Lehmer's continued fraction factorization algorithm offers a way to find factors of a composite integer n. It leverages the convergents of the continued fraction expansion of √n to generate pairs of integers x and y such that x² ≡ y² (mod n). If x is not congruent to ±y (mod n), then gcd(x-y, n) and gcd(x+y, n) will yield non-trivial factors of n. While not as efficient as more advanced methods like the general number field sieve, it provides a relatively simple approach to factorization and serves as a stepping stone towards understanding more complex techniques.
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.
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.
This blog post introduces Differentiable Logic Cellular Automata (DLCA), a novel approach to creating cellular automata (CA) that can be trained using gradient descent. Traditional CA use discrete rules to update cell states, making them difficult to optimize. DLCA replaces these discrete rules with continuous, differentiable logic gates, allowing for smooth transitions between states. This differentiability allows for the application of standard machine learning techniques to train CA for specific target behaviors, including complex patterns and computations. The post demonstrates DLCA's ability to learn complex tasks, such as image classification and pattern generation, surpassing the capabilities of traditional, hand-designed CA.
HN users discussed the potential of differentiable logic cellular automata, expressing excitement about its applications in areas like program synthesis and hardware design. Some questioned the practicality given current computational limitations, while others pointed to the innovative nature of embedding logic within a differentiable framework. The concept of "soft" logic gates operating on continuous values intrigued several commenters, with some drawing parallels to analog computing and fuzzy logic. A few users desired more details on the training process and specific applications, while others debated the novelty of the approach compared to existing techniques like neural cellular automata. Several commenters expressed interest in exploring the code and experimenting with the ideas presented.
This Google Form poses a series of questions to William J. Rapaport regarding his views on the possibility of conscious AI. It probes his criteria for consciousness, asking him to clarify the necessary and sufficient conditions for a system to be considered conscious, and how he would test for them. The questions specifically explore his stance on computational theories of mind, the role of embodiment, and the relevance of subjective experience. Furthermore, it asks about his interpretation of specific thought experiments related to consciousness and AI, including the Chinese Room Argument, and solicits his opinions on the potential implications of creating conscious machines.
The Hacker News comments on the "Questions for William J. Rapaport" post are sparse and don't offer much substantive discussion. A couple of users express skepticism about the value or seriousness of the questionnaire, questioning its purpose and suggesting it might be a student project or even a prank. One commenter mentions Rapaport's work in cognitive science and AI, suggesting a potential connection to the topic of consciousness. However, there's no in-depth engagement with the questionnaire itself or Rapaport's potential responses. Overall, the comment section provides little insight beyond a general sense of skepticism.
Roger Penrose argues that Gödel's incompleteness theorems demonstrate that human mathematical understanding transcends computation and therefore, strong AI, which posits that consciousness is computable, is fundamentally flawed. He asserts that humans can grasp the truth of Gödelian sentences (statements unprovable within a formal system yet demonstrably true outside of it), while a computer bound by algorithms within that system cannot. This, Penrose claims, illustrates a non-computable element in human consciousness, suggesting we understand truth through means beyond mere calculation.
Hacker News users discuss Penrose's argument against strong AI, with many expressing skepticism. Several commenters point out that Gödel's incompleteness theorems don't necessarily apply to the way AI systems operate, arguing that AI doesn't need to be consistent or complete in the same way as formal mathematical systems. Others suggest Penrose misinterprets or overextends Gödel's work. Some users find Penrose's ideas intriguing but remain unconvinced, while others find his arguments simply wrong. The concept of "understanding" is a key point of contention, with some arguing that current AI models only simulate understanding, while others believe that sophisticated simulation is indistinguishable from true understanding. A few commenters express appreciation for Penrose's thought-provoking perspective, even if they disagree with his conclusions.
This 2018 paper demonstrates how common spreadsheet software can be used to simulate neural networks, offering a readily accessible and interactive educational tool. It details the implementation of a multilayer perceptron (MLP) within a spreadsheet, using built-in functions to perform calculations for forward propagation, backpropagation, and gradient descent. The authors argue that this approach allows for a deeper understanding of neural network mechanics due to its transparent and step-by-step nature, which can be particularly beneficial for teaching purposes. They provide examples of classification and regression tasks, showcasing the spreadsheet's capability to handle different activation functions and datasets. The paper concludes that spreadsheet-based simulations, while not suitable for large-scale applications, offer a valuable pedagogical alternative for introducing and exploring fundamental neural network concepts.
