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! "#$#"&%' ()+* ),- /.

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T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >

I Σ Ò Ω kΓJc armk v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-


T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >


I Pk

Q Ò P ∪ Q = ∅RÓ>_ansdk v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-


T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >


I Pk


I i ∈ PBdcÔ@c dnfÓ>_ansdk` v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-


T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >


I Pk



I ∆ : P × Σ→ P ∪ QJbrdmkMdacfh`&Õsn i;ka_aò&ÓHh&Õsr

Pv

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-


T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >


I Pk




Pv

I d : Q × Γ× 0, 1 → P ∪ QRbrdmkªdacfh`&Õ nsi;ka_aò&ÓHh#Õsr

Qv

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-


T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >


I Pk




Pv


Qv

I λ : P × Σ→ Γ∗ Ò Dom(λ) = Dom(∆)Rbrdmk Ò scko cÖcoTnac×Vmcs_c v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-


T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >


I Pk




Pv


Qv


I out : Q × Γ× 0, 1 → Ω∗ Ò Dom(out) = Dom(d)k&Õsn i;dacbrdmk v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-


T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >


I Pk




Pv


Qv



I φ : P → Q

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-


T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >


I Pk




Pv


Qv



I φ : P → Q

I ψ : Q → Ω∗vw a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-


T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >


I Pk




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I φ : P → Q

I ψ : Q → Ω∗v

I

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I T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ > ÒI κ = (p, α, γ, ω)

` WJ<;NOG9Ü¦L8;:9QRGS Ò mÓ£iE`_an

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w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-




I α ∈ Σ∗

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-




I α ∈ Σ∗

I γ ∈ Γ∗v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-




I α ∈ Σ∗

I γ ∈ Γ∗v

I ω ∈ Ω∗v

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à m&nκ′ = (p′, α′, γ′, ω′)

kκ′′ = (p′′, α′′, γ′′, ω′′) Ò _an Z98>Dß£W? κ′`κ′′

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-


à m&nκ′ = (p′, α′, γ′, ω′)


I p′ ∈ P Ò α′ = σα′′ Ò σ ∈ Σ kp′′ = ∆(p′, σ) Ò γ′′ = γ′λ(p′, σ) Ò

ω′′ = ω′v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-


à m&nκ′ = (p′, α′, γ′, ω′)



ω′′ = ω′v

I p′ ∈ Q Ò γ′ = bγ′′ Ò b ∈ Γ Ò γ′′ 6= εk

p′′ = d(p′, b, 1) Ò α′′ = α′ Òω′′ = ω out(p′, b, 1)

v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-


à m&nκ′ = (p′, α′, γ′, ω′)



ω′′ = ω′v

I p′ ∈ Q Ò γ′ = bγ′′ Ò b ∈ Γ Ò γ′′ 6= εk


vI p′ ∈ Q Ò γ′ = b ∈ Γ

kγ′′ = ε Ò p′′ = d(p′, b, 0) Ò α′′ = α′ Ò

ω′′ = ω′ out(p′, b, 0)v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-


à m&nκ′ = (p′, α′, γ′, ω′)



ω′′ = ω′v

I p′ ∈ Q Ò γ′ = bγ′′ Ò b ∈ Γ Ò γ′′ 6= εk


vI p′ ∈ Q Ò γ′ = b ∈ Γ

kγ′′ = ε Ò p′′ = d(p′, b, 0) Ò α′′ = α′ Ò

ω′′ = ω′ out(p′, b, 0)v

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à m&nκ′ = (p′, α′, γ′, ω′)

kκ′′ = (p′′, α′′, γ′′, ω′′) Ò _an G;\4§W?κ′²κ′′7L8;:9<\4G>7LGP§><W G 8>DENJDF =FGP§;<W \:£7L§E:98SB<>Dâ<F:

Ìä ÉJå ¬ ¿ ¬Mæ Éçº « ÀBÍÏÁè¿¾B¸XÐfÁVÑ ÀÉ· «æ ÉØTÉ « ¼ ´

Á ²2æR«2¬é

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! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. Kê9¢OGP<;:Þ<;:ÞG;\4§W?

