Left-bracketed Ackermann function also not primitive recursive? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T05:40:18Z http://mathoverflow.net/feeds/question/62019 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/62019/left-bracketed-ackermann-function-also-not-primitive-recursive Left-bracketed Ackermann function also not primitive recursive? bo198214 2011-04-17T10:15:05Z 2011-04-20T20:18:38Z <p>The original Ackermann function $\varphi\colon \mathbb{N}\times\mathbb{N}\times\mathbb{N}_0\to \mathbb{N}$ as defined in [1] was invented to prove that there is a function that is recursive but not primitive recursive. It can be given by the following recursion:</p> <ul> <li>$\varphi(a,b,0) = a+b$</li> <li>$\varphi(a,b,n+1) = (x\mapsto \varphi(a,x,n))^b(\alpha(a,n))$</li> </ul> <p>Where $\alpha(a,0)=0$, $\alpha(a,1)=1$ and $\alpha(a,n)=a$ for $n\ge 2$ are initial values and $(x\mapsto f(x))^k$ is the k-times composition of the function $x\mapsto f(x)$. The function $n\mapsto \varphi(n,n,n)$ is not primitive recursive because - informally speaking - it grows too quickly.</p> <p>These operations are right-bracketed, this does not matter for $\varphi(a,b,1)=ab$ and $\varphi(a,b,2)=a^b$, but for the next higher rank it is important $\varphi(a,b,3)=\underbrace{a^\land (a^\land (...(a^\land a)))}_{b+1\; \text{occurences of}\; a}$ where $a^\land b:=a^b$.</p> <p>If we would choose left-bracketing the functions would not grow so quickly. The left-bracketed operations would be defined as:</p> <ul> <li>$\psi(a,b,0) = a+b$</li> <li>$\psi(a,b,n+1) = (x\mapsto \psi(x,a,n))^b(\alpha(a,n))$</li> </ul> <p>Again $\psi(a,b,1)=ab$ and $\psi(a,b,2)=a^b$, but here the forth operation would be $\psi(a,b,3)=a^{a^b}$</p> <p>My question is now whether the left-bracketed operations still grow fast enough for not being primitive recursive, i.e. is $n\mapsto \psi(n,n,n)$ still not primitive recursive?</p> <p>[1] Ackermann, W. (1928 ). Zum Hilbertschen Aufbau der reellen Zahlen. Math. Ann., 99, 118–133.</p>