[1] In his 1947 paper,[2] R. L. Goodstein introduced the specific sequence of operations that are now called hyperoperations.
Goodstein also suggested the Greek names tetration, pentation, etc., for the extended operations beyond exponentiation.
The sequence starts with a unary operation (the successor function with n = 0), and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n = 3), tetration (n = 4), pentation (n = 5), etc.
For example: The general definition of the up-arrow notation is as follows (for
The hyperoperations naturally extend the arithmetic operations of addition and multiplication as follows.
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, 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.
But many environments — such as programming languages and plain-text e-mail — do not support superscript typesetting.
doesn't lend itself well to generalization, which explains why Knuth chose to work from the inline notation
using the familiar superscript notation gives a power tower.
could be written with a stack of such power towers, each describing the size of the one above it.
is a variable or is too large, the stack might be written using dots and a note indicating its height.
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
Finally, as an example, the fourth Ackermann number
could be represented as: 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.
(Petillion) Even faster-growing functions can be categorized using an ordinal analysis called the fast-growing hierarchy.
is comparable to tetrational growth and is upper-bounded by a function involving the first four hyperoperators;.
is already beyond the reach of indexed arrows but can be used to approximate Graham's number, and
is comparable to arbitrarily-long Conway chained arrow notation.
Even faster computable functions, such as the Goodstein sequence and the TREE sequence require the usage of large ordinals, may occur in certain combinatorical and proof-theoretic contexts.
There exist functions which grow uncomputably fast, such as the Busy Beaver, whose very nature will be completely out of reach from any up-arrow, or even any ordinal-based analysis.
Without reference to hyperoperation the up-arrow operators can be formally defined by for all integers
This is equivalent to the hyperoperation sequence except it omits the three more basic operations of succession, addition and multiplication.
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,
in the top row, and fill the left column with values 2.
in the top row, and fill the left column with values 3.
in the top row, and fill the left column with values 4.
in the top row, and fill the left column with values 10.