The Halting Problem is frequently cited as an example of an NP-hard problem, but this is misleading. While both are "hard" problems, the nature of their difficulty is fundamentally different. NP-hard problems deal with the difficulty of finding a solution among a vast number of possibilities, where verifying a given solution is relatively easy. The Halting Problem, however, is about the impossibility of determining whether a program will even finish, regardless of how long we wait. This undecidability is a stronger statement than NP-hardness, as it asserts that no algorithm can solve the problem for all inputs, not just that efficient algorithms are unknown. Using the Halting Problem to introduce NP-hardness confuses computational complexity (how long a problem takes to solve) with computability (whether a problem can even be solved). A better introductory example would be something like the Traveling Salesperson Problem, which highlights the search for an optimal solution within a large, but finite, search space.
An undergraduate student, Noah Stephens-Davidowitz, has disproven a longstanding conjecture in computer science related to hash tables. He demonstrated that "linear probing," a simple hash table collision resolution method, can achieve optimal performance even with high load factors, contradicting a 40-year-old assumption. His work not only closes a theoretical gap in our understanding of hash tables but also introduces a new, potentially faster type of hash table based on "robin hood hashing" that could improve performance in databases and other applications.
Hacker News commenters discuss the surprising nature of the discovery, given the problem's long history and apparent simplicity. Some express skepticism about the "disproved" claim, suggesting the Kadane algorithm is a more efficient solution for the original problem than the article implies, and therefore the new hash table isn't a direct refutation. Others question the practicality of the new hash table, citing potential performance bottlenecks and the limited scenarios where it offers a significant advantage. Several commenters highlight the student's ingenuity and the importance of revisiting seemingly solved problems. A few point out the cyclical nature of computer science, with older, sometimes forgotten techniques occasionally finding renewed relevance. There's also discussion about the nature of "proof" in computer science and the role of empirical testing versus formal verification in validating such claims.
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.
Sublinear time algorithms provide a way to glean meaningful information from massive datasets too large to examine fully. They achieve this by cleverly sampling or querying only small portions of the input, allowing for approximate solutions or property verification in significantly less time than traditional algorithms. These techniques are crucial for handling today's ever-growing data, enabling applications like quickly estimating the average value of elements in a database or checking if a graph is connected without examining every edge. Sublinear algorithms often rely on randomization and probabilistic guarantees, accepting a small chance of error in exchange for drastically improved efficiency. They are a vital tool in areas like graph algorithms, statistics, and database management.
Hacker News users discuss the linked resource on sublinear time algorithms, primarily focusing on its practical applications. Several commenters express surprise and interest in the concept of algorithms that don't require reading all input data, with examples like property testing and finding the median element cited. Some question the real-world usefulness, while others point to applications in big data analysis, databases, and machine learning where processing the entire dataset is infeasible. There's also discussion about the trade-offs between accuracy and speed, with some suggesting these algorithms provide "good enough" solutions for certain problems. Finally, a few comments highlight specific sublinear algorithms and their associated use cases, further emphasizing the practicality of the subject.
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.
A Brown University undergraduate, Noah Golowich, disproved a long-standing conjecture in data science related to the "Kadison-Singer problem." This problem, with implications for signal processing and quantum mechanics, asked about the possibility of extending certain "frame" functions while preserving their key properties. A 2013 proof showed this was possible in specific high dimensions, leading to the conjecture it was true for all higher dimensions. Golowich, building on recent mathematical tools, demonstrated a counterexample, proving the conjecture false and surprising experts in the field. His work, conducted under the mentorship of Assaf Naor, highlights the potential of exploring seemingly settled mathematical areas.
Hacker News users discussed the implications of the undergraduate's discovery, with some focusing on the surprising nature of such a significant advancement coming from an undergraduate researcher. Others questioned the practicality of the new algorithm given its computational complexity, highlighting the trade-off between statistical accuracy and computational feasibility. Several commenters also delved into the technical details of the conjecture and its proof, expressing interest in the specific mathematical techniques employed. There was also discussion regarding the potential applications of the research within various fields and the broader implications for data science and machine learning. A few users questioned the phrasing and framing in the original Quanta Magazine article, finding it slightly sensationalized.
