Быстрорастущая иерархия (также называемая расширенной иерархией Гржегорчика ) — это семейство быстрорастущих функций, индексированных ординалами . Наиболее известным частным случаем быстрорастущей иерархии является иерархия Лёба -Вайнера.
Быстрорастущая иерархия определяется следующими правилами:
f
0
(
n
)
=
n
+
1
{\displaystyle f_{0}(n)=n+1}
f
0
(
n
)
{\displaystyle f_{0}(n)}
f
α
+
1
(
n
)
=
f
α
n
(
n
)
=
f
α
(
f
α
(
⋯
(
f
α
(
f
α
(
⏟
n
раз
n
)
)
⋯
)
)
{\displaystyle f_{\alpha +1}(n)=f_{\alpha }^{n}(n)=\underbrace {f_{\alpha }(f_{\alpha }(\cdots (f_{\alpha }(f_{\alpha }(} _{n{\mbox{ раз}}}n))\cdots ))}
f
α
(
n
)
=
f
α
[
n
]
(
n
)
{\displaystyle f_{\alpha }(n)=f_{\alpha [n]}(n)}
α
{\displaystyle \alpha }
где
α
[
n
]
{\displaystyle \alpha [n]}
n -м элементом фундаментальной последовательности, установленной для некого предельного ординала
α
{\displaystyle \alpha }
Существуют различные версии быстрорастущей иерархии, однако наиболее известной является иерархия Лёба-Вайнера, в которой фундаментальные последовательности для предельных ординалов, записанных в нормальной форме Кантора, определяются следующими правилами:
ω
[
n
]
=
n
{\displaystyle \omega [n]=n}
(
ω
α
1
+
ω
α
2
+
⋯
+
ω
α
k
−
1
+
ω
α
k
)
[
n
]
=
ω
α
1
+
ω
α
2
+
⋯
+
ω
α
k
−
1
+
ω
α
k
[
n
]
{\displaystyle (\omega ^{\alpha _{1}}+\omega ^{\alpha _{2}}+\cdots +\omega ^{\alpha _{k-1}}+\omega ^{\alpha _{k}})[n]=\omega ^{\alpha _{1}}+\omega ^{\alpha _{2}}+\cdots +\omega ^{\alpha _{k-1}}+\omega ^{\alpha _{k}}[n]}
для
α
1
≥
α
2
≥
⋯
≥
α
k
{\displaystyle \alpha _{1}\geq \alpha _{2}\geq \cdots \geq \alpha _{k}}
ω
α
+
1
[
n
]
=
ω
α
n
{\displaystyle \omega ^{\alpha +1}[n]=\omega ^{\alpha }n}
ω
α
[
n
]
=
ω
α
[
n
]
{\displaystyle \omega ^{\alpha }[n]=\omega ^{\alpha [n]}}
α
{\displaystyle \alpha }
ε
0
[
0
]
=
0
{\displaystyle \varepsilon _{0}[0]=0}
ε
0
[
n
+
1
]
=
ω
ε
0
[
n
]
=
ω
↑↑
n
{\displaystyle \varepsilon _{0}[n+1]=\omega ^{\varepsilon _{0}[n]}=\omega \uparrow \uparrow n}
Фундаментальные последовательности для предельных ординалов свыше
ε
0
{\displaystyle \varepsilon _{0}}
функциях Веблена и функциях Бухгольца .
f
1
(
n
)
=
f
0
n
(
n
)
=
f
0
(
f
0
(
⋯
(
f
0
(
f
0
(
⏟
n
f
0
n
)
)
⋯
)
)
=
2
×
n
{\displaystyle f_{1}(n)=f_{0}^{n}(n)=\underbrace {f_{0}(f_{0}(\cdots (f_{0}(f_{0}(} _{n\quad f_{0}}n))\cdots ))=2\times n}
f
2
(
n
)
=
f
1
n
(
n
)
=
f
1
(
f
1
(
⋯
(
f
1
(
f
1
(
⏟
n
f
1
n
)
)
⋯
)
)
=
2
n
×
n
{\displaystyle f_{2}(n)=f_{1}^{n}(n)=\underbrace {f_{1}(f_{1}(\cdots (f_{1}(f_{1}(} _{n\quad f_{1}}n))\cdots ))=2^{n}\times n}
Для функций, индексированных конечными ординалами
α
>
1
{\displaystyle \alpha >1}
f
m
(
n
)
>
2
↑
m
−
1
n
{\displaystyle f_{m}(n)>2\uparrow ^{m-1}n}
В частности, при n =10:
f
3
(
10
)
=
f
2
10
(
10
)
≈
10
10
10
⋯
10
10
3.489
⏟
10
десяток
≈
10
↑↑
11
{\displaystyle f_{3}(10)=f_{2}^{10}(10)\approx \underbrace {10^{10^{10^{\cdots {^{10^{10^{3.489}}}}}}}} _{10\quad {\text{десяток}}}\approx 10\uparrow \uparrow 11}
f
4
(
10
)
=
f
3
10
(
10
)
≈
10
10
10
⋯
10
10
3.489
⏟
10
10
10
⋯
10
10
3.489
⏟
⋮
⏟
10
10
10
⋯
10
10
3.489
⏟
10
десяток
}
10 нижних фигурных скобок
≈
10
↑↑↑
11
{\displaystyle f_{4}(10)=f_{3}^{10}(10)\approx \left.{\begin{matrix}&&\underbrace {10^{10^{10^{\cdots {^{10^{10^{3.489}}}}}}}} \\&&\underbrace {10^{10^{10^{\cdots {^{10^{10^{3.489}}}}}}}} \\&&\underbrace {\qquad \;\;\vdots \qquad \;\;} \\&&\underbrace {10^{10^{10^{\cdots {^{10^{10^{3.489}}}}}}}} \\&&10\quad {\text{десяток}}\end{matrix}}\right\}{\text{10 нижних фигурных скобок}}\approx 10\uparrow \uparrow \uparrow 11}
f
m
(
10
)
≈
10
↑
m
−
1
11
{\displaystyle f_{m}(10)\approx 10\uparrow ^{m-1}11}
Таким образом, уже первый трансфинитный ординал
ω
{\displaystyle \omega }
стрелочной нотации Кнута .
Знаменитое число Грэма меньше, чем
f
ω
+
1
(
64
)
{\displaystyle f_{\omega +1}(64)}
Благодаря простоте и ясности определения быстрорастущая иерархия применяется для анализа различных нотаций для записи больших чисел .
