Strutural properties of bounded languages with respect to multiplication by a constant

(1)

! "# $% & $' () " $* % " $+ ,-.& +/ # - + $ 0 ' (1 + 2 3 4 & #$ # 3 + ' & " 5 ' , ' 5 5 6 7 +7 % ' () " $* % " $+ , -8 # +# 9

(2)

S

B

`

λ = β

`

(3)

! " # $ !

S = (L, Σ, <)

L

%

(Σ, <)

&

L

<

%

rep

_S

:

N

_{→ L}

_val

_S

_{= rep}

−1

(4)

a

∗

n 0 1

2

3

4 · · ·

rep(n) ε a aa aaa aaaa · · ·

{a, b}

∗

_{, a < b}

n 0 1 2

3

4

5

6

7 · · ·

rep(n) ε a b aa ab ba bb aaa · · ·

a

∗

_b

∗

_{, a < b}

n 0 1 2

3

4

5

6 · · ·

rep(n) ε a b aa ab bb aaa · · ·

(5)

X

⊆

N

S

! # " "!

rep

S

(X) ⊆ Σ

∗

" ! ! ! ! $

(6)

S = (L, Σ, <)

! # ! ! ! $ " ! # " " # #

λ

∈

N

# $

S

! # " ! !

X

! !

λX

" "

S

! # " !

(7)

S

S = (L, Σ, <)

&

S

&

u

_L

(n)

n

L

&

L

⊆ Σ

∗

u

_L

(n) ∈ Θ(n

k

)

k

∈

N

S = (L, Σ, <)

& %

S

λ

λ = β

k+1

β

∈

N

&

(8)

L

! !

u

L

(n) ∈ O(1).

L

⊂ Σ

∗

S = (L, Σ, <)

&

X

⊆

N

S

X

&

S

&

X

⊆

N

S

λX

S

λ

∈

N

&

(9)

β > 0

&

S = (a

∗

b

∗

,

{a < b}),

β

2

%

S

β

&

(10)

B

`

= a

∗

1

· · · a

∗

`

%

Σ

`

= {a

1

< . . . < a

`

}

`

≥ 1.

(B

`

, Σ

`

)

rep

`

val

`

&

X

⊆

N

B

`

! # " !

rep

`

(X)

%

Σ

`

.

(11)

λ

%

f

_λ

: B

_`

→ B

_`

.

! ! ! " ! # ! ! ! $ " ! # " " # ! " " # " # ! ! ! ! ! #

B

`

(12)

` = 2

Σ

2

= {a, b}

λ = 25

&

8 −−−→ 200

×25

rep

₂

↓

↓ rep

₂

a b

2

−−−→ a

×25

9

b

10

N

_−−→

×λ

N

rep

_`

↓

↓ rep

_`

B

_`

−→ B

f

λ

_`

λ = 25

f

λ

B

2

w, w

0

_{∈ B}

2

f

λ

(w) = w

0

val

2

(w

0

_{) = 25 val}

2

(w)

&

(13)

B

`

u

_`

(n) :=

u

_B

`

(n) = #(B

`

∩ Σ

n

`

)

v

`

(n) := #(B

`

∩ Σ

≤n

_`

) =

n

X

i=0

u

`

(i).

`

≥ 1

n

≥ 0

%

u

_`+1

(n) =

v

_`

(n)

u

`

(n) =

n + `

− 1

`

− 1

.

(14)

S = (a

∗

1

· · · a

∗

`

,

{a

1

<

· · · < a

`

})

& %

val

_`

(a

n

₁

1

· · · a

n

`

) =

`

X

i=1

n

_i

+ · · · + n

_`

+ ` − i

`

− i + 1

.

n

∈

N

|rep

_`

(n)| = k ⇔

k + `

− 1

`

|

{z

}

val

_`

(a

k

₁

)

≤ n ≤

`

X

i=1

k + i

− 1

i

|

{z

}

val

_`

(a

k

_`

)

.

(15)

3 a

∗

_b

∗

_c

∗

aaa < aab < aac < abb < abc < acc < bbb < bbc < bcc < ccc.

%

val

3

(aaa) =

₅

3

= 10

val

3

(acc) = 15

&

ϕ :

{a, b, c} → {a, b, c}

∗

ϕ(a) = ε, ϕ(b) = b, ϕ(c) = c

3 ε < b < c < bb < bc < cc < bbb < bbc < bcc < ccc.

%

val

3

(acc) = val

3

(aaa) + val

2

(cc)

val

2

b

∗

_c

∗

&

(16)

%

u

B

`

(n) ∈ Θ(n

`−1

_).

%

λ = β

`

.

(17)

`

∈

N

\ {0}

&

n

n =

z

`

+

z

`−1

`

− 1

+ · · · +

z

1

1 z

`

> z

`−1

>

· · · > z

1

≥ 0

&

(18)

rep

`

(n).

% &

t

i

≤

n

_<

t

+1

i

←

t

i

←

%

α

1

, . . . , α

`

α

_i

+ · · · + α

_`

=

z

(` − i + 1) − ` + i,

_{i = 1, . . . , `.}

rep

`

(

n

) = a

α

₁

1

· · · a

α

_`

`

&

(19)

` = 3

12345678901234567890 =

4199737

3 +

3803913

2 +

1580642

1

%











n

₁

+ n

₂

+ n

₃

= 4199737 − 2

n

₂

+ n

₃

= 3803913 − 1

n

₃

= 1580642

⇔ (n

₁

, n

₂

, n

₃

) = (395823, 2223270, 1580642),

%

rep

₃

(12345678901234567890) = a

395823

b

2223270

c

1580642

.

(20)

λ = β

`

S = (a

∗

b

∗

c

∗

,

{a < b < c}),

β

∈

N

\ {0, 1}

β

6≡ ±1 (mod 6)

β

3

%

S

&

(21)

β

`

%

S

S = (a

∗

₁

· · · a

∗

_`

,

{a

₁

<

· · · < a

_`

})

B

`

%

`

β =

k

Y

i=1

p

θ

_i

i

p

1

, . . . , p

k

`

&

(22)

n

∈

N

%

|rep

_`

(β

`

n)

| = β |rep

_`

(n)| +

(β − 1)(` − 1)

2 + i

i

∈ {−1, 0, . . . , β − 1}

&