FANDOM


xSMN
Based onNested Parentheses
Growth rate\(f_{\zeta_0}(n)\)

Extended Separator Matrix Notation, abbreviated as xSMN, is the second component of Separator Matrix Notation.

Hence the name, xSMN is a basic extension to bSMN, using nested parentheses and separating symbols.

xSMN Ruleset

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

Elipses between two symbols can also indicate that there are only two symbols in total in that group of symbols.

Rule 1: All rules of bSMN apply to xSMN.

Rule 2: a(...(b)...)a w/ n ()s = a(...(0)...)a w/ n+1 ()s*

Rule 3a: a(...(a)...)a w/ b parenthesis nests = a(0(0))b

Rule 3b: a(c(...(a)...))a w/ b parenthesis nests = a(c+1(0))b

Rule 3c: a(b(0))a = a((0)(0))b

Rule 3d: a(...(b)(b)(b)...(b...))a w/ c b's = a(...(b+1)"...)c

Rule 3e: b((a+1)(a)(a)...(a))b w/ c a's = b((a+1)(a)(a+1))c

Rule 3f: b(@(a+1)(a)(a)...(a))b w/c a's = b(@(a+1)(a)(a+1))c

Examples & Analysis

xSMN Expression Approximate Fast-Growing Hierarchy
a((0))b \(f_{\varepsilon_0}\)
a((0))b((0))c \(f_{\varepsilon_{0}2}\)
a((0))(0)b \(f_{\varepsilon_{0}\omega}\)
a((0))((0))b \(f_{\varepsilon_{0}^2}\)
a((1))b \(f_{\varepsilon_{0}^\omega}\)
a((2))b \(f_{\varepsilon_0^{\omega^{\omega}}}\)
a(((0)))b \(f_{\varepsilon_0^{\varepsilon_0}}\)
a((((0))))b \(f_{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}\)
a(0(0))b \(f_{\varepsilon_1}\)
a(0(0))(0(0))b \(f_{\varepsilon_1^2}\)
a(0(1))b \(f_{\varepsilon_1^\omega}\)
a(0((0)))b \(f_{\varepsilon_1^{\varepsilon_0}}\)
a(0(((0))))b \(f_{\varepsilon_1^{\varepsilon_0^{\varepsilon_0}}}\)
a(1(0))b \(f_{\varepsilon_1^{\varepsilon_1}}\)
a(2(0))b \(f_{\varepsilon_1^{\varepsilon_1^{\varepsilon_1}}}\)
a((0)(0))b \(f_{\varepsilon_2}\)
a(1(0)(0))b \(f_{\varepsilon_2^{\varepsilon_1}}\)
a((0)(0)(0))b \(f_{\varepsilon_2^{\varepsilon_2}}\)
a((1)")b \(f_{\varepsilon_3}\)
a((1)(0))b \(f_{\varepsilon_3^{\varepsilon_2}}\)
a((1)(0)(1))b \(f_{\varepsilon_3^{\varepsilon_3}}\)
a((1)(1))b \(f_{\varepsilon_4}\)
a((1)(1)(1))b \(f_{\varepsilon_5}\)
a((2)")b \(f_{\varepsilon_\omega}\)
a(((0)"))b \(f_{\varepsilon_{\varepsilon_0}}\)
a((((0)")))b \(f_{\varepsilon_{\varepsilon_{\varepsilon_0}}}\)
Limit

\(f_{\zeta_0}\)

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