нотация Кнута
нотация Конвея
нотация Бауэрса
предел нотации
ω
{\displaystyle \omega }
ω
2
{\displaystyle \omega ^{2}}
ω
ω
{\displaystyle \omega ^{\omega }}
примеры
f
ω
(
10
)
≈
10
↑
9
11
=
10
↑↑↑↑↑↑↑↑↑
11
{\displaystyle f_{\omega }(10)\approx 10\uparrow ^{9}11=10\uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow 11}
f
ω
2
(
n
)
≈
n
→
n
→
n
⋯
→
n
⏟
n
+
1
→
{\displaystyle f_{\omega ^{2}}(n)\approx \underbrace {n\rightarrow n\rightarrow n\cdots \rightarrow n} _{n+1\quad \rightarrow }}
f
ω
ω
(
n
)
≈
{
n
,
n
,
n
,
⋯
,
n
,
n
}
⏟
n
+
2
n
′
s
{\displaystyle f_{\omega ^{\omega }}(n)\approx \underbrace {\{n,n,n,\cdots ,n,n\}} _{n+2\quad n's}}
f
ω
+
1
(
10
)
≈
10
↑↑
⋯
↑↑
11
⏟
10
↑↑
⋯
↑↑
11
⏟
⋮
⏟
10
↑↑
⋯
↑↑
11
⏟
9 стрелок
}
10
{\displaystyle f_{\omega +1}(10)\approx \left.{\begin{matrix}&&\underbrace {10\uparrow \uparrow \cdots \uparrow \uparrow 11} \\&&\underbrace {10\uparrow \uparrow \cdots \uparrow \uparrow 11} \\&&\underbrace {\quad \quad \;\;\vdots \quad \quad \;\;} \\&&\underbrace {10\uparrow \uparrow \cdots \uparrow \uparrow 11} \\&&{\text{9 стрелок}}\end{matrix}}\right\}{\text{10}}}
f
ω
2
+
1
(
n
)
≈
n
→
n
→
⋯
n
→
n
⏟
n
→
n
→
⋯
n
→
n
⏟
⋮
⏟
n
→
n
→
⋯
n
→
n
⏟
n+1 стрелка
}
n
{\displaystyle f_{\omega ^{2}+1}(n)\approx \left.{\begin{matrix}&&\underbrace {n\rightarrow n\rightarrow \cdots n\rightarrow n} \\&&\underbrace {n\rightarrow n\rightarrow \cdots n\rightarrow n} \\&&\underbrace {\quad \quad \quad \;\;\vdots \quad \quad \quad \;\;} \\&&\underbrace {n\rightarrow n\rightarrow \cdots n\rightarrow n} \\&&{\text{n+1 стрелка}}\end{matrix}}\right\}{\text{n}}}
f
ω
ω
+
1
(
n
)
≈
{
n
,
n
,
n
,
⋯
,
n
,
n
}
⏟
{
n
,
n
,
n
,
⋯
,
n
,
n
}
⏟
⋮
⏟
{
n
,
n
,
n
,
⋯
,
n
,
n
}
⏟
n+2 n's
}
n
{\displaystyle f_{\omega ^{\omega }+1}(n)\approx \left.{\begin{matrix}&&\underbrace {\{n,n,n,\cdots ,n,n\}} \\&&\underbrace {\{n,n,n,\cdots ,n,n\}} \\&&\underbrace {\quad \quad \quad \;\;\vdots \quad \quad \quad \;\;} \\&&\underbrace {\{n,n,n,\cdots ,n,n\}} \\&&{\text{n+2 n's}}\end{matrix}}\right\}{\text{n}}}
Данная выше дефиниция определяет быстрорастущую иерархию до
ε
0
=
ω
↑↑
n
{\displaystyle \varepsilon _{0}=\omega \uparrow \uparrow n}
функцию Веблена и другие, ещё более мощные нотации для ординалов[ 1]
f
0
(
n
)
=
n
+
1
{\displaystyle f_{0}(n)=n+1}
f
1
(
n
)
=
2
n
{\displaystyle f_{1}(n)=2n}
f
2
(
n
)
=
2
n
n
{\displaystyle f_{2}(n)=2^{n}n}
f
3
(
n
)
>
2
↑↑
n
{\displaystyle f_{3}(n)>2\uparrow \uparrow n}
Стрелочная нотация Кнута )
f
4
(
n
)
>
2
↑↑↑
n
{\displaystyle f_{4}(n)>2\uparrow \uparrow \uparrow n}
f
m
(
n
)
>
2
↑
m
−
1
n
{\displaystyle f_{m}(n)>2\uparrow ^{m-1}n}
f
ω
(
n
)
>
2
↑
n
−
1
n
=
{
n
,
n
,
n
−
1
}
{\displaystyle f_{\omega }(n)>2\uparrow ^{n-1}n=\{n,n,n-1\}}
Массивная нотация Бауэрса )
f
ω
+
1
(
n
)
=
{
n
,
n
,
1
,
2
}
≈
G
(
n
)
{\displaystyle f_{\omega +1}(n)=\{n,n,1,2\}\approx G(n)}
Число Грэма )
f
ω
+
2
(
n
)
=
{
n
,
n
,
2
,
2
}
{\displaystyle f_{\omega +2}(n)=\{n,n,2,2\}}
f
ω
+
m
(
n
)
=
{
n
,
n
,
m
,
2
}
{\displaystyle f_{\omega +m}(n)=\{n,n,m,2\}}
f
ω
2
(
n
)
=
{
n
,
n
,
n
,
2
}
{\displaystyle f_{\omega 2}(n)=\{n,n,n,2\}}
f
ω
2
+
1
(
n
)
=
{
n
,
n
,
1
,
3
}
{\displaystyle f_{\omega 2+1}(n)=\{n,n,1,3\}}
f
ω
3
(
n
)
=
{
n
,
n
,
n
,
3
}
{\displaystyle f_{\omega 3}(n)=\{n,n,n,3\}}
f
ω
m
(
n
)
=
{
n
,
n
,
n
,
m
}
{\displaystyle f_{\omega {m}}(n)=\{n,n,n,m\}}
f
ω
m
+
k
(
n
)
=
{
n
,
n
,
k
,
m
+
1
}
{\displaystyle f_{\omega {m+k}}(n)=\{n,n,k,m+1\}}
f
ω
2
(
n
)
=
{
n
,
n
,
n
,
n
}
{\displaystyle f_{\omega ^{2}}(n)=\{n,n,n,n\}}
f
ω
3
(
n
)
=
{
n
,
n
,
n
,
n
,
n
}
{\displaystyle f_{\omega ^{3}}(n)=\{n,n,n,n,n\}}
f
ω
m
(
n
)
=
{
n
,
n
,
n
,
.
.
.
,
n
}
{\displaystyle f_{\omega ^{m}}(n)=\{n,n,n,...,n\}}
f
ω
ω
(
n
)
=
{
n
,
n
,
n
,
.
.
.
,
n
}
{\displaystyle f_{\omega ^{\omega }}(n)=\{n,n,n,...,n\}}
=
{
n
,
n
(
1
)
2
}
{\displaystyle =\{n,n(1)2\}}
f
ω
ω
+
1
(
n
)
=
{
n
,
n
(
1
)
1
,
2
}
{\displaystyle f_{\omega ^{\omega +1}}(n)=\{n,n(1)1,2\}}
f
ω
ω
+
2
(
n
)
=
{
n
,
n
(
1
)
1
,
1
,
2
}
{\displaystyle f_{\omega ^{\omega +2}}(n)=\{n,n(1)1,1,2\}}
f
ω
ω
2
(
n
)
=
{
n
,
n
(
1
)
1
(
1
)
2
}
{\displaystyle f_{\omega ^{\omega 2}}(n)=\{n,n(1)1(1)2\}}
f
ω
ω
2
+
1
(
n
)
=
{
n
,
n
(
1
)
1
(
1
)
1
,
2
}
{\displaystyle f_{\omega ^{\omega 2+1}}(n)=\{n,n(1)1(1)1,2\}}
f
ω
ω
3
(
n
)
=
{
n
,
n
(
1
)
1
(
1
)
1
(
1
)
2
}
{\displaystyle f_{\omega ^{\omega 3}}(n)=\{n,n(1)1(1)1(1)2\}}
f
ω
ω
2
(
n
)
=
{
n
,
n
(
2
)
2
}
{\displaystyle f_{\omega ^{\omega ^{2}}}(n)=\{n,n(2)2\}}
f
ω
ω
3
(
n
)
=
{
n
,
n
(
3
)
2
}
{\displaystyle f_{\omega ^{\omega ^{3}}}(n)=\{n,n(3)2\}}
f
ω
ω
m
(
n
)
=
{
n
,
n
(
m
)
2
}
{\displaystyle f_{\omega ^{\omega ^{m}}}(n)=\{n,n(m)2\}}
f
ω
ω
ω
(
n
)
=
{
n
,
n
[
1
,
2
]
2
}
{\displaystyle f_{\omega ^{\omega ^{\omega }}}(n)=\{n,n[1,2]2\}}
Bird's Array Notation )
f
ω
ω
ω
2
(
n
)
=
{
n
,
n
[
1
,
1
,
2
]
2
}
{\displaystyle f_{\omega ^{\omega ^{\omega ^{2}}}}(n)=\{n,n[1,1,2]2\}}
f
ω
ω
ω
ω
(
n
)
=
{
n
,
n
[
1
[
2
]
2
]
2
}
{\displaystyle f_{\omega ^{\omega ^{\omega ^{\omega }}}}(n)=\{n,n[1[2]2]2\}}
f
ϵ
0
(
n
)
=
{
n
,
n
[
1
∖
2
]
2
}
{\displaystyle f_{\epsilon _{0}}(n)=\{n,n[1{\backslash }2]2\}}
f
ω
ω
ϵ
0
+
1
(
n
)
=
{
n
,
n
[
1
[
1
∖
2
]
2
∖
2
]
2
}
{\displaystyle f_{\omega ^{\omega ^{\epsilon _{0}+1}}}(n)=\{n,n[1[1{\backslash }2]2{\backslash }2]2\}}
f
ω
ω
ω
ϵ
0
+
1
(
n
)
=
{
n
,
n
[
1
[
1
[
1
∖
2
]
2
∖
2
]
2
∖
2
]
2
}
{\displaystyle f_{\omega ^{\omega ^{\omega ^{\epsilon _{0}+1}}}}(n)=\{n,n[1[1[1{\backslash }2]2{\backslash }2]2{\backslash }2]2\}}
