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Matters Computational (2010) [pdf]

Posted: 2025-03-07 10:06:38

Jürgen Schmidhuber's "Matters Computational" provides a comprehensive overview of computer science, spanning its theoretical foundations and practical applications. It delves into topics like algorithmic information theory, computability, complexity theory, and the history of computation, including discussions of Turing machines and the Church-Turing thesis. The book also explores the nature of intelligence and the possibilities of artificial intelligence, covering areas such as machine learning, neural networks, and evolutionary computation. It emphasizes the importance of self-referential systems and universal problem solvers, reflecting Schmidhuber's own research interests in artificial general intelligence. Ultimately, the book aims to provide a unifying perspective on computation, bridging the gap between theoretical computer science and the practical pursuit of artificial intelligence.

Jürgen Schmidhuber's "Matters Computational (2010)" presents a comprehensive, and at times highly technical, overview of theoretical computer science, with a particular focus on its intersection with artificial intelligence and machine learning, specifically Schmidhuber's own contributions to these fields. The book, structured as a collection of previously published papers and new material, explores fundamental computational principles through the lens of algorithmic information theory, Kolmogorov complexity, and the concept of universal problem solvers.

The narrative begins with an exploration of the theoretical foundations of computation, delving into the nature of computability, Turing machines, and the limits of what can be computed. It emphasizes the importance of universality and self-reference in computation, drawing connections to Gödel's incompleteness theorems and the halting problem. Schmidhuber introduces the speed prior, a concept central to his research, which favors programs that not only solve a given problem but do so efficiently. This leads into a discussion of optimal universal search algorithms, such as Levin Search, and their implications for artificial general intelligence.

A substantial portion of the book is dedicated to algorithmic information theory and its application to inductive inference and learning. Schmidhuber argues that the shortest program capable of generating a given data sequence is the best explanation for that sequence, aligning with Occam's Razor and the principle of Minimum Description Length. This theoretical framework is used to analyze the problem of learning from data, including supervised learning, unsupervised learning, and reinforcement learning. He discusses various algorithms for finding short programs, including variations of Levin Search and methods based on Monte Carlo sampling.

The text also delves into the practical implications of these theoretical concepts, examining the design and implementation of artificial neural networks, recurrent neural networks, and other learning systems. Schmidhuber meticulously details his own contributions to the field, including the development of Long Short-Term Memory (LSTM) networks, a type of recurrent neural network architecture designed to address the vanishing gradient problem, which had previously hampered the training of deep networks. He also discusses other architectures like self-referential neural networks and the concept of Gödel machines, self-improving AI systems inspired by Gödel's incompleteness theorems.

Throughout the book, Schmidhuber emphasizes the importance of formal mathematical frameworks for understanding intelligence and the potential for creating truly intelligent machines. He rigorously analyzes the computational complexity of various learning algorithms and discusses the limitations imposed by finite resources. The text is replete with mathematical formulas, algorithms, and proofs, providing a deep dive into the theoretical underpinnings of the field. He concludes with a forward-looking perspective on the future of artificial intelligence, emphasizing the potential for creating increasingly sophisticated and capable learning systems based on the principles outlined in the book. The overarching theme is a pursuit of general purpose artificial intelligence through a formal, mathematically grounded approach centered on optimal program search and algorithmic information theory.

Summary of Comments ( 11 )
https://news.ycombinator.com/item?id=43288861

HN users discuss the density and breadth of "Matters Computational," praising its unique approach to connecting diverse computational topics. Several commenters highlight the book's treatment of randomness, floating-point arithmetic, and the FFT as particularly insightful. The author's background in physics is noted, contributing to the book's distinct perspective. Some find the book challenging, requiring multiple readings to fully grasp the concepts. The free availability of the PDF is appreciated, and its enduring relevance a decade after publication is also remarked upon. A few commenters express interest in a physical copy, while others suggest potential updates or expansions on certain topics.

The Hacker News post titled "Matters Computational (2010) [pdf]" linking to a PDF of Jörg Fliege's book "Matters Computational" has a moderate number of comments, discussing various aspects of the book and computational mathematics in general.

Several commenters praise the book's comprehensive nature and clarity. One user highlights its value as a reference for "all sorts of basic algorithms and data structures," appreciating the detailed explanations and pseudocode provided. They specifically mention its usefulness for understanding fundamental concepts like numerical stability.

