DeepSeek has introduced DeepSeek-Prover-V2, a groundbreaking open-source large language model (LLM) meticulously crafted for the intricate domain of formal theorem proving within the Lean 4 framework. This novel model leverages a recursive theorem proving pipeline, harnessing the power of DeepSeek’s cutting-edge DeepSeek-V3 foundation model. Lean 4, the latest iteration of the Lean theorem prover, stands as an interactive proof assistant developed by Microsoft Research. This sophisticated functional programming language and interactive theorem proving system empowers mathematicians and computer scientists to construct formal proofs with unparalleled machine-checked verification.
The project signifies a monumental stride towards bridging the gap between formal and informal mathematical reasoning. By capitalizing on the inherent capabilities of general-purpose LLMs, it seeks to effectively address the highly structured domain of formal theorem proving. The DeepSeek research team posits that their innovative approach mirrors the cognitive processes employed by human mathematicians when constructing proofs, meticulously dissecting complex theorems into more manageable and comprehensible components.
Expanding the Evaluation Framework: Introducing ProverBench
In a significant move to enhance the rigor of their research, the DeepSeek team has significantly expanded their evaluation framework with the introduction of ProverBench, an all-new benchmark collection meticulously designed specifically for the comprehensive assessment of formal theorem proving capabilities. This comprehensive collection serves as a valuable resource for evaluating the performance of LLMs in the context of formal mathematics.
"Beyond the conventional benchmarks, we proudly introduce ProverBench, a meticulously curated collection of 325 formalized problems, to enrich our evaluation process. This collection includes 15 carefully selected problems sourced directly from the recent American Invitational Mathematics Examination (AIME) competitions, specifically from the years 24-25," the researchers elaborated.
The inclusion of AIME problems in the ProverBench dataset is particularly noteworthy, as it introduces a set of challenging and well-established mathematical problems that are widely recognized within the mathematical community. This provides a standardized and rigorous basis for evaluating the performance of DeepSeek-Prover-V2 and comparing it to other approaches. The AIME problems are crafted to test mathematical problem-solving skills, requiring deep understanding of mathematical concepts and the ability to apply them creatively. They often involve number theory, algebra, combinatorics, and geometry, providing a comprehensive assessment of a system’s mathematical aptitude.
Promising Initial Results: Tackling AIME Problems
The initial results stemming from rigorous testing on these challenging AIME problems have revealed exceptionally promising performance from their meticulously designed specialized theorem proving model. The DeepSeek team proudly reports that DeepSeek-Prover-V2 demonstrated its prowess by successfully solving an impressive 6 out of the 15 AIME problems presented to it. In comparison, the general-purpose DeepSeek-V3 model, when employing majority votingtechniques, managed to successfully solve 8 problems.
These findings highlight the potential of both specialized and general-purpose LLMs in tackling complex mathematical problems. While the general-purpose model exhibited a slightly higher success rate in this particular benchmark, the specialized theorem proving model demonstrated its proficiency in formal mathematical reasoning. The results suggest that specialized models can excel at specific tasks, while general-purpose models offer broader capabilities. The difference in performance underscores the tradeoff between specialization and generalization in LLMs. Further research is needed to determine how to best combine the strengths of both types of models. The fact that both models could solve a significant number of AIME problems demonstrates the rapid progress that has been made in the field of AI-assisted mathematical problem-solving.
Mimicking Human Proof Construction: A Chain-of-Thought Approach
"Given the well-documented challenges that general-purpose models often encounter when attempting to produce complete Lean proofs, we strategically instructed DeepSeek-V3 to generate only a high-level proof sketch, deliberately omitting the intricate details. The resulting chain of thought culminates in a Lean theorem composed of a sequence of have statements, each meticulously concluded with a sorry placeholder, effectively indicating a subgoal that needs to be resolved. This innovative approach elegantly mirrors the human style of proof construction, in which a complex theorem is incrementally reduced to a sequence of more manageable lemmas," the DeepSeek team elaborated.
This innovative approach of generating high-level proof sketches aligns with how mathematicians often approach complex proofs. By focusing on the overall structure and key steps, the model can effectively guide the subsequent refinement and completion of the proof. It helps the model avoid getting bogged down in the details and allows it to focus on the big picture. The use of "sorry" placeholders is a clever way to indicate where additional reasoning is needed. It creates a roadmap for the model to follow as it works to complete the proof. It mirrors the way humans often attack complex problems, by breaking them down into smaller, more manageable subproblems. This step-by-step approach makes the problem easier to address.
A Methodical Strategy: Addressing Each Proof Component Individually
The system then meticulously employs a methodical and structured strategy to address each individual component of the proof. This systematic approach ensures that every aspect of the proof is carefully considered and addressed in a logical and coherent manner. The system creates a highly structured approach to theorem proving, building upon previously established results to ensure a solid foundation for each subsequent step. In mathematics and particularly in formal theorem proving, relying on previously confirmed truths is vital for ensuring the validity of the final proof.
"Leveraging the subgoals generated by DeepSeek-V3, we adopt a recursive solving strategy to systematically resolve each intermediate proof step. We extract subgoal expressions from have statements to substitute them for the original goals in the given problems and then incorporate the preceding subgoals as premises. This construction enables subsequent subgoals to be resolved using the intermediate results of earlier steps, thereby promoting a more localized dependency structure and facilitating the development of simpler lemmas," the researchers detailed.
The recursive solving strategy is a key aspect of the system’s ability to handle complex proofs. By breaking down the problem into smaller, more manageable subgoals, the system can effectively apply its reasoning capabilities to each individual component. It is especially useful when dealing with complex mathematical operations that might seem difficult to comprehend to a system that is not designed to handle math. This way each mini-problem can be solved, and all solutions can be stacked into one to derive the final answer. This process is akin to using previously solved lemmas, which allows the system to progressively build upon the results it has already established.