f
ϵ
1
(
n
)
=
{
n
,
n
[
1
∖
3
]
2
}
{\displaystyle f_{\epsilon _{1}}(n)=\{n,n[1{\backslash }3]2\}}
f
ϵ
2
(
n
)
=
{
n
,
n
[
1
∖
4
]
2
}
{\displaystyle f_{\epsilon _{2}}(n)=\{n,n[1{\backslash }4]2\}}
f
ϵ
ω
(
n
)
=
{
n
,
n
[
1
∖
1
,
2
]
2
}
{\displaystyle f_{\epsilon _{\omega }}(n)=\{n,n[1{\backslash }1,2]2\}}
f
ϵ
ω
ω
(
n
)
=
{
n
,
n
[
1
∖
1
[
2
]
2
]
2
}
{\displaystyle f_{\epsilon _{\omega ^{\omega }}}(n)=\{n,n[1{\backslash }1[2]2]2\}}
f
ϵ
ϵ
0
(
n
)
=
{
n
,
n
[
1
∖
1
[
1
∖
2
]
2
]
2
}
{\displaystyle f_{\epsilon _{\epsilon _{0}}}(n)=\{n,n[1{\backslash }1[1{\backslash }2]2]2\}}
f
ϵ
ϵ
ϵ
0
(
n
)
=
{
n
,
n
[
1
∖
1
[
1
∖
1
[
1
∖
2
]
2
]
2
]
2
}
{\displaystyle f_{\epsilon _{\epsilon _{\epsilon _{0}}}}(n)=\{n,n[1{\backslash }1[1{\backslash }1[1{\backslash }2]2]2]2\}}
f
ζ
0
(
n
)
=
{
n
,
n
[
1
∖
1
∖
2
]
2
}
{\displaystyle f_{\zeta _{0}}(n)=\{n,n[1{\backslash }1{\backslash }2]2\}}
f
ϵ
ζ
0
+
1
(
n
)
=
{
n
,
n
[
1
∖
2
∖
2
]
2
}
{\displaystyle f_{\epsilon _{\zeta _{0}+1}}(n)=\{n,n[1{\backslash }2{\backslash }2]2\}}
f
ϵ
ζ
0
+
ω
(
n
)
=
{
n
,
n
[
1
∖
1
,
2
∖
2
]
2
}
{\displaystyle f_{\epsilon _{\zeta _{0}+\omega }}(n)=\{n,n[1{\backslash }1,2{\backslash }2]2\}}
f
ϵ
ζ
0
+
ϵ
0
(
n
)
=
{
n
,
n
[
1
∖
1
[
1
∖
2
]
2
∖
2
]
2
}
{\displaystyle f_{\epsilon _{\zeta _{0}+\epsilon _{0}}}(n)=\{n,n[1{\backslash }1{[1{\backslash }2]}2{\backslash }2]2\}}
f
ϵ
ζ
0
2
(
n
)
=
{
n
,
n
[
1
∖
1
[
1
∖
1
∖
2
]
2
∖
2
]
2
}
{\displaystyle f_{\epsilon _{\zeta _{0}2}}(n)=\{n,n[1{\backslash }1{[1{\backslash }1{\backslash }2]}2{\backslash }2]2\}}
f
ϵ
ϵ
ζ
0
+
1
(
n
)
=
{
n
,
n
[
1
∖
1
[
1
∖
2
∖
2
]
2
∖
2
]
2
}
{\displaystyle f_{\epsilon _{\epsilon _{\zeta _{0}+1}}}(n)=\{n,n[1{\backslash }1{[1{\backslash }2{\backslash }2]}2{\backslash }2]2\}}
f
ζ
1
(
n
)
=
{
n
,
n
[
1
∖
1
∖
3
]
2
}
{\displaystyle f_{\zeta _{1}}(n)=\{n,n[1{\backslash }1{\backslash }3]2\}}
f
ζ
ω
(
n
)
=
{
n
,
n
[
1
∖
1
∖
1
,
2
]
2
}
{\displaystyle f_{\zeta _{\omega }}(n)=\{n,n[1{\backslash }1{\backslash }1,2]2\}}
f
ζ
ϵ
0
(
n
)
=
{
n
,
n
[
1
∖
1
∖
1
[
1
∖
2
]
2
]
2
}
{\displaystyle f_{\zeta _{\epsilon _{0}}}(n)=\{n,n[1{\backslash }1{\backslash }1[1{\backslash }2]2]2\}}
f
ζ
ζ
0
(
n
)
=
{
n
,
n
[
1
∖
1
∖
1
[
1
∖
1
∖
2
]
2
]
2
}
{\displaystyle f_{\zeta _{\zeta _{0}}}(n)=\{n,n[1{\backslash }1{\backslash }1[1{\backslash }1{\backslash }2]2]2\}}
f
η
0
(
n
)
=
{
n
,
n
[
1
∖
1
∖
1
∖
2
]
2
}
{\displaystyle f_{\eta _{0}}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }2]2\}}
f
φ
(
4
,
0
)
(
n
)
=
{
n
,
n
[
1
∖
1
∖
1
∖
1
∖
2
]
2
}
{\displaystyle f_{\varphi (4,0)}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }1{\backslash }2]2\}}
f
φ
(
m
,
0
)
(
n
)
=
{
n
,
n
[
1
∖
1
∖
1
⋯
1
∖
2
]
2
}
{\displaystyle f_{\varphi (m,0)}(n)=\{n,n[1{\backslash }1{\backslash }1{\cdots }1{\backslash }2]2\}}
f
φ
(
ω
,
0
)
(
n
)
=
{
n
,
n
[
1
[
2
¬
2
]
2
]
2
}
{\displaystyle f_{\varphi (\omega ,0)}(n)=\{n,n[1[2{\neg }2]2]2\}}
f
ϵ
φ
(
ω
,
0
)
+
1
(
n
)
=
{
n
,
n
[
1
∖
2
[
2
¬
2
]
2
]
2
}
{\displaystyle f_{\epsilon _{\varphi (\omega ,0)+1}}(n)=\{n,n[1{\backslash }2[2{\neg }2]2]2\}}
f
ζ
φ
(
ω
,
0
)
+
1
(
n
)
=
{
n
,
n
[
1
∖
1
∖
2
[
2
¬
2
]
2
]
2
}
{\displaystyle f_{\zeta _{\varphi (\omega ,0)+1}}(n)=\{n,n[1{\backslash }1{\backslash }2[2{\neg }2]2]2\}}
f
η
φ
(
ω
,
0
)
+
1
(
n
)
=
{
n
,
n
[
1
∖
1
∖
1
∖
2
[
2
¬
2
]
2
]
2
}
{\displaystyle f_{\eta _{\varphi (\omega ,0)+1}}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }2[2{\neg }2]2]2\}}
f
φ
(
ω
,
1
)
(
n
)
=
{
n
,
n
[
1
[
2
¬
2
]
3
]
2
}
{\displaystyle f_{\varphi (\omega ,1)}(n)=\{n,n[1[2{\neg }2]3]2\}}
f
φ
(
ω
,
2
)
(
n
)
=
{
n
,
n
[
1
[
2
¬
2
]
4
]
2
}
{\displaystyle f_{\varphi (\omega ,2)}(n)=\{n,n[1[2{\neg }2]4]2\}}
f
φ
(
ω
,
ω
)
(
n
)
=
{
n
,
n
[
1
[
2
¬
2
]
1
,
2
]
2
}
{\displaystyle f_{\varphi (\omega ,\omega )}(n)=\{n,n[1[2{\neg }2]1,2]2\}}
f
φ
(
ω
,
ϵ
0
)
(
n
)
=
{
n
,
n
[
1
[
2
¬
2
]
1
[
1
∖
2
]
2
]
2
}
{\displaystyle f_{\varphi (\omega ,\epsilon _{0})}(n)=\{n,n[1[2{\neg }2]1[1{\backslash }2]2]2\}}
f
φ
(
ω
,
ζ
0
)
(
n
)
=
{
n
,
n
[
1
[
2
¬
2
]
1
[
1
∖
1
∖
2
]
2
]
2
}
{\displaystyle f_{\varphi (\omega ,\zeta _{0})}(n)=\{n,n[1[2{\neg }2]1[1{\backslash }1{\backslash }2]2]2\}}
f
φ
(
ω
,
φ
(
ω
,
0
)
)
(
n
)
=
{
n
,
n
[
1
[
2
¬
2
]
1
[
1
[
2
¬
2
]
2
]
2
]
2
}
{\displaystyle f_{\varphi (\omega ,\varphi (\omega ,0))}(n)=\{n,n[1[2{\neg }2]1[1[2{\neg }2]2]2]2\}}
f
φ
(
ω
,
φ
(
ω
,
φ
(
ω
,
0
)
)
)
(
n
)
=
{
n
,
n
[
1
[
2
¬
2
]
1
[
1
[
2
¬
2
]
1
[
1
[
2
¬
2
]
2
]
2
]
2
]
2
}
{\displaystyle f_{\varphi (\omega ,\varphi (\omega ,\varphi (\omega ,0)))}(n)=\{n,n[1[2{\neg }2]1[1[2{\neg }2]1[1[2{\neg }2]2]2]2]2\}}
f
φ
(
ω
+
1
,
0
)
(
n
)
=
{
n
,
n
[
1
[
2
¬
2
]
1
∖
2
]
2
}
{\displaystyle f_{\varphi (\omega +1,0)}(n)=\{n,n[1[2{\neg }2]1{\backslash }2]2\}}
f
φ
(
ω
+
2
,
0
)
(
n
)
=
{
n
,
n
[
1
[
2
¬
2
]
1
∖
1
∖
2
]
2
}
{\displaystyle f_{\varphi (\omega +2,0)}(n)=\{n,n[1[2{\neg }2]1{\backslash }1{\backslash }2]2\}}
f
φ
(
ω
2
,
0
)
(
n
)
=
{
n
,
n
[
1
[
2
¬
2
]
1
[
2
¬
2
]
2
]
2
}
{\displaystyle f_{\varphi (\omega 2,0)}(n)=\{n,n[1[2{\neg }2]1[2{\neg }2]2]2\}}
f
φ
(
ω
2
,
0
)
(
n
)
=
{
n
,
n
[
1
[
3
¬
2
]
2
]
2
}
{\displaystyle f_{\varphi (\omega ^{2},0)}(n)=\{n,n[1[3{\neg }2]2]2\}}
f
φ
(
ω
3
,
0
)
(
n
)
=
{
n
,
n
[
1
[
4
¬
2
]
2
]
2
}
{\displaystyle f_{\varphi (\omega ^{3},0)}(n)=\{n,n[1[4{\neg }2]2]2\}}
f
φ
(
ω
ω
,
0
)
(
n
)
=
{
n
,
n
[
1
[
1
,
2
¬
2
]
2
]
2
}
{\displaystyle f_{\varphi (\omega ^{\omega },0)}(n)=\{n,n[1[1,2{\neg }2]2]2\}}
f
φ
(
ω
ω
ω
,
0
)
(
n
)
=
{
n
,
n
[
1
[
1
[
2
]
2
¬
2
