Picture yourself in a grocery store, confronted with the common challenge of choosing the checkout line. Your goal is to minimize waiting time. You quickly assess the queues and, based on intuition, you estimate the line with the least number of people. This process exemplifies a heuristic—an intuitive, mental shortcut enabling swift decisions and problem-solving. Heuristics prioritize speed and efficiency, but can sometimes yield errors, such as selecting a short line with a time-consuming shopper or cashier dispute.

On the other hand, imagine long division—a methodical, provably correct algorithm with the goal of accurately dividing numbers. By following a systematic, step-by-step procedure, you can achieve the correct outcome, ensuring that the quotient and remainder are accurate. Provably correct solutions emphasize accuracy and reliability, but may demand considerable time and effort in real-world applications.

Intuitively, we recognize these two concepts in our everyday lives. We understand that, while using heuristics like estimating the shortest line, there’s a chance we may end up in a slower line. However, we also expect that by following the methodical steps of long division we learned in school, we will always obtain the correct result. In summary, heuristics are intuitive shortcuts that prioritize speed and efficiency but can lead to errors, while provably correct algorithms are methodical approaches that guarantee accuracy and reliability but can be time-consuming.

In the rapidly evolving landscape of artificial intelligence (AI), these distinctions between heuristics and provably correct algorithms become even more crucial. AI systems, particularly large language models (LLMs), essentially function as advanced heuristics, generating solutions with impressive speed but lacking a guarantee of correctness. As AI applications proliferate, it becomes increasingly important to verify and establish trust in the solutions that these systems provide.

This is where the synergy between AI and proof assistants comes into play. Proof assistants are powerful tools that have been used for decades within academia to provide a rigorous, step-by-step approach for verifying the correctness of mathematical proofs and other formal systems.

A proof assistant is a sophisticated software tool that aids in the development and verification of formal proofs, which are rigorous mathematical arguments that establish the truth of a given statement. At the heart of a proof assistant is a type checker, a component that may not possess the creativity to generate novel ideas but can meticulously and reliably verify the logical soundness of even the most complex arguments. This verification ability makes the type checker a vital tool in establishing the correctness and dependability of AI-generated solutions.

By combining the advanced heuristics of AI with the dependability of proof assistants, we can harness the strengths of both approaches.

Envision an AI system that generates not only a solution but also a proof of its correctness. This proof can then be scrutinized by a trustworthy type checker—a component of the proof assistant toolkit—to ensure the solution’s validity. By utilizing AI’s advanced heuristics to swiftly generate potential solutions and employing proof assistants to rigorously verify their correctness, we can create a heightened level of confidence in AI systems’ output.

This remarkable fusion of AI and proof assistants has the potential to unlock a wide array of applications, ranging from intricate engineering tasks to addressing the AI alignment problem, which seeks to ensure that AI systems align with human values and intentions.