Stories with Tag automated theorem proving

• A Mechanically Verified Garbage Collector for OCaml [pdf]

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  Posted: 2025-02-27 05:38:07

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  This paper details the formal verification of a garbage collector for a substantial subset of OCaml, including higher-order functions, algebraic data types, and mutable references. The collector, implemented and verified using the Coq proof assistant, employs a hybrid approach combining mark-and-sweep with Cheney's copying algorithm for improved performance. A key achievement is the proof of correctness showing that the garbage collector preserves the semantics of the original OCaml program, ensuring no unintended behavior alterations due to memory management. This verification increases confidence in the collector's reliability and serves as a significant step towards a fully verified implementation of OCaml.

  This paper details the design, implementation, and formal verification of a new garbage collector for the OCaml programming language, aiming to improve performance and provide strong guarantees about its correctness. The existing OCaml runtime utilizes the "incremental major collector" known as the ZGC, which, while effective, presents challenges for formal verification due to its complexity. This new garbage collector, named “MLgc,” employs a concurrent, multi-core-friendly mark-and-sweep algorithm with a focus on simplicity and verifiability.

  The authors highlight the significance of mechanical verification in ensuring the garbage collector's reliability, preventing potentially disastrous bugs that can be difficult to detect and diagnose in complex memory management systems. They employ the Coq proof assistant to formally verify key properties of the garbage collector, assuring that it preserves memory safety and satisfies essential invariants. This rigorous verification process provides a high level of confidence in the collector's correctness, going beyond traditional testing methodologies.

  The MLgc design is rooted in the "Beltway" algorithm, which partitions the heap into regions and employs a concurrent marking phase. A key innovation is the use of a "snapshot-at-the-beginning" (SATB) marking scheme, allowing the collector to accurately track live objects even as the mutator (the main program) continues execution. This concurrent operation minimizes pauses and improves overall performance, especially in multi-core environments. The sweeping phase reclaims unreachable memory regions, making them available for allocation.

  The paper emphasizes the challenges involved in verifying the concurrent nature of the collector. Reasoning about concurrent algorithms is inherently complex due to the potential for interleavings and race conditions. The authors leverage Coq's capabilities to formally model the concurrency and prove the absence of data races and other concurrency-related errors. The verification focuses on key properties, including ensuring that all live objects are preserved, no dangling pointers are created, and the heap remains consistent throughout the garbage collection process.

  The implementation of MLgc is integrated into the Multicore OCaml runtime system, allowing for practical evaluation. While performance results are not the primary focus of this paper, preliminary benchmarks suggest that MLgc achieves competitive throughput and latency compared to existing OCaml garbage collectors. Furthermore, the simplified design and formal verification contribute to increased maintainability and confidence in the long-term stability of the runtime.

  In conclusion, the paper presents a significant advancement in garbage collection for OCaml by introducing a formally verified, concurrent mark-and-sweep collector. The use of Coq provides strong guarantees about the collector's correctness, addressing the complexities of concurrent memory management. This work lays a foundation for more reliable and performant OCaml runtimes, paving the way for broader adoption of formal verification in language runtime systems.

  • OCaml
  • Garbage Collection
  • Formal Verification
  • Mechanized Proofs
  • compiler construction
  • Programming Languages
  • memory management
  • software verification
  • automated theorem proving
  • formal methods

  Summary of Comments ( 0 )
  https://news.ycombinator.com/item?id=43191667

  Verbose tl;dr

  Hacker News users discuss a mechanically verified garbage collector for OCaml, focusing on the practical implications of such verification. Several commenters express skepticism about the real-world performance impact, questioning whether the verification translates to noticeable improvements in speed or reliability for average users. Some highlight the trade-offs between provable correctness and potential performance limitations. Others note the significance of the work for critical systems where guaranteed safety and predictable behavior are paramount, even at the cost of some performance. The discussion also touches on the complexity of garbage collection and the challenges in achieving both efficiency and correctness. Some commenters raise concerns about the applicability of the specific approach to other languages or garbage collection algorithms.

  The Hacker News post discussing the mechanically verified garbage collector for OCaml has several comments exploring various aspects of the work.

  Several commenters express appreciation for the accomplishment of verifying a garbage collector, acknowledging the complexity and difficulty inherent in such an undertaking. They see this as a significant step towards more reliable and robust software, particularly in areas where memory safety is critical.

  One commenter delves into the specifics of the Coq proof assistant, used for the verification, mentioning the challenges associated with its steep learning curve and the significant time investment required to become proficient. They further highlight the value of Coq in ensuring the correctness of complex systems like garbage collectors.

