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.
A Brown University undergraduate, Noah Solomon, disproved a long-standing conjecture in data science known as the "conjecture of Kahan." This conjecture, which had puzzled researchers for 40 years, stated that certain algorithms used for floating-point computations could only produce a limited number of outputs. Solomon developed a novel geometric approach to the problem, discovering a counterexample that demonstrates these algorithms can actually produce infinitely many outputs under specific conditions. His work has significant implications for numerical analysis and computer science, as it clarifies the behavior of these fundamental algorithms and opens new avenues for research into improving their accuracy and reliability.
Hacker News commenters generally expressed excitement and praise for the undergraduate student's achievement. Several questioned the "40-year-old conjecture" framing, pointing out that the problem, while known, wasn't a major focus of active research. Some highlighted the importance of the mentor's role and the collaborative nature of research. Others delved into the technical details, discussing the specific implications of the findings for dimensionality reduction techniques like PCA and the difference between theoretical and practical significance in this context. A few commenters also noted the unusual amount of media attention for this type of result, speculating about the reasons behind it. A recurring theme was the refreshing nature of seeing an undergraduate making such a contribution.
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.
A new algorithm for the "pancake sorting problem" — sorting a disordered stack by repeatedly flipping sections of it — has achieved near-optimal efficiency. While the minimal number of flips required to sort any stack remains unknown, the new algorithm, developed by researchers at MIT and other institutions, guarantees completion within 1.375 times the theoretical minimum. This represents a significant improvement over previous algorithms, edging closer to a perfect solution for a problem that has puzzled computer scientists for decades. The researchers employed a recursive strategy that breaks down large stacks into smaller, more manageable substacks, optimizing the flipping process and setting a new benchmark for pancake sorting efficiency.
Hacker News users discussed the practicality and significance of the new book-sorting algorithm. Some questioned the real-world applicability given the specialized constraints, like pre-sorted sections and a single robot arm. Others debated the definition of "perfection" in sorting, pointing out that minimizing the arm's travel distance might not be the only relevant metric. The algorithm's novelty and mathematical elegance were acknowledged, but skepticism remained about its potential impact beyond theoretical computer science. Several commenters highlighted the existing highly optimized solutions for real-world sorting problems and suggested that this new algorithm is more of an interesting theoretical exercise than a practical breakthrough. There was also discussion about the difference between this algorithm and existing techniques like Timsort, with some arguing the new algorithm addresses a distinctly different problem.
Summary of Comments ( 74 )
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.