The blog post details a formal verification of the standard long division algorithm using the Dafny programming language and its built-in Hoare logic capabilities. It walks through the challenges of representing and reasoning about the algorithm within this formal system, including defining loop invariants and handling edge cases like division by zero. The core difficulty lies in proving that the quotient and remainder produced by the algorithm are indeed correct according to the mathematical definition of division. The author meticulously constructs the necessary pre- and post-conditions, and elaborates on the specific insights and techniques required to guide the verifier to a successful proof. Ultimately, the post demonstrates the power of formal methods to rigorously verify even relatively simple, yet subtly complex, algorithms.
This blog post explores methods for proving false statements within formal systems like logic and mathematics. It focuses on proof by contradiction, where you assume the statement is true and then demonstrate that this assumption leads to a logical inconsistency, thereby proving the original statement false. The post uses the example of proving the irrationality of √2, illustrating how assuming its rationality (expressibility as a fraction) ultimately contradicts the fundamental theorem of arithmetic. It highlights the importance of clearly defining the terms and axioms of the system within which the proof operates.
Hacker News users discuss the potential misuse of zero-knowledge proofs (ZKPs), expressing concern that they could be used to convincingly lie or create fraudulent attestations. Some commenters highlight the importance of distinguishing between a ZKP verifying a computation versus verifying a real-world fact. They argue that while ZKPs can prove the correct execution of a program on given inputs, they cannot inherently prove the veracity of those inputs. Others discuss the "garbage in, garbage out" principle in this context, suggesting the need for robust, real-world verification methods alongside ZKPs to prevent their misuse. The trustworthiness of the prover remains crucial, and ZKPs alone cannot bridge the gap between computation and reality. A few comments also touch upon the complexity of understanding and implementing ZKPs correctly, potentially leading to vulnerabilities.
Summary of Comments ( 1 )
https://news.ycombinator.com/item?id=43185059
Hacker News users discussed the application of Hoare logic to verify long division, with several expressing appreciation for the clear explanation and visualization of the algorithm. Some commenters debated the practical benefits of formal verification for such a well-established algorithm, questioning the likelihood of uncovering unknown bugs. Others highlighted the educational value of the exercise, emphasizing the importance of understanding foundational algorithms. A few users delved into the specifics of the chosen proof method and its implications. One commenter suggested exploring alternative verification approaches, while another pointed out the potential for applying similar techniques to other arithmetic operations.
The Hacker News post "Long division verified via Hoare logic" discussing the article about verifying long division using Hoare logic sparked a small but focused conversation. Several commenters express appreciation for the clear explanation of both Hoare logic and its application to a concrete example like long division. One commenter highlights the pedagogical value of such demonstrations, suggesting it's a good way to teach people about formal verification methods. They appreciate the author's approach of starting with a simple example and gradually introducing complexities, making the concepts more accessible.
Another commenter delves into the practical implications, pondering whether such verified algorithms could find their way into real-world applications like optimizing compilers. They acknowledge the potential benefits of guaranteed correctness for critical operations like division, especially in performance-sensitive contexts.
A different user questions the choice of long division as the example, wondering if simpler algorithms might have served the illustrative purpose equally well while requiring less intricate proofs. This commenter suggests that the complexity of the long division algorithm might overshadow the core principles of Hoare logic being demonstrated.
Finally, a comment points out the historical context, mentioning Edsger W. Dijkstra's early work on formal program verification and how the article's approach aligns with Dijkstra's vision. This comment connects the present work to the foundational ideas in the field.
Overall, the comments demonstrate a positive reception of the article, praising its clarity and educational value. The discussion also touches upon practical considerations and historical context, enriching the understanding of the presented work.