  Discussion arises around the practicality and performance implications of verified software. Some commenters question whether the performance overhead introduced by the verification process is acceptable, while others express optimism about the potential for future optimizations and the long-term benefits of increased reliability.

  The topic of formal verification in general is also touched upon, with commenters discussing its growing importance in various fields and the potential for broader adoption in the future. The complexities and trade-offs of formal methods are acknowledged, but the overall sentiment appears to be one of encouragement for continued research and development in this area.

  One commenter specifically points out the significance of verifying a concurrent garbage collector, highlighting the added difficulty this presents due to the intricate interactions and potential race conditions inherent in concurrent systems.

  The use of OCaml as the target language is also mentioned, with some commenters expressing interest in the implications for the OCaml ecosystem and the potential for wider adoption of verified components within the language.

  Finally, a commenter questions the extent of the verification, asking whether the entire garbage collector or only specific properties were verified. This highlights the importance of clearly defining the scope and limitations of formal verification efforts. Another commenter mentions that the work is being done in the context of the "Verdi" framework, which is itself formally verified, adding another layer of confidence to the results.

• Large Language Models for Mathematicians

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  Posted: 2025-02-01 15:41:08

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  This paper explores the potential of Large Language Models (LLMs) as tools for mathematicians. It examines how LLMs can assist with tasks like generating conjectures, finding proofs, simplifying expressions, and translating between mathematical formalisms. While acknowledging current limitations such as occasional inaccuracies and a lack of deep mathematical understanding, the authors demonstrate LLMs' usefulness in exploring mathematical ideas, automating tedious tasks, and providing educational support. They argue that future development focusing on formal reasoning and symbolic computation could significantly enhance LLMs' capabilities, ultimately leading to a more symbiotic relationship between mathematicians and AI. The paper also discusses the ethical implications of using LLMs in mathematics, including concerns about plagiarism and the potential displacement of human mathematicians.

  The arXiv preprint titled "Large Language Models for Mathematicians" explores the potential utility and current limitations of Large Language Models (LLMs) within the domain of mathematical research and practice. The authors meticulously examine how these powerful language models, trained on vast datasets of text and code, can be leveraged by mathematicians across various aspects of their work. This includes, but is not limited to, tasks such as generating code for mathematical computations, translating mathematical ideas between formal and informal language, assisting in the exploration of mathematical concepts, and even aiding in the generation of conjectures or proofs.

  The paper provides a comprehensive overview of the current state-of-the-art in applying LLMs to mathematical problems. It delves into specific examples demonstrating how LLMs can be utilized for tasks like symbolic computation, numerical calculation, and the generation of mathematical text in different styles and levels of formality. Furthermore, the authors discuss the capabilities of LLMs to interact with specialized mathematical software systems, thereby extending their potential impact on mathematical workflows.

  A significant portion of the preprint is devoted to a nuanced discussion of the limitations and potential pitfalls associated with employing LLMs in mathematical contexts. The authors acknowledge the inherent limitations of these models, including their tendency to generate plausible-sounding yet incorrect mathematical statements, their occasional struggle with complex logical reasoning, and their dependence on the quality and scope of the training data. They emphasize the crucial role of human oversight and critical evaluation when using LLMs for mathematical work, cautioning against blind reliance on the output generated by these models.

  The preprint also explores the broader implications of LLMs for the future of mathematical research and education. It considers the potential for LLMs to democratize access to mathematical knowledge and tools, enabling wider participation in mathematical exploration and discovery. Furthermore, it examines the ethical considerations surrounding the use of LLMs in mathematics, highlighting the importance of responsible development and deployment of these powerful technologies.

  In conclusion, the paper "Large Language Models for Mathematicians" provides a detailed and balanced assessment of the current capabilities and limitations of LLMs in the realm of mathematics. It offers a valuable resource for mathematicians interested in exploring the potential of these models to enhance their work, while also emphasizing the importance of critical evaluation and responsible usage in this context. The authors suggest that LLMs, while not a replacement for human mathematical ingenuity, can serve as powerful tools that augment and amplify human capabilities in the pursuit of mathematical understanding.

