FANDOM


bSMN
Based onHyperoperation and Goucher Strings
Growth rate\(f_{\varepsilon_0}(n)\)

Basic Separator Matrix Notation, abbreviated as bSMN, is the first component of Separator Matrix Notation.

Hence the name, the basic concept of bSMN works with a hierarchy of many different types of separators.

bSMN Ruleset

@ = Any separator of the same kind in any one expression.

Rules numbered 1 are full expression rules. Rules numbered 2 are separator rules. Those separators are put into expressions, just like the ones in rules numbered 1. Rules numbered 3 are lexical rules.

Rule 1a: a(0)b = a^^b

Rule 1b: a@a@a...@a w/ b a's =  a@(0)b


In the rules below, the indeterminately long strings of parasegs (each item that is enclosed in parentheses along with the parantheses themselves is called a paraseg) are all part of the expression a@b, so the ellipses denote that the string is b parasegs long.

Rule 2a: (a)(a)...(a)(a) = (a+1)

Rule 2b: (a+1)(a)(a)...(a)(a) = (a+1)(a)(a+1)

Rule 2c: @(a+1)(a)(a)...(a)(a) = @(a+1)(a)(a+1)

Rule 3a: All numbers within parasegs (paraseg numbers) must be whole numbers.

Rule 3b: The first paraseg number in a sequence must be larger than the one directly proceeding it.

Rule 3c: If paraseg number x, in a sequence, is larger than the one before it (paraseg number y), then x must be y+1. This can only occur when the parasegs are in a sequence (x)(y)(x), @(x)(y)(x), or @(x)(y)(x)@. Rule 4: All seperator rules apply to separators in any scenario. This means that if separators are standalone, in an array, in a matrix, etc., the standalone rules always apply.

Rule 4: ANY RULES REGARDING SEPARATORS APPLY TO ANY SEPARATOR NO MATTER WHETHER THE SEPARATOR IS STANDALONE. THIS MEANS THAT IF THE SEPARATOR IS IN AN ARRAY OR OTHERWISE, THE RULES STILL APPLY.

Examples & Analysis

bSMN Expression Approximate Fast-Growing Hierarchy
a(0)b

\(f_4(n)\)

a(0)b(0)c

\(f_5(n)\)

a(0)b(0)c(0)d

\(f_6(n)\)

a(0)b(0)c(0)d(0)e

\(f_7(n)\)

a(0)(0)b

\(f_\omega(n)\)

a(0)(0)b(0)c

\(f_{\omega+1}(n)\)

a(0)(0)b(0)c(0)d

\(f_{\omega+2}(n)\)

a(0)(0)b(0)c(0)d(0)e

\(f_{\omega+3}(n)\)

a(0)(0)b(0)(0)c \(f_{\omega2}(n)\)
a(0)(0)b(0)(0)c(0)d \(f_{\omega2+1}(n)\)
a(0)(0)b(0)(0)c(0)(0)d \(f_{\omega3}(n)\)
a(0)(0)b(0)(0)c(0)(0)d(0)(0)e \(f_{\omega4}(n)\)
a(0)(0)(0)b \(f_{\omega^2}(n)\)
a(0)(0)(0)b(0)c \(f_{\omega^{2}+1}(n)\)
a(0)(0)(0)b(0)(0)c \(f_{\omega^{2}+\omega}(n)\)
a(0)(0)(0)b(0)(0)(0)c \(f_{\omega^{2}2}(n)\)
a(0)(0)(0)(0)b \(f_{\omega^{3}}(n)\)
a(0)(0)(0)(0)(0)b

\(f_{\omega^4}(n)\)

a(0)(0)(0)(0)(0)(0)b \(f_{\omega^5}(n)\)
a(1)b \(f_{\omega^{\omega}}(n)\)
a(1)b(0)c \(f_{\omega^{\omega}+1}(n)\)
a(1)b(0)(0)c \(f_{\omega^{\omega}+\omega}(n)\)
a(1)b(0)(0)(0)c \(f_{\omega^{\omega}+\omega^{2}}(n)\)
a(1)b(1)c \(f_{\omega^{\omega}2}(n)\)
a(1)b(1)c(1)d \(f_{\omega^{\omega}3}(n)\)
a(1)(0)b \(f_{\omega^{\omega+1}}(n)\)
a(1)(0)(0)b \(f_{\omega^{\omega+2}}(n)\)
a(1)(0)(0)(0)b