2Ó£jlk(dac dacκ′ ² κ′′

I κ′=κ′′ Ò iÓ£jlk(dac 0 v

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2Ó£jlk(dac dacκ′ ² κ′′

I κ′=κ′′ Ò iÓ£jlk(dac 0 vI κ′`κ ² κ′′ Ò iÓ£jlk(dac 1+ iÓ£jlk(dacs_cCdac

κ ² κ′′v

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T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >ÈJ_hacdi;R `h ví k(m

L(T ) = (α, ω) î

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! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. ëP\4GP Æ<;:ì02104365 798;:9<>=@?BAC=EDF8


L(T ) = (α, ω) îï@v∃f ∈ P, ∃γ ∈ Γ∗, ∃ω1 ∈ Ω

∗ Ò²< f , ε, γ, ω1 >

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

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L(T ) = (α, ω) îï@v∃f ∈ P, ∃γ ∈ Γ∗, ∃ω1 ∈ Ω

∗ Ò²< f , ε, γ, ω1 >pv

∃q ∈ Q, ∃ω2 ∈ Ω∗ î

< φ(f ), ε, γ, ω1 >²< q, ε, ε, ω1ω2 >

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. ëP\4GP Æ<;:ì02104365 798;:9<>=@?BAC=EDF8


L(T ) = (α, ω) îï@v∃f ∈ P, ∃γ ∈ Γ∗, ∃ω1 ∈ Ω

∗ Ò²< f , ε, γ, ω1 >pv

∃q ∈ Q, ∃ω2 ∈ Ω∗ î

< φ(f ), ε, γ, ω1 >²< q, ε, ε, ω1ω2 >ðvω = ω1ω2ψ(q)

v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. ëP\4GP Æ<;:ì02104365 798;:9<>=@?BAC=EDF8


L(T ) = (α, ω) îï@v∃f ∈ P, ∃γ ∈ Γ∗, ∃ω1 ∈ Ω

∗ Ò²< f , ε, γ, ω1 >pv

∃q ∈ Q, ∃ω2 ∈ Ω∗ î

< φ(f ), ε, γ, ω1 >²< q, ε, ε, ω1ω2 >ðvω = ω1ω2ψ(q)

vL(T )

` u hacbk(m&cfdacbrdmkfT ñ

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T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >ÈJ_hacdi;R `h v

l = max |λ(p, σ)|,

Ià m&n

q′, q′′ ∈ Q_an o `mk kon i

< q′, α, γ, ω >²< q′′, α, γ′, ω′ >k(4c iÓ£jlk(dac î

|γ′| − |γ|.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-



l = max |λ(p, σ)|,

Ià m&n



|γ′| − |γ|.

I ó d irmkn[hn dgh`&Õ nsi;ka_a` v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-



l = max |λ(p, σ)|,

Ià m&n



|γ′| − |γ|.

Ià m&n

p′, p′′ ∈ P Ò _an o `mk konsi< p′, α′, γ′, ω′ >²< p′′, α′′, γ′′, ω′′ >

k(4c iÓ£jlk(dac î≤ (α′ − α′′)(l + 1) + |γ′| − |γ′′|.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-



l = max |λ(p, σ)|,

Ià m&n



|γ′| − |γ|.

Ià m&n

p′, p′′ ∈ P Ò _an o `mk konsi< p′, α′, γ′, ω′ >²< p′′, α′′, γ′′, ω′′ >

k(4c iÓ£jlk(dac î≤ (α′ − α′′)(l + 1) + |γ′| − |γ′′|.

I ó d irmkn[hn ∆gh`&Õsn i;ka_a` vw a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. ôõ¦L<9 FQRGSL7F:

irI

I ö rdmk(_c ir : (Σ ∪ $)∗ → (Σ ∪ $)∗ Òir(α) =

α,cm&n

α = ω$ωscfd m#ns[ir4c

ω ∈ Σ∗

¬!k(dacÔ`

.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. ôõ¦L<9 FQRGSL7F:

irI

I ö rdmk(_c ir : (Σ ∪ $)∗ → (Σ ∪ $)∗ Òir(α) =

α,cm&n

α = ω$ωscfd m#ns[ir4c

ω ∈ Σ∗

¬!k(dacÔ`

.

I

a ↓ a, b ↓ b

b : b

$ ↓ $

a ↓ a

b ↓ b

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rep0,1

I ö rdmk(_c rep0,1 : a, b, 0, 1∗ → A, a, b∗ Ò

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-


rep0,1


I

rep0,1(ci ) =

ε,cm&n

ci ∈ 0, 1

¬!k(dacÔ`

.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-


rep0,1


I

rep0,1(ci ) =

ε,cm&n

ci ∈ 0, 1

¬!k(dacÔ`

.

