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 Iterated nand tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 03/23/2017, 01:27 PM (03/12/2017, 03:08 PM)Xorter Wrote: I am interested in iterates (and uniterates) of logical operators, like nand, xor ... etc. . Let's investigate the nand op: x↑y = ¬(x&y) Iterating x O y means this: x O x O ... O x (y-times) Thus x ↑ x ↑ ... ↑ x := x ↑↑ y x ↑↑ 1 = x x ↑↑ 2 = x ↑ x = ¬x x ↑↑ 3 = x ↑ ¬x = ¬(x & ¬x) = x v ¬x x ↑↑ 4 = x ↑ (x v ¬x) = ¬(x & (x v ¬x)) = x & ¬x x ↑↑ 5 = x ↑ (x & ¬x) = ¬(x & (x & ¬x)) = x v ¬x x ↑↑ 6 = x & ¬x x ↑↑ 7 = x v ¬x ... etc. So x ↑↑ 2k = x & ¬x x ↑↑ 2k-1 = x v ¬x where k is bigger integer than 2 My question: Can k be any real or complex number? (In my view to do this, we should know what "between" & and v is.) --- I like the basic idea to do dynamics in set theory , Logic and the alike. However i see many issues. First you write alot " x and not x " and "x or not x" and they are ( in boolean ) either trivial or paradoxical ! Like " this sentense is false ". or " this is true and false " , " this is true or false. Secondly suppose we let times -1 mean not. Than iterations of not give the unit circle in the complex plane. but what does it mean " i " ? What does it mean to half-iterate NOT ? 0r the pi th iteration of x OR y ??? you only have a few things like and or not true etc. but you want continue iterations ?? or if you introduce new things like " i " above you need to Well define it !! so as of now im very skeptical for continue iterations. as for integer iterates that might work. i think one then needs to associate it with groups or rings. maybe modular arithmetic too. Regards tommy1729 the master « Next Oldest | Next Newest »

 Messages In This Thread Iterated nand - by Xorter - 03/12/2017, 03:08 PM RE: Iterated nand - by tommy1729 - 03/23/2017, 01:27 PM RE: Iterated nand - by Xorter - 03/27/2017, 06:51 PM

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