Conway chained arrow notation


Conway chained arrow notation, created by mathematician John Horton Conway, is a means of expressing certain extremely large numbers. It is simply a finite sequence of positive integers separated by rightward arrows, e.g..
As with most combinatorial notations, the definition is recursive. In this case the notation eventually resolves to being the leftmost number raised to some integer power.

Definition and overview

A "Conway chain" is defined as follows:
Any chain represents an integer, according to the five rules below. Two chains are said to be equivalent if they represent the same integer.
If,, and are positive integers, and is a subchain, then:
  1. An empty chain is equal to, and the chain represents the number.
  2. is equivalent to.
  3. is equivalent to.
  4. is equivalent to
.
  1. Because is equivalent to, and also to = , we can define to equal
Note that the fourth rule can be replaced by repeatedly applying two rules to avoid the ellipses:

Properties

  1. A chain evaluates to a perfect power of its first number
  2. Therefore, is equal to
  3. is equivalent to
  4. is equal to
  5. is equivalent to

    Interpretation

One must be careful to treat an arrow chain as a whole. Arrow chains do not describe the iterated application of a binary operator. Whereas chains of other infixed symbols can often be considered in fragments + 5 + ) without a change of meaning, or at least can be evaluated step by step in a prescribed order, e.g. 34567 from right to left, that is not so with Conway's arrow chains.
For example:
The fourth rule is the core: A chain of 4 or more elements ending with 2 or higher becomes a chain of the same length with a increased penultimate element. But its ultimate element is decremented, eventually permitting the second rule to shorten the chain. After, to paraphrase Knuth, "much detail", the chain is reduced to three elements and the third rule terminates the recursion.

Examples

Examples get quite complicated quickly. Here are some small examples:

Systematic examples

The simplest cases with four terms are:
We can see a pattern here. If, for any chain, we let then .
Applying this with, then and
Thus, for example,.
Moving on:
Again we can generalize. When we write we have, that is,. In the case above, and, so

Ackermann function

The Ackermann function may be expressed using Conway chained arrow notation:
hence

Graham's number

itself cannot be expressed concisely in Conway chained arrow notation, but it is bounded by the following:
Proof: We first define the intermediate function, which can be used to define Graham's number as.
By applying rule 2 and rule 4 backwards, we simplify:
Since f is strictly increasing,
which is the given inequality.
With chained arrows, it is very easy to specify a number much greater than, for example,.
which is much greater than Graham's number, because the number is much greater than.

CG function

Conway and Guy created a simple, single-argument function that diagonalizes over the entire notation, defined as:
meaning the sequence is:
...
This function, as one might expect, grows extraordinarily fast.

Extension by Peter Hurford

Peter Hurford, a web developer and statistician, has defined an extension to this notation:
All normal rules are unchanged otherwise.
is already equal to the aforementioned, and the function is much faster growing than Conway and Guy's.
Note that expressions like are illegal if and are different numbers; one chain must only have one type of right-arrow.
However, if we modify this slightly such that:
then not only does become legal, but the notation as a whole becomes much stronger.