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.

Read more about this topic:  Knuth's Up-arrow Notation

Famous quotes containing the word definition:

    “Mothers often are too easily intimidated by their children’s negative reactions...When the child cries or is unhappy, the mother reads this as meaning that she is a failure. This is why it is so important for a mother to know...that the process of growing up involves by definition things that her child is not going to like. Her job is not to create a bed of roses, but to help him learn how to pick his way through the thorns.”
    —Elaine Heffner (20th century)

    “Was man made stupid to see his own stupidity?
    Is God by definition indifferent, beyond us all?
    Is the eternal truth man’s fighting soul
    Wherein the Beast ravens in its own avidity?”
    —Richard Eberhart (b. 1904)

    “The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
    —Jean Baudrillard (b. 1929)