Another commenter focuses on the book's treatment of linear algebra, noting its depth and accessibility, even for those without a strong mathematical background. They contrast it with other resources they found less helpful.

A few comments delve into specific topics covered in the book. One user discusses the exploration of floating-point arithmetic and its associated challenges, acknowledging the importance of understanding these concepts for anyone working with numerical computations. Another highlights the chapter on optimization, mentioning its practical value and the inclusion of various optimization algorithms.

Some commenters offer broader perspectives on computational mathematics and its role in computer science. One reflects on the importance of a strong mathematical foundation for software engineers, advocating for more emphasis on these concepts in education.

The discussion also touches on the book's availability. The author's decision to make it freely available is commended, with some users expressing gratitude for open access to such valuable educational resources. A link to the author's webpage is shared, offering further context.

While a number of commenters express interest in the book based on the description and other comments, there isn't extensive engagement in deep technical discussions. The overall sentiment is positive, with the comments primarily focusing on the book's breadth, clarity, and value as a resource for understanding fundamental computational concepts.

A brief meditation on formal systems and lying goblins

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Posted: 2025-03-06 22:00:38

The blog post explores the limitations of formal systems, particularly in discerning truth. It uses the analogy of two goblins, one always truthful and one always lying, to demonstrate how relying solely on a system's rules, without external context or verification, can lead to accepting falsehoods as truths. Even with additional rules added to account for the goblins' lying, clever manipulation can still exploit the system. The post concludes that formal systems, while valuable for structuring thought, are ultimately insufficient for determining truth without external validation or a connection to reality. This highlights the need for critical thinking and skepticism even when dealing with seemingly rigorous systems.

This blog post, titled "A Brief Meditation on Formal Systems and Lying Goblins," delves into the fascinating intersection of formal systems, logic, and the challenges of extracting truthful information from unreliable sources. The author constructs a thought experiment involving a population of goblins, some of whom invariably speak the truth, and others who consistently lie. The central problem revolves around devising a strategy to reliably determine the veracity of statements made by these goblins, given that one cannot ascertain a priori which type of goblin one is interacting with.

The author meticulously lays out the formal framework for this puzzle. They introduce the concept of truth-telling goblins, represented symbolically, and lying goblins, similarly represented. This symbolic representation allows for the manipulation of statements and the exploration of logical consequences within a clearly defined system. The post then proceeds to examine the complexities introduced by the goblins' deceptive nature. A simple question, such as inquiring about the nature of the goblin itself (truth-teller or liar), proves insufficient, as both types would provide the same answer if attempting to deceive.

The core of the post focuses on the development of a questioning strategy that can circumvent the goblins' inherent deceptiveness. The author explores the idea of nested questions, effectively asking a goblin about what another goblin would say. This indirect approach leverages the predictable nature of the goblins' lies: a liar will always lie about what a truth-teller would say, and vice-versa. By carefully constructing these nested inquiries, one can effectively force the goblins to reveal the truth, irrespective of their individual predilection for truth or falsehood.

The author further elaborates on the underlying principles at play, highlighting the power of formal systems in dissecting and solving such puzzles. By representing the problem within a formal framework, one can systematically explore potential solutions and rigorously test their validity. The post concludes by emphasizing the broader implications of this thought experiment. It serves as a microcosm of the challenges we face in the real world when dealing with information from potentially unreliable sources. The ability to construct robust questioning strategies, analogous to the nested inquiries used with the goblins, becomes crucial for navigating a world awash in misinformation and discerning truth from falsehood. The seemingly simple puzzle of the lying goblins thus provides a valuable lesson in the application of formal systems to complex problems of knowledge and belief.

Summary of Comments ( 0 )
https://news.ycombinator.com/item?id=43285485

The Hacker News comments generally praise the clarity and engaging presentation of the article's topic (formal systems and the halting problem, illustrated by a lying goblin puzzle). Several commenters discuss the philosophical implications of the piece, particularly regarding the nature of truth and provability within defined systems. Some draw parallels to Gödel's incompleteness theorems, while others offer alternate goblin scenarios or slight modifications to the puzzle's rules. A few commenters suggest related resources, such as Raymond Smullyan's work, which explores similar logical puzzles. There's also a short thread discussing the potential applicability of these concepts to legal systems and contract interpretation.

