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. In some cases however, I have been unable to formulate a formal definition, as in the case of Rule 4, and so I have tried to make the rule as clear as possible.

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.

Planar Level Arrays
Rule 4: The Planar Base Case

$$\dbinom{a}{b}=(a,a,a,...,a,a,a)$$ where there are b a's.

Rule 5a: Two Arguments in the First Row, One in the Second

$$\dbinom{a,b}{c}=\dbinom{a}{\dbinom{a}{\dbinom{a}{...}}}$$ where there are b a's. Note too that there is to be a $$\dbinom{a}{c}$$ in the 'center' of this completed nesting.

Ellipsis-less definition:

$$\dbinom{a,b}{c}=Z_b$$, where $$Z_{n+1}=\dbinom{a}{Z_n}$$, and $$Z_1=\dbinom{a}{c}$$.

Rule 5b: Three Arguments in the First Row, One in the Second

$$\dbinom{a,b,c}{d}=\dbinom{a,\dbinom{a,\dbinom{a,...}{d}}{d}}{d}$$ where there are c a's. There is to be a $$\dbinom{a,b}{d}$$ in the 'center' of this nesting.

Ellipsis-less definition:

$$\dbinom{a,b,c}{d}=Z_c$$, where $$Z_{n+1}=\dbinom{a,Z_n}{d}$$, and $$Z_1=\dbinom{a,b}{d}$$.

Rule 6: Generalized Case for Two-Row Planar Arrays

$$\dbinom{a_1,a_2,a_3,...,a_n,a_{n+1},a_{n+2}}{b}=\dbinom{a_1,a_2,a_3,...,a_n,\dbinom{a_1,a_2,a_3,...,a_n,\dbinom{a_1,a_2,a_3,...,a_n,...}{b}}{b}}{b}$$ where there are $$a_{n+2}$$ $$a_1$$'s. A $$\dbinom{a_1,a_2,a_3,...,a_n,a_{n+1}}{b}$$ is in the 'center' of this nesting.

Revisionary Section
To make my definitions of the arrays clearer I've decided to use matrices instead of binomial brackets, which can beconfusing. This section is being used for that purpose so that anyone chancing upon this blog post will not be confused as to why the definitions look inconsistent, when in fact I would have just been revising them.

\[ \left[ {\begin{array}{c} a \\ b \\ \end{array} } \right]=M \]

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