These lecture notes provide a concise introduction to domain theory, focusing on its applications in computer science, particularly denotational semantics. They cover core concepts like partially ordered sets, complete partial orders (cpos), continuous functions, and the fixed-point theorem, explaining how these tools can be used to model computation and give meaning to recursive programs. The notes also touch on more advanced topics such as algebraic cpos and function spaces, providing a solid foundation for further exploration of the subject. The emphasis is on clear explanations and practical examples, making it accessible to those with a background in basic set theory and logic.
Nordström, Petersson, and Smith's "Programming in Martin-Löf's Type Theory" provides a comprehensive introduction to Martin-Löf's constructive type theory, emphasizing its practical application as a programming language. The book covers the foundational concepts of type theory, including dependent types, inductive definitions, and universes, demonstrating how these powerful tools can be used to express mathematical proofs and develop correct-by-construction programs. It explores various programming paradigms within this framework, like functional programming and modular development, and provides numerous examples to illustrate the theory in action. The focus is on demonstrating the expressive power and rigor of type theory for program specification, verification, and development.
Hacker News users discuss the linked book, "Programming in Martin-Löf's Type Theory," primarily focusing on its historical significance and influence on functional programming and dependent types. Some commenters note its dense and challenging nature, even for those familiar with type theory, but acknowledge its importance as a foundational text. Others highlight the book's role in shaping languages like Agda and Idris, and its impact on the development of theorem provers. The practicality of dependent types in everyday programming is also debated, with some suggesting their benefits remain largely theoretical while others point to emerging use cases. Several users express interest in revisiting or finally tackling the book, prompted by the discussion.
Philip Wadler's "Propositions as Types" provides a concise overview of the Curry-Howard correspondence, which reveals a deep connection between logic and programming. It explains how logical propositions can be viewed as types in a programming language, and how proofs of those propositions correspond to programs of those types. Specifically, implication corresponds to function types, conjunction to product types, disjunction to sum types, universal quantification to dependent product types, and existential quantification to dependent sum types. This correspondence allows programmers to reason about programs using logical tools, and conversely, allows logicians to use computational tools to reason about proofs. The paper illustrates these connections with clear examples, demonstrating how a proof of a logical formula can be directly translated into a program, and vice-versa, solidifying the idea that proofs are programs and propositions are the types they inhabit.
Hacker News users discuss Wadler's "Propositions as Types," mostly praising its clarity and accessibility in explaining the Curry-Howard correspondence. Several commenters share personal anecdotes about how the paper illuminated the connection between logic and programming for them, highlighting its effectiveness as an introductory text. Some discuss the broader implications of the correspondence and its relevance to type theory, automated theorem proving, and functional programming. A few mention related resources, like Software Foundations, and alternative presentations of the concept. One commenter notes the paper's omission of linear logic, while another suggests its focus is intentionally narrow for pedagogical purposes.
This blog post introduces an algebraic approach to representing and manipulating knitting patterns. It defines a knitting algebra based on two fundamental operations: knit and purl, along with transformations like increase and decrease, capturing the essential structure of stitch manipulations. These operations are combined with symbolic variables representing yarn colors and stitch types, allowing for formal representation of complex patterns and transformations like mirroring or rotating designs. The algebra enables automated manipulation and analysis of knitting instructions, potentially facilitating the generation of new patterns and supporting tools for knitters to explore variations and verify their designs. This formal, mathematical framework provides a powerful basis for developing software tools that can bridge the gap between abstract design and physical realization in knitting.
HN users were generally impressed with the algebraic approach to knitting, finding it a novel and interesting application of formal methods. Several commenters with knitting experience appreciated the potential for generating complex patterns and automating aspects of the design process. Some discussed the possibility of using similar techniques for other crafts like crochet or weaving. A few questioned the practicality for everyday knitters, given the learning curve involved in understanding the algebraic notation. The connection to functional programming was also noted, with comparisons made to Haskell and other declarative languages. Finally, there was some discussion about the limitations of the current implementation and potential future directions, like incorporating color changes or more complex stitch types.
Verus is a Rust verification framework designed for low-level systems programming. It extends Rust with features like specifications (preconditions, postconditions, and invariants) and data-race freedom proofs, allowing developers to formally verify the correctness and safety of their code. Verus integrates with existing Rust tools and aims to be practical for real-world systems development, leveraging SMT solvers to automate the verification process. It specifically targets areas like cryptography, operating systems kernels, and concurrent data structures, where rigorous correctness is paramount.
Hacker News users discussed Verus's potential and limitations. Some expressed excitement about its ability to verify low-level code, seeing it as a valuable tool for critical systems. Others questioned its practicality, citing the complexity of verification and the potential for performance overhead. The discussion also touched on the trade-offs between verification and traditional testing, with some arguing that testing remains essential even with formal verification. Several comments highlighted the challenge of balancing the strictness of verification with the flexibility needed for practical systems programming. Finally, some users were curious about Verus's performance characteristics and its suitability for real-world projects.
