Быстрорастущая иерархия
(также называемая
расширенной иерархией Гржегорчика
) — это семейство быстрорастущих функций, индексированных
ординалами
. Наиболее известным частным случаем быстрорастущей иерархии является иерархия
Лёба
-Вайнера.
Определение
Быстрорастущая иерархия определяется следующими правилами:
f
0
(
n
)
=
n
+
1
{\displaystyle f_{0}(n)=n+1}
(в общем случае
f
0
(
n
)
{\displaystyle f_{0}(n)}
может быть любой растущей функцией),
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)}
если
α
{\displaystyle \alpha }
предельный ординал,
где
α
[
n
]
{\displaystyle \alpha [n]}
является
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]}}
если
α
{\displaystyle \alpha }
предельный ординал,
ε
0
[
0
]
=
0
{\displaystyle \varepsilon _{0}[0]=0}
и
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}}
приведены в статьях о
функциях Веблена
и
функциях Бухгольца
.
Примеры
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}
верно
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 }
соответствует пределу
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}
. Для дальнейшего роста можно использовать
функцию Веблена
и другие, еще более мощные нотации для ординалов
.
Примеры
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}
(см.
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\}}
(см.
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)}
(см.
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\}}
(m раз)
f
ω
ω
(
n
)
=
{
n
,
n
,
n
,
.
.
.
,
n
}
{\displaystyle f_{\omega ^{\omega }}(n)=\{n,n,n,...,n\}}
(n раз)
=
{
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\}}
(см.
)
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\}}
(m обратных слэшей)
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)}
(см.
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]}
(см.
)
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]}
См. также
Примечания
Kerr, Josh
(неопр.)
.
Дата обращения: 7 октября 2016.
13 июля 2019 года.
Ссылки
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
:
,
ISSN
,
MR
Gallier, Jean H. (1991),
,
Ann. Pure Appl. Logic
,
53
(3): 199—260,
doi
:
,
MR
(недоступная ссылка)
PDF's:
. (In particular part 3, Section 12, pp. 59–64, "A Glimpse at Hierarchies of Fast and Slow Growing Functions".)
(1981), "Π
1
2
-logic. I. Dilators",
Annals of Mathematical Logic
,
21
(2): 75—219,
doi
:
,
ISSN
,
MR
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
:
, Part 2
doi
:
, Corrections
doi
:
.
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
:
.
Wainer, S.S (1989), "
".
54
(2): 608-614.
Числа
Функции
Нотации
![{\displaystyle f_{\alpha }(n)=f_{\alpha [n]}(n)}](/images/005/608/5608030/4.jpg?rand=92597)
![{\displaystyle \alpha [n]}](/images/005/608/5608030/6.jpg?rand=289648)
![{\displaystyle \omega [n]=n}](/images/005/608/5608030/8.jpg?rand=616099)
![{\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]}](/images/005/608/5608030/9.jpg?rand=154678)
![{\displaystyle \omega ^{\alpha +1}[n]=\omega ^{\alpha }n}](/images/005/608/5608030/11.jpg?rand=365243)
![{\displaystyle \omega ^{\alpha }[n]=\omega ^{\alpha [n]}}](/images/005/608/5608030/12.jpg?rand=14203)
![{\displaystyle \varepsilon _{0}[0]=0}](/images/005/608/5608030/14.jpg?rand=630942)
![{\displaystyle \varepsilon _{0}[n+1]=\omega ^{\varepsilon _{0}[n]}=\omega \uparrow \uparrow n}](/images/005/608/5608030/15.jpg?rand=83403)
