Can AI Automate the Art of Formal Proof?

Author: Denis Avetisyan

A new study explores whether automated optimization can create effective agentic systems for verifying software correctness.

Researchers evaluate automatic optimization methods for agentic systems using the Rocq framework and find few-shot learning offers the most consistent performance, though it still lags behind hand-engineered approaches.

Achieving robust formal verification often demands painstaking, manual tuning of agentic systems. The work presented in ‘RocqSmith: Can Automatic Optimization Forge Better Proof Agents?’ investigates whether automated optimization techniques can effectively enhance the performance of such agents within the challenging domain of automated theorem proving using the Rocq system. Results demonstrate that while several optimization methods yield measurable gains, simple few-shot bootstrapping proves the most consistently effective approach-though still falling short of expertly engineered proof agents. Could further advancements in automated optimization bridge this gap and unlock truly autonomous, high-performance formal verification?

The Illusion of Automated Proof

Despite the promise of absolute certainty, the field of formal methods and automated theorem proving currently faces a significant practical hurdle: the persistent need for substantial human input. While these techniques aim to rigorously verify the correctness of systems – from software code to mathematical proofs – the process often requires experts to guide the automated provers, translating high-level goals into a format the machine can understand and strategically directing the search for a solution. This reliance on manual effort limits scalability; complex systems demand extensive human time and expertise, effectively creating a bottleneck in the verification pipeline. Consequently, although capable of establishing unassailable correctness, these powerful tools remain challenging to deploy broadly, hindering their potential to revolutionize software development and mathematical discovery.

The increasing sophistication of modern software and mathematical formulations presents a significant hurdle for formal verification techniques. Traditional automated reasoning systems, while robust on simpler problems, often falter when confronted with the sheer scale and intricacy of contemporary statements – think of verifying the correctness of a modern operating system kernel or a complex cryptographic protocol. This limitation arises because the computational resources required to explore the vast search spaces inherent in these problems grow exponentially with their size. Consequently, a bottleneck emerges, hindering the widespread adoption of formal methods; developers are often forced to rely on less rigorous testing procedures due to the impracticality of exhaustive, automated verification. Addressing this challenge necessitates the development of novel algorithms and heuristics capable of navigating these complex logical landscapes more efficiently, potentially through techniques like machine learning or more advanced forms of symbolic reasoning to prune irrelevant search paths and focus on critical areas of concern within the Σ-calculus.

Mimicking the Mathematician: The Rise of Agentic Systems

Agentic systems represent a departure from traditional automated theorem proving (ATP) methods by emulating the problem-solving process of a human mathematician. Unlike ATP systems which typically execute a predefined search strategy, agentic systems function through iterative cycles of reasoning and action. This approach allows the system to dynamically adjust its strategy based on the current ‘proof state’ and the outcomes of applied tactics. The analogy to human programmers lies in the ability to decompose a complex proof into manageable subgoals, attempt solutions using various techniques (tactics), and refine the approach based on successes and failures – a process mirrored in the agent’s continuous loop of observation, reasoning, and action application. This iterative refinement is intended to overcome limitations of exhaustive search and static strategies commonly found in conventional theorem provers.

Agentic theorem proving systems function by maintaining a ‘Proof State’ which encapsulates the current progress and outstanding challenges in a given proof. This state is iteratively updated through the application of ‘Tactic’ operations – formally defined transformations that attempt to resolve one or more subgoals within the Proof State. Each tactic application modifies the Proof State, potentially simplifying existing subgoals, introducing new ones, or ultimately achieving a complete proof. The system continues this cycle of tactic application and Proof State refinement until a desired outcome – typically a verified theorem or a determination that the theorem cannot be proven – is reached. The effectiveness of this approach relies on the system’s ability to select and apply appropriate tactics to effectively reduce the complexity of the Proof State over time.