Optimizing Computational Resources: A Specialized 7B Parameter Model
To effectively optimize computational resources and ensure efficient processing, the system strategically employs a smaller, highly specialized 7B parameter model for processing the decomposed lemmas. This approach is crucial for effectively managing the computational demands associated with extensive proof searches, ensuring that the system can operate efficiently without being overwhelmed by the complexity of the search space. The approach ultimately culminates in an automatically derived complete proof when all decomposed steps are successfully resolved. Large models are able to grasp the fundamentals but are computationally very intensive. The 7B parameter model offers enough power to create a valid proof while not draining as much power.
"The algorithmic framework operates in two distinct stages, leveraging two complementary models: DeepSeek-V3 for lemma decomposition and a 7B prover model to complete the corresponding formal proof details," the researchers described.
This two-stage approach allows the system to leverage the strengths of both a large general-purpose model and a smaller specialized model. The large model is used to generate high-level proof sketches, while the smaller model is used to fill in the details and complete the formal proof. The separation of concerns allows the DeepSeek team to optimize the performance and efficiency of the overall system. It shows that general-purpose models can prepare the work for smaller and fast specialized models. This can be applied in other fields to save costs and solve the same level of difficult questions.
Synthesizing Formal Reasoning Data: A Natural Pathway
This meticulously designed architecture effectively establishes a natural and intuitive pathway for synthesizing formal reasoning data, seamlessly merging high-level mathematical reasoning with the stringent and rigorous requirements of formal verification. This integration is essential for ensuring the reliability and trustworthiness of the system’s results. Formal verification ensures that the proofs are not incorrect, and prevents the LLM from hallucinating proofs.
"We curate a subset of challenging problems that remain unsolved by the 7B prover model in an end-to-end manner, but for which all decomposed sub-goals have been successfully resolved. By composing the proofs of all sub-goals, we construct a complete-formal proof for the original problem," the researchers explained.
This approach allows the system to learn from its mistakes and improve its ability to solve complex problems. By identifying the specific subgoals that are causing difficulties, the system can focus its efforts on improving its performance in those areas. The capacity to learn from its own mistakes allows the tool to refine itself further with each question. This enables it to become more accurate in producing formal reasoning data. By using the already approved proofs the system is able to increase the chance of the answer being correct.
Concerns and Challenges: Implementation Details Under Scrutiny
Despite the undeniable technical achievements demonstrated by DeepSeek-Prover-V2, some experts in the field have raised pertinent concerns regarding certain implementation details. Elliot Glazer, a highly respected Lead mathematician at Epoch AI, has pointed out potential issues that warrant further investigation. It is always important that when releasing a model, other experts check to see if the model has vulnerabilities or inaccuracies.
Some concerns about the DeepSeek-Prover-V2 paper. Potentially misformalized examples, and discussion on the Lean zulip suggests the PutnamBench proofs are nonsense and use an implicit sorry (possibly hidden in the apply? tactic) not reported in their read-eval-print-loop.
These concerns vividly highlight the ongoing challenges inherent in the formal verification space, where even the most minute and seemingly insignificant implementation details can wield a disproportionately large impact on the overall validity and reliability of the results. The formal verification process demands unwavering attention to detail and meticulous adherence to established standards. A single misplaced symbol can invalidate an entire proof.
The potential for misformalized examples and the possibility of hidden "sorry" tactics in the PutnamBench proofs raise important questions about the rigor and completeness of the verification process. These concerns underscore the need for continued scrutiny and independent verification of the results. Before adopting the DeepSeek-Prover-V2, it would be best to make sure all the hidden "sorry" tactics have been exposed and corrected. This may prove that some of the prior results may be inaccurate.
Availability and Resources: Democratizing Access to Formal Theorem Proving
DeepSeek has made its Prover-V2 available in two distinct model sizes, catering to a diverse range of computational resources and research objectives. The first version is a 7B parameter model built upon their previous Prover-V1.5-Base, featuring an extended context length of up to 32K tokens. The second version is a significantly larger 671B parameter model trained on DeepSeek-V3-Base. Both models are now readily accessible on HuggingFace, a leading platform for sharing and collaborating on machine learning models. This helps foster the tool to be updated and be improved upon, leading to potentially more accurate proofs and solutions. By sharing these tools with the community it will help democratize the world of formal theorem proving and will offer other researchers to build upon it.
In addition to the models themselves, DeepSeek has also made the full ProverBench dataset, containing 325 meticulously formalized problems for evaluation purposes, available on HuggingFace. This comprehensive dataset provides researchers and developers with a valuable resource for evaluating the performance of their models and comparing them to DeepSeek-Prover-V2. The dataset becomes a resource for other developers and researchers to fine tune models that can also have formal verification. This helps with making models that are more accurate and may be able to handle the AIME.
By making these resources freely available, DeepSeek is democratizing access to formal theorem proving technology and fostering collaboration within the research community. This open-source approach is likely to accelerate progress in the field and lead to new breakthroughs in automated reasoning and verification. This also builds more trust in the models provided due to the fact that the models are open source.
This release empowers researchers and developers with the resources needed to delve into the capabilities and limitations of this technology. By providing open access to the models and the ProverBench dataset, DeepSeek encourages further exploration and collaborative efforts to address the concerns raised by experts in the field. This collaborative approach holds the key to unraveling the complexities of formal theorem proving and solidifying the reliability of these groundbreaking advancements. This collaborative approach is an important thing to have. This open-source collaboration can help to make sure the code is accurate and it creates accountability that normally isn’t there with closed systems.