  • Large Language Models
  • LLMs
  • mathematics
  • Mathematical Reasoning
  • Formal Mathematics
  • theorem proving
  • automated theorem proving
  • Symbolic Computation
  • Formal Verification
  • machine learning
  • artificial intelligence
  • AI
  • arXiv
  • Preprint
  • Mathematical Software
  • Computer-Assisted Proof

  Summary of Comments ( 4 )
  https://news.ycombinator.com/item?id=42899184

  Verbose tl;dr

  Hacker News users discussed the potential for LLMs to assist mathematicians, but also expressed skepticism. Some commenters highlighted LLMs' current weaknesses in formal logic and rigorous proof construction, suggesting they're more useful for brainstorming or generating initial ideas than for producing finalized proofs. Others pointed out the importance of human intuition and creativity in mathematics, which LLMs currently lack. The discussion also touched upon the potential for LLMs to democratize access to mathematical knowledge and the possibility of future advancements enabling more sophisticated mathematical reasoning by AI. There was some debate about the specific examples provided in the paper, with some users questioning their significance. Overall, the sentiment was cautiously optimistic, acknowledging the potential but emphasizing the limitations of current LLMs in the field of mathematics.

  The Hacker News post titled "Large Language Models for Mathematicians," linking to the arXiv preprint "Large Language Models for Mathematicians," has generated a moderate discussion with several insightful comments.

  Several commenters discuss the potential benefits and drawbacks of using LLMs for mathematical research. One commenter points out that LLMs could be useful for "grunt work" like writing boilerplate code or checking basic calculations, freeing up mathematicians to focus on more creative tasks. However, they also caution against relying too heavily on LLMs for proofs, as they may not be fully reliable. Another commenter echoes this sentiment, suggesting that LLMs might be more helpful for generating "ideas or conjectures" rather than rigorously proving them. They highlight the importance of human oversight and critical thinking when using these tools.

  One thread focuses on the specific examples provided in the paper. A commenter questions the validity of claiming an LLM "solved" a problem if it simply recognized a known solution from its training data. They argue that true mathematical understanding involves more than pattern matching. Another commenter challenges this, suggesting that even recognizing and applying known solutions to new problems is a valuable skill.

  The discussion also touches on the broader implications of LLMs for the field of mathematics. One commenter speculates about the future role of mathematicians, wondering if LLMs could eventually automate significant portions of mathematical research. They express both excitement and concern about this possibility. Another commenter raises the question of whether LLMs could discover genuinely new mathematical concepts or theorems, or if they are fundamentally limited to recombining existing knowledge. This leads to a brief discussion of the nature of mathematical creativity and the potential for LLMs to contribute to it.

  Finally, some commenters offer more practical perspectives. One suggests that LLMs could be particularly useful for educational purposes, helping students learn and practice mathematical concepts. Another commenter mentions the potential for LLMs to assist with literature reviews, enabling mathematicians to more easily access and synthesize relevant research.

  Overall, the comments reflect a nuanced perspective on the potential of LLMs in mathematics. While acknowledging the limitations and potential risks, many commenters express optimism about the ways in which these tools could enhance mathematical research and education in the future. The discussion highlights the ongoing debate about the role of AI in scientific discovery and the evolving relationship between humans and machines in the pursuit of knowledge.

• Anatomy of a Formal Proof

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  Posted: 2025-01-24 18:19:35

  Verbose tl;dr

  This article dissects the structure of a formal mathematical proof, illustrating it with a simple example about even and odd numbers. It emphasizes the distinction between informal proofs aimed at human understanding and formal proofs designed for automated verification. Formal proofs meticulously lay out every logical step, referencing specific axioms and inference rules within a chosen formal system. This detailed approach, while tedious for humans, enables computer-assisted verification and eliminates ambiguity, ensuring absolute rigor. The article highlights the importance of choosing appropriate axioms and the role of proof assistants in constructing and checking these complex formal structures, ultimately increasing confidence in mathematical results.

  The American Mathematical Society's Notices article, "Anatomy of a Formal Proof," delves into the intricate process of constructing a formal mathematical proof, contrasting it with the more informal proofs typically encountered in mathematical publications. It emphasizes that formal proofs, unlike their informal counterparts, are meticulously detailed and rigorously structured to be verifiable by automated proof assistants, also known as proof checkers. These software tools require a level of precision far exceeding human expectations in traditional mathematical discourse.

  The article elucidates this distinction by dissecting a specific example: the formalization of a theorem concerning Cauchy sequences in metric spaces. This theorem, relatively simple in its informal presentation, becomes considerably more complex when formalized. The formalization process necessitates explicitly stating and proving many foundational concepts that are often implicitly assumed in informal proofs. This includes defining fundamental notions like equality, ordered pairs, functions, Cartesian products, and the real numbers, all within the specific logical framework of the proof assistant. The article highlights the substantial effort required to build this foundational layer, illustrating the "iceberg phenomenon" where a concise informal proof rests upon a vast, submerged body of underlying definitions and lemmas.

