Various notations have been used to represent hyperoperations. One such notation is . Another notation is , an infix notation which is convenient for ASCII. The notation is known as 'square bracket notation'.
Knuth's up-arrow notation is an alternative notation. It is obtained by replacing in the square bracket notation by arrows.
For example:
the single arrow represents exponentiation (iterated multiplication)
the double arrow represents tetration (iterated exponentiation)
the triple arrow represents pentation (iterated tetration)
The general definition of the up-arrow notation is as follows (for ):
Exponentiation for a natural power is defined as iterated multiplication, which Knuth denoted by a single up-arrow:
For example,
Tetration is defined as iterated exponentiation, which Knuth denoted by a “double arrow”:
For example,
Expressions are evaluated from right to left, as the operators are defined to be right-associative.
According to this definition,
etc.
This already leads to some fairly large numbers, but the hyperoperator sequence does not stop here.
Pentation, defined as iterated tetration, is represented by the “triple arrow”:
Hexation, defined as iterated pentation, is represented by the “quadruple arrow”:
and so on. The general rule is that an -arrow operator expands into a right-associative series of ()-arrow operators. Symbolically,
Examples:
Notation
In expressions such as , the notation for exponentiation is usually to write the exponent as a superscript to the base number . But many environments — such as programming languages and plain-text e-mail — do not support superscript typesetting. People have adopted the linear notation for such environments; the up-arrow suggests 'raising to the power of'. If the character set does not contain an up arrow, the caret (^) is used instead.
The superscript notation doesn't lend itself well to generalization, which explains why Knuth chose to work from the inline notation instead.
is a shorter alternative notation for n uparrows. Thus .
Knuth's arrows have become quite popular, maybe because is a stronger logo than for instance .[original research?]
Writing out up-arrow notation in terms of powers
Attempting to write using the familiar superscript notation gives a power tower.
For example:
If b is a variable (or is too large), the power tower might be written using dots and a note indicating the height of the tower.
Continuing with this notation, could be written with a stack of such power towers, each describing the size of the one above it.
Again, if b is a variable or is too large, the stack might be written using dots and a note indicating its height.
Furthermore, might be written using several columns of such stacks of power towers, each column describing the number of power towers in the stack to its left:
And more generally:
This might be carried out indefinitely to represent as iterated exponentiation of iterated exponentiation for any a, n and b (although it clearly becomes rather cumbersome).
Using tetration
The tetration notation allows us to make these diagrams slightly simpler while still employing a geometric representation (we could call these tetration towers).
Finally, as an example, the fourth Ackermann number could be represented as:
Generalizations
Some numbers are so large that multiple arrows of Knuth's up-arrow notation become too cumbersome; then an n-arrow operator is useful (and also for descriptions with a variable number of arrows), or equivalently, hyper operators.
Some numbers are so large that even that notation is not sufficient. The Conway chained arrow notation can then be used: a chain of three elements is equivalent with the other notations, but a chain of four or more is even more powerful.
Definition
Without reference to Hyperoperation the up-arrow operators can be formally defined by
One can alternatively choose multiplication as the base case and iterate from there. Then exponentiation becomes repeated multiplication. The formal definition would be
for all integers with .
Remark however that Knuth did not define the "nil-arrow" (). One could extend the notation to negative indices (n ≥ -2) in such a way as to agree with the entire hyperoperation sequence, except for the lag in the indexing:
The up-arrow operation is a right-associative operation, that is, is understood to be , instead of . If ambiguity is not an issue parentheses are sometimes dropped.
Computing can be restated in terms of an infinite table. We place the numbers in the top row, and fill the left column with values 2. 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.
The table is the same as that of the Ackermann function, except for a shift in and , and an addition of 3 to all values.
Computing 3↑nb
We place the numbers in the top row, and fill the left column with values 3. 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.
We place the numbers in the top row, and fill the left column with values 4. 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.
We place the numbers in the top row, and fill the left column with values 10. 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.
For 2 ≤ b ≤ 9 the numerical order of the numbers is the lexicographical order with n as the most significant number, so for the numbers of these 8 columns the numerical order is simply line-by-line. The same applies for the numbers in the 97 columns with 3 ≤ b ≤ 99, and if we start from n = 1 even for 3 ≤ b ≤ 9,999,999,999.
^R. L. Goodstein (Dec 1947). "Transfinite Ordinals in Recursive Number Theory". Journal of Symbolic Logic. 12 (4): 123–129. doi:10.2307/2266486. JSTOR2266486.