\(f_{\omega^{\omega+3}}(n)\)

a(1)(0)(1)b

\(f_{\omega^{\omega2}}(n)\)
a(1)(0)(1)(0)b \(f_{\omega^{\omega2+1}}(n)\)
a(1)(0)(1)(0)(0)b \(f_{\omega^{\omega2+2}}(n)\)
a(1)(0)(1)(0)(1)b \(f_{\omega^{\omega3}}(n)\)
a(1)(0)(1)(0)(1)(0)(1)b \(f_{\omega^{\omega4}}(n)\)
a(1)(1)b

\(f_{\omega^{\omega^{2}}}(n)\)

a(1)(1)(0)b \(f_{\omega^{\omega^{2}+1}}(n)\)
a(1)(1)(0)(1)b \(f_{\omega^{\omega^{2}+\omega}}(n)\)
a(1)(1)(0)(1)(1)b \(f_{\omega^{\omega^{2}2}}(n)\)
a(1)(1)(0)(1)(1)(0)(1)(1)b \(f_{\omega^{\omega^{2}3}}(n)\)
a(1)(1)(1)b \(f_{\omega^{\omega^{3}}}(n)\)
a(1)(1)(1)(0)(1)(1)(1)b \(f_{\omega^{\omega^{3}2}}(n)\)
a(1)(1)(1)(1)b \(f_{\omega^{\omega^{4}}}(n)\)
a(1)(1)(1)(1)(1)b \(f_{\omega^{\omega^{5}}}(n)\)
a(2)b \(f_{\omega^{\omega^{\omega}}}(n)\)
a(2)(0)b \(f_{\omega^{\omega^{\omega}+1}}(n)\)
a(2)(1)b \(f_{\omega^{\omega^{\omega}+\omega}}(n)\)
a(2)(0)(2)b \(f_{\omega^{\omega^{\omega+1}}}(n)\)
a(2)(0)(2)(0)(2)b

\(f_{\omega^{\omega^{\omega+2}}}(n)\)

a(2)(1)(2)b \(f_{\omega^{\omega^{\omega2}}}(n)\)
a(2)(1)(2)(1)(2)b \(f_{\omega^{\omega^{\omega3}}}(n)\)
a(2)(2)b \(f_{\omega^{\omega^{\omega^2}}}(n)\)
a(2)(2)(2)b \(f_{\omega^{\omega^{\omega^3}}}(n)\)
a(3)b \(f_{\omega^{\omega^{\omega^{\omega}}}}(n)\)
a(3)(3)b \(f_{\omega^{\omega^{\omega^{\omega^2}}}}(n)\)
a(4)b \(f_{\omega^{\omega^{\omega^{\omega^{\omega}}}}}(n)\)
a(5)b \(f_{\omega^{\omega^{\omega^{\omega^{\omega^{\omega}}}}}}(n)\)
a(n)b \(f_{\omega\uparrow\uparrow{n}}(n)\)
Limit \(f_{\varepsilon_0}\)

bSMN Numbers

My Numbers

bSMN base suffix = -ax

(0)s on their own = -im-

Multiple parasegs separated by numbers

S = 2, Tr = 3, Qu = 4, Pr = 5, Set = 6, Het = 7, Od = 8, Nov = 9

Multiple (0)s in a row

Bil = 2, Tril = 3, Quil = 4, Pril = 5, Setil = 6, Hetil = 7, Otil = 8, Novil = 9

Paraseg Numbers

Nu = 0, Uka = 1, Kaka = 2, Kolma = 3, Nela = 4, Visa = 5, Kusa = 6, Sesta = 7, Heksa = 8, Usta = 9

Multiple (n)s in a row when there are more parasegs than just (0)s

Ni = 2, Sana = 3, Shi = 4, Go = 5, Roku = 6, Nana = 7, Hachi = 8, Ku = 9

Vowels may be removed.


