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.
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.