  Furthermore, the article explores the challenges of translating informal mathematical language, rich with nuances and implicit understandings, into the rigid and unambiguous syntax demanded by formal proof systems. This translation process requires a meticulous deconstruction of the informal argument, meticulously filling in all the implicit steps and justifications. The article underscores that this often reveals hidden complexities and ambiguities in the informal proof, forcing mathematicians to confront subtle assumptions they might have unconsciously made.

  The authors describe the iterative nature of formal proof development. The process typically involves writing a formal proof sketch, attempting to verify it with the proof assistant, addressing the resulting errors and gaps, and repeating this cycle until the entire proof is formally verified. This iterative refinement, aided by the precise feedback from the proof assistant, contributes to an exceptionally high level of certainty in the correctness of the final formalized proof.

  The article concludes by reflecting on the broader implications of formalization for mathematical practice. While acknowledging the significant investment of time and effort required, it highlights the benefits of increased confidence in the validity of complex mathematical arguments, the potential for discovering new mathematical insights through the formalization process, and the role formalization plays in bridging the gap between human mathematical reasoning and computational verification. The authors suggest that while full formalization of all mathematical results is likely impractical, strategically formalizing key theorems and foundational concepts can significantly enhance the rigor and reliability of mathematics as a whole.

  • formal proof
  • mathematics
  • proof theory
  • Logic
  • formal systems
  • deductive reasoning
  • mathematical logic
  • theorem proving
  • automated theorem proving
  • Formal Verification
  • computer-assisted proofs
  • proof assistants
  • Notices of the AMS

  Summary of Comments ( 0 )
  https://news.ycombinator.com/item?id=42815755

  Verbose tl;dr

  HN commenters discuss the accessibility of formal proof systems, particularly referencing Lean. Some express excitement about the potential of formal proofs to revolutionize mathematics, while others are more skeptical, citing the steep learning curve and questioning the practical benefits for most mathematicians. Several commenters debate the role of intuition versus rigor in mathematical practice, with some arguing that formalization can enhance understanding and others suggesting it might stifle creativity. The feasibility of formalizing existing mathematical knowledge is also discussed, with varying opinions on the timescale and resources required for such a project. Some users highlight the potential of AI in assisting with formalization efforts, while others remain cautious about its current capabilities. The overall tone is one of cautious optimism, acknowledging the challenges but also recognizing the potential transformative power of formal proof systems.

  The Hacker News post "Anatomy of a Formal Proof" (linking to an American Mathematical Society article detailing a formal proof of the Central Limit Theorem) generated a moderate discussion with several interesting points.

  A few commenters discussed the practical implications and applications of formal proofs. One noted the potential for increased trust in critical systems, suggesting that formal proofs could eliminate bugs and vulnerabilities in areas like flight control software. They highlighted the importance of this, given the increasing complexity and reliance on software in critical systems. Another commenter pondered the future impact on mathematics education, speculating that tools and techniques from formal proof systems might eventually filter down to the undergraduate or even high school level, changing how math is taught.

  The conversation also touched upon the evolution and accessibility of formal proof tools. One commenter, familiar with older systems like Mizar, expressed pleasant surprise at the relative readability and clarity of the Isabelle/HOL proof presented in the article. They viewed this as a significant advancement in making formal methods more approachable. Another commenter pointed out the existing applications of formal methods in hardware verification, suggesting that the software world could learn from the hardware industry's experience.

  Some comments delved into the philosophical implications of formal proofs. One commenter questioned the ultimate value of formalization, arguing that informal proofs, while potentially flawed, still hold significant value due to their accessibility and explanatory power. They suggested that the effort involved in formalization might outweigh the benefits in some cases. This sparked a counter-argument emphasizing that informal proofs can hide subtle errors, and the rigor of formalization provides a higher level of certainty, even if it comes at a cost in terms of complexity.

  Finally, several comments focused on the specific tools and techniques used in the proof. Commenters mentioned specific proof assistants like Lean, Coq, and Isabelle/HOL, comparing their features and discussing their respective communities. There was also some discussion of the trade-offs between different approaches to formalization, with some commenters expressing preferences for particular styles or methods.

  In summary, the comments on the Hacker News post explored the practical, pedagogical, philosophical, and technical aspects of formal proofs, reflecting the diverse interests of the Hacker News community. The discussion provided a nuanced perspective on the potential benefits and challenges of formalization in mathematics and beyond.