I

rep0,1(αcβ) =

αA rep0,1(β)cm&n

α ∈ a, b∗, c = 1

α rep0,1(β)cm&n

α ∈ a, b∗, c = 0

¬!k(dacaÔ`

,

αA

a → A

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rep0,1<;:Þ :98£7LGP<9 (:

a ↓ a, b ↓ b

0 ↓ 0

1 ↓ 1

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q1

f

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I ö rdmk(_c ins0,1 : a, b+ → a, b, 0, 1∗ Ò

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. áøD?BGP<ùYl:EB W¡ZW>5GP<£7;DF8>DE=ED;<ù02104365 7L8;:H<>=?BAC=DF8PIPôõ¦L<9 FQRGSL74:ins0,1


I

ins0,1

(k∏

i=0

(ami bni )

)=

k∏

i=0

(ciami bni )0,

ci ≡k∑

j=i

nj( mod 2).

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. ¤Xê89§W¡ZW£?BZWH=(RD?PWJ§E:£7;DRDF<úZ98>DW4¥J8;:\¦L§E:£7;DÞ\:Z98>D?R=7F:H§; FG>7>DLûÎôõ¦L<9 FQRGSL7F:

pref0,1

I ö rdmk(_c pref0,1 : a, b+ → a, b, 0, 1∗ Ò

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. ¤Xê89§W¡ZW£?BZWH=(RD?PWJ§E:£7;DRDF<úZ98>DW4¥J8;:\¦L§E:£7;DÞ\:Z98>D?R=7F:H§; FG>7>DLûÎôõ¦L<9 FQRGSL7F:

pref0,1

I ö rdmk(_c pref0,1 : a, b+ → a, b, 0, 1∗ Ò

I

pref0,1

(k∏

i=0

(ami bni )

)=

k∏

i=0

(ciami bni )0,

ci ≡i−1∑

j=0

nj( mod 2).

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. ôõ¦L<9 FQRGSL7F:

pref0,1<;:ù :98£7LGP<9 :

q0

0

B0

0

B0

1

EA0

0

EA0

1

OA0

0

OA0

1

a : 0a

a : 1a

b : 0b

b : b

a : aa : a

a : a

b : b

b : b

a : 0a

b : b

a : a

b : b

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! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. ôõ¦L<9 FQRGSL7F:

ins0,1<;:ù :98£7LGP<9 :

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φ φ

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0

B1

1

EA1

0

EA1

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OA1

1

b : b

a : 1a

a : 0a

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a : a

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a : a

b : b

OA1

0

q0

0

B0

0

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EA0

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OA0

0

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ôOWJ89Y¦BGP8WJ§; (:ì<;:Þ8>D\¦@L7F:£7F:

ÿ É « ¾BÉ ¸ T =< Σ× Ω∗, Γ∗,P,Q, s,∆, d , λ, out, φ, ψ >EJ_hacdi;R `h Ò

T ′ =< Ω× Ξ∗,Q ′, s ′, δ′, λ′, φ′ >Bn i;n@&`&iEno cs_a`&`d_hacdgi;R `h v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


ÿ É « ¾BÉ ¸ T =< Σ× Ω∗, Γ∗,P,Q, s,∆, d , λ, out, φ, ψ >EJ_hacdi;R `h Ò

T ′ =< Ω× Ξ∗,Q ′, s ′, δ′, λ′, φ′ >Bn i;n@&`&iEno cs_a`&`d_hacdgi;R `h v⇒

∃T E@ _hacdgi;R `h î

fT= fT ′ fT

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

áX?RD(STIêT=7>WESB<LG£S

^9k(>rgck dacT ′o&ÓHh#Õsr k&Õsn i;ka_a`dac

T

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.



T

⇒`mchs_anonhnko`#iE`dk`[dacÓ>_an dk î

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.



T

⇒`mchs_anonhnko`#iE`dk`[dacÓ>_an dk îI P = P × Q ′

vI Q = Q × Q ′

vI i =< s, s ′ >

v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

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I ÓHhono&ÓHh#Õsr

P = P × Q ′ î∆(< p, q′ >, σ) =< ∆(p, σ), q′ >λ(< p, q′ >, σ) = λ(p, σ)

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

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I ÓHhono&ÓHh#Õsr

P = P × Q ′ î∆(< p, q′ >, σ) =< ∆(p, σ), q′ >λ(< p, q′ >, σ) = λ(p, σ)

I . . .cfn@&`k6o&ÓHh#Õsr

Q = Q × Q ′

d(< p, q′ >, b, j) =< d(p, b, j), (δ′)∗(q′, out(p, b, j)) > Ò scj ∈ 0, 1

vout(< p, q′ >, b, j) = (λ′)∗(q′, out(p, b, j)) Ò sc j ∈ 0, 1

v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

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I ÓHhon[_a` kCo&ÓHh&Õsr

P = P × Q ′ îφ(< p, q′ >) =< φ(p), q′ >

v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

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I ÓHhon[_a` kCