User blog:Gonz0TheGreatt/My new array notation

This is my new array notation that I've been developing for the past few weeks. I'll edit this blog post as I add extensions.

The arrays are of the form

$$b[n_1, n_2, ...]$$ . b is called the base of the array, and the rest is called the body.

Basic Notation: There are three rules for evaluating Basic Notation: Let # represent the rest of the array. B1: If there is only one entry n in the body, return b*n.

$$b[n] = b \cdot n$$ .

B2: If the last entry is a 0, return the second to last entry.

$$b[#, n, 0] = n$$ .

B3: Otherwise, replace the second to last entry with the array with the last entry cut off, and decrement the last entry.

$$b[#, m, n] = b[#, b[#, m], n - 1]$$ .

Some theorems: 1s at the end of the array can be cut off. If the first entry is 0, the value of the whole array is 0. Every function defined from basic notation that doesn't change the array size is primitive recursive.

$$b[1, 1, ..., c] = b \uparrow ^{n} c$$ where n is the number of 1s before c (every entry before c has to be a 1 for this to work).

$$2[c, c, ...] = f_{n}(c)$$ in FGH, where n is the number of cs in the body.

Colon Notation:

Here we introduce the : separator.

$$b[#:n]$$ basically means you put n 1s before #.

By the previous theorem,

$$b[c:n] = b \uparrow ^{n} c$$ .

C1: If the last entry is 0 and it is after a :, it can be removed.

$$b[# : 0] = b[#]$$ .

C2: If there is only one : and only one entry afterwards, place a 1 before the rest of the body, and decrement the last entry.

$$b[# : n] = b[1, # : n - 1]$$

C3 and C4 are just B2 and B3.

C5: If there is more than one : and only one entry after the last :, place a 1 after the previous :.

$$b[# : @ : n] = b[# : 1, @ : n - 1]$$ where @ does not contain any :s. Graham's number can be written in my array notation as

$$3[3 : 4, 64]$$ . The zilla function that I defined in an earlier blog post and its extensions can also be expressed in my array notation.

Colon notation has the same strength as Conway's chained arrow notation.

Double colon notation, Multiple colon notation, and Ordinal-separator notation:

Coming soon in an edit