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How to show that the inverse of a primitive recursive permutation of $\mathbb{N}$ is not necessarily a primitive recursive function?

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    $\begingroup$ Does it necessarily have to be a bijection? The inverse of the Ackermann function is known to be primitive recursive (since Ackermann has a primitive recursive graph). (I guess maybe you can pad this out into a primitive recursive bijection too). $\endgroup$
    – Anupam Das
    May 9 at 14:15
  • $\begingroup$ @AnupamDas: yes. It is easy to show that recursive permutations form a group under composition. But primitive recursive permutations do not; that is why I would like to show (or see a proof...) that the inverse of a primitive recursive permutation might be non-primitive recursive. Thank you for pointing out that property of Ackermann function. $\endgroup$
    – ijon
    May 10 at 9:30
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    $\begingroup$ I had not seen this before, but a quick search led to descriptions of (essentially the same) counterexamples here (math.stackexchange.com/questions/2975305/…) and here (ncatlab.org/nlab/show/partial+recursive+function#inverse). $\endgroup$ May 10 at 20:11
  • $\begingroup$ @NoamZeilberger Thanks! $\endgroup$
    – ijon
    May 14 at 20:32

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Given a set $X\not\ni 0$, let $\pi_X$ be the permutation of $\mathbb{N}$ defined as follows: $$\pi_X(x)=\begin{cases} x+1 & \mbox{ if }x\not\in X,\\ \max(\{y\in X: y<x\}\cup \{0\}) & \mbox{ if }x\in X.\\ \end{cases}$$

This basically breaks $\mathbb{N}$ into finite loops. Now if the characteristic function of $X$ is primitive recursive, so is the function $\pi_X$. However, if $X$ is extremely sparse then $\pi_X^{-1}$ will fail to be primitive recursive since it will have to grow extremely quickly.

In particular, if we let $X$ be the range of the Ackermann function, we get $\pi_X$ primitive recursive but $\pi_X^{-1}$ not primitive recursive.

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