![{\displaystyle f_{\omega ^{\omega ^{\omega }}}(n)=\{n,n[1,2]2\}}](/images/005/608/5608030/65.jpg?rand=168905)
![{\displaystyle f_{\omega ^{\omega ^{\omega ^{2}}}}(n)=\{n,n[1,1,2]2\}}](/images/005/608/5608030/66.jpg?rand=557573)
![{\displaystyle f_{\omega ^{\omega ^{\omega ^{\omega }}}}(n)=\{n,n[1[2]2]2\}}](/images/005/608/5608030/67.jpg?rand=341086)
![{\displaystyle f_{\epsilon _{0}}(n)=\{n,n[1{\backslash }2]2\}}](/images/005/608/5608030/68.jpg?rand=12708)
![{\displaystyle f_{\omega ^{\omega ^{\epsilon _{0}+1}}}(n)=\{n,n[1[1{\backslash }2]2{\backslash }2]2\}}](/images/005/608/5608030/69.jpg?rand=231147)
![{\displaystyle f_{\omega ^{\omega ^{\omega ^{\epsilon _{0}+1}}}}(n)=\{n,n[1[1[1{\backslash }2]2{\backslash }2]2{\backslash }2]2\}}](/images/005/608/5608030/70.jpg?rand=486675)
![{\displaystyle f_{\epsilon _{1}}(n)=\{n,n[1{\backslash }3]2\}}](/images/005/608/5608030/71.jpg?rand=703622)
![{\displaystyle f_{\epsilon _{2}}(n)=\{n,n[1{\backslash }4]2\}}](/images/005/608/5608030/72.jpg?rand=762819)
![{\displaystyle f_{\epsilon _{\omega }}(n)=\{n,n[1{\backslash }1,2]2\}}](/images/005/608/5608030/73.jpg?rand=696765)
![{\displaystyle f_{\epsilon _{\omega ^{\omega }}}(n)=\{n,n[1{\backslash }1[2]2]2\}}](/images/005/608/5608030/74.jpg?rand=234561)
![{\displaystyle f_{\epsilon _{\epsilon _{0}}}(n)=\{n,n[1{\backslash }1[1{\backslash }2]2]2\}}](/images/005/608/5608030/75.jpg?rand=86009)
![{\displaystyle f_{\epsilon _{\epsilon _{\epsilon _{0}}}}(n)=\{n,n[1{\backslash }1[1{\backslash }1[1{\backslash }2]2]2]2\}}](/images/005/608/5608030/76.jpg?rand=553446)
![{\displaystyle f_{\zeta _{0}}(n)=\{n,n[1{\backslash }1{\backslash }2]2\}}](/images/005/608/5608030/77.jpg?rand=102035)
![{\displaystyle f_{\epsilon _{\zeta _{0}+1}}(n)=\{n,n[1{\backslash }2{\backslash }2]2\}}](/images/005/608/5608030/78.jpg?rand=660955)
![{\displaystyle f_{\epsilon _{\zeta _{0}+\omega }}(n)=\{n,n[1{\backslash }1,2{\backslash }2]2\}}](/images/005/608/5608030/79.jpg?rand=895942)
![{\displaystyle f_{\epsilon _{\zeta _{0}+\epsilon _{0}}}(n)=\{n,n[1{\backslash }1{[1{\backslash }2]}2{\backslash }2]2\}}](/images/005/608/5608030/80.jpg?rand=57117)
![{\displaystyle f_{\epsilon _{\zeta _{0}2}}(n)=\{n,n[1{\backslash }1{[1{\backslash }1{\backslash }2]}2{\backslash }2]2\}}](/images/005/608/5608030/81.jpg?rand=432614)
![{\displaystyle f_{\epsilon _{\epsilon _{\zeta _{0}+1}}}(n)=\{n,n[1{\backslash }1{[1{\backslash }2{\backslash }2]}2{\backslash }2]2\}}](/images/005/608/5608030/82.jpg?rand=441734)
![{\displaystyle f_{\zeta _{1}}(n)=\{n,n[1{\backslash }1{\backslash }3]2\}}](/images/005/608/5608030/83.jpg?rand=544246)
![{\displaystyle f_{\zeta _{\omega }}(n)=\{n,n[1{\backslash }1{\backslash }1,2]2\}}](/images/005/608/5608030/84.jpg?rand=203998)
![{\displaystyle f_{\zeta _{\epsilon _{0}}}(n)=\{n,n[1{\backslash }1{\backslash }1[1{\backslash }2]2]2\}}](/images/005/608/5608030/85.jpg?rand=717197)
![{\displaystyle f_{\zeta _{\zeta _{0}}}(n)=\{n,n[1{\backslash }1{\backslash }1[1{\backslash }1{\backslash }2]2]2\}}](/images/005/608/5608030/86.jpg?rand=402166)
![{\displaystyle f_{\eta _{0}}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }2]2\}}](/images/005/608/5608030/87.jpg?rand=945416)
![{\displaystyle f_{\varphi (4,0)}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }1{\backslash }2]2\}}](/images/005/608/5608030/88.jpg?rand=290523)
![{\displaystyle f_{\varphi (m,0)}(n)=\{n,n[1{\backslash }1{\backslash }1{\cdots }1{\backslash }2]2\}}](/images/005/608/5608030/89.jpg?rand=123847)
