FANDOM


blaSMN
Based onTwo-Entry Array Notation
Growth rate\(f_{\Gamma_0}(n)\)
 Basic Linear Array Separator Matrix Notation, abbreviated as blaSMN, is the third component of Separator Matrix Notation.

blaSMN features groups of two numbers separated by a comma as separators.

blaSMN 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 xSMN apply to blaSMN.

Rule 2a: a(0,0)b = a((...(a)...)")a w/ b () layers.

Rule 2b: a(d,c)b = a(d,c-1)(d,c-1)...(d,c-1)a w/ b parasegs.

(Ex: @(a+1)(a)(a)...(a)(a) = @(a+1)(a)(a+1) also applies to (b,@(a+1)(a)(a)...(a)(a) = (b,@(a+1)(a)(a+1))

Rule 3a: (@,(@,(@,(...,(@,0)...) = (@,(@+1,0))

Rule 3b: (@,(0,0),0),0)...),0) = (@+1,0)

Rule 4a: ((...(0,0),0)...) = ((0,0))

Rule 4b: ((...(..(0,0),0)..)...) w/ n () two-dot nests = ((...((0,0))...)) w/ n+1 () nests

Rule 4c: (...(0,0)...) = (0,0(0))

Rule 4d: (...(0,0(@))...) = (0,0(@+1))

Examples & Analysis

blaSMN Expression Approximate Fast-Growing Hierarchy
(0,0) \(f_{\zeta_0}(n)\)
(0,0)(0,0) \(f_{\zeta_0^2}(n)\)
(0,1) \(f_{\zeta_0^{\omega}}(n)\)
(0,2) \(f_{\zeta_0^{\omega^\omega}}(n)\)
(0,(0)) \(f_{\zeta_0^{\epsilon_0}}(n)\)
(0,(0,0)) \(f_{\zeta_0^{\zeta_0}}(n)\)
(0,(0,(0,0))) \(f_{\zeta_0^{\zeta_0^{\zeta_0}}}(n)\)
(0,(1,0)) \(f_{\varepsilon_{\zeta_0+1}}(n)\)
(0,(1,(0,0))) \(f_{\varepsilon_{\zeta_0+1}^{\zeta_0}}(n)\)
(0,(1,(1,0))) \(f_{\varepsilon_{\zeta_0+1}^{\varepsilon_{\zeta_0+1}}}(n)\)
(0,(2,0)) \(f_{\varepsilon_{\zeta_0+2}}(n)\)
(0,((0),0)) \(f_{\varepsilon_{\zeta_0+\omega}}(n)\)
(0,((0,0),0)) \(f_{\varepsilon_{\zeta_02}}(n)\)
(0,(((0,0),0),0)) \(f_{\varepsilon_{\zeta_03}}(n)\)
(1,0) \(f_{\varepsilon_{\zeta_0^2}}(n)\)
(2,0) \(f_{\varepsilon_{\zeta_0^22}}(n)\)
((0),0) \(f_{\varepsilon_{\zeta_0^2\omega}}(n)\)
((0,0),0) \(f_{\varepsilon_{\zeta_0^3}}(n)\)
(((0,0),0),0) \(f_{\varepsilon_{\zeta_0^\omega}}(n)\)
((0,((0,0)))) \(f_{\varepsilon_{\zeta_0^{\zeta_0}}}(n)\)
((0,((0,((0,0)))))) \(f_{\varepsilon_{\zeta_0^{\zeta_0^{\zeta_0}}}}(n)\)
((0,((1,0)))) \(f_{\varepsilon_{\varepsilon_{\zeta_0+1}}}(n)\)
((0,((2,0)))) \(f_{\varepsilon_{\varepsilon_{\zeta_0+2}}}(n)\)
((0,(((0,0),0)))) \(f_{\varepsilon_{\varepsilon_{\zeta_02}}}(n)\)
((0,((((0,0)),0)))) \(f_{\varepsilon_{\varepsilon_{\zeta_0^\omega}}}(n)\)
((1,0)) \(f_{\zeta_1}(n)\)
((2,0)) \(f_{\zeta_2}(n)\)
(((0,0),0)) \(f_{\zeta_{\zeta_0}}(n)\)
(((1,0),0)) \(f_{\zeta_{\zeta_1}}(n)\)
(((0,0))) \(f_{\eta_0}(n)\)
(((0,(((1,0)))))) \(f_{\varepsilon_{\eta_0}}(n)\)
(((1,0))) \(f_{\eta_1}(n)\)
((((((0,0))),0))) \(f_{\eta_{\eta_0}}(n)\)
((((0,0)))) \(f_{\phi(4,0)}(n)\)
(((((0,0))))) \(f_{\phi(5,0)}(n)\)
(0,0(0)) \(f_{\phi(\omega,0)}(n)\)
(0,0(0,0)) \(f_{\phi({\zeta_0},0)}(n)\)
(0,0(0,0(0))) \(f_{\phi(\phi(\omega,0))}(n)\)
Limit of blaSMN \(f_{\Gamma_0}(n)\)

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