Stories with Tag halting problem

• Surprises in Logic (2016)

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  Posted: 2025-04-22 15:24:53

  Verbose tl;dr

  John Baez's post "Surprises in Logic" explores counterintuitive results within mathematical logic. It highlights the unexpected power of first-order logic, capable of expressing sophisticated concepts like finiteness and the natural numbers despite its seemingly simple structure. Conversely, it demonstrates limitations, such as the inability of first-order theories of the natural numbers to capture all true statements about them (Gödel's incompleteness theorem). The post emphasizes the surprising disconnect between a theory's ability to define a concept and its ability to characterize it completely, using examples like Peano arithmetic. This leads to the exploration of second-order logic and its increased expressive power, though at the cost of losing the completeness and compactness theorems enjoyed by first-order logic. The overall message is that even seemingly basic logical systems can harbor deep and often unintuitive complexities.

  John Baez's 2016 blog post, "Surprises in Logic," delves into the fascinating realm of logical systems and their sometimes counterintuitive properties. The post begins by acknowledging the common perception of logic as a dry, rigid field, yet argues that this view overlooks the rich tapestry of unexpected results and intriguing paradoxes that lie beneath the surface. Baez sets the stage by highlighting the fundamental role of logic in mathematics, computer science, and philosophy, emphasizing its power to formalize reasoning and deduce truths from given premises.

  He then proceeds to introduce several examples of "surprises" within logic, starting with the Löwenheim–Skolem theorem. This theorem, Baez explains, asserts that if a first-order theory with an infinite model has a model of any infinite cardinality. This can lead to counterintuitive situations where a theory intended to describe, say, the real numbers, also has models containing elements beyond the real numbers. This challenges our naive intuition about the power of first-order logic to uniquely pin down specific mathematical structures.

  The exploration continues with Gödel's completeness theorem, a cornerstone of mathematical logic. Baez explains how this theorem establishes a fundamental connection between syntactic provability and semantic truth in first-order logic. In essence, it states that any statement that is true in all models of a theory can be formally proven within that theory. This powerful result demonstrates the remarkable expressive power of first-order logic.

  However, the narrative takes a dramatic turn with the introduction of Gödel's incompleteness theorems. These theorems, Baez explains, reveal inherent limitations in formal systems powerful enough to encompass arithmetic. The first incompleteness theorem demonstrates that any consistent formal system capable of expressing basic arithmetic will contain true statements that are unprovable within the system. The second incompleteness theorem builds upon this by showing that such a system cannot prove its own consistency. These theorems shattered the hopes of establishing a complete and consistent axiomatization of all of mathematics, as envisioned by Hilbert's program.

  Baez further elaborates on the ramifications of Gödel's incompleteness theorems, highlighting their philosophical implications for the nature of truth and provability. He emphasizes the distinction between truth and provability, demonstrating how a statement can be true even if it cannot be formally proven within a given system.

  The post concludes with a reflection on the ongoing quest to understand the foundations of mathematics and logic. Baez emphasizes the dynamic nature of the field, with ongoing research continuously revealing new insights and challenges. He encourages readers to embrace the surprises and paradoxes of logic, recognizing them not as flaws but as opportunities to deepen our understanding of the intricate workings of formal systems and the very nature of reasoning itself.

  • Logic
  • mathematics
  • mathematical logic
  • paradoxes
  • Set Theory
  • foundations of mathematics
  • philosophy of mathematics
  • Godel
  • incompleteness theorem
  • Turing
  • halting problem
  • computability theory
  • surprises
  • 2016

  Summary of Comments ( 6 )
  https://news.ycombinator.com/item?id=43763291

  Verbose tl;dr

  Hacker News users discuss various aspects of the surprises in mathematical logic presented in the linked article. Several commenters delve into the implications of Gödel's incompleteness theorems, with some highlighting the distinction between truth and provability. The concept of "surprising" itself is debated, with some arguing that the listed examples are well-known within the field and therefore not surprising to experts. Others point out the connection between logic and computation, referencing Turing machines and the halting problem. The role of axioms in shaping mathematical systems is also mentioned, alongside the challenge of finding "natural" axioms that accurately reflect our intuitive understanding of mathematics. A few commenters express appreciation for the article's clear explanations of complex topics.

  The Hacker News post titled "Surprises in Logic (2016)" linking to John Baez's article has generated several comments discussing various aspects of logic, set theory, and their implications.

  One commenter highlights the significance of Löwenheim-Skolem theorem, which states that if a first-order theory has an infinite model, then it has a model of every infinite cardinality. They explain how this theorem can be counterintuitive, especially when applied to theories intended to describe a unique structure, like the real numbers. They suggest it implies there are "countable models of set theory that think they contain uncountable sets," a concept they find fascinating and paradoxical.