]
2
]
2
}
{\displaystyle f_{\varphi (\omega ^{\omega ^{\omega }},0)}(n)=\{n,n[1[1[2]2{\neg }2]2]2\}}
f
φ
(
ϵ
0
,
0
)
(
n
)
=
{
n
,
n
[
1
[
1
[
1
∖
2
]
2
¬
2
]
2
]
2
}
{\displaystyle f_{\varphi ({\epsilon _{0}},0)}(n)=\{n,n[1[1[1{\backslash }2]2{\neg }2]2]2\}}
f
Γ
0
(
n
)
=
{
n
,
n
[
1
[
1
∖
2
¬
2
]
2
]
2
}
{\displaystyle f_{\Gamma _{0}}(n)=\{n,n[1[1{\backslash }2{\neg }2]2]2\}}
f
Γ
1
(
n
)
=
{
n
,
n
[
1
[
1
∖
2
¬
2
]
3
]
2
}
{\displaystyle f_{\Gamma _{1}}(n)=\{n,n[1[1{\backslash }2{\neg }2]3]2\}}
f
Γ
Γ
0
(
n
)
=
{
n
,
n
[
1
[
1
∖
2
¬
2
]
1
[
1
[
1
∖
2
¬
2
]
2
]
2
]
2
}
{\displaystyle f_{\Gamma _{\Gamma _{0}}}(n)=\{n,n[1[1{\backslash }2{\neg }2]1[1[1{\backslash }2{\neg }2]2]2]2\}}
f
φ
(
1
,
1
,
0
)
(
n
)
=
{
n
,
n
[
1
[
1
∖
2
¬
2
]
1
∖
2
]
2
}
{\displaystyle f_{\varphi (1,1,0)}(n)=\{n,n[1[1{\backslash }2{\neg }2]1{\backslash }2]2\}}
f
φ
(
1
,
2
,
0
)
(
n
)
=
{
n
,
n
[
1
[
1
∖
2
¬
2
]
1
∖
1
∖
2
]
2
}
{\displaystyle f_{\varphi (1,2,0)}(n)=\{n,n[1[1{\backslash }2{\neg }2]1{\backslash }1{\backslash }2]2\}}
f
φ
(
2
,
0
,
0
)
(
n
)
=
{
n
,
n
[
1
[
1
∖
2
¬
2
]
1
[
1
∖
2
¬
2
]
2
]
2
}
{\displaystyle f_{\varphi (2,0,0)}(n)=\{n,n[1[1{\backslash }2{\neg }2]1[1{\backslash }2{\neg }2]2]2\}}
f
φ
(
ω
,
0
,
0
)
(
n
)
=
{
n
,
n
[
1
[
2
∖
2
¬
2
]
2
]
2
}
{\displaystyle f_{\varphi (\omega ,0,0)}(n)=\{n,n[1[2{\backslash }2{\neg }2]2]2\}}
f
φ
(
Γ
0
,
0
,
0
)
(
n
)
=
{
n
,
n
[
1
[
1
[
1
[
1
∖
2
¬
2
]
2
]
2
∖
2
¬
2
]
2
]
2
}
{\displaystyle f_{\varphi (\Gamma _{0},0,0)}(n)=\{n,n[1[1[1[1{\backslash }2{\neg }2]2]2{\backslash }2{\neg }2]2]2\}}
f
φ
(
1
,
0
,
0
,
0
)
(
n
)
=
{
n
,
n
[
1
[
1
∖
3
¬
2
]
2
]
2
}
{\displaystyle f_{\varphi (1,0,0,0)}(n)=\{n,n[1[1{\backslash }3{\neg }2]2]2\}}
f
φ
(
1
,
0
,
0
,
0
,
0
)
(
n
)
=
{
n
,
n
[
1
[
1
∖
4
¬
2
]
2
]
2
}
{\displaystyle f_{\varphi (1,0,0,0,0)}(n)=\{n,n[1[1{\backslash }4{\neg }2]2]2\}}
f
ψ
(
Ω
Ω
ω
)
(
n
)
=
{
n
,
n
[
1
[
1
∖
1
,
2
¬
2
]
2
]
2
}
≈
T
R
E
E
(
n
)
{\displaystyle f_{\psi (\Omega ^{\Omega ^{\omega }})}(n)=\{n,n[1[1{\backslash }1,2{\neg }2]2]2\}\approx TREE(n)}
TREE(3) )
f
ψ
(
Ω
Ω
ψ
(
Ω
Ω
)
)
(
n
)
=
{
n
,
n
[
1
[
1
∖
1
[
1
[
1
∖
2
¬
2
]
2
]
2
¬
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega ^{\Omega ^{\psi (\Omega ^{\Omega })}})}(n)=\{n,n[1[1{\backslash }1[1[1{\backslash }2{\neg }2]2]2{\neg }2]2]2\}}
f
ψ
(
Ω
Ω
Ω
)
(
n
)
=
{
n
,
n
[
1
[
1
∖
1
∖
2
¬
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega }})}(n)=\{n,n[1[1{\backslash }1{\backslash }2{\neg }2]2]2\}}
f
ψ
(
Ω
Ω
Ω
2
)
(
n
)
=
{
n
,
n
[
1
[
1
∖
1
∖
1
∖
2
¬
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega ^{2}}})}(n)=\{n,n[1[1{\backslash }1{\backslash }1{\backslash }2{\neg }2]2]2\}}
f
ψ
(
Ω
Ω
Ω
ω
)
(
n
)
=
{
n
,
n
[
1
[
1
[
2
¬
2
]
2
¬
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega ^{\omega }}})}(n)=\{n,n[1[1[2{\neg }2]2{\neg }2]2]2\}}
f
ψ
(
Ω
Ω
Ω
ψ
(
Ω
Ω
Ω
)
)
(
n
)
=
{
n
,
n
[
1
[
1
[
1
[
1
[
1
∖
1
∖
2
¬
2
]
2
]
2
¬
2
]
2
¬
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega ^{\psi (\Omega ^{\Omega ^{\Omega }})}}})}(n)=\{n,n[1[1[1[1[1{\backslash }1{\backslash }2{\neg }2]2]2{\neg }2]2{\neg }2]2]2\}}
f
ψ
(
Ω
Ω
Ω
Ω
)
(
n
)
=
{
n
,
n
[
1
[
1
[
1
∖
2
¬
2
]
2
¬
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega ^{\Omega }}})}(n)=\{n,n[1[1[1{\backslash }2{\neg }2]2{\neg }2]2]2\}}
f
ψ
(
Ω
2
)
(
n
)
=
{
n
,
n
[
1
[
1
¬
3
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2})}(n)=\{n,n[1[1{\neg }3]2]2\}}
f
ψ
(
Ω
2
+
1
)
(
n
)
=
{
n
,
n
[
1
∖
2
[
1
¬
3
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}+1)}(n)=\{n,n[1{\backslash }2[1{\neg }3]2]2\}}
f
ψ
(
Ω
2
+
ω
)
(
n
)
=
{
n
,
n
[
1
∖
1
,
2
[
1
¬
3
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}+\omega )}(n)=\{n,n[1{\backslash }1,2[1{\neg }3]2]2\}}
f
ψ
(
Ω
2
+
ψ
(
Ω
Ω
)
)
(
n
)
=
{
n
,
n
[
1
∖
1
[
1
[
1
∖
2
¬
2
]
2
]
2
[
1
¬
3
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega ^{\Omega }))}(n)=\{n,n[1{\backslash }1[1[1{\backslash }2{\neg }2]2]2[1{\neg }3]2]2\}}
f
ψ
(
Ω
2
+
ψ
(
Ω
Ω
ω
)
)
(
n
)
=
{
n
,
n
[
1
∖
1
[
1
[
1
∖
1
,
2
¬
2
]
2
]
2
[
1
¬
3
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega ^{\Omega ^{\omega }}))}(n)=\{n,n[1{\backslash }1[1[1{\backslash }1,2{\neg }2]2]2[1{\neg }3]2]2\}}
f
ψ
(
Ω
2
+
ψ
(
Ω
Ω
Ω
)
)
(
n
)
=
{
n
,
n
[
1
∖
1
[
1
[
1
∖
1
∖
2
¬
2
]
2
]
2
[
1
¬
3
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega ^{\Omega ^{\Omega }}))}(n)=\{n,n[1{\backslash }1[1[1{\backslash }1{\backslash }2{\neg }2]2]2[1{\neg }3]2]2\}}
f
ψ
(
Ω
2
+
ψ
(
Ω
Ω
Ω
Ω
)
)
(
n
)
=
{
n
,
n
[
1
∖
1
[
1
[
1
[
1
∖
2
¬
2
]
2
¬
2
]
2
]
2
[
1
¬
3
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega ^{\Omega ^{\Omega ^{\Omega }}}))}(n)=\{n,n[1{\backslash }1[1[1[1{\backslash }2{\neg }2]2{\neg }2]2]2[1{\neg }3]2]2\}}
f
ψ
(
Ω
2
+
ψ
(
Ω
2
)
)
(
n
)
=
{
n
,
n
[
1
∖
1
[
1
[
1
¬
3
]
2
]
2
[
1
¬
3
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega _{2}))}(n)=\{n,n[1{\backslash }1[1[1{\neg }3]2]2[1{\neg }3]2]2\}}
f
ψ
(
Ω
2
+
Ω
)
(
n
)
=
{
n
,
n
[
1
∖
1
∖
2
[
1
¬
3
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}+\Omega )}(n)=\{n,n[1{\backslash }1{\backslash }2[1{\neg }3]2]2\}}
f
ψ
(
Ω
2
+
Ω
2
)
(
n
)
=
{
n
,
n
[
1
∖
1
∖
1
∖
2
[
1
¬
3
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}+\Omega ^{2})}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }2[1{\neg }3]2]2\}}
f
ψ
(
Ω
2
+
Ω
ω
)
(
n
)
=
{
n
,
n
[
1
[
2
¬
2
]
2
[
1
¬
3
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}+\Omega ^{\omega })}(n)=\{n,n[1[2{\neg }2]2[1{\neg }3]2]2\}}
