User blog:Boboris02/Redefining ABHAN

Considering I've spent the past few days measuring the strength of a notation that was first defined on a website,that no longer exists,I've decided to redefine it to avoid confusion.

For this definition I am going to use these symbols a lot:

A - An array. Can have one or any finite amount of entries.


 * 1) - A segment of an array. It can also be nothing,for example b(a,b,#,c) = b(a,b,c).

a,b,c,d,.....x,y,z - Arbitrary integers.

/A/ - The array "A" is solved "Normally",aka the same way it's solved in b(a,b,A).

ABHAN (Another Boris' Hyper Array Notation) is expressed in the form "b(A)",where "A" is an array.

Arrays and array seperators have ranks.

\(A_1 < A_2 \iff b(a,b,A_1) < b(a,b,A_2)\)

Seperator ranks are measured in R.

\(R(\{A_1\}) < R(\{A_2\}) \iff A_1 < A_2 iff b(a,b,A_1) < b(a,b,A_2))\)

Linear arrays
Rule A1: b(a,b) = ab (as in a times b.)

Rule A2: b(a,1,#) = a

Rule A3: b(a,b,0,......,0,c+1,#) = b(a,a,0,.......,b,c,#) (as in "0,.......,0" can be any finite string of zeros,that can even be one zero.)

Rule A4: b(a,b,#,0) = b(a,b,#)

Rule A5: b(a,b,c,#) = b(a,b(a,b-1,c,#),c-1,#)

"," is shorthand for {0}.

Multidimentional arrays
Rule B1: b(a,b{n}c,#) = b(a,a{n-1}a{n-1}......{n-1}a{n-1}a{n}c-1,#) with a "b+1" total amount of "a"s.

Rule B2: b(a,b,A1{n}A2) = b(a,b,/A1/{n}/A2/)

Rule B3: b(a,#1{#2}0) = b(a,#1)

Also ",{#}" = "{#}". This will be important later.

Hyperdimentional arrays
The only new rule in hd-ABHAN is Rule C1.

Rule C1: b(a,b,#{A}#) = b(a,b,#{/A/}#)

This rule allows arrays longer than one entry to exist within seperators.

At this level of ABHAN,rule B1 remains but changes slightly to  b(a,b{n,#1}c,#2) = b(a,a{n-1,#1}a{n-1,#1}......{n-1,#1}a{n-1,#1}a{n,#1}c-1,#2).

Nested arrays
Rule D1: b(a,b{#1,0\c,#2}1) = b(a,a{#1,0{#1,0{......{#1,0\c-1,#2}.......}1\c-1,#2}1\c-1,#2}1) with "b" amount of "\"s.

Rule D2: b(a,b{#1\{A}#2}1) = b(a,b\{#1\{/A/}#2}1)

"\" has a higher rank than all "{A}" seperators.

\(\forall A: R(\{A\}) < R(\)\)

Hyper-Nested arrays
Rule E1: b(a,b{#1,0\(:_n\)c,#2}1) = b(a,a{#1,0\(:_{n-1}\){#1,0\(:_{n-1}\){......\(:_{n-1}\){#1,0\(:_{n}\)c-1,#2}.......}1\(:_n\)c-1,#2}1\(:_n\)c-1,#2}1) with "b" amount of "\(:_n\)"s.

Rule E2: {#1{#3}#2:A} = {#1\(:_A\){#3}#2}

"\" is another way of writing \(:_1\) and "/" is another way of writing "\(:_2\)"

"," is shorthand for "\(:_0\)"

\(\forall A_1,A_2,A_3,A_4 ((A_1 < A_3) \lor (A_1 = A_3 \land A_2 < A_4)): R(:_{A_1}\{A_2\}) < R(:_{A_3}\{A_4\})\)

Nested Hyper-Nested arrays
Rule F1: {#1{#3}#2{0\({\bullet}_{n+1}\)d,#4}A} = {#1{#3\({\bullet}_n\)A}#2{0\({\bullet}_{n+1}d-1,#4\}1}

Rule F2: {#1{#3\({\bullet}_0\)#4}#2} = {#1}

Rule F3: {#1\({\bullet}_1\)#2} = \(:_{\#2}\){#1}

Rule F4: {#1{A\(\bullet_{\#4}\)#3}#2} = {#1{/A/\(\bullet_{\#4}\)#3}#2}

\(\forall A,\#,n > 1: R(:_{A_1}\{A_2\}) < R(\#\{\#\bullet_{n}\#\}\#)\)