The ReAct paradigm, combining reasoning and action, directly addresses the challenges of interactive theorem proving by enabling an agent to dynamically adjust its approach based on the current ‘Proof State’. This is achieved through iterative cycles of generating thoughts – outlining the rationale for a tactic selection – and actions – applying the chosen tactic to modify the proof state. Crucially, the agent’s observations of the resulting proof state inform subsequent reasoning, allowing it to correct errors, explore alternative strategies, and ultimately converge on a solution without requiring explicit external guidance. This contrasts with traditional approaches where a fixed strategy is applied, and is especially beneficial in complex proofs where the optimal path is not immediately apparent.

Automated Tweaking: Optimizing Agents Without a Human in the Loop

The creation of effective proof agents traditionally requires significant manual effort in defining prompting strategies, tool usage, and overall agent behavior. This manual design process is both time-consuming and necessitates substantial expertise in both theorem proving and prompt engineering. Automatic Agent Optimization addresses this limitation by employing algorithms to search for and refine agent configurations without direct human intervention. This automation aims to reduce development time, improve agent performance beyond what manual tuning can achieve, and facilitate broader accessibility to automated theorem proving techniques by lowering the barrier to entry for creating high-performing agents.

Automated agent optimization for theorem proving utilizes a variety of techniques to enhance performance. MIPROv2 employs mixed-integer programming to refine agent behavior, while BootstrapFewShot leverages few-shot learning with large language models to improve initial prompting and response generation. GEPA (Genetic Evolution for Prompting Agents) utilizes evolutionary algorithms, iteratively modifying and evaluating prompts to identify those yielding the highest success rates in theorem proving. These approaches differ significantly in their underlying mechanisms – ranging from deterministic optimization through MIPROv2 to stochastic search with GEPA and data-driven refinement with BootstrapFewShot – but all aim to automate the process of creating effective proof agents without extensive manual intervention.

DSPy and Koog are software frameworks designed to facilitate the development and comparative analysis of automated agent optimization techniques for theorem proving. DSPy focuses on compositional program synthesis, enabling modular agent construction and evaluation through a declarative interface. Koog, conversely, emphasizes evolutionary optimization, employing genetic algorithms to refine agent strategies. Both frameworks provide standardized evaluation environments, including benchmark datasets like Rocq, and metrics for assessing agent performance – such as success rate and proof length. This infrastructure allows researchers to systematically compare different optimization methods – including instruction tuning, bootstrapping, and evolutionary algorithms – and track progress in automating the design of effective proof agents.

Integration of large language models, specifically GPT-4.1, has demonstrably improved the performance of automated theorem proving agents. Recent evaluations using the Rocq dataset indicate that employing a simple few-shot bootstrapping technique significantly increases success rates. Baseline performance, established using the DSPy and Koog frameworks, was 19% and 20% respectively. Following the implementation of few-shot bootstrapping, DSPy achieved a 40% success rate, while Koog improved to 33%. These results highlight the potential of leveraging large language models to enhance automated agent capabilities in formal verification tasks.

The Illusion of Intelligence: Context and the Limits of Automation

Successfully navigating intricate mathematical proofs demands more than just immediate calculation; an agent’s capacity to effectively retrieve and apply lessons from previously solved problems is paramount. This ability to learn from experience allows the agent to recognize patterns, avoid repeating unsuccessful strategies, and build upon existing knowledge when confronted with novel challenges. Without robust memory and recall mechanisms, even a highly capable agent may struggle with complexity, repeatedly encountering obstacles that have already been overcome. Consequently, significant research focuses on equipping these systems with the tools to not simply process information, but to truly learn from it, creating a cumulative advantage as they tackle increasingly difficult proofs and ultimately pushing the boundaries of automated reasoning.

