User blog:Edwin Shade/Elementary Array Notations

It is time, I have decided, to construct my own set of array notations. However, I will not pretend that these array notations are extremely powerful, or even that they supersede BEAF or ExE, in actuality it probably will be one of the weakest array notations you have ever seen, and will be full of naive extensions. The name is, after all, Elementary Array Notations.

I am warning you of this, because this blog post is going to be a place where I can develop various array notations and compare their strength to various hierarchies until I feel I have developed a notation sufficiently powerful enough to merit it's own blog post. In short, this is a sandpit where I can work out my ideas, so that I won't fill the blog-post main-space with trivial notations under $$f_{\epsilon_0}(n)$$ in strength.

My Little Array Notation (MLAN)
MLAN is my first array notation, and is currently under development. Note I have provided definitions without ellipsis' whenever possible. This is to give a formal definition below the more intuitive one.

Linear Level Arrays
Rule 1: Base Case

$$[a,b]=a^b$$

Rule 2a: Three-Argument Case

$$[a,b,c]=[a,[a,[a,...[a,b]...]]]$$ where there are c a's.

Ellipsis-less definition:

$$[a,b,c]=Z_c$$, where $$Z_{n+1}=[a,Z_n]$$ and $$Z_1=[a,b]$$.

Rule 2b: Four-Argument Case

$$[a,b,c,d]=[a,b,[a,b,[a,b,...[a,b,c]...]]]$$ where there are c a's.

Ellipsis-less definition:

$$[a,b,c,d]=Z_d$$, where $$Z_{n+1}=[a,b,Z_n]$$ and $$Z_1=[a,b,c]$$.

Rule 3: Generalized Case for Linear Arrays

$$[a_1,a_2,a_3,...,a_n,a_{n+1}]=[a_1,a_1,a_3,...,a_{n-1},[a_1,a_1,a_3,...,a_{n-1},[a_1,a_1,a_3,...,a_{n-1},...a_1,a_1,a_3,...,a_{n-1},a_n]...]$$, where there are $$a_{n+1}$$ $$a_1$$'s.

Ellipsis-less definition:

$$[a_1,a_2,a_3,...,a_n,a_{n+1}]=Z_{a_{n+1}}$$, where $$Z_{m+1}=[a_1,a_2,a_3,...,Z_m]$$ and $$Z_1=[a_1,a_2,a_3,...,a_n]$$.

Linear arrays are equal in strength to $$f_{\alpha}(n)$$ such that $$\alpha<\omega$$, and their limit is $$f_{\omega}(n)$$; in both cases the fast-growing hierarchy was used.

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