yes... there's also another definition (recursive) called hyperoperation. https://en.wikipedia.org/wiki/Hyperoperation
2015-02-18 18:20 GMT+01:00 R.E. Boss <[email protected]>: > Link? > > Notice that Conway (who else?) in The Book of Numbers wrote a generalization > of Knuth's up-notation (actually the Ackermann notation), his chained arrow > notation. > > > R.E. Boss > > >> -----Original Message----- >> From: [email protected] [mailto:programming- >> [email protected]] On Behalf Of John Baker >> Sent: woensdag 18 februari 2015 16:15 >> To: [email protected] >> Subject: Re: [Jprogramming] Fwd: Hello all! >> >> Very slick. I was just reading Scott Aronson's fine blog post about the > Busy >> Beaver problem and he commented on Knuth's up up notation. If anyone's >> interested in very large numbers Aronson's post is a superb overview. >> >> Sent from my iPhone >> >> > On Feb 17, 2015, at 3:05 PM, Jose Mario Quintana >> <[email protected]> wrote: >> > >> > This a way to produce numbers using the Knuth up arrow notation in J: >> > >> > Knuth=. &* NB. (adv) >> > up=. &1 NB. (adv) >> > >> > 2x Knuth up up 4 5 >> > 65536 >> > >> 2003529930406846464979072351560255750447825475569751419265016973710 >> 8940595563114530895061308809333481010382343429072631818229493821188 >> 1266886950636476154702916504187191635158796634721944293092798208430 >> 9104855990570159318959639524863372367203002916969... >> > >> > # @: ": 2x Knuth up up 5 >> > 19729 >> > >> > 6x Knuth up up 3 >> > >> 2659119772153226779682489404387918594905342200269924300660432789497 >> 0735598738829091213422929061755830324406828265067234256016357755902 >> 7938964261261109302039893034777446061389442537960087466214788422902 >> 2133853819192905427915750759274952935109319020362271989... >> > #@: ": 6x Knuth up up 3 >> > 36306 >> > >> > 3x Knuth up up up 0 1 2 >> > 1 3 7625597484987 >> > >> > 2x Knuth up up 6 NB. It is toooooooooooo big! >> > >> > On Tue, Feb 17, 2015 at 2:23 PM, Fausto Saporito >> <[email protected]> >> > wrote: >> > >> >> Hello, >> >> >> >> yes the number is very big, but why if I don't use the extended >> >> precision I have "infinity" as result, and if I use it I got an error >> >> ? >> >> >> >> I should get infinity anyways. >> >> >> >> this is my J session: >> >> >> >> ^/ 2 2 2 2 >> >> >> >> 65536 >> >> >> >> ^/ 2 2 2 2 2 NB. do not use extended precision and I have "+inf" >> >> >> >> _ >> >> >> >> ^/ 2 2 2 2 2 2 NB. do not use extended precision and I have "+inf" >> >> >> >> _ >> >> >> >> ^/ x: 2 2 2 2 2 NB. using extended precision I have the result (part > of >> >> it) >> >> >> >> >> >> >> 2003529930406846464979072351560255750447825475569751419265016973710 >> 8940595563114530895061308809333481010382343429072631818229493821188 >> 1266886950636476154702916504187191635158796634721944293092798208430 >> 9104855990570159318959639524863372367203002916969592156... >> >> >> >> ^/ x: 2 2 2 2 2 2 NB. using extended precision I have error... not > "+inf" >> >> >> >> |limit error >> >> >> >> | ^/x:2 2 2 2 2 2 >> >> >> >> https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation >> >> >> >> thanks, >> >> Fausto >> >> >> >> 2015-02-17 18:55 GMT+01:00 'Pascal Jasmin' via Programming >> >> <[email protected]>: >> >>> 2 ^. ^/ 5 # 2x >> >>> 65536 >> >>> >> >>> so at just 5, it is a 65k bit number >> >>> >> >>> at 6, the 2log of that number would be that 65kbit number. The number >> >> of atoms in the universe is an 80 bit number. >> >>> >> >>> >> >>> >> >>> >> >>> >> >>> ----- Original Message ----- >> >>> From: Raul Miller <[email protected]> >> >>> To: Programming forum <[email protected]> >> >>> Cc: >> >>> Sent: Tuesday, February 17, 2015 12:32 PM >> >>> Subject: Re: [Jprogramming] Fwd: Hello all! >> >>> >> >>> I would guess that the number you are generating is too big to be >> >>> represented using J's data structures (which would also suggest that >> >>> it would be too big to fit into memory). >> >>> >> >>> Thanks, >> >>> >> >>> -- >> >>> Raul >> >>> >> >>> On Tue, Feb 17, 2015 at 12:12 PM, Fausto Saporito >> >>> <[email protected]> wrote: >> >>>> HI! >> >>>> >> >>>> I'm a new J user with a little experience of APL and LISP. >> >>>> >> >>>> In these days I'm playing with big numbers... very big indeed, and I >> >>>> found a bug (?) in the exteded precision implementation of J. >> >>>> >> >>>> I'm not sure if I can call it a bug, but if I use the standard >> >>>> precision number I got a "infinity" as result... as should be. >> >>>> >> >>>> I'm talking about knuth-up-arrow notation, to build the "tower of >> >>>> power". In J the syntax is amazingly simple : ^/ 2 2 2 2 >> >>>> >> >>>> 2^^4 is 2 * (2* (2* 2)) = 65536 >> >>>> >> >>>> Now 2^^5 is _ with standard precision... but if I use x: (i.e. ^/ x: >> >>>> 2 2 2 2 2) can get most of number... it's quite big indeed. >> >>>> >> >>>> The problem arises with 2^^6 or 3^^4 I get "limit error" instead of _ >> >> ... why ? >> >>>> >> >>>> Is it an expected behaviour ? >> >>>> >> >>>> thanks in advance, >> >>>> Fausto >> >>>> > ---------------------------------------------------------------------- >> >>>> For information about J forums see >> http://www.jsoftware.com/forums.htm >> >>> >> >>> ---------------------------------------------------------------------- >> >>> For information about J forums see >> http://www.jsoftware.com/forums.htm >> >>> ---------------------------------------------------------------------- >> >>> For information about J forums see >> http://www.jsoftware.com/forums.htm >> >> ---------------------------------------------------------------------- >> >> For information about J forums see >> http://www.jsoftware.com/forums.htm >> > ---------------------------------------------------------------------- >> > For information about J forums see >> http://www.jsoftware.com/forums.htm >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