Jonathan Protzenko announced the release of Evercrypt 1.0 for Python, providing a high-assurance cryptography library with over 15,000 lines of formally verified code. This release leverages the HACL* cryptographic library, which has been mathematically proven correct, and makes it readily available for Python developers through a simple and performant interface. Evercrypt aims to bring robust, verified cryptographic primitives to a wider audience, improving security and trustworthiness for applications that depend on strong cryptography. It offers a drop-in replacement for existing libraries, significantly enhancing the security guarantees without requiring extensive code changes.
Hacker News users discussed the implications of having 15,000 lines of verified cryptography in Python, focusing on the trade-offs between verification and performance. Some expressed skepticism about the practical benefits of formal verification for cryptographic libraries, citing the difficulty of verifying real-world usage and the potential performance overhead. Others emphasized the importance of correctness in cryptography, arguing that verification offers valuable guarantees despite its limitations. The performance costs were debated, with some suggesting that the overhead might be acceptable or even negligible in certain scenarios. Several commenters also discussed the challenges of formal verification in general, including the expertise required and the limitations of existing tools. The choice of Python was also questioned, with some suggesting that a language like OCaml might be more suitable for this type of project.
Edsger Dijkstra argues against "natural language programming," believing it a foolish endeavor. He contends that natural language's inherent ambiguity and imprecision make it unsuitable for expressing the rigorous logic required in programming. Instead of striving for superficial readability through natural language, Dijkstra advocates for focusing on developing formal notations and abstractions that are clear, concise, and verifiable, even if they appear less "natural" initially. He emphasizes that programming requires a level of precision and unambiguity that natural language simply cannot provide, and attempting to bridge this gap will ultimately lead to more confusion and less reliable software.
HN commenters generally agree with Dijkstra's skepticism of "natural language programming." Some highlight the ambiguity inherent in natural language as fundamentally incompatible with the precision required for programming. Others point out the success of domain-specific languages (DSLs) as a middle ground, offering a more human-readable syntax without sacrificing clarity. One commenter suggests Dijkstra's critique is more aimed at vague specifications disguised as programs rather than genuinely well-defined natural language programming. Several commenters mention the value of formal methods and mathematical notation for clear program design, echoing Dijkstra's sentiments. A few offer historical context, suggesting the "natural language programming" Dijkstra criticized likely refers to early, overly ambitious attempts, and that modern NLP advancements might warrant revisiting the concept.
Deduce is a proof checker designed specifically for educational settings. It aims to bridge the gap between informal mathematical reasoning and formal proof construction by providing a simple, accessible interface and a focused set of logical connectives. Its primary goal is to teach the core concepts of formal logic and proof techniques without overwhelming users with complex syntax or advanced features. The system supports natural deduction style proofs and offers immediate feedback, guiding students through the process of building valid arguments step-by-step. Deduce prioritizes clarity and ease of use to make learning formal logic more engaging and less daunting.
Hacker News users discussed the educational value of the Deduce proof checker. Several commenters appreciated its simplicity and accessibility compared to other systems like Coq, finding its focus on propositional and first-order logic suitable for introductory logic courses. Some suggested potential improvements, such as adding support for natural deduction and incorporating a more interactive tutorial. Others debated the pedagogical merits of different proof styles and the balance between automated assistance and requiring students to fill in proof steps themselves. The overall sentiment was positive, with many seeing Deduce as a promising tool for teaching logic.
This 1986 paper explores representing the complex British Nationality Act 1981 as a Prolog program. It demonstrates how Prolog's declarative nature and built-in inference mechanisms can effectively encode the Act's intricate rules regarding citizenship acquisition and loss. The authors translate legal definitions of British citizenship, descent, and residency into Prolog clauses, showcasing the potential of logic programming to represent and reason with legal statutes. While acknowledging the limitations of this initial attempt, such as simplifying certain aspects of the Act and handling time-dependent clauses, the paper highlights the potential of using Prolog for legal expert systems and automated legal reasoning. It ultimately serves as an early exploration of applying computational logic to the domain of law.
Hacker News users discussed the ingenuity of representing the British Nationality Act as a Prolog program, highlighting the elegance of Prolog for handling complex logic and legal rules. Some expressed nostalgia for the era's focus on symbolic AI and rule-based systems. Others debated the practicality and maintainability of such an approach for real-world legal applications, citing the potential difficulty of updating and debugging the code as laws change. The discussion also touched on the broader implications of encoding law in a computationally interpretable format, considering the benefits for automated legal reasoning and the potential risks of bias and misinterpretation. Some users shared their own experiences with Prolog and other logic programming languages, and pondered the reasons for their decline in popularity despite their inherent strengths for certain problem domains.