HN users discuss the practicality and educational value of simulating neural networks in spreadsheets. Some find it a clever way to visualize and understand the underlying mechanics, especially for beginners, while others argue its limitations make it unsuitable for real-world applications. Several commenters point out the computational constraints of spreadsheets, making them inefficient for larger networks or datasets. The discussion also touches on alternative tools for learning and experimenting with neural networks, like Python libraries, which offer greater flexibility and power. A compelling point raised is the potential for oversimplification, potentially leading to misconceptions about the complexities of real-world neural network implementations.
Catalytic computing, a new theoretical framework, aims to overcome the limitations of traditional computing by leveraging the entire storage capacity of a device, such as a hard drive, for computation. Instead of relying on limited working memory, catalytic computing treats the entire memory system as a catalyst, allowing data to transform itself through local interactions within the storage itself. This approach, inspired by chemical catalysts, could drastically expand the complexity and scale of computations possible, potentially enabling the efficient processing of massive datasets that are currently intractable for conventional computers. While still theoretical, catalytic computing represents a fundamental shift in thinking about computation, promising to unlock the untapped potential of existing hardware.
Hacker News users discussed the potential and limitations of catalytic computing. Some expressed skepticism about the practicality and scalability of the approach, questioning the overhead and energy costs involved in repeatedly reading and writing data. Others highlighted the potential benefits, particularly for applications involving massive datasets that don't fit in RAM, drawing parallels to memory mapping and virtual memory. Several commenters pointed out that the concept isn't entirely new, referencing existing techniques like using SSDs as swap space or leveraging database indexing. The discussion also touched upon the specific use cases where catalytic computing might be advantageous, like bioinformatics and large language models, while acknowledging the need for further research and development to overcome current limitations. A few commenters also delved into the theoretical underpinnings of the concept, comparing it to other computational models.
Summary of Comments ( 3 )
https://news.ycombinator.com/item?id=43524385
Hacker News users discuss Lehmer's algorithm, mostly focusing on its impracticality despite its mathematical elegance. Several commenters point out the exponential complexity, making it slower than trial division for realistically sized numbers. The discussion touches upon the algorithm's reliance on finding small quadratic residues, a process that becomes computationally expensive quickly. Some express interest in its historical significance and connection to other factoring methods, while others question the article's claim of it being "simple" given its actual complexity. A few users note the lack of practical applications, emphasizing its theoretical nature. The overall sentiment leans towards appreciation of the mathematical beauty of the algorithm but acknowledges its limited real-world use.
The Hacker News post titled "Lehmer's Continued Fraction Factorization Algorithm" linking to a Substack article on the same topic has generated several comments discussing various aspects of the algorithm and its historical context.
Several commenters discuss the practical efficiency of Lehmer's algorithm. One commenter points out that although it's not the fastest known factorization method, it held the record for some time and is still a relatively straightforward algorithm to understand and implement. Another clarifies that while trial division is generally faster for very small factors, continued fraction methods like Lehmer's become more efficient for composite numbers with larger factors. The discussion also touches on the computational complexity, with a commenter noting that continued fraction factorization methods, including Lehmer's, fall into a subexponential but superpolynomial runtime category.
A thread delves into the history of factoring algorithms and their relation to cryptography, highlighting that while Lehmer's algorithm isn't competitive with modern methods like the general number field sieve, it was a significant advancement in its time. The conversation then expands to include mentions of Fermat's factorization method and how these older algorithms contributed to the foundation of more sophisticated techniques.
The performance of Lehmer's algorithm relative to other historical methods is another topic of discussion. One commenter mentions the quadratic sieve as a successor to continued fraction methods, offering a substantial speed improvement. Another clarifies that while the general number field sieve is the most efficient algorithm known for very large numbers, the quadratic sieve still performs better for numbers within a certain range. This comparison provides context for how Lehmer's algorithm fits within the broader landscape of factoring algorithms.
Some comments offer practical perspectives. One points out the potential use of these older algorithms for educational purposes or as a starting point for understanding more complex methods. This echoes the sentiment that even though superseded by more powerful algorithms, Lehmer's method holds historical and pedagogical value. Finally, at least one commenter provides links to additional resources on factoring algorithms, allowing readers to further explore the topic if they wish.