[HN Gopher] Ninth Dedekind number found by two independent groups ___________________________________________________________________ Ninth Dedekind number found by two independent groups Author : beefman Score : 39 points Date : 2023-11-19 20:50 UTC (2 hours ago) (HTM) web link (www.quantamagazine.org) (TXT) w3m dump (www.quantamagazine.org) | dang wrote: | Related: | | _Ninth Dedekind number discovered: long-known problem in | mathematics solved_ - | https://news.ycombinator.com/item?id=36491677 - June 2023 (26 | comments) | | Also https://www.sciencealert.com/mathematicians-have-found- | the-n... (via https://news.ycombinator.com/item?id=38333952, but | no comments there) | nullc wrote: | All monotone functions can be represented as a tree of threshold | gates (or, more extreme, as a tree of and & or gates-- themselves | just the extreme cases for thresholds). But this doesn't result | in useful way of counting the monotone functions due to | duplicates. | | A great many monotone functions have a very natural | representation as a threshold gate or a simple composition of | threshold gates-- like a majority of three majorities. But I | wonder what monotone functions are the least threshold-like. | Unfortunately I don't know of a useful way to ask the question | because at the extreme they can all be constructed from threshold | gates, so it's not useful e.g. to ask for monotone functions that | can't be constructed from them. | | A related question might be what are the N monotone functions | that can be combined to most efficiently represent all monotone | functions up to size M? Does the selection of these best basis | functions change for different sizes? (or fall into a finite set | of groups like odd and even M?). | jacquesm wrote: | > But I wonder what monotone functions are the least threshold- | like. | | The list of primes? | non- wrote: | The last line is very humorous "Jakel said he would have written | up his thesis two years ago if not for the distraction of d(9). | "I think I need a break," he said." | | If there are any mathematicians in the comments I'd love some | layman's insight into why solving this problem is not itself | worthy of a doctoral thesis in Maths. ___________________________________________________________________ (page generated 2023-11-19 23:00 UTC)