The Hacker News post "A brief meditation on formal systems and lying goblins" has generated several comments discussing the article's premise and exploring related concepts.

Several commenters engage with the core idea of the article, which uses the analogy of lying goblins to illustrate how seemingly sound logical systems can lead to incorrect conclusions if based on false premises. One commenter points out the parallel to real-world scenarios where misinformation or flawed assumptions can corrupt a system, regardless of its internal consistency. They mention how this applies to areas like political discourse and conspiracy theories.

Another commenter delves deeper into the concept of formal systems, emphasizing the distinction between validity and soundness. They clarify that a valid argument can still be untrue if its premises are false, echoing the goblin analogy. They also introduce the idea of Gödel's incompleteness theorems, suggesting that even consistent formal systems can contain unprovable truths.

The discussion extends to the practical implications of these ideas. One commenter reflects on the challenge of identifying false premises in real-world situations, highlighting the importance of critical thinking and questioning assumptions. Another commenter draws a connection to Bayesian reasoning, suggesting that incorporating prior probabilities can help mitigate the risk of being misled by false information.

Further comments explore related philosophical themes, touching on the nature of truth and the limits of knowledge. One commenter mentions the concept of "unknown unknowns," emphasizing the difficulty of accounting for information that we are not even aware of.

Some commenters also offer alternative analogies to illustrate the same principles. One suggests the image of a perfectly functioning calculator that produces incorrect results due to a user inputting the wrong numbers.

Overall, the comments on the Hacker News post provide a thoughtful and engaging discussion of the article's core ideas, exploring their implications in various contexts and connecting them to broader philosophical and mathematical concepts. They highlight the importance of critical thinking, the limitations of formal systems, and the challenges of navigating a world filled with potentially misleading information.

Translating Natural Language to First-Order Logic for Logical Fallacy Detection

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Posted: 2025-03-04 17:36:23

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.

The arXiv preprint "Translating Natural Language to First-Order Logic for Logical Fallacy Detection" by Liu et al. explores a novel approach to identifying logical fallacies within natural language arguments. The authors posit that current methods for fallacy detection, which largely rely on surface-level linguistic features or shallow semantic analysis, are insufficient for capturing the underlying logical structure necessary for robust fallacy identification. They propose instead a method grounded in formal logic, specifically first-order logic (FOL), which allows for a more rigorous and precise representation of argumentative structures.

The core of their proposed methodology lies in translating natural language arguments into FOL representations. This translation process involves several intricate steps. First, the argumentative text is parsed to identify individual premises and the conclusion. Subsequently, these components are subjected to semantic parsing, transforming them into logical forms expressible within FOL. This necessitates the identification of entities, predicates, and quantifiers present in the natural language, and their subsequent mapping to corresponding elements within the FOL framework. The authors acknowledge the inherent complexity and ambiguity of natural language, which poses a significant challenge for accurate translation. To address this, they employ a combination of existing semantic parsing techniques and introduce novel strategies tailored to the specific requirements of fallacy detection.

Once the argument is represented in FOL, the authors leverage the power of automated theorem provers to assess the argument's validity. By attempting to prove the conclusion from the premises within the FOL framework, they can determine whether the argument is logically sound. If the conclusion cannot be derived from the premises, this suggests the potential presence of a logical fallacy. However, the mere failure of a proof does not definitively indicate a fallacy; it could simply reflect limitations in the translation process or the theorem prover's capabilities.

Therefore, the authors introduce a further layer of analysis based on fallacy templates. These templates represent common logical fallacies, such as ad hominem, straw man, or false dilemma, formalized within the FOL framework. By matching the FOL representation of the argument against these pre-defined fallacy templates, the system can identify instances where the argument's structure aligns with a known fallacious pattern. This template-matching approach provides a more targeted and nuanced mechanism for fallacy detection, going beyond the simple binary classification of valid or invalid.

The paper details experiments conducted on established fallacy datasets, comparing their proposed FOL-based method against existing state-of-the-art techniques. The authors report promising results, demonstrating that their approach achieves improved accuracy in identifying various types of logical fallacies. They further analyze the strengths and limitations of their methodology, acknowledging the ongoing challenges in accurately translating complex natural language arguments into FOL and the need for more comprehensive fallacy templates. The research concludes by emphasizing the potential of FOL-based approaches for advancing the field of automated logical fallacy detection and suggests future research directions, such as incorporating more sophisticated semantic parsing techniques and expanding the library of formalized fallacy templates.

