Large language models have made significant progress in mathematical reasoning, which serves as an important testbed for AI and could impact scientific research if further advanced. By scaling reasoning with reinforcement learning that rewards correct final answers, LLMs have improved from poor performance to saturating quantitative reasoning competitions like AIME and HMMT in one year. However, this approach faces fundamental limitations. Pursuing higher final answer accuracy doesn't address a key issue: correct answers don't guarantee correct reasoning. Moreover, many mathematical tasks like theorem proving require rigorous step-by-step derivation rather than numerical answers, making final answer rewards inapplicable. To push the limits of deep reasoning, we believe it is necessary to verify the comprehensiveness and rigor of mathematical reasoning. Self-verification is particularly important for scaling test-time compute, especially for open problems without known solutions. Towards self-verifiable mathematical reasoning, we investigate how to train an accurate and faithful LLM-based verifier for theorem proving. We then train a proof generator using the verifier as the reward model, and incentivize the generator to identify and resolve as many issues as possible in their own proofs before finalizing them. To maintain the generation-verification gap as the generator becomes stronger, we propose to scale verification compute to automatically label new hard-to-verify proofs, creating training data to further improve the verifier. Our resulting model, DeepSeekMath-V2, demonstrates strong theorem-proving capabilities, achieving gold-level scores on IMO 2025 and CMO 2024 and a near-perfect 118/120 on Putnam 2024 with scaled test-time compute.

Developing the logic necessary to solve mathematical problems or write mathematical proofs is one of the more difficult objectives for large language models (LLMS). Currently, the most popular methods in literature consists of fine-tuning the model on written mathematical content such as academic publications and textbooks, so that the model can learn to emulate the style of mathematical writing. In this project, we explore the effectiveness of using retrieval augmented generation (RAG) to address gaps in the mathematical reasoning of LLMs. We develop LemmaHead, a RAG knowledge base that supplements queries to the model with relevant mathematical context, with particular focus on context from published textbooks. To measure our model's performance in mathematical reasoning, our testing paradigm focuses on the task of automated theorem proving via generating proofs to a given mathematical claim in the Lean formal language.

We consider the scenario of ontology-based query answering. It is generally accepted that true scalability in this setting can only be achieved via query rewriting, which in turn allows for the exploitation of standard RDBMSs. In this work, we close two open fundamental questions related to query rewriting. We establish that linear existential rules are polynomially combined rewritable, while full linear rules are polynomially (purely) rewritable; in both cases, the target query language consists of first-order or non-recursive Datalog queries. An immediate consequence of our results is that DLR-Lite_R, the extension of DL-Lite_R with n-ary roles, is polynomially combined rewritable.