f
ψ
(
Ω
2
+
Ω
Ω
)
(
n
)
=
{
n
,
n
[
1
[
1
∖
2
¬
2
]
2
[
1
¬
3
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}+\Omega ^{\Omega })}(n)=\{n,n[1[1{\backslash }2{\neg }2]2[1{\neg }3]2]2\}}
f
ψ
(
Ω
2
+
ψ
1
(
Ω
2
)
)
(
n
)
=
{
n
,
n
[
1
[
1
¬
3
]
3
]
2
}
{\displaystyle f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}))}(n)=\{n,n[1[1{\neg }3]3]2\}}
f
ψ
(
Ω
2
+
ψ
1
(
Ω
2
+
ψ
1
(
Ω
2
)
)
)
(
n
)
=
{
n
,
n
[
1
[
1
¬
3
]
1
[
1
¬
3
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}(\Omega _{2})))}(n)=\{n,n[1[1{\neg }3]1[1{\neg }3]2]2\}}
f
ψ
(
Ω
2
+
ψ
1
(
Ω
2
+
ψ
1
(
Ω
2
+
1
)
)
)
(
n
)
=
{
n
,
n
[
1
[
2
¬
3
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}(\Omega _{2}+1)))}(n)=\{n,n[1[2{\neg }3]2]2\}}
f
ψ
(
Ω
2
+
ψ
1
(
Ω
2
+
ψ
1
(
Ω
2
+
ψ
1
(
Ω
2
)
)
)
)
(
n
)
=
{
n
,
n
[
1
[
1
[
1
¬
3
]
2
¬
3
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}(\Omega _{2}))))}(n)=\{n,n[1[1[1{\neg }3]2{\neg }3]2]2\}}
f
ψ
(
Ω
2
2
)
(
n
)
=
{
n
,
n
[
1
[
1
¬
4
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}2)}(n)=\{n,n[1[1{\neg }4]2]2\}}
f
ψ
(
Ω
2
3
)
(
n
)
=
{
n
,
n
[
1
[
1
¬
5
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}3)}(n)=\{n,n[1[1{\neg }5]2]2\}}
f
ψ
(
Ω
2
ω
)
(
n
)
=
{
n
,
n
[
1
[
1
¬
1
,
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}\omega )}(n)=\{n,n[1[1{\neg }1,2]2]2\}}
f
ψ
(
Ω
2
ψ
(
0
)
)
(
n
)
=
{
n
,
n
[
1
[
1
¬
1
[
1
∖
2
]
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}\psi (0))}(n)=\{n,n[1[1{\neg }1[1{\backslash }2]2]2]2\}}
f
ψ
(
Ω
2
ψ
(
Ω
Ω
)
)
(
n
)
=
{
n
,
n
[
1
[
1
¬
1
[
1
[
1
∖
2
¬
2
]
2
]
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}\psi (\Omega ^{\Omega }))}(n)=\{n,n[1[1{\neg }1[1[1{\backslash }2{\neg }2]2]2]2]2\}}
f
ψ
(
Ω
2
ψ
(
Ω
2
)
)
(
n
)
=
{
n
,
n
[
1
[
1
¬
1
[
1
[
1
¬
3
]
2
]
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}\psi (\Omega _{2}))}(n)=\{n,n[1[1{\neg }1[1[1{\neg }3]2]2]2]2\}}
f
ψ
(
Ω
2
Ω
)
(
n
)
=
{
n
,
n
[
1
[
1
¬
1
∖
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}\Omega )}(n)=\{n,n[1[1{\neg }1{\backslash }2]2]2\}}
f
ψ
(
Ω
2
Ω
Ω
)
(
n
)
=
{
n
,
n
[
1
[
1
¬
1
[
1
∖
2
¬
2
]
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}\Omega ^{\Omega })}(n)=\{n,n[1[1{\neg }1[1{\backslash }2{\neg }2]2]2]2\}}
f
ψ
(
Ω
2
ψ
1
(
Ω
2
)
)
(
n
)
=
{
n
,
n
[
1
[
1
¬
1
[
1
¬
3
]
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}\psi _{1}(\Omega _{2}))}(n)=\{n,n[1[1{\neg }1[1{\neg }3]2]2]2\}}
f
ψ
(
Ω
2
2
)
(
n
)
=
{
n
,
n
[
1
[
1
¬
1
¬
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}^{2})}(n)=\{n,n[1[1{\neg }1{\neg }2]2]2\}}
f
ψ
(
Ω
2
ω
)
(
n
)
=
{
n
,
n
[
1
[
1
[
2
∖
3
2
]
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}^{\omega })}(n)=\{n,n[1[1[2{\backslash }_{3}2]2]2]2\}}
f
ψ
(
Ω
2
Ω
2
)
(
n
)
=
{
n
,
n
[
1
[
1
[
1
∖
2
2
∖
3
2
]
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{2}^{\Omega _{2}})}(n)=\{n,n[1[1[1{\backslash }_{2}2{\backslash }_{3}2]2]2]2\}}
f
ψ
(
Ω
3
)
(
n
)
=
{
n
,
n
[
1
[
1
[
1
∖
3
3
]
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{3})}(n)=\{n,n[1[1[1{\backslash }_{3}3]2]2]2\}}
f
ψ
(
Ω
3
Ω
3
)
(
n
)
=
{
n
,
n
[
1
[
1
[
1
[
1
∖
3
2
∖
4
2
]
2
]
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{3}^{\Omega _{3}})}(n)=\{n,n[1[1[1[1{\backslash }_{3}2{\backslash }_{4}2]2]2]2]2\}}
f
ψ
(
Ω
4
)
(
n
)
=
{
n
,
n
[
1
[
1
[
1
[
1
∖
4
3
]
2
]
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{4})}(n)=\{n,n[1[1[1[1{\backslash }_{4}3]2]2]2]2\}}
f
ψ
(
Ω
ω
)
(
n
)
=
{
n
,
n
[
1
[
2
∖
1
,
2
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{\omega })}(n)=\{n,n[1[2{\backslash }_{1,2}2]2]2\}}
f
ψ
(
Ω
ψ
(
Ω
)
)
(
n
)
=
{
n
,
n
[
1
[
2
∖
1
[
1
∖
2
]
2
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{\psi (\Omega )})}(n)=\{n,n[1[2{\backslash }_{1[1{\backslash }2]2}2]2]2\}}
f
ψ
(
Ω
ψ
(
Ω
ψ
(
Ω
)
)
)
(
n
)
=
{
n
,
n
[
1
[
2
∖
1
[
2
∖
1
[
1
∖
2
]
2
2
]
2
2
]
2
]
2
}
{\displaystyle f_{\psi (\Omega _{\psi (\Omega _{\psi (\Omega )})})}(n)=\{n,n[1[2{\backslash }_{1[2{\backslash _{1[1{\backslash }2]2}}2]2}2]2]2\}}
f
ψ
(
Ω
Ω
)
(
n
)
=
{
n
,
n
[
1
[
2
∖
1
∖
2
2
]
2
]
2
}
=
(
0
,
0
,
0
)
(
1
,
1
,
1
)
(
2
,
1
,
1
)
(
3
,
1
,
0
)
[
n
]
{\displaystyle f_{\psi (\Omega _{\Omega })}(n)=\{n,n[1[2{\backslash }_{1{\backslash }2}2]2]2\}=(0,0,0)(1,1,1)(2,1,1)(3,1,0)[n]}
Bashicu Matrix System )
f
ψ
(
Ω
Ω
2
)
(
n
)
=
(
0
,
0
,
0
)
(
1
,
1
,
1
)
(
2
,
1
,
1
)
(
3
,
1
,
0
)
(
1
,
1
,
0
)
(
2
,
2
,
1
)
(
3
,
2
,
1
)
(
4
,
2
,
0
)
[
n
]
{\displaystyle f_{\psi ({\Omega _{\Omega _{2}}})}(n)=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,0)(2,2,1)(3,2,1)(4,2,0)[n]}
f
ψ
(
Ω
Ω
ω
)
(
n
)
=
(
0
,
0
,
0
)
(
1
,
1
,
1
)
(
2
,
1
,
1
)
(
3
,
1
,
0
)
(
1
,
1
,
1
)
[
n
]
{\displaystyle f_{\psi ({\Omega _{\Omega _{\omega }}})}(n)=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,1)[n]}
f
ψ
(
Ω
Ω
Ω
)
(
n
)
=
(
0
,
0
,
0
)
(
1
,
1
,
1
)
(
2
,
1
,
1
)
(
3
,
1
,
0
)
(
1
,
1
,
1
)
(
2
,
1
,
1
)
(
3
,
1
,
0
)
[
n
]
{\displaystyle f_{\psi ({\Omega _{\Omega _{\Omega }}})}(n)=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,1)(2,1,1)(3,1,0)[n]}
f
ψ
(
ψ
I
(
0
)
)
(
n
)
=
(
0
,
0
,
0
)
(
1
,
1
,
1
)
(
2
,
1
,
1
)
(
3
,
1
,
0
)
(
2
,
0
,
0
)
[
n
]
{\displaystyle f_{\psi (\psi _{I}(0))}(n)=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)[n]}
Buchholz, W.; Wainer, S.S (1987). "Provably Computable Functions and the Fast Growing Hierarchy". Logic and Combinatorics , edited by S. Simpson, Contemporary Mathematics, Vol. 65, AMS, 179-198.