This paper proposes a new quantum Fourier transform (QFT) algorithm that significantly reduces the circuit depth compared to the standard implementation. By leveraging a recursive structure and exploiting the symmetries inherent in the QFT matrix, the authors achieve a depth of O(log * n + log log n), where n is the number of qubits and log * denotes the iterated logarithm. This improvement represents an exponential speedup in depth compared to the O(log² n) depth of the standard QFT while maintaining the same asymptotic gate complexity. The proposed algorithm promises faster and more efficient quantum computations that rely on the QFT, particularly in near-term quantum computers where circuit depth is a crucial limiting factor.
Hacker News users discussed the potential impact of a faster Quantum Fourier Transform (QFT). Some expressed skepticism about the practicality due to the significant overhead of classical computation still required and questioned if this specific improvement truly addressed the bottleneck in quantum algorithms. Others were more optimistic, highlighting the mathematical elegance of the proposed approach and its potential to unlock new applications if the classical overhead can be mitigated in the future. Several commenters also debated the relevance of asymptotic complexity improvements given the current state of quantum hardware, with some arguing that more practical advancements are needed before these theoretical gains become significant. There was also a brief discussion regarding the paper's notation and clarity.
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https://news.ycombinator.com/item?id=43714041
HN commenters largely agree with the author's premise that the halting problem is a poor example for explaining NP-hardness. Many point out that the halting problem is about undecidability, a distinct concept from computational complexity which NP-hardness addresses. Some suggest better examples for illustrating NP-hardness, such as the traveling salesman problem or SAT. A few commenters argue that the halting problem is a valid, albeit confusing, example because all NP-hard problems can be reduced to it. However, this view is in the minority, with most agreeing that the difference between undecidability and intractability should be emphasized when teaching these concepts. One commenter clarifies the author's critique: it's not that the halting problem isn't NP-hard, but rather that its undecidability overshadows its NP-hardness, making it a pedagogically poor example. Another thread discusses the nuances of Turing completeness in relation to the discussion.
The Hacker News post titled "The Halting Problem is a terrible example of NP-Harder" spawned a lively discussion with several compelling comments. Many commenters agreed with the author's central thesis that the Halting Problem is a poor pedagogical tool for introducing NP-hardness. They argued that its undecidability overshadows the nuances of NP-hardness, which deals with decidable but computationally expensive problems. The inherent complexity of the Halting Problem makes it difficult for newcomers to grasp the core concepts of NP-hardness.
Several commenters suggested alternative examples that they found more effective in teaching these concepts. Suggestions included the Traveling Salesperson Problem, Sudoku, and Boolean satisfiability (SAT). These problems, while still complex, are more relatable and easier to visualize, allowing students to develop an intuitive understanding of computational complexity before delving into the abstract realm of undecidability.
Some commenters pushed back against the author's assertion. They argued that the Halting Problem, while complex, serves as a useful upper bound of computational difficulty, demonstrating that some problems are simply unsolvable by any algorithm. They believed this provides valuable context for understanding the limitations of computation.
A few commenters pointed out that the choice of example depends on the specific audience and learning objectives. For introductory courses, simpler, more concrete examples like the Traveling Salesperson are indeed preferable. However, for more advanced students, the Halting Problem could be a valuable tool for exploring the theoretical boundaries of computation.
One commenter offered a nuanced perspective, suggesting that the halting problem might be suitable after an initial introduction to NP-hardness using more accessible examples. This approach would allow students to first grasp the core concepts of NP-hardness before confronting the more abstract notion of undecidability.
The discussion also touched on the importance of clear and precise language when teaching complex topics like computational complexity. Some commenters noted that the misuse of terminology, like conflating "hard" with "impossible," can further contribute to student confusion.
Finally, a few comments explored the broader implications of the Halting Problem, connecting it to other fundamental concepts in computer science such as Gödel's incompleteness theorems.