![{\displaystyle f_{\varphi (\omega ,0)}(n)=\{n,n[1[2{\neg }2]2]2\}}](/images/005/608/5608030/90.jpg?rand=707542)
![{\displaystyle f_{\epsilon _{\varphi (\omega ,0)+1}}(n)=\{n,n[1{\backslash }2[2{\neg }2]2]2\}}](/images/005/608/5608030/91.jpg?rand=251004)
![{\displaystyle f_{\zeta _{\varphi (\omega ,0)+1}}(n)=\{n,n[1{\backslash }1{\backslash }2[2{\neg }2]2]2\}}](/images/005/608/5608030/92.jpg?rand=415421)
![{\displaystyle f_{\eta _{\varphi (\omega ,0)+1}}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }2[2{\neg }2]2]2\}}](/images/005/608/5608030/93.jpg?rand=126723)
![{\displaystyle f_{\varphi (\omega ,1)}(n)=\{n,n[1[2{\neg }2]3]2\}}](/images/005/608/5608030/94.jpg?rand=77418)
![{\displaystyle f_{\varphi (\omega ,2)}(n)=\{n,n[1[2{\neg }2]4]2\}}](/images/005/608/5608030/95.jpg?rand=861635)
![{\displaystyle f_{\varphi (\omega ,\omega )}(n)=\{n,n[1[2{\neg }2]1,2]2\}}](/images/005/608/5608030/96.jpg?rand=809844)
![{\displaystyle f_{\varphi (\omega ,\epsilon _{0})}(n)=\{n,n[1[2{\neg }2]1[1{\backslash }2]2]2\}}](/images/005/608/5608030/97.jpg?rand=161006)
![{\displaystyle f_{\varphi (\omega ,\zeta _{0})}(n)=\{n,n[1[2{\neg }2]1[1{\backslash }1{\backslash }2]2]2\}}](/images/005/608/5608030/98.jpg?rand=564696)
![{\displaystyle f_{\varphi (\omega ,\varphi (\omega ,0))}(n)=\{n,n[1[2{\neg }2]1[1[2{\neg }2]2]2]2\}}](/images/005/608/5608030/99.jpg?rand=586468)
![{\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\}}](/images/005/608/5608030/100.jpg?rand=949547)
![{\displaystyle f_{\varphi (\omega +1,0)}(n)=\{n,n[1[2{\neg }2]1{\backslash }2]2\}}](/images/005/608/5608030/101.jpg?rand=588275)
![{\displaystyle f_{\varphi (\omega +2,0)}(n)=\{n,n[1[2{\neg }2]1{\backslash }1{\backslash }2]2\}}](/images/005/608/5608030/102.jpg?rand=702888)
![{\displaystyle f_{\varphi (\omega 2,0)}(n)=\{n,n[1[2{\neg }2]1[2{\neg }2]2]2\}}](/images/005/608/5608030/103.jpg?rand=259287)
![{\displaystyle f_{\varphi (\omega ^{2},0)}(n)=\{n,n[1[3{\neg }2]2]2\}}](/images/005/608/5608030/104.jpg?rand=576623)
![{\displaystyle f_{\varphi (\omega ^{3},0)}(n)=\{n,n[1[4{\neg }2]2]2\}}](/images/005/608/5608030/105.jpg?rand=207192)
![{\displaystyle f_{\varphi (\omega ^{\omega },0)}(n)=\{n,n[1[1,2{\neg }2]2]2\}}](/images/005/608/5608030/106.jpg?rand=413850)
![{\displaystyle f_{\varphi (\omega ^{\omega ^{\omega }},0)}(n)=\{n,n[1[1[2]2{\neg }2]2]2\}}](/images/005/608/5608030/107.jpg?rand=755960)
![{\displaystyle f_{\varphi ({\epsilon _{0}},0)}(n)=\{n,n[1[1[1{\backslash }2]2{\neg }2]2]2\}}](/images/005/608/5608030/108.jpg?rand=979770)
![{\displaystyle f_{\Gamma _{0}}(n)=\{n,n[1[1{\backslash }2{\neg }2]2]2\}}](/images/005/608/5608030/109.jpg?rand=560092)
![{\displaystyle f_{\Gamma _{1}}(n)=\{n,n[1[1{\backslash }2{\neg }2]3]2\}}](/images/005/608/5608030/110.jpg?rand=838675)
![{\displaystyle f_{\Gamma _{\Gamma _{0}}}(n)=\{n,n[1[1{\backslash }2{\neg }2]1[1[1{\backslash }2{\neg }2]2]2]2\}}](/images/005/608/5608030/111.jpg?rand=154191)
![{\displaystyle f_{\varphi (1,1,0)}(n)=\{n,n[1[1{\backslash }2{\neg }2]1{\backslash }2]2\}}](/images/005/608/5608030/112.jpg?rand=313077)
![{\displaystyle f_{\varphi (1,2,0)}(n)=\{n,n[1[1{\backslash }2{\neg }2]1{\backslash }1{\backslash }2]2\}}](/images/005/608/5608030/113.jpg?rand=161205)
![{\displaystyle f_{\varphi (2,0,0)}(n)=\{n,n[1[1{\backslash }2{\neg }2]1[1{\backslash }2{\neg }2]2]2\}}](/images/005/608/5608030/114.jpg?rand=340580)
