Bowers's operators

Bowers' operators (BEAF) is a notation for writing large numbers proposed by the American mathematician Jonathan Bowers in 2002. This notation is a generalization of the preceding 4-argument notation (known as Bowers' operators).

Rules

The Bowers notation for a linear array includes the following rules:

${a} = a$ and ${a, b} = a^{b}$
${a, b, c, \dots, n, 1} = {a, b, c, \dots, n}$
${a, 1, b, c, \dots, n} = a$
${a, b, 1, \dots, 1, c, d, \dots, n} = {a, a, a, \dots, {a, b - 1, 1, \dots, 1, c, d, \dots, n}, c - 1, d, \dots, n}$ .
If rules 1-4 do not apply, ${a, b, c, d, \dots, n} = {a, {a, b - 1, c, d, \dots, n}, c - 1, d, \dots, n}$

Examples

The array includes 2 elements

${10, 100} = 1 0^{100} = 10 ↑ 100$ (rule 1 applied)

The array includes 3 elements

${10, 100, 1} = {10, 100}$ (rule 2 applied)
${10, 100, 2} = {10, {10, 99, 2}} = {10, {10, {10, 98, 2}}} = \underset{100 t e n s}{\underset{⏟}{1 0^{1 0^{1 0^{\dots^{1 0^{10}}}}}}} = 10 ↑ ↑ 100$ (rule 5 applied)
${10, 100, 3} = {10, {10, 99, 3}, 2} = {10, {10, {10, 98, 3}, 2}, 2} = 10 ↑ ↑ ↑ 100$ (rule 5 applied)

In general, for a three-element array, ${a, b, m} = a ↑^{m} b$ is true according to Knuth's up-arrow notation.

The array includes 4 elements

${10, 100, 1, 1} = {10, 100}$ (rule 2 applied)
${10, 100, 1, 2} = {10, 10, {10, 99, 1, 2}} = {10, 10, {10, 10, {10, 98, 1, 2}}} = \begin{matrix} \underset{⏟}{10 ↑ ↑ \dots ↑ ↑ 10} \\ \underset{⏟}{10 ↑ ↑ \dots ↑ ↑ 10} \\ \underset{⏟}{⋮} \\ \underset{⏟}{10 ↑ ↑ \dots ↑ ↑ 10} \\ 10 arrows \end{matrix}} 100 \approx 10 \to 10 \to 100 \to 2$ (rule 4 is applied)

and this is already more than Graham number (the Graham number itself is somewhere between {3,64,1,2} and {3,65,1,2}).

${10, 100, 2, 2} = {10, {10, 99, 2, 2}, 1, 2} = {10, {10, {10, 98, 2, 2}, 1, 2}, 1, 2} \approx 10 \to 10 \to 100 \to 3$ (rule 5 applied)
${10, 100, m, 2} \approx 10 \to 10 \to 100 \to (m + 1)$

In general, for a four-element array, the following is true

{a, b, c, d} > \underset{d - 1 arrow}{\underset{⏟}{a \to a \to \dots a \to a}} \to (b - 1) \to (c + 1)

according to Conway chained arrow notation.

Thus, if the Bowers array, which includes 3 elements, has the power of Knuth's up-arrow notation (fEdlimit $ω$ ), then the four-element array already has the power of Conway notation (limit $ω^{2}$ ), and so on with the addition of each new element. The Bowers notation for a linear array including a finite number of elements has a limit $ω^{ω}$ in the terminology of fast-growing hierarchy.

Notes

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Bowers's operators

Rules

Examples

Notes

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