PSB Notation

## Info

Powerful Scientific B Notation (PSB Notation for short) fuses the properties of Hyper-B Notation and Powerful Scientific Notation.

## Definition

B
P
[
0

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m

(
n
# )
0

{\displaystyle BP[0]\_{m}(n)=0}

B
P
[
1

]

0

(
n
# )
n

{\displaystyle BP[1]\_{0}(n)=n}

B
P
[
1

]

1

(
n
# )
n
!

{\displaystyle BP[1]\_{1}(n)=n!}

B
P
[
1

]

2

(
n
# )
(
n
!
)
!

{\displaystyle BP[1]\_{2}(n)=(n!)!}

B
P
[
1

]

m

(
n
# )

(
(
⋯
(
(
n
!
)
!
)
⋯
)
!
)
!

⏟

m

{\displaystyle BP[1]\_{m}(n)=\underbrace{((\cdots((n!)!)\cdots)!)!}\_{m}}

B
P
[
2

]

m

(
n
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B
P
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1

]

m

n

(
n

n

n
)

{\displaystyle BP[2]\_{m}(n)=BP[1]\_{m^{n}}(n\uparrow^{n}n)}

B
P
[
3

]

m

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B
P
[
2

]

m

n

(
n

n

n
)

{\displaystyle BP[3]\_{m}(n)=BP[2]\_{m^{n}}(n\uparrow^{n}n)}

if x > 1

B
P
[
x

]

m

(
n
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B
P
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x
−
1

]

m

n

(
n

n

n
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{\displaystyle BP[x]\_{m}(n)=BP[x-1]\_{m^{n}}(n\uparrow^{n}n)}

B
P
[
ω

]

1

(
n
# )
B
P
[
B
P
[
n

]

1

]

1

{\displaystyle BP[\omega]\_{1}(n)=BP[BP[n]\_{1}]\_{1}}

B
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ω
2

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1

(
n
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B
P
[
B
P
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B
P
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n

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1

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1

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1

{\displaystyle BP[\omega2]\_{1}(n)=BP[BP[BP[n]\_{1}]\_{1}]\_{1}}

B
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ω
x

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⋯
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⏟

x

{\displaystyle BP[\omega x]\_{m}(n)=\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{x}}

B
P
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ω

2

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1

(
n
# )
B
P
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ω

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n

(
B
P
[
m

]

n

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n
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{\displaystyle BP[\omega^{2}]\_{1}(n)=BP[\omega]\_{n}(BP[m]\_{n}(n))}

B
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ω

x

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B
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ω

x
−
1

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(
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m

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n
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{\displaystyle BP[\omega^{x}]\_{m}(n)=BP[\omega^{x-1}]\_{n}(BP[m]\_{n}(n))}

B
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ϵ

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⋯
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⏟

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{\displaystyle BP[\epsilon\_{\omega}]\_{m}(n)=\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{BP[\omega]\_{m}(n)}}

B
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ζ

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⏟

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ϵ

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n
)

{\displaystyle BP[\zeta\_{\omega}]\_{m}(n)=\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{BP[\epsilon\_{\omega}]\_{m}(n)}}

B
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η

ω

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⏟

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(
n
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{\displaystyle BP[\eta\_{\omega}]\_{m}(n)=\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{BP[\zeta\_{\omega}]\_{m}(n)}}

B
P
[
φ
(
4
,
ω
)

]

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# )

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⋯
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⏟

B
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η

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{\displaystyle BP[\varphi(4,\omega)]\_{m}(n)=\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{BP[\eta\_{\omega}]\_{m}(n)}}

B
P
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φ
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5
,
ω
)

]

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⏟

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{\displaystyle BP[\varphi(5,\omega)]\_{m}(n)=\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{BP[\varphi(4,\omega)]\_{m}(n)}}

B
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⏟

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x
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1
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(
n
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{\displaystyle BP[\varphi(x,\omega]\_{m}(n)=\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{BP[\varphi(x-1,\omega]\_{m}(n)}}

B
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φ
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1
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)

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⏟

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⏟

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n
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{\displaystyle BP[\varphi(1,0,\omega)]\_{m}(n)=\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{BP[\omega]\_{m}(n)}}}

B
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,
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)

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⏟

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⏟

B
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[

ϵ

ω

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m

(
n
)

{\displaystyle BP[\varphi(1,1,\omega)]\_{m}(n)=\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{BP[\epsilon\_{\omega}]\_{m}(n)}}}

B
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φ
(
1
,
2
,
ω
)

]

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# )

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⏟

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⏟

B
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[

ζ

ω

]

m

(
n
)

{\displaystyle BP[\varphi(1,2,\omega)]\_{m}(n)=\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{BP[\zeta\_{\omega}]\_{m}(n)}}}

B
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φ
(
1
,
3
,
ω
)

]

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# )

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⏟

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⏟

B
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[

η

ω

]

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(
n
)

{\displaystyle BP[\varphi(1,3,\omega)]\_{m}(n)=\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{BP[\eta\_{\omega}]\_{m}(n)}}}

B
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,
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,
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)

]

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# )

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⏟

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n

]

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m

⏟

B
P
[
φ
(
4
,
ω
)

]

m

(
n
)

{\displaystyle BP[\varphi(1,4,\omega)]\_{m}(n)=\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{\underbrace{BP[BP[\cdots[BP[BP[n]\_{m}]\_{m}]\cdots]\_{m}]\_{m}}\_{BP[\varphi(4,\omega)]\_{m}(n)}}}

B
P
[
ϑ
(

Ω

ω

)

]

m

(
n
# )
{
B
P
[
ω

]

m

(
n
)
,
B
P
[
ω

]

m

(
n
)
}

{\displaystyle BP[\vartheta(\Omega^{\omega})]\_{m}(n)=\{ BP[\omega]\_{m}(n),BP[\omega]\_{m}(n) \}}

B
P
[
ϑ
(

Ω

Ω

)

]

m

(
n
# )
{
B
P
[
φ
(
n
,
ω
)

]

m

(
n
)
,
B
P
[
φ
(
n
,
ω
)

]

m

(
n
)
}

{\displaystyle BP[\vartheta(\Omega^{\Omega})]\_{m}(n)=\{ BP[\varphi(n,\omega)]\_{m}(n),BP[\varphi(n,\omega)]\_{m}(n) \}}

B
P
[
ϑ
(

Ω

Ω

ω

)

]

m

(
n
# )
{
B
P
[
ω

]

m

(
n
)
,
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P
[
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]

m

(
n
)
,
B
P
[
ω

]

m

(
n
)
}

{\displaystyle BP[\vartheta(\Omega^{\Omega^{\omega}})]\_{m}(n)=\{ BP[\omega]\_{m}(n),BP[\omega]\_{m}(n),BP[\omega]\_{m}(n) \}}

B
P
[
Ω
↑↑
Ω

]

m

(
n
# )
{

B
P
[
φ
(
n
,
ω
)

]

m

(
n
)
,
B
P
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φ
(
n
,
ω
)

]

m

(
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)
,
⋯
,
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,
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)

]

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(
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)
,
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,
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)

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(
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)

⏟

B
P
[
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(
n
,
ω
)

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m

(
n
)

}

{\displaystyle BP[\Omega\uparrow\uparrow\Omega]\_{m}(n)=\{ \underbrace{BP[\varphi(n,\omega)]\_{m}(n),BP[\varphi(n,\omega)]\_{m}(n),\cdots,BP[\varphi(n,\omega)]\_{m}(n),BP[\varphi(n,\omega)]\_{m}(n)}\_{BP[\varphi(n,\omega)]\_{m}(n)} \}}