Summary of Comments ( 68 )
https://news.ycombinator.com/item?id=43257719

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.

The Hacker News post titled "Translating Natural Language to First-Order Logic for Logical Fallacy Detection" (linking to arXiv paper 2405.02318) has a modest number of comments, sparking a discussion around the practicality and challenges of using formal logic for fallacy detection.

One commenter expresses skepticism about the real-world applicability of this approach. They argue that logical fallacies in everyday discourse often hinge on implicit premises and contextual nuances that are difficult to capture in formal logic. They suggest that focusing on these implicit elements, which the current approach seems to bypass, is crucial for effective fallacy detection. This commenter also points out the challenge of translating the richness and ambiguity of natural language into the rigid structure of first-order logic, questioning the feasibility of achieving high accuracy in this translation process.

Another commenter builds on this skepticism by highlighting the issue of ambiguity inherent in natural language. They provide the example of the phrase "most people," which can have different interpretations depending on the context, and how formalizing such a phrase would necessitate making assumptions about the intended quantifier. This emphasizes the difficulty of creating a universally applicable system, as the interpretation of such phrases would need to be tailored to specific domains or contexts.

A different commenter suggests an alternative perspective, mentioning a different approach to fallacy detection that utilizes large language models (LLMs). They point to a paper where LLMs are used to identify fallacies without explicit formalization. This comment implies that perhaps direct application of statistical methods via LLMs could be a more promising avenue for fallacy detection than attempting the complex task of translating natural language into formal logic.

Another commenter echoes the concern about the limitations of formal logic in capturing the subtleties of natural language arguments, particularly those involving informal fallacies. They also touch upon the issue of computational complexity associated with logical reasoning, suggesting that practical implementations might face performance bottlenecks.

Finally, one commenter asks a clarifying question about the specific types of logical fallacies the research addresses, indicating a desire to understand the scope and limitations of the proposed approach. This highlights the importance of clearly defining the target fallacies when evaluating the effectiveness of such systems.

In summary, the comments largely express reservations about the practicality of the approach outlined in the linked paper, focusing on the difficulties of translating nuanced natural language into formal logic and the potential computational complexities. Alternatives using LLMs are suggested, and the need for careful consideration of the target fallacies is highlighted.

Efficient Reasoning with Hidden Thinking

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Posted: 2025-02-03 16:06:48

The paper "Efficient Reasoning with Hidden Thinking" introduces Hidden Thinking Networks (HTNs), a novel architecture designed to enhance the efficiency of large language models (LLMs) in complex reasoning tasks. HTNs augment LLMs with a differentiable "scratchpad" that allows them to perform intermediate computations and logical steps, mimicking human thought processes during problem-solving. This hidden thinking process is learned through backpropagation, enabling the model to dynamically adapt its reasoning strategies. By externalizing and making the reasoning steps differentiable, HTNs aim to improve transparency, controllability, and efficiency compared to standard LLMs, which often struggle with multi-step reasoning or rely on computationally expensive prompting techniques like chain-of-thought. The authors demonstrate the effectiveness of HTNs on various reasoning tasks, showcasing their potential for more efficient and interpretable problem-solving with LLMs.

The arXiv preprint "Efficient Reasoning with Hidden Thinking" introduces a novel approach to enhance the efficiency and reasoning capabilities of large language models (LLMs). The authors posit that current LLMs, while demonstrating impressive performance on various tasks, often struggle with complex reasoning problems that require multiple steps or the derivation of intermediate conclusions. They argue that this limitation stems from the direct generation of output without explicitly representing the underlying thought process, akin to a "black box" approach.

The paper proposes "Hidden Thinking" as a solution, a technique that encourages LLMs to explicitly generate intermediate reasoning steps before producing the final answer. This is achieved by prompting the model to first generate a sequence of hidden thoughts, represented as natural language sentences, that reflect the logical deductions and intermediate conclusions necessary to solve the given problem. These hidden thoughts are not directly included in the final output but serve as an internal scaffold to guide the model's reasoning process. Subsequently, the model uses these hidden thoughts as the basis for generating the final answer.