Cichon, E. A.; Wainer, S. S. (1983), "The slow-growing and the Grzegorczyk hierarchies", The Journal of Symbolic Logic , 48 (2): 399—408, doi :10.2307/2273557 , ISSN 0022-4812 , MR 0704094 Gallier, Jean H. (1991), "What's so special about Kruskal's theorem and the ordinal Γ0 ? A survey of some results in proof theory" , Ann. Pure Appl. Logic , 53 (3): 199—260, doi :10.1016/0168-0072(91)90022-E , MR 1129778 (недоступная ссылка) PDF's: part 1 2 3 . (In particular part 3, Section 12, pp. 59–64, "A Glimpse at Hierarchies of Fast and Slow Growing Functions".)Girard, Jean-Yves (1981), "Π1 2 -logic. I. Dilators", Annals of Mathematical Logic , 21 (2): 75—219, doi :10.1016/0003-4843(81)90016-4 , ISSN 0003-4843 , MR 0656793 Löb, M.H.; Wainer, S.S. (1970), "Hierarchies of number theoretic functions", Arch. Math. Logik , 13. Correction, Arch. Math. Logik , 14, 1971. Part I doi :10.1007/BF01967649 , Part 2 doi :10.1007/BF01973616 , Corrections doi :10.1007/BF01991855 .
Prömel, H. J.; Thumser, W.; Voigt, B. "Fast growing functions based on Ramsey theorems", Discrete Mathematics , v.95 n.1-3, p. 341-358, Dec. 1991 doi :10.1016/0012-365X(91)90346-4 .
Wainer, S.S (1989), "Slow Growing Versus Fast Growing ". Journal of Symbolic Logic 54 (2): 608-614. 
![{\displaystyle f_{\alpha }(n)=f_{\alpha [n]}(n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a206f2343740c51355fec5b30979922b98ef77a)
![{\displaystyle \alpha [n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6afd165397678256dc95ef2843142ed69e38680)
![{\displaystyle \omega [n]=n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d56cec7cc0e38615faa684337979c21c95d8fb5)
![{\displaystyle (\omega ^{\alpha _{1}}+\omega ^{\alpha _{2}}+\cdots +\omega ^{\alpha _{k-1}}+\omega ^{\alpha _{k}})[n]=\omega ^{\alpha _{1}}+\omega ^{\alpha _{2}}+\cdots +\omega ^{\alpha _{k-1}}+\omega ^{\alpha _{k}}[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8769ca9ce5db3eb7a2d36d5c0ce1ed5aed6c760b)
![{\displaystyle \omega ^{\alpha +1}[n]=\omega ^{\alpha }n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68a0d5a520ec15e1d18e9b9594a5a529ff65fe16)
![{\displaystyle \omega ^{\alpha }[n]=\omega ^{\alpha [n]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f70df39328ecb42897bf325f6cf299ab7993762)
![{\displaystyle \varepsilon _{0}[0]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8395ad7004fbbc304b896906d9299dc75dc7799)
![{\displaystyle \varepsilon _{0}[n+1]=\omega ^{\varepsilon _{0}[n]}=\omega \uparrow \uparrow n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f764230e8ae639cdea743822e4ef40007ecf4c1)
![{\displaystyle f_{\omega ^{\omega ^{\omega }}}(n)=\{n,n[1,2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6f9a47e174198b1fd1ec80c832fa5f5e2a6c9bc)
![{\displaystyle f_{\omega ^{\omega ^{\omega ^{2}}}}(n)=\{n,n[1,1,2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54dcad4d66d863fc475ee22b688b80339c2223fe)
![{\displaystyle f_{\omega ^{\omega ^{\omega ^{\omega }}}}(n)=\{n,n[1[2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b66188d185afe8721da132d06e0ae9847a46683a)
![{\displaystyle f_{\epsilon _{0}}(n)=\{n,n[1{\backslash }2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a568f93fa207c75691d666c05b412e2031072f9f)
![{\displaystyle f_{\omega ^{\omega ^{\epsilon _{0}+1}}}(n)=\{n,n[1[1{\backslash }2]2{\backslash }2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcd0e95f7e3ee0ac9810d13af1128c5fad50cc43)
![{\displaystyle f_{\omega ^{\omega ^{\omega ^{\epsilon _{0}+1}}}}(n)=\{n,n[1[1[1{\backslash }2]2{\backslash }2]2{\backslash }2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc92b2fb568efa19306c90caf52a1a87efa5fc68)
![{\displaystyle f_{\epsilon _{1}}(n)=\{n,n[1{\backslash }3]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a1a27cbe0934ca8ae3d3d2c67b3a34e3ec5a50c)
![{\displaystyle f_{\epsilon _{2}}(n)=\{n,n[1{\backslash }4]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4f7d08eca53c881a59189914f1718b99a8c9992)
![{\displaystyle f_{\epsilon _{\omega }}(n)=\{n,n[1{\backslash }1,2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8189566d5ae09b60c6b5ffcff341e3319b38e08d)
![{\displaystyle f_{\epsilon _{\omega ^{\omega }}}(n)=\{n,n[1{\backslash }1[2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f8d1a8684459255238d75e96064965497ff26a)
![{\displaystyle f_{\epsilon _{\epsilon _{0}}}(n)=\{n,n[1{\backslash }1[1{\backslash }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4128e83f0c7781c45122f93bae784865811bfa6)
![{\displaystyle f_{\epsilon _{\epsilon _{\epsilon _{0}}}}(n)=\{n,n[1{\backslash }1[1{\backslash }1[1{\backslash }2]2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbf8a7a7b512942208491e4f344db72c913f0796)
![{\displaystyle f_{\zeta _{0}}(n)=\{n,n[1{\backslash }1{\backslash }2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68ccc31a1ae2e3157cbfd762fe5b9578dbd63872)
![{\displaystyle f_{\epsilon _{\zeta _{0}+1}}(n)=\{n,n[1{\backslash }2{\backslash }2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9df387b36e2f54f2cc963763dc33b829dc0530cf)
![{\displaystyle f_{\epsilon _{\zeta _{0}+\omega }}(n)=\{n,n[1{\backslash }1,2{\backslash }2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a19094d38ce288f0ff3752e63601c805caae670)
![{\displaystyle f_{\epsilon _{\zeta _{0}+\epsilon _{0}}}(n)=\{n,n[1{\backslash }1{[1{\backslash }2]}2{\backslash }2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9be9157b94dac5e4d319bd523a52d412536ceba)
![{\displaystyle f_{\epsilon _{\zeta _{0}2}}(n)=\{n,n[1{\backslash }1{[1{\backslash }1{\backslash }2]}2{\backslash }2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77de79b77248ad2284d690ae703d6428b325fe91)
![{\displaystyle f_{\epsilon _{\epsilon _{\zeta _{0}+1}}}(n)=\{n,n[1{\backslash }1{[1{\backslash }2{\backslash }2]}2{\backslash }2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cfca72e3d6084d6e9429964538d671395379fcc)
![{\displaystyle f_{\zeta _{1}}(n)=\{n,n[1{\backslash }1{\backslash }3]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ffe30602e8eec3a57a6a23a3e1330cbd82490efc)
![{\displaystyle f_{\zeta _{\omega }}(n)=\{n,n[1{\backslash }1{\backslash }1,2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a208c01de2e6db081ed5010c64250a71017a8ebf)
![{\displaystyle f_{\zeta _{\epsilon _{0}}}(n)=\{n,n[1{\backslash }1{\backslash }1[1{\backslash }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd1038b1b3480bb38db37221dccae2feae68944b)
![{\displaystyle f_{\zeta _{\zeta _{0}}}(n)=\{n,n[1{\backslash }1{\backslash }1[1{\backslash }1{\backslash }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb5a9192107a091308201dfbb018594ed571fce8)
![{\displaystyle f_{\eta _{0}}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8e288abece476be9fd37523c1ea7f36a5af5a7e)
![{\displaystyle f_{\varphi (4,0)}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }1{\backslash }2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7522a4262b25b43e6782b128092ee81294fd68fe)
![{\displaystyle f_{\varphi (m,0)}(n)=\{n,n[1{\backslash }1{\backslash }1{\cdots }1{\backslash }2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5f9577e4f8ac27466f19846c95325cef3df47f9)
