[HN Gopher] Ninth Dedekind number found by two independent groups
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       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.
        
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