally coherent responses.

### Evaluating Large Language Models Trained on Code

- Computer ScienceArXiv
- 2021

It is found that repeated sampling from the GPT language model is a surprisingly effective strategy for producing working solutions to difﬁcult prompts, and the potential broader impacts of deploying powerful code generation technologies, covering safety, security, and economics are discussed.

### Memorizing Transformers

- Computer ScienceICLR
- 2022

It is demonstrated that an approximate kNN lookup into a non-differentiable memory of recent (key, value) pairs improves language modeling across various benchmarks and tasks, including generic webtext, math papers, books, code, as well as formal theorems (Isabelle).

### COLD Decoding: Energy-based Constrained Text Generation with Langevin Dynamics

- Computer ScienceArXiv
- 2022

This paper presents Energy-based Constrained Decoding with Langevin Dynamics (C OLD), a decoding framework which describes constrained generation as specifying constraints through an energy function, then performing differentiable reasoning over the constraints through gradient-based sampling.

### Formal Mathematics Statement Curriculum Learning

- Computer ScienceArXiv
- 2022

It is shown that at same compute budget, expert iteration, by which the authors mean proof search interleaved with learning, dramatically outperforms proof search only and is capable of finding and solving a curriculum of increasingly difficult problems, without the need for associated ground-truth proofs.

### Competition-level code generation with AlphaCode

- BiologyScience
- 2022

AlphaCode is introduced, a system for code generation that achieved an average ranking in the top 54.3% in simulated evaluations on recent programming competitions on the Codeforces platform, marking the first time an artificial intelligence system has performed competitively in programming competitions.

### Chain of Thought Prompting Elicits Reasoning in Large Language Models

- Computer ScienceArXiv
- 2022

Experiments on three large language models show that chain-of-thought prompting improves performance on a range of arithmetic, commonsense, and symbolic reasoning tasks.