The authors hypothesize that this approach offers several advantages. First, it forces the model to decompose complex reasoning problems into smaller, more manageable steps, making the overall reasoning process more transparent and potentially easier to learn. Second, it allows the model to leverage intermediate conclusions, preventing errors that might arise from attempting to generate the final answer directly. Third, it provides a mechanism for incorporating external knowledge or constraints into the reasoning process, as these can be integrated into the hidden thoughts.

The effectiveness of Hidden Thinking is evaluated through experiments on several reasoning benchmarks, including multi-hop question answering and mathematical reasoning. The results demonstrate that augmenting LLMs with Hidden Thinking leads to significant improvements in accuracy compared to baseline models that do not utilize this technique. The authors further analyze the generated hidden thoughts to gain insights into the model's reasoning process and demonstrate that Hidden Thinking encourages more structured and logical reasoning pathways. Furthermore, they explore different prompting strategies for eliciting effective hidden thoughts and investigate the impact of the number of hidden thoughts on performance.

In conclusion, the paper presents Hidden Thinking as a promising method for enhancing the reasoning abilities of LLMs by encouraging them to explicitly generate intermediate reasoning steps. The empirical results suggest that this approach leads to improved performance on complex reasoning tasks and offers a more transparent and interpretable view into the model's internal thought processes. This opens up avenues for future research on incorporating more structured reasoning mechanisms into LLMs and developing more effective prompting strategies for eliciting high-quality hidden thoughts.

Summary of Comments ( 27 )
https://news.ycombinator.com/item?id=42919597

Hacker News users discussed the practicality and implications of the "Hidden Thinking" paper. Several commenters expressed skepticism about the real-world applicability of the proposed method, citing concerns about computational cost and the difficulty of accurately representing complex real-world problems within the framework. Some questioned the novelty of the approach, comparing it to existing techniques like MCTS (Monte Carlo Tree Search) and pointing out potential limitations in scaling and handling uncertainty. Others were more optimistic, seeing potential applications in areas like game playing and automated theorem proving, while acknowledging the need for further research and development. A few commenters also discussed the philosophical implications of machines engaging in "hidden thinking," raising questions about transparency and interpretability.

The Hacker News post titled "Efficient Reasoning with Hidden Thinking" (linking to arXiv paper 2501.19201) has generated several comments discussing the concept of "hidden thinking" in large language models and its potential implications.

Several commenters delve into the idea of LLMs exhibiting behavior reminiscent of "thinking" or internal deliberation, even though their underlying mechanism is statistical pattern matching. One commenter points out the distinction between "thinking" as traditionally understood (conscious, deliberate reasoning) and the emergent behavior of LLMs, suggesting the term "thinking" may be misleading. They acknowledge the impressive capabilities of these models while emphasizing the need for a more precise understanding of their internal processes.

The discussion also touches upon the computational cost associated with this "hidden thinking." Commenters speculate about whether the observed "thinking" is an emergent property or a result of specific architectural choices within the LLMs. One user raises the question of whether this apparent deliberation is an efficient strategy for problem-solving, considering the computational resources required.

Another commenter highlights the importance of understanding how these models arrive at their outputs, regardless of whether we label it "thinking" or not. They emphasize the need for greater transparency and interpretability in LLMs.

One commenter draws a parallel to human cognition, suggesting that the distinction between explicit and implicit processing might be relevant to understanding LLMs. They propose that while LLMs don't have conscious thought, their complex internal processing could be analogous to the unconscious processing that occurs in the human brain.

The concept of "chain-of-thought prompting" is mentioned, highlighting a technique where the model is prompted to explicitly lay out its reasoning steps. This is contrasted with the "hidden thinking" discussed in the paper, where the internal reasoning process is not directly observable.

Finally, some comments express skepticism about the novelty of the "hidden thinking" concept, suggesting that similar observations have been made previously in the field of machine learning. They question whether the paper presents genuinely new insights or simply repackages existing ideas.

Overall, the comments reflect a mixture of fascination and skepticism regarding the idea of "hidden thinking" in LLMs. While acknowledging the impressive capabilities of these models, commenters emphasize the need for a more nuanced understanding of their internal processes and caution against anthropomorphizing their behavior. The discussion highlights ongoing debates within the AI community about interpretability, efficiency, and the very nature of intelligence in artificial systems.

