Conway chained arrow notation

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Conway chained arrow notation, created by mathematician John Horton Conway, is a means of expressing certain extremely large numbers.[1] 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 (usually enormous) integer power.

Definition and overview[edit]

A "Conway chain" is defined as follows:

  • Any positive integer is a chain of length .
  • A chain of length n, followed by a right-arrow → and a positive integer, together form a chain of length .

Any chain represents an integer, according to the five (technically four) rules below. Two chains are said to be equivalent if they represent the same integer.

Let denote positive integers and let denote the unchanged remainder of the chain. Then:

  1. An empty chain (or a chain of length 0) is equal to
  2. The chain represents the number .
  3. The chain represents the number .
  4. The chain represents the number (see Knuth's up-arrow notation)
  5. The chain represents the same number as the chain
  6. Else, the chain represents the same number as the chain .

Properties[edit]

  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 (not to be confused with )

Interpretation[edit]

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 (e.g. 3 + 4 + 5 + 6 + 7) can often be considered in fragments (e.g. (3 + 4) + 5 + (6 + 7)) without a change of meaning (see associativity), 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 (usually vastly) 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[edit]

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

(By rule 1)

(By rule 5)
Thus,

(By rule 3)

(By rule 3)
(see Knuth's up arrow notation)

(By rule 3)
(see tetration)

(By rule 4)
(By rule 5)
(By rule 2)
(By rule 3)
= much larger than previous number

(By rule 4)
(By rule 5)
(By rule 2)
(By rule 3)
= much, much larger than previous number

Systematic examples[edit]

The simplest cases with four terms (containing no integers less than 2) are:





(equivalent to the last-mentioned property)






We can see a pattern here. If, for any chain , we let then (see functional powers).

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[edit]

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

for (Since in hyperoperation)

hence

for
( and would correspond with and , which could logically be added).

Graham's number[edit]

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 . (The superscript 64 denotes a functional power.)

By applying rule 2 and rule 4 backwards, we simplify:

(with 64 's)

(with 64 's)

(with 64 's)
(with 65 's)
(computing as above).

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[edit]

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[edit]

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.[2]

See also[edit]

References[edit]

  1. ^ John H. Conway & Richard K. Guy, The Book of Numbers, 1996, p.59-62
  2. ^ "Large Numbers, Part 2: Graham and Conway - Greatplay.net". archive.is. 2013-06-25. Archived from the original on 2013-06-25. Retrieved 2018-02-18.

External links[edit]