10(0)10 = Simpax

10(0)10(0)10 = Trimpax

10(0)10(0)10(0)10 = Quimpax

10(0)10(0)10(0)10(0)10 = Primpax

10(0)10(0)10(0)10(0)10(0)10 = Setimpax

10(0)10(0)10(0)10(0)10(0)10(0)10 = Hetimpax

10(0)(0)8 = Odimpax

10(0)(0)9 = Novimpax

10(0)(0)10 = Simbilax

10(0)(0)10(0)10 = Simbilasimpax

10(0)(0)10(0)10(0)10 = Simbilatrimpax

10(0)(0)10(0)10(0)10(0)10 = Simbilaquimpax

10(0)(0)10(0)(0)5 = Simbilaprimpax

10(0)(0)10(0)(0)6 = Simbilasetimpax

10(0)(0)10(0)(0)7 = Simbilahetimpax

10(0)(0)10(0)(0)8 = Simbilodimpax

10(0)(0)10(0)(0)9 = Simbilanovimpax

10(0)(0)10(0)(0)10 = Trimbilax

10(0)(0)10(0)(0)10(0)(0)10 = Quimbilax

10(0)(0)(0)5 = Primbilax


10(0)(0)(0)6 = Setimbilax


10(0)(0)(0)7 = Hetimbilax


10(0)(0)(0)8 = Odimbilax


10(0)(0)(0)9 = Novimbilax


10(0)(0)(0)10 = Simtrilax


10(0)(0)(0)10(0)(0)(0)10 = Trimtrilax


10(0)(0)(0)(0)4 = Quimtrilax

10(0)(0)(0)(0)10 = Simquilax

10(0)(0)(0)(0)(0)10 = Simprilax


10(1)6 = Simsetilax


10(1)7 = Simhetilax


10(1)8 = Simodilax


10(1)9 = Simnovilax

10(1)10 = Ukapax

10(1)(0)10 = Ukanupax

10(1)(0)(0)10 = Ukaninupax

10(1)(0)(0)(0)10 = Ukasananupax

10(1)(0)(1)4 = Ukashinupax

10(1)(0)(1)5 = Ukagonupax

10(1)(0)(1)6 = Ukarokunupax

10(1)(0)(1)7 = Ukanananupax

10(1)(0)(1)8 = Ukahachinupax

10(1)(0)(1)9 = Ukakunupax

10(1)(0)(1)10 = Ukanukapax

10(1)(0)(1)(0)10 = Ukanukanupax

10(1)(0)(1)(0)(1)10 = Ukanukanukapax

10(1)(1)10 = Nikapax

10(1)(1)(0)(1)(1)10 = Nikanunikapax

10(1)(1)(1)10 = Sanakapax

10(1)(1)(1)(1)10 = Shikapax

10(2)5 = Gokapax

10(2)6 = Rokukapax

10(2)7 = Nanukapax

10(2)8 = Hachikapax

10(2)9 = Kukapax

10(2)10 = Kakapax

10(2)(0)10 = Kakanupax

10(2)(1)10 = Kakukapax

10(2)(1)(2)10 = Kakukakakapax

10(2)(2)10 = Nikakapax

10(2)(2)(2)10 = Sanakakapax

10(3)10 = Kolmapax

10(4)10 = Nelapax

10(5)10 = Visapax

10(6)10 = Kusapax

10(7)10 = Sestapax

10(8)10 = Heksapax

10(9)10 = Ustapax

10(10)10 = Xapax

Aarex's Numbers

10(0)100 = Starol

10(0)10(0)100 = Starolplex

10(0)(0)100 = Starolex

10(0)(0)10(0)(0)100 = Starolexplex

10(0)(0)(0)100 = Starolduex

10(0)(0)(0)(0)100 = Staroltriex

10(1)100 = Starolunex

10(1)10(1)100 = Starolunexplex

10(1)(0)100 = Starolunexex

10(1)(0)(0)100 = Starolunexduex

10(1)(0)(1)100 = Starolunexunexiex

10(1)(0)(1)(0)100 = Starolunexexunexiex

10(1)(0)(1)(0)(1)100 = Starolunexunexiduex

10(1)(0)(1)(0)(1)(0)(1)100 = Starolunexunexitriex

10(1)(1)100 = Starolduunex

10(2)100 = Starolduex

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.