![{\displaystyle f_{\varphi (\omega ,0)}(n)=\{n,n[1[2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/caf3a52e2a92ac44554db49b14d2474953444532)
![{\displaystyle f_{\epsilon _{\varphi (\omega ,0)+1}}(n)=\{n,n[1{\backslash }2[2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41dff54cf6811e1ca8c74bcab0c74ad67da77568)
![{\displaystyle f_{\zeta _{\varphi (\omega ,0)+1}}(n)=\{n,n[1{\backslash }1{\backslash }2[2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58298c0f8f46d03dfb5173d305ccaa2466e8c352)
![{\displaystyle f_{\eta _{\varphi (\omega ,0)+1}}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }2[2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e2a7906758a3e20350a0f24c91d4c6d0b18de1)
![{\displaystyle f_{\varphi (\omega ,1)}(n)=\{n,n[1[2{\neg }2]3]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4e2d9dab541e99336d7008c7a1838b58a47bbb8)
![{\displaystyle f_{\varphi (\omega ,2)}(n)=\{n,n[1[2{\neg }2]4]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68361e524a79f6dc6717202b852e405e609a4917)
![{\displaystyle f_{\varphi (\omega ,\omega )}(n)=\{n,n[1[2{\neg }2]1,2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/842585099bbcb4287db2a3647f3ae0ab48ced4d8)
![{\displaystyle f_{\varphi (\omega ,\epsilon _{0})}(n)=\{n,n[1[2{\neg }2]1[1{\backslash }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1696dd23837c1c5f2362426c3687707dc52ff718)
![{\displaystyle f_{\varphi (\omega ,\zeta _{0})}(n)=\{n,n[1[2{\neg }2]1[1{\backslash }1{\backslash }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52fc11ca34ab97fecfdba2cec3fb01879a6da160)
![{\displaystyle f_{\varphi (\omega ,\varphi (\omega ,0))}(n)=\{n,n[1[2{\neg }2]1[1[2{\neg }2]2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0934aab329e362d18e0ffb60ca232bd420abbd)
![{\displaystyle f_{\varphi (\omega ,\varphi (\omega ,\varphi (\omega ,0)))}(n)=\{n,n[1[2{\neg }2]1[1[2{\neg }2]1[1[2{\neg }2]2]2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63df2466f603db3522005590a0b81122eda57a9e)
![{\displaystyle f_{\varphi (\omega +1,0)}(n)=\{n,n[1[2{\neg }2]1{\backslash }2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/beeb500574fd0b61c6ca836c65162dbad2702c86)
![{\displaystyle f_{\varphi (\omega +2,0)}(n)=\{n,n[1[2{\neg }2]1{\backslash }1{\backslash }2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c4ec0868bb1a7bb47ebb615c1fa329ab3cb675d)
![{\displaystyle f_{\varphi (\omega 2,0)}(n)=\{n,n[1[2{\neg }2]1[2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14c47f80a7d94a2fe9ab43da85e30c682748e84c)
![{\displaystyle f_{\varphi (\omega ^{2},0)}(n)=\{n,n[1[3{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c73d78a983f67781cff40d582e39443301d7fc2)
![{\displaystyle f_{\varphi (\omega ^{3},0)}(n)=\{n,n[1[4{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/801c00c69e061569b984b905831eb8dddb700722)
![{\displaystyle f_{\varphi (\omega ^{\omega },0)}(n)=\{n,n[1[1,2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c0f0f60be020d5ef0ce8c34648aa380bc83dfe8)
![{\displaystyle f_{\varphi (\omega ^{\omega ^{\omega }},0)}(n)=\{n,n[1[1[2]2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa7b54dfc97b93059ebaeecb53311327eb703f6b)
![{\displaystyle f_{\varphi ({\epsilon _{0}},0)}(n)=\{n,n[1[1[1{\backslash }2]2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a798efbbf22654a0eabaac88f31dd0ca6d7d50e2)
![{\displaystyle f_{\Gamma _{0}}(n)=\{n,n[1[1{\backslash }2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02aab8b5801f8da817294304ebeb374be2e49453)
![{\displaystyle f_{\Gamma _{1}}(n)=\{n,n[1[1{\backslash }2{\neg }2]3]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/819733d3c5d40152b9298fdb256eef404d46a011)
![{\displaystyle f_{\Gamma _{\Gamma _{0}}}(n)=\{n,n[1[1{\backslash }2{\neg }2]1[1[1{\backslash }2{\neg }2]2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0291974f5960cabd6bce20c728e085a365992be)
![{\displaystyle f_{\varphi (1,1,0)}(n)=\{n,n[1[1{\backslash }2{\neg }2]1{\backslash }2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de3dbdfd62658c06010dce48665793a37b914680)
![{\displaystyle f_{\varphi (1,2,0)}(n)=\{n,n[1[1{\backslash }2{\neg }2]1{\backslash }1{\backslash }2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf40f9c6af7dd13e97689e831607180c1bc1f7f)
![{\displaystyle f_{\varphi (2,0,0)}(n)=\{n,n[1[1{\backslash }2{\neg }2]1[1{\backslash }2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d996278ad44661addb439b607c4d1dfb4c018d0c)
![{\displaystyle f_{\varphi (\omega ,0,0)}(n)=\{n,n[1[2{\backslash }2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75d8a36ca368a0e641078cec0a500ec8028a4af0)
![{\displaystyle f_{\varphi (\Gamma _{0},0,0)}(n)=\{n,n[1[1[1[1{\backslash }2{\neg }2]2]2{\backslash }2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb9636b5e3a92ae829e439b3fdb013b43bd19492)
![{\displaystyle f_{\varphi (1,0,0,0)}(n)=\{n,n[1[1{\backslash }3{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb0a988566c985f8737bbbf96de7f17cfda955ef)
![{\displaystyle f_{\varphi (1,0,0,0,0)}(n)=\{n,n[1[1{\backslash }4{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e763c65ab1e84edb2178fd2c31ecb529f19420)
![{\displaystyle f_{\psi (\Omega ^{\Omega ^{\omega }})}(n)=\{n,n[1[1{\backslash }1,2{\neg }2]2]2\}\approx TREE(n)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f261e6bdb37273499edd2d3d51d58408e425c10)
![{\displaystyle f_{\psi (\Omega ^{\Omega ^{\psi (\Omega ^{\Omega })}})}(n)=\{n,n[1[1{\backslash }1[1[1{\backslash }2{\neg }2]2]2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c4adcc3aa0c19c9354ae2b6685344b8ec1a706f)
![{\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega }})}(n)=\{n,n[1[1{\backslash }1{\backslash }2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21f665fee1035bf918bb7d00d197eceafe3462f3)
![{\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega ^{2}}})}(n)=\{n,n[1[1{\backslash }1{\backslash }1{\backslash }2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/675de27d63cb62f01bcf2485c6edcd4381792efe)
![{\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega ^{\omega }}})}(n)=\{n,n[1[1[2{\neg }2]2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a4b45be7a9b43f18dd273bd10f89696f7771489)
![{\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega ^{\psi (\Omega ^{\Omega ^{\Omega }})}}})}(n)=\{n,n[1[1[1[1[1{\backslash }1{\backslash }2{\neg }2]2]2{\neg }2]2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c64adef9639248827feb48a1ea9f66977eda6f8d)
![{\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega ^{\Omega }}})}(n)=\{n,n[1[1[1{\backslash }2{\neg }2]2{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99be6883a7500b239aa2c9e19d0e62d13788bf52)
![{\displaystyle f_{\psi (\Omega _{2})}(n)=\{n,n[1[1{\neg }3]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c4c50ab1b13836d449532675d462d9ebfbc785c5)
![{\displaystyle f_{\psi (\Omega _{2}+1)}(n)=\{n,n[1{\backslash }2[1{\neg }3]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/775516fe2920b0a73a0e49a2dbe6880f807badf0)
![{\displaystyle f_{\psi (\Omega _{2}+\omega )}(n)=\{n,n[1{\backslash }1,2[1{\neg }3]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc2b571772c0f8feb27a155e036391ab348b06f8)
