Knuth's Up-arrow Notation - Definition

Definition

The up-arrow notation is formally defined by

a\uparrow^n b= \left\{ \begin{matrix} a^b, & \mbox{if }n=1; \\ 1, & \mbox{if }b=0; \\ a\uparrow^{n-1}(a\uparrow^n(b-1)), & \mbox{otherwise} \end{matrix} \right.

for all integers with .

All up-arrow operators (including normal exponentiation, ) are right associative, i.e. evaluation is to take place from right to left in an expression that contains two or more such operators. For example, not ; for example
is not

There is good reason for the choice of this right-to-left order of evaluation. If we used left-to-right evaluation, then would equal, so that would not be an essentially new operation. Right associativity is also natural because we can rewrite the iterated arrow expression that appears in the expansion of as, so that all the s appear as left operands of arrow operators. This is significant since the arrow operators are not commutative.

Writing for the bth functional power of the function we have .

The definition could be extrapolated one step, starting with if n = 0, because exponentiation is repeated multiplication starting with 1. Extrapolating one step more, writing multiplication as repeated addition, is not as straightforward because multiplication is repeated addition starting with 0 instead of 1. "Extrapolating" again one step more, writing addition of n as repeated addition of 1, requires starting with the number a. Compare the definition of the hyper operator, where the starting values for addition and multiplication are also separately specified.

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