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Yesterday I got sucked into a bingewatch of Computerphile's and Numberphile's videos on youtube. In particular I ended up watching some on Ackermann's function. While I knew already this function (and the effect it can have on mathematicians if called with Graham's number as the arguments), I did not know that it was designed to be computable only through recursion (or, at least, that is what the video I watched claims).

The thing is that in a follow-up video is shown that you can the results can be computed through exponentiation, and Wikipedia's page says that it is really easy to write doen the result using Knuth's up-arrow notation.

So here's lie my problem: [stretching for a moment the imagination and assuming that all integers are representable into a computer, avoiding the risk of overflows, that are outside the scope of this question] if it can be represented through Knuth's up-arrow notation, can't a program that does not use recursion be written to compute the value?

Is it any different for the original Ackermann's function? (the one that Wikipedia says it has 3 arguments instead of 2)

Federico
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  • How would you implement the up-arrow operation without recursion? – celtschk Aug 19 '15 at 06:50
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    You might want to have a look at http://math.stackexchange.com/questions/96483/ackermann-function-primitive-recursive , whose top answer gives a pretty good characterization of 'primitive recursive' that might offer an answer to your question. – Steven Stadnicki Aug 19 '15 at 06:50
  • @StevenStadnicki oh, thanks, I missed that one. flagged as duplicate. – Federico Aug 19 '15 at 06:54

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