This 1987 paper by Dybvig explores three distinct implementation models for Scheme: compilation to machine code, abstract machine interpretation, and direct interpretation of source code. It argues that while compilation offers the best performance for finished programs, the flexibility and debugging capabilities of interpreters are crucial for interactive development environments. The paper details the trade-offs between these models, emphasizing the advantages of a mixed approach that leverages both compilation and interpretation techniques. It concludes that an ideal Scheme system would utilize compilation for optimized execution and interpretation for interactive use, debugging, and dynamic code loading, hinting at a system where the boundaries between compiled and interpreted code are blurred.
HN commenters discuss the historical significance of the paper in establishing Scheme's minimalist design and portability. They highlight the cleverness of the three implementations, particularly the threaded code interpreter, and its influence on later languages like Lua. Some note the paper's accessibility and clarity, even for those unfamiliar with Scheme, while others reminisce about using the techniques described. A few comments delve into technical details like register allocation and garbage collection, comparing the approaches to modern techniques. The overall sentiment is one of appreciation for the paper's contribution to computer science and programming language design.
This paper explores using first-order logic (FOL) to detect logical fallacies in natural language arguments. The authors propose a novel approach that translates natural language arguments into FOL representations, leveraging semantic role labeling and a defined set of predicates to capture argument structure. This structured representation allows for the application of automated theorem provers to evaluate the validity of the arguments, thus identifying potential fallacies. The research demonstrates improved performance compared to existing methods, particularly in identifying fallacies related to invalid argument structure, while acknowledging limitations in handling complex linguistic phenomena and the need for further refinement in the translation process. The proposed system provides a promising foundation for automated fallacy detection and contributes to the broader field of argument mining.
Hacker News users discussed the potential and limitations of using first-order logic (FOL) for fallacy detection as described in the linked paper. Some praised the approach for its rigor and potential to improve reasoning in AI, while also acknowledging the inherent difficulty of translating natural language to FOL perfectly. Others questioned the practical applicability, citing the complexity and ambiguity of natural language as major obstacles, and suggesting that statistical/probabilistic methods might be more robust. The difficulty of scoping the domain knowledge necessary for FOL translation was also brought up, with some pointing out the need for extensive, context-specific knowledge bases. Finally, several commenters highlighted the limitations of focusing solely on logical fallacies for detecting flawed reasoning, suggesting that other rhetorical tactics and nuances should also be considered.
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.
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 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.
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.
Hillel Wayne's post dissects the concept of "nondeterminism" in computer science, arguing that it's often used ambiguously and encompasses five distinct meanings. These are: 1) Implementation-defined behavior, where the language standard allows for varied outcomes. 2) Unspecified behavior, similar to implementation-defined but offering even less predictability. 3) Error/undefined behavior, where anything could happen, often leading to crashes. 4) Heisenbugs, which are bugs whose behavior changes under observation (e.g., debugging). 5) True nondeterminism, exemplified by hardware randomness or concurrency races. The post emphasizes that these are fundamentally different concepts with distinct implications for programmers, and understanding these nuances is crucial for writing robust and predictable software.
Hacker News users discussed various aspects of nondeterminism in the context of Hillel Wayne's article. Several commenters highlighted the distinction between predictable and unpredictable nondeterminism, with some arguing the author's categorization conflated the two. The importance of distinguishing between sources of nondeterminism, such as hardware, OS scheduling, and program logic, was emphasized. One commenter pointed out the difficulty in achieving true determinism even with seemingly simple programs due to factors like garbage collection and just-in-time compilation. The practical challenges of debugging nondeterministic systems were also mentioned, along with the value of tools that can help reproduce and analyze nondeterministic behavior. A few comments delved into specific types of nondeterminism, like data races and the nuances of concurrency, while others questioned the usefulness of the proposed categorization in practice.
Dusa is a logic programming language based on finite-choice logic, designed for declarative problem solving and knowledge representation. It emphasizes simplicity and approachability, with a Python-inspired syntax and built-in support for common data structures like lists and dictionaries. Dusa programs define relationships between facts and rules, allowing users to describe problems and let the system find solutions. Its core features include backtracking search, constraint satisfaction, and a type system based on logical propositions. Dusa aims to be both a practical tool for everyday programming tasks and a platform for exploring advanced logic programming concepts.
Hacker News users discussed Dusa's novel approach to programming with finite-choice logic, expressing interest in its potential for formal verification and constraint solving. Some questioned its practicality and performance compared to established Prolog implementations, while others highlighted the benefits of its clear semantics and type system. Several commenters drew parallels to miniKanren, another logic programming language, and discussed the trade-offs between Dusa's finite-domain focus and the more general approach of Prolog. The static typing and potential for compile-time optimization were seen as significant advantages. There was also a discussion about the suitability of Dusa for specific domains like game AI and puzzle solving. Some expressed skepticism about the claim of "blazing fast performance," desiring benchmarks to validate it. Overall, the comments reflected a mixture of curiosity, cautious optimism, and a desire for more information, particularly regarding real-world applications and performance comparisons.
