Table of contents |
2 Simple examples 3 More typical examples 4 See also 5 External |
One must be careful to treat an arrow chain as a whole. Whereas chains of other infixed symbols (eg 3+4+5+6+7) can often be considered in fragments (eg (3+4)+5+(6+7)) without a change of meaning (see associativity), that is not so with Conway's arrow.
As with most combinatorial symbologies, the definition is recursive. Here, p and q are positive integers, and X is a chain (possibly of only one element) substituted textually.
The first rule is the core: A chain of 3 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 two elements and the third rule terminates the recursion.
Knuth's arrow and the hyper operators equate to 3-element chains:
It is impossible to give a fully worked-out interesting example since at least 4 elements are required. However 1-, 2- and 3-length chains, which are subsumed in other notations, are expanded here as illustrated examples.
n
4→3→2
Conway's arrow doesn't help to express Graham's number G succinctly, but:
Note that 3→3→3→3 is much greater than Graham's number.Interpretation/Expansion
The last two rules do not conflict, since p→1 = p¹ = p
X→(X→(...X→(X)→q...)→q)→q (with p copies of X).
Simple examples
p→q
1→(any arrowed expression)
(Indeed, any chain containing a 1 can be truncated just before that 1; ie X→1→Y=X for any (embedded) chains X,Y.)
4→3→2 alternatively analysed
2→2→4
2→4→3
2→3→2→2
3→2→2→2
More typical examples
3→3→64→2 < G < 3→3→65→2
Applying the first rule above,
3→3→64→2 = 3→3→(3→3→(...3→3→(3→3)→1...)→1)→1 with 128 threes.
Applying the second rule,
3→3→64→2 = 3→3→(3→3→(...3→3→(3→3)...))
If, for convenience, we define
f(n)=3→3→n
then
3→3→64→2 = f64(1) (see
functional powers),
G=f64(4), and
3→3→65→2 = f65(1) = f64(27)
Since f is strictly increasing,
f64(1) < f64(4) < f64(27)
which is the given inequality.