[HN Gopher] Complexity theory's 50-year journey to the limits of... ___________________________________________________________________ Complexity theory's 50-year journey to the limits of knowledge Author : nsoonhui Score : 39 points Date : 2023-08-18 05:07 UTC (17 hours ago) (HTM) web link (www.quantamagazine.org) (TXT) w3m dump (www.quantamagazine.org) | PartiallyTyped wrote: | Is there a missed connection between entropy and circuit | complexity? It looks like one might be able to draw some | parallels between the two. | | Other thoughts; it's interesting that PvsNP is essentially about | the lengths of proofs, and thus it is the task of deciding in | poly time whether you can decide all decidable problems whose | proofs are polynomial to their encoding. | PartiallyTyped wrote: | This is easily one of the most well written articles published by | quanta. Truly a pleasure to read. | xmonkee wrote: | Yeah, a lot of love went into this. Could be made into a book, | i would buy. | guga42k wrote: | nitpick: I wonder how TSP became NP complete. It would require | to polynomial time algorithm to validate the solution. I doubt | such algorithm does exist. | bumbledraven wrote: | > Suppose that P = NP. To prove it, researchers would need to | find a fast algorithm for an NP-complete problem, which might be | hiding in some obscure corner of that vast landscape. | | They wouldn't need to _find_ such an algorithm. They would just | need to prove that such an algorithm _exists_. | wddkcs wrote: | Can you really separate the two? For something like NP- | completeness, I'm having trouble conceptualizing how a proof | for existence would not require demonstration. | linkgoron wrote: | An example for such a proof would be using the probabilistic | method. However, even if we are ignoring some artificial | problems, there are some "natural" problems that are known to | be in P that we do not have algorithms for. | | https://en.wikipedia.org/wiki/Non- | constructive_algorithm_exi... | | Also see: | | https://cs.stackexchange.com/questions/92087/are-there- | any-p... | fooker wrote: | Yes, proofs don't have to be constructive. | | Consider proofs by contradiction, you could potentially show | that if such an algorithm does not exist some important true | statement would be rendered false. | viscountchocula wrote: | Sure. There is definitely a googolth digit of pi. Computing | what the digit is, however, is not necessary to prove that | such a digit exists. | tetrazine wrote: | This is an aside. But I twigged on a caption for one of the | figures: "Every computational problem takes longer to solve as | its input grows larger, but not all problems grow harder at the | same rate." | | It's obvious from CS201 that this phrase is generally true and I | have no pedantic objection to its inclusion in the article. | However, I'm curious if this is strictly true or if we can find a | (nontrivial) counterexample. Are there any algorithms that run | faster as input grows? My first intuition is some kind of | probabilistic question solved to a given correctness bound. | | Edit: it is trivial to construct an algorithm with this property. | My precise question is whether any meaningful problems have | optimal algorithms with this property. ___________________________________________________________________ (page generated 2023-08-18 23:00 UTC)