![{\displaystyle f_{\varphi (\omega ,0,0)}(n)=\{n,n[1[2{\backslash }2{\neg }2]2]2\}}](/images/005/608/5608030/115.jpg?rand=417119)
![{\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\}}](/images/005/608/5608030/116.jpg?rand=391350)
![{\displaystyle f_{\varphi (1,0,0,0)}(n)=\{n,n[1[1{\backslash }3{\neg }2]2]2\}}](/images/005/608/5608030/117.jpg?rand=227135)
![{\displaystyle f_{\varphi (1,0,0,0,0)}(n)=\{n,n[1[1{\backslash }4{\neg }2]2]2\}}](/images/005/608/5608030/118.jpg?rand=320883)
![{\displaystyle f_{\psi (\Omega ^{\Omega ^{\omega }})}(n)=\{n,n[1[1{\backslash }1,2{\neg }2]2]2\}\approx TREE(n)}](/images/005/608/5608030/119.jpg?rand=480963)
![{\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\}}](/images/005/608/5608030/120.jpg?rand=254973)
![{\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega }})}(n)=\{n,n[1[1{\backslash }1{\backslash }2{\neg }2]2]2\}}](/images/005/608/5608030/121.jpg?rand=433461)
![{\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega ^{2}}})}(n)=\{n,n[1[1{\backslash }1{\backslash }1{\backslash }2{\neg }2]2]2\}}](/images/005/608/5608030/122.jpg?rand=344720)
![{\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega ^{\omega }}})}(n)=\{n,n[1[1[2{\neg }2]2{\neg }2]2]2\}}](/images/005/608/5608030/123.jpg?rand=781415)
![{\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\}}](/images/005/608/5608030/124.jpg?rand=600907)
![{\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega ^{\Omega }}})}(n)=\{n,n[1[1[1{\backslash }2{\neg }2]2{\neg }2]2]2\}}](/images/005/608/5608030/125.jpg?rand=936918)
![{\displaystyle f_{\psi (\Omega _{2})}(n)=\{n,n[1[1{\neg }3]2]2\}}](/images/005/608/5608030/126.jpg?rand=133293)
![{\displaystyle f_{\psi (\Omega _{2}+1)}(n)=\{n,n[1{\backslash }2[1{\neg }3]2]2\}}](/images/005/608/5608030/127.jpg?rand=984522)
![{\displaystyle f_{\psi (\Omega _{2}+\omega )}(n)=\{n,n[1{\backslash }1,2[1{\neg }3]2]2\}}](/images/005/608/5608030/128.jpg?rand=905565)
![{\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\}}](/images/005/608/5608030/129.jpg?rand=548726)
![{\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\}}](/images/005/608/5608030/130.jpg?rand=334476)
![{\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\}}](/images/005/608/5608030/131.jpg?rand=678462)
![{\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\}}](/images/005/608/5608030/132.jpg?rand=640774)
![{\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\}}](/images/005/608/5608030/133.jpg?rand=622437)
![{\displaystyle f_{\psi (\Omega _{2}+\Omega )}(n)=\{n,n[1{\backslash }1{\backslash }2[1{\neg }3]2]2\}}](/images/005/608/5608030/134.jpg?rand=17268)
![{\displaystyle f_{\psi (\Omega _{2}+\Omega ^{2})}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }2[1{\neg }3]2]2\}}](/images/005/608/5608030/135.jpg?rand=186567)
![{\displaystyle f_{\psi (\Omega _{2}+\Omega ^{\omega })}(n)=\{n,n[1[2{\neg }2]2[1{\neg }3]2]2\}}](/images/005/608/5608030/136.jpg?rand=850392)
![{\displaystyle f_{\psi (\Omega _{2}+\Omega ^{\Omega })}(n)=\{n,n[1[1{\backslash }2{\neg }2]2[1{\neg }3]2]2\}}](/images/005/608/5608030/137.jpg?rand=939274)
![{\displaystyle f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}))}(n)=\{n,n[1[1{\neg }3]3]2\}}](/images/005/608/5608030/138.jpg?rand=439117)
![{\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\}}](/images/005/608/5608030/139.jpg?rand=50841)