  Another comment dives into the implications of Gödel's incompleteness theorems, specifically focusing on their impact on Hilbert's program. They mention how Gödel's work demonstrated the inherent limitations of formal systems in proving all true statements within a given set of axioms. This commenter further connects this to the concept of truth being "larger" than provability, emphasizing that there will always be true statements that are unprovable within a given formal system.

  Further discussion revolves around the nature of infinity and its various interpretations within set theory. One comment clarifies the distinction between countable and uncountable infinities, using the analogy of integers versus real numbers. They point out that while both sets are infinite, the real numbers are "denser" than the integers, leading to the concept of uncountability.

  The conversation also touches upon the implications of the Axiom of Choice, a fundamental principle in set theory that allows for making infinitely many arbitrary choices. Some comments express how counterintuitive this axiom can be, even though it's necessary for many important mathematical theorems. They mention how it can lead to seemingly paradoxical results, like the Banach-Tarski paradox, which demonstrates how a sphere can be decomposed and reassembled into two spheres identical to the original.

  A few commenters also delve into the philosophical implications of these mathematical concepts, questioning the nature of mathematical truth and its relationship to reality. They discuss whether mathematical structures are discovered or invented, and whether the limitations of formal systems reflect limitations on our ability to understand the universe.

  Finally, some comments offer additional resources for exploring these topics further, including links to relevant Wikipedia pages, books, and online lectures. These recommendations provide avenues for those interested in gaining a deeper understanding of the discussed concepts.

• The Halting Problem is a terrible example of NP-Harder

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  Posted: 2025-04-17 07:34:08

  Verbose tl;dr

  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.

  Hillel Wayne's blog post, "The Halting Problem is a terrible example of NP-Hard," argues that while technically correct, classifying the Halting Problem as NP-hard is misleading and pedagogically unhelpful, especially for those first learning about computational complexity. The core issue lies in the vastly different natures of the Halting Problem and typical NP-hard problems, which obscures the practical implications of NP-hardness.

  Wayne begins by acknowledging that the Halting Problem is technically NP-hard under the strictest definition. Given a magical oracle that could instantly solve any problem in NP, one could theoretically use it to solve the Halting Problem. Constructing a specific instance of an NP problem (like SAT) that encodes the behavior of a given Turing machine and then querying the oracle about its satisfiability would reveal whether the Turing machine halts. Therefore, the Halting Problem meets the criteria for NP-hardness.

  However, the post emphasizes that this technical correctness misses the practical significance of NP-hardness. NP-hard problems are typically characterized by their exponential growth in computational complexity as the input size increases. This makes them practically unsolvable for sufficiently large inputs, necessitating approximations and heuristics. The Halting Problem, on the other hand, is undecidable – meaning there is no algorithm, regardless of its complexity, that can solve it for all possible inputs. This inherent unsolvability is a fundamentally different kind of difficulty than the practical intractability of NP-hard problems.

  Furthermore, the reduction used to prove the Halting Problem's NP-hardness relies on a hypothetical, all-powerful oracle for NP problems. This is unlike typical NP-hardness reductions, which demonstrate relationships between realistically solvable (though computationally expensive) problems. These reductions allow us to understand the relative difficulty of problems within NP and to leverage existing algorithms and heuristics. The reduction used for the Halting Problem provides no such practical insights or algorithmic leverage.

  The post also addresses the common misconception that NP-hardness implies exponential runtime. While many NP-hard problems do exhibit exponential behavior, this is not a defining characteristic. The Halting Problem, being undecidable, doesn't even have a defined runtime since it can never be solved algorithmically. This further reinforces the idea that categorizing the Halting Problem as NP-hard obfuscates the key features of both NP-hardness and undecidability.

  In conclusion, Wayne contends that while technically accurate, classifying the Halting Problem as NP-hard is a poor pedagogical choice. It confuses the practical implications of NP-hardness with the absolute unsolvability of undecidable problems. This confusion can hinder a true understanding of computational complexity, especially for learners encountering these concepts for the first time. A more effective approach would be to treat the Halting Problem as a separate category of difficulty, emphasizing its unique nature and avoiding potentially misleading comparisons to NP-hard problems.

  • Computer Science
  • computability theory
  • complexity theory
  • halting problem
  • NP-hard
  • NP-complete
  • Turing machines
  • decidability
  • undecidability
  • reductions
  • Theoretical Computer Science
  • algorithms

  Summary of Comments ( 74 )
  https://news.ycombinator.com/item?id=43714041

  Verbose tl;dr

  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.