Ackermann Function - Table of Values

Table of Values

Computing the Ackermann function can be restated in terms of an infinite table. We place the natural numbers along the top row. To determine a number in the table, take the number immediately to the left, then look up the required number in the previous row, at the position given by the number just taken. If there is no number to its left, simply look at the column headed "1" in the previous row. Here is a small upper-left portion of the table:

Values of A(m, n)
m\n 0 1 2 3 4 n
0 1 2 3 4 5
1 2 3 4 5 6
2 3 5 7 9 11
3 5 13 29 61 125
4 13

=
65533

=
265536 − 3

=


=


=

The numbers listed here in a recursive reference are very large and cannot be easily notated in some other form.

Despite the large values occurring in this early section of the table, some even larger numbers have been defined, such as Graham's number, which cannot be written with any small number of Knuth arrows. This number is constructed with a technique similar to applying the Ackermann function to itself recursively.

This is a repeat of the above table, but with the values replaced by the relevant expression from the function definition to show the pattern clearly:

Values of A(m, n)
m\n 0 1 2 3 4 n
0 0+1 1+1 2+1 3+1 4+1
1 A(0,1) A(0,A(1,0)) A(0,A(1,1)) A(0,A(1,2)) A(0,A(1,3))
2 A(1,1) A(1,A(2,0)) A(1,A(2,1)) A(1,A(2,2)) A(1,A(2,3))
3 A(2,1) A(2,A(3,0)) A(2,A(3,1)) A(2,A(3,2)) A(2,A(3,3))
4 A(3,1) A(3,A(4,0)) A(3,A(4,1)) A(3,A(4,2)) A(3,A(4,3))

5 A(4,1) A(4,A(5,0)) A(4,A(5,1)) A(4,A(5,2)) A(4,A(5,3))

A(4, A(5, n-1))

6 A(5,1) A(5,A(6,0)) A(5,A(6,1)) A(5,A(6,2)) A(5,A(6,3))

A(5, A(6, n-1))

Read more about this topic:  Ackermann Function

Famous quotes containing the words table and/or values:

    The salt person and blasted place
    I furnish with the meat of a fable;
    If the dead starve, their stomachs turn to tumble
    An upright man in the antipodes
    Or spray-based and rock-chested sea:
    Over the past table I repeat this present grace.
    Dylan Thomas (1914–1953)

    Any relation to the land, the habit of tilling it, or mining it, or even hunting on it, generates the feeling of patriotism. He who keeps shop on it, or he who merely uses it as a support to his desk and ledger, or to his manufactory, values it less.
    Ralph Waldo Emerson (1803–1882)