Dear Andrew!
concerning:
Ack(n) = n[n]n.
Of course, the much better known sequence defined by the Knuth's up-arrows notation is starting by the exponentiation rank 1[3]1 = 1^1 = 1 and it defines the "Sequence of the Ackermann Numbers", something like:
An(n) = n[n+2]n.
In my opinion, the first sequence (the Ackermann Sequence, proposed by Prof. Scott Aaronson, MIT) is more compatible with the subject that we are studying. The second and better known version is strongly influenced by the Knuth's arrow notation and gives a kind of ultra-exponential sequence, completely ignoring hyperranks 1 and 2 (not to mention ... 0!).
It always goes without saying that the "Ackermann Function" is a function of two variables (attention please: not a two-valued function, because they are strictly forbidden
in this Forum), noted as:
A(0, n) = n+1
A(m, 0) = A(m-1, 1)
A(m, n) = A(m-1, A(m, n-1)).
As we can see the situation is both extremely clear and very confused, from the terminological point of view.
Unfortunately, due to other personal and family priorities, from to-day I am obliged to give much lesser time to these important and intersting subjects.
It was nice to discuss with you.
GFR
concerning:
andydude Wrote:No, the Ackermann numbers (from MathWorld) are defined as \( n[n+2]n \), because they are defined by arrow notation, and not box notation.Actually, in the Department of Engineering and Computer Science of the MIT there is a "research line" covering a rapidly increasing sequence of numbers, called "The Ackermann Sequence", defined as :
The numbers defined by \( n[n]n \) have no name.
Ack(n) = n[n]n.
Of course, the much better known sequence defined by the Knuth's up-arrows notation is starting by the exponentiation rank 1[3]1 = 1^1 = 1 and it defines the "Sequence of the Ackermann Numbers", something like:
An(n) = n[n+2]n.
In my opinion, the first sequence (the Ackermann Sequence, proposed by Prof. Scott Aaronson, MIT) is more compatible with the subject that we are studying. The second and better known version is strongly influenced by the Knuth's arrow notation and gives a kind of ultra-exponential sequence, completely ignoring hyperranks 1 and 2 (not to mention ... 0!).
It always goes without saying that the "Ackermann Function" is a function of two variables (attention please: not a two-valued function, because they are strictly forbidden
in this Forum), noted as:A(0, n) = n+1
A(m, 0) = A(m-1, 1)
A(m, n) = A(m-1, A(m, n-1)).
As we can see the situation is both extremely clear and very confused, from the terminological point of view.
Unfortunately, due to other personal and family priorities, from to-day I am obliged to give much lesser time to these important and intersting subjects.
It was nice to discuss with you.
GFR