This paper introduces Crusade, a formally verified translation from a subset of C to safe Rust. Crusade targets a memory-safe dialect of C, excluding features like arbitrary pointer arithmetic and casts. It leverages the Coq proof assistant to formally verify the translation's correctness, ensuring that the generated Rust code behaves identically to the original C, modulo non-determinism inherent in C. This rigorous approach aims to facilitate safe integration of legacy C code into Rust projects without sacrificing confidence in memory safety, a critical aspect of modern systems programming. The translation handles a substantial subset of C, including structs, unions, and functions, and demonstrates its practical applicability by successfully converting real-world C libraries.
HN commenters discuss the challenges and nuances of formally verifying the C to Rust transpiler, Cracked. Some express skepticism about the practicality of fully verifying such a complex tool, citing the potential for errors in the formal proofs themselves and the inherent difficulty of capturing all undefined C behavior. Others question the performance impact of the generated Rust code. However, many commend the project's ambition and see it as a significant step towards safer systems programming. The discussion also touches upon the trade-offs between a fully verified transpiler and a more pragmatic approach focusing on common C patterns, with some suggesting that prioritizing practical safety improvements could be more beneficial in the short term. There's also interest in the project's handling of concurrency and the potential for integrating Cracked with existing Rust tooling.
Rishi Mehta reflects on the key contributions and learnings from AlphaProof, his AI research project focused on automated theorem proving. He highlights the successes of AlphaProof in tackling challenging mathematical problems, particularly in abstract algebra and group theory, emphasizing its unique approach of combining language models with symbolic reasoning engines. The post delves into the specific techniques employed, such as the use of chain-of-thought prompting and iterative refinement, and discusses the limitations encountered. Mehta concludes by emphasizing the significant progress made in bridging the gap between natural language and formal mathematics, while acknowledging the open challenges and future directions for research in automated theorem proving.
Hacker News users discuss AlphaProof's approach to testing, questioning its reliance on property-based testing and mutation testing for catching subtle bugs. Some commenters express skepticism about the effectiveness of these techniques in real-world scenarios, arguing that they might not be as comprehensive as traditional testing methods and could lead to a false sense of security. Others suggest that AlphaProof's methodology might be better suited for specific types of problems, such as concurrency bugs, rather than general software testing. The discussion also touches upon the importance of code review and the potential limitations of automated testing tools. Some commenters found the examples provided in the original article unconvincing, while others praised AlphaProof's innovative approach and the value of exploring different testing strategies.
Summary of Comments ( 4 )
https://news.ycombinator.com/item?id=44084577
HN users generally praised the clarity and accessibility of the lecture notes, particularly for beginners. Several appreciated the focus on intuition and practicality over strict formalism, making the often-dense subject matter easier to grasp. One commenter pointed out the helpful use of diagrams and examples, while others highlighted the effective explanation of core concepts like directed sets and continuous functions. Some suggested additional topics or resources that could further enhance the notes, such as exploring the connection between domain theory and denotational semantics, or including more advanced topics like powerdomains. A few commenters with prior experience in the field expressed renewed appreciation for the foundational material presented in a refreshingly clear way.
The Hacker News post titled "Domain Theory Lecture Notes" with the ID 44084577 has a modest number of comments, sparking a focused discussion around the presented lecture notes on domain theory. Notably, several commenters express appreciation for the clarity and accessibility of the notes, contrasting them with the often-perceived density and difficulty of the subject matter.
One commenter highlights the value of the notes for programmers, emphasizing the connection between domain theory and practical programming concepts like lazy evaluation and memoization. They suggest that understanding domain theory can provide deeper insights into these common programming techniques.
Another commenter points out the author's successful approach of presenting the material in a digestible way, particularly praising the use of Haskell code examples. They feel this practical implementation helps solidify the theoretical concepts and makes the topic more approachable for those unfamiliar with domain theory.
The discussion also touches upon the historical significance and theoretical underpinnings of domain theory. One comment mentions its origins in denotational semantics and its relevance to understanding the mathematical foundations of programming language semantics. This adds context to the notes and underscores their importance within the broader field of computer science.
A few comments offer specific feedback on the content, suggesting minor improvements or pointing out areas where further clarification could be beneficial. This demonstrates an engaged readership actively working through the material and offering constructive criticism.
While the overall volume of comments isn't extensive, the discussion is substantial, revealing a shared appreciation for the resource being shared and demonstrating its potential value to both seasoned computer scientists and those newer to the field. The comments avoid delving into tangential topics and remain focused on the quality and utility of the lecture notes themselves.