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Mary Star
Mary Star
1:22 AM
I have also an other question... Are you familiar with Knuth's up-arrow notation?? @Axoren
 
Axoren
Axoren
1:40 AM
Yes.
The Rozsa Peter one looks right.
Normally, it's written as a piece-wise function.
 
Mary Star
Mary Star
Ok...

Knuth's up-arrow notation defined as followed:

$a\uparrow b=a^b$

$a \uparrow \uparrow b=a^{a^{a^{ .^{.^{.^{a}}}}}}$

In general, $a \uparrow^m=a \uparrow^{m-1} a \uparrow^{m-1} \dots \uparrow^{m-1}a$

Right??
The idea is that the function at level k+1 is a repeated application of the function at level k, right?? @Axoren
 
Axoren
Axoren
2:07 AM
Correct. It's used to denote hyper-binary operations
But normally, you use the up arrow notation as a binary operation, not a unary operation
For example, when you did $a \uparrow \uparrow b$, that's actually $a$ taken to the $a$ power, $b$ times.
So, the general case would be $a \uparrow^n b = a \uparrow_1^{n-1} a \uparrow_2^{n-1} ... \uparrow_b^{n-1} a$
Specifically $b$ copies of the earlier hyper operator.
I used subscript on the arrows just so it was clear there were $b$ of them, but note that that's a non-standard notation.
If you do want to stray into hyper-operators, you can look into infinite tetration.
Which is $x^{x^{x^{.^{.^.}}}}$
Infinitely.
For example, it's possible to solve equations involving infinite tetration.
$f(x) = x^{x^{x^{.^{.^.}}}}$
$$f(x) = x^{x^{x^{.^{.^.}}}} = c \\ x^{x^{x^{.^{.^.}}}} = x^c = c \\ x = \sqrt[c]{c}$$
In the second step, you raise $x$ to both sides and the left side remains an infinite tetration of $x$.
It all depends on how strictly you need to adhere to presenting the Ackermann Function
 
 
7 hours later…
Mary Star
Mary Star
9:41 AM
Ahaa.. Ok!!
Could you explain further to me how Knuth's up-arrow notation is related to the Ackermann's function?? @Axoren
 
 
7 hours later…
Mary Star
Mary Star
4:43 PM
For the original version of the Ackermann's function $\phi$ we have the following:

$$\phi(m,n,0)=m + n \\ \phi (m,n,1)=m \cdot n \\ \phi (m,n,2)=m^n$$


The idea of Knuth's up-arrow notatuion is the following:

addition is a repeated addition of 1
multiplication is repeated addition
exponentiation is repeated multiplication

$$a \cdot b=a+a+ \dots +a \ \ \ \ (b \text{ Kopien von } a) \\ a \uparrow b=a^b =a \cdot a \cdot \dots \cdot a \ \ \ \ (b \text{ Kopien von } a) $$


So, does this definition has something in common with the definition of the Ackermann's function ?? @Axoren
 

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