![{\displaystyle f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}(\Omega _{2}+1)))}(n)=\{n,n[1[2{\neg }3]2]2\}}](/images/005/608/5608030/140.jpg?rand=387480)
![{\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\}}](/images/005/608/5608030/141.jpg?rand=920705)
![{\displaystyle f_{\psi (\Omega _{2}2)}(n)=\{n,n[1[1{\neg }4]2]2\}}](/images/005/608/5608030/142.jpg?rand=598533)
![{\displaystyle f_{\psi (\Omega _{2}3)}(n)=\{n,n[1[1{\neg }5]2]2\}}](/images/005/608/5608030/143.jpg?rand=823295)
![{\displaystyle f_{\psi (\Omega _{2}\omega )}(n)=\{n,n[1[1{\neg }1,2]2]2\}}](/images/005/608/5608030/144.jpg?rand=918175)
![{\displaystyle f_{\psi (\Omega _{2}\psi (0))}(n)=\{n,n[1[1{\neg }1[1{\backslash }2]2]2]2\}}](/images/005/608/5608030/145.jpg?rand=956445)
![{\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\}}](/images/005/608/5608030/146.jpg?rand=810998)
![{\displaystyle f_{\psi (\Omega _{2}\psi (\Omega _{2}))}(n)=\{n,n[1[1{\neg }1[1[1{\neg }3]2]2]2]2\}}](/images/005/608/5608030/147.jpg?rand=966889)
![{\displaystyle f_{\psi (\Omega _{2}\Omega )}(n)=\{n,n[1[1{\neg }1{\backslash }2]2]2\}}](/images/005/608/5608030/148.jpg?rand=467867)
![{\displaystyle f_{\psi (\Omega _{2}\Omega ^{\Omega })}(n)=\{n,n[1[1{\neg }1[1{\backslash }2{\neg }2]2]2]2\}}](/images/005/608/5608030/149.jpg?rand=212355)
![{\displaystyle f_{\psi (\Omega _{2}\psi _{1}(\Omega _{2}))}(n)=\{n,n[1[1{\neg }1[1{\neg }3]2]2]2\}}](/images/005/608/5608030/150.jpg?rand=258020)
![{\displaystyle f_{\psi (\Omega _{2}^{2})}(n)=\{n,n[1[1{\neg }1{\neg }2]2]2\}}](/images/005/608/5608030/151.jpg?rand=80954)
![{\displaystyle f_{\psi (\Omega _{2}^{\omega })}(n)=\{n,n[1[1[2{\backslash }_{3}2]2]2]2\}}](/images/005/608/5608030/152.jpg?rand=782404)
![{\displaystyle f_{\psi (\Omega _{2}^{\Omega _{2}})}(n)=\{n,n[1[1[1{\backslash }_{2}2{\backslash }_{3}2]2]2]2\}}](/images/005/608/5608030/153.jpg?rand=313427)
![{\displaystyle f_{\psi (\Omega _{3})}(n)=\{n,n[1[1[1{\backslash }_{3}3]2]2]2\}}](/images/005/608/5608030/154.jpg?rand=268988)
![{\displaystyle f_{\psi (\Omega _{3}^{\Omega _{3}})}(n)=\{n,n[1[1[1[1{\backslash }_{3}2{\backslash }_{4}2]2]2]2]2\}}](/images/005/608/5608030/155.jpg?rand=250541)
![{\displaystyle f_{\psi (\Omega _{4})}(n)=\{n,n[1[1[1[1{\backslash }_{4}3]2]2]2]2\}}](/images/005/608/5608030/156.jpg?rand=296859)
![{\displaystyle f_{\psi (\Omega _{\omega })}(n)=\{n,n[1[2{\backslash }_{1,2}2]2]2\}}](/images/005/608/5608030/157.jpg?rand=95244)
![{\displaystyle f_{\psi (\Omega _{\psi (\Omega )})}(n)=\{n,n[1[2{\backslash }_{1[1{\backslash }2]2}2]2]2\}}](/images/005/608/5608030/158.jpg?rand=752741)
![{\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\}}](/images/005/608/5608030/159.jpg?rand=650941)
![{\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]}](/images/005/608/5608030/160.jpg?rand=764673)
![{\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]}](/images/005/608/5608030/161.jpg?rand=213450)
![{\displaystyle f_{\psi ({\Omega _{\Omega _{\omega }}})}(n)=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(1,1,1)[n]}](/images/005/608/5608030/162.jpg?rand=229252)
![{\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]}](/images/005/608/5608030/163.jpg?rand=717740)
![{\displaystyle f_{\psi (\psi _{I}(0))}(n)=(0,0,0)(1,1,1)(2,1,1)(3,1,0)(2,0,0)[n]}](/images/005/608/5608030/164.jpg?rand=390694)
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