Anatomy of a Formal Proof

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

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.

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

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.

Show HN: SudokuVariants – play and construct different variants of Sudoku

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Posted: 2025-01-21 16:49:32

SudokuVariants.com lets you play and create a wide variety of Sudoku puzzles beyond the classic 9x9 grid. The website offers different grid sizes, shapes, and rule sets, including variations like Killer Sudoku, Irregular Sudoku, and even custom rule combinations. Users can experiment with existing variants or design their own unique Sudoku challenges using a visual editor, and then share their creations with others via a generated link. The site aims to provide a comprehensive platform for both playing and exploring the vast possibilities within the Sudoku puzzle format.

Summary of Comments ( 11 )
https://news.ycombinator.com/item?id=42782246

Hacker News users generally expressed interest in the SudokuVariants website. Several praised its clean design and the variety of puzzles offered. Some found the "construct your own variant" feature particularly appealing, and one user suggested adding a difficulty rating system for user-created puzzles. A few commenters mentioned specific variant recommendations, including "Killer Sudoku" and a variant with prime number constraints. There was also a brief discussion about the underlying logic and algorithms involved in generating and solving these puzzles. One user pointed out that some extreme variants might be NP-complete, implying significant computational challenges for larger grids or complex rules.

The Hacker News post titled "Show HN: SudokuVariants – play and construct different variants of Sudoku" linking to sudokuvariants.com has generated several comments discussing various aspects of the website and Sudoku variants in general.

Several commenters praise the website's clean design and user-friendly interface. One user appreciates the clear instructions and the ability to easily understand the rules of different Sudoku variations. Another commenter highlights the value of the "Construct" feature, allowing users to create their own puzzles, as a significant addition to the site's appeal. The ease of sharing custom-created puzzles is also mentioned as a positive aspect.

A discussion arises around the complexity and enjoyment derived from different Sudoku variants. Some commenters share their preferences for specific types, like "Killer Sudoku" and "Arrow Sudoku," while others express interest in exploring less common variations. One user mentions the satisfaction of solving particularly challenging puzzles, while another points out the potential for frustration if a variant's rules are too complex or poorly explained.

The technical aspects of the website are also touched upon. One commenter inquires about the implementation details, specifically regarding the solver algorithm used. Another user discusses the responsiveness of the site on mobile devices, suggesting potential improvements for smaller screens.

There's a brief exchange about the broader appeal of logic puzzles and the role of websites like SudokuVariants in providing a platform for both playing and creating them. One user suggests the possibility of adding a feature for users to rate or review different variants.

A few commenters share their personal experiences with Sudoku, some recounting their long-standing enjoyment of the puzzle, while others express a renewed interest sparked by the website's diverse offerings.

Finally, the creator of the website engages with the commenters, responding to queries about technical details, acknowledging feedback, and expressing gratitude for the positive reception. They also mention future plans for the website, including potential additions of more Sudoku variants and features.

Stories with Tag Logic

Matters Computational (2010) [pdf]

Summary of Comments ( 11 ) https://news.ycombinator.com/item?id=43288861

A brief meditation on formal systems and lying goblins

Summary of Comments ( 0 ) https://news.ycombinator.com/item?id=43285485

Translating Natural Language to First-Order Logic for Logical Fallacy Detection

Summary of Comments ( 68 ) https://news.ycombinator.com/item?id=43257719

Efficient Reasoning with Hidden Thinking

Summary of Comments ( 27 ) https://news.ycombinator.com/item?id=42919597

Anatomy of a Formal Proof

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

Show HN: SudokuVariants – play and construct different variants of Sudoku

Summary of Comments ( 11 ) https://news.ycombinator.com/item?id=42782246

Summary of Comments ( 11 )
https://news.ycombinator.com/item?id=43288861

Summary of Comments ( 0 )
https://news.ycombinator.com/item?id=43285485

Summary of Comments ( 68 )
https://news.ycombinator.com/item?id=43257719

Summary of Comments ( 27 )
https://news.ycombinator.com/item?id=42919597

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

Summary of Comments ( 11 )
https://news.ycombinator.com/item?id=42782246