Recent advancements in artificial intelligence demonstrate that an agent’s capacity to effectively generalize hinges on a robust foundation of contextual knowledge. Systems such as ReasoningBank and ACE prioritize optimizing this knowledge base, not simply by increasing its size, but by refining how information is stored and retrieved. These methods move beyond rote memorization, instead focusing on structuring experiences in a way that allows the agent to identify relevant patterns and apply previously successful strategies to novel situations. This structured approach enables the agent to draw analogies, adapt to unforeseen challenges, and ultimately, perform more effectively on complex tasks that demand flexible problem-solving abilities – a significant leap toward true artificial general intelligence.

The Automated Deductive Agent System (ADAS) represents a significant leap towards fully automating the process of theorem proving by dynamically generating executable agent code. Rather than relying on pre-programmed strategies, ADAS constructs its own problem-solving routines during execution, allowing for adaptation to novel proof challenges. This approach moves beyond simply applying existing knowledge to actively creating the tools needed to solve a given problem. By generating code that directly instructs the agent’s actions, ADAS facilitates a level of flexibility and efficiency previously unattainable, opening avenues for tackling increasingly complex mathematical problems with minimal human intervention and paving the way for self-improving, autonomous reasoning systems.

Recent advancements in automated reasoning have spurred the development of specialized agentic systems, notably RocqStar, designed to excel at specific proof tasks. These systems don’t merely apply general problem-solving techniques; instead, they integrate and refine contextual knowledge – leveraging past experiences and successful strategies – to navigate the unique challenges of formal verification. RocqStar, and similar platforms, build upon methods like ReasoningBank and ACE, concentrating computational resources on the most relevant information and optimizing the agent’s ability to generalize from previous proofs. This focused approach allows for significantly improved performance and efficiency when tackling complex mathematical problems, effectively tailoring the agent’s skillset to a particular domain and demonstrating the power of specialization in artificial intelligence.

The pursuit of automated proof agents, as outlined in this evaluation of RocqSmith, feels predictably optimistic. The study highlights the limitations of even sophisticated optimization techniques when faced with the brutal realities of formal verification. It’s a dance between elegant theory and the inevitable messiness of implementation. As Paul Erdős observed, “A mathematician knows all the answers… but sometimes doesn’t know where to find them.” This rings true; the paper demonstrates that while these agentic systems can find proofs, consistently outperforming hand-engineered approaches remains elusive. The best results, surprisingly, came from simple few-shot learning, a testament to the fact that even basic approaches can offer resilience when faced with production-level complexity. Every abstraction, even a proof agent, dies in production, though perhaps with a little more structured panic than usual.

What’s Next?

The pursuit of automated proof assistants, as evidenced by this work, inevitably bumps against the inconvenient truth of software engineering: anything self-healing just hasn’t broken yet. RocqSmith demonstrates that even optimized agentic systems, leveraging the current enthusiasm for large language models and ReAct, remain stubbornly inferior to solutions painstakingly crafted by hand. The consistently ‘best’ approach-simple few-shot learning-is less a triumph of automation and more a testament to the limits of ambition. It works, predictably, because it mimics what a human expert already does, albeit with diminished capacity.

The logical next step, of course, is more automation. But a more honest appraisal would acknowledge that documentation is collective self-delusion. The real challenge isn’t scaling proof generation, it’s managing the inevitable cascade of edge cases and unforeseen interactions. A system that flawlessly handles 99% of verification tasks is still a liability; the remaining 1% will define its failure. If a bug is reproducible, it suggests a stable system, not a clever one.

Future work will undoubtedly explore increasingly sophisticated optimization algorithms and larger models. However, the underlying problem-mapping the infinite complexity of formal systems onto finite computational resources-isn’t solvable with cleverness. It’s a matter of accepting inherent limitations and prioritizing robustness over elegance. The eventual outcome will not be a perfect proof assistant, but a predictable pattern of escalating tech debt.

Original article: https://arxiv.org/pdf/2602.05762.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

The Illusion of Automated Proof

Mimicking the Mathematician: The Rise of Agentic Systems

Automated Tweaking: Optimizing Agents Without a Human in the Loop

The Illusion of Intelligence: Context and the Limits of Automation

What’s Next?

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