![{\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega ^{\Omega }))}(n)=\{n,n[1{\backslash }1[1[1{\backslash }2{\neg }2]2]2[1{\neg }3]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82fcc4ddc8a7bb96f0155de41741dd86b39db317)
![{\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega ^{\Omega ^{\omega }}))}(n)=\{n,n[1{\backslash }1[1[1{\backslash }1,2{\neg }2]2]2[1{\neg }3]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5454ef75d1db01e0c1bfa761d7cf9313468202ad)
![{\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega ^{\Omega ^{\Omega }}))}(n)=\{n,n[1{\backslash }1[1[1{\backslash }1{\backslash }2{\neg }2]2]2[1{\neg }3]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a736a13fdfb46c1d4242809c2bb162e72acc739)
![{\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega ^{\Omega ^{\Omega ^{\Omega }}}))}(n)=\{n,n[1{\backslash }1[1[1[1{\backslash }2{\neg }2]2{\neg }2]2]2[1{\neg }3]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3792ee2ac2f5edbef2479457433bed3ab81de999)
![{\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega _{2}))}(n)=\{n,n[1{\backslash }1[1[1{\neg }3]2]2[1{\neg }3]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d29088d281678e6ade613cbf292a47f82e911e6)
![{\displaystyle f_{\psi (\Omega _{2}+\Omega )}(n)=\{n,n[1{\backslash }1{\backslash }2[1{\neg }3]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d375ebff67449849dfe572b0c4226d3d11d9b0e)
![{\displaystyle f_{\psi (\Omega _{2}+\Omega ^{2})}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }2[1{\neg }3]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e4edf9ee77b25bedab79938ae1ccb3a12a0af38)
![{\displaystyle f_{\psi (\Omega _{2}+\Omega ^{\omega })}(n)=\{n,n[1[2{\neg }2]2[1{\neg }3]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc02f9eb5489b96f1ffb2971a282a580d1a9a0f3)
![{\displaystyle f_{\psi (\Omega _{2}+\Omega ^{\Omega })}(n)=\{n,n[1[1{\backslash }2{\neg }2]2[1{\neg }3]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4de78cd9ef8caecfad93cd4deb9d042b8780af)
![{\displaystyle f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}))}(n)=\{n,n[1[1{\neg }3]3]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf984185dca6a31c64c4d959d0702441ca9bb06f)
![{\displaystyle f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}(\Omega _{2})))}(n)=\{n,n[1[1{\neg }3]1[1{\neg }3]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/721362e13c1a80ce931992346bb3af5e8c474ead)
![{\displaystyle f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}(\Omega _{2}+1)))}(n)=\{n,n[1[2{\neg }3]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b65c3728a151d3a85e0533b55eb865736f624978)
![{\displaystyle f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}(\Omega _{2}))))}(n)=\{n,n[1[1[1{\neg }3]2{\neg }3]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7fc27b2e16d57930f82f5caa2a0c17ca01ffb390)
![{\displaystyle f_{\psi (\Omega _{2}2)}(n)=\{n,n[1[1{\neg }4]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a16efaefede901a3f25841423e6c3f3ab40ffc4)
![{\displaystyle f_{\psi (\Omega _{2}3)}(n)=\{n,n[1[1{\neg }5]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c632f758157d16acd1a458cf4596f04a3a8df9c)
![{\displaystyle f_{\psi (\Omega _{2}\omega )}(n)=\{n,n[1[1{\neg }1,2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0bfe176d1f84a1504b74b3818b0504f86a17aa2)
![{\displaystyle f_{\psi (\Omega _{2}\psi (0))}(n)=\{n,n[1[1{\neg }1[1{\backslash }2]2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7614f6d9ca2a3dd4ae80b732ce592d167edd7d94)
![{\displaystyle f_{\psi (\Omega _{2}\psi (\Omega ^{\Omega }))}(n)=\{n,n[1[1{\neg }1[1[1{\backslash }2{\neg }2]2]2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/323490efeee9d73c237fb9cfe631272d6564a284)
![{\displaystyle f_{\psi (\Omega _{2}\psi (\Omega _{2}))}(n)=\{n,n[1[1{\neg }1[1[1{\neg }3]2]2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aaccff636609fed6e0ffae003ebf7a7769a9c26a)
![{\displaystyle f_{\psi (\Omega _{2}\Omega )}(n)=\{n,n[1[1{\neg }1{\backslash }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b9bae03f1986666de2a350a7848e0f62652eb97)
![{\displaystyle f_{\psi (\Omega _{2}\Omega ^{\Omega })}(n)=\{n,n[1[1{\neg }1[1{\backslash }2{\neg }2]2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1efe02a0c8a4cc6531e067f6b453e7b47aaa910)
![{\displaystyle f_{\psi (\Omega _{2}\psi _{1}(\Omega _{2}))}(n)=\{n,n[1[1{\neg }1[1{\neg }3]2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a56880423cb1f1a24ffd35ae0775332fd9b493ad)
![{\displaystyle f_{\psi (\Omega _{2}^{2})}(n)=\{n,n[1[1{\neg }1{\neg }2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5927ab859e27eff27588f2ce36c421d7d5f6861)
![{\displaystyle f_{\psi (\Omega _{2}^{\omega })}(n)=\{n,n[1[1[2{\backslash }_{3}2]2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f515e3d4aed3831729d411ff8cc6cadbf7168c28)
![{\displaystyle f_{\psi (\Omega _{2}^{\Omega _{2}})}(n)=\{n,n[1[1[1{\backslash }_{2}2{\backslash }_{3}2]2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e5192cda0e63874f1cbad4fc1badc6dfe353fdd)
![{\displaystyle f_{\psi (\Omega _{3})}(n)=\{n,n[1[1[1{\backslash }_{3}3]2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39ab6198390be4624978d6cac2199bdc1b6e4275)
![{\displaystyle f_{\psi (\Omega _{3}^{\Omega _{3}})}(n)=\{n,n[1[1[1[1{\backslash }_{3}2{\backslash }_{4}2]2]2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af42af3b9147f8dda794735d5d537aaca168d561)
![{\displaystyle f_{\psi (\Omega _{4})}(n)=\{n,n[1[1[1[1{\backslash }_{4}3]2]2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9d8dd1222b5c76684253e6989c4b5a45123f474)
![{\displaystyle f_{\psi (\Omega _{\omega })}(n)=\{n,n[1[2{\backslash }_{1,2}2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3e9e8dc8eb1e287ed0e7b821cba0133c86a5f7)
![{\displaystyle f_{\psi (\Omega _{\psi (\Omega )})}(n)=\{n,n[1[2{\backslash }_{1[1{\backslash }2]2}2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b225383e599bc38eea533e9d9aecbd733cc96821)
![{\displaystyle f_{\psi (\Omega _{\psi (\Omega _{\psi (\Omega )})})}(n)=\{n,n[1[2{\backslash }_{1[2{\backslash _{1[1{\backslash }2]2}}2]2}2]2]2\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d86cb32275524bdc886265a6cb49e93bc0f48f57)
![{\displaystyle f_{\psi (\Omega _{\Omega })}(n)=\{n,n[1[2{\backslash }_{1{\backslash }2}2]2]2\}=(0,0,0)(1,1,1)(2,1,1)(3,1,0)[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dac13b4350ffa191fd4a4b2ed200af972ea290ec)
![{\displaystyle f_{\psi ({\Omega _{\Omega _{2}}})}(n)=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,0)(2,2,1)(3,2,1)(4,2,0)[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dd6d7a56658fcbcb2c0d8a99c83270f9e4bf48d)
![{\displaystyle f_{\psi ({\Omega _{\Omega _{\omega }}})}(n)=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,1)[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32e180398977b058b96c377761877935245e5adf)
![{\displaystyle f_{\psi ({\Omega _{\Omega _{\Omega }}})}(n)=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,1)(2,1,1)(3,1,0)[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/536c0ce71b1a4914a4fab85a984549b4944a7384)
![{\displaystyle f_{\psi (\psi _{I}(0))}(n)=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a47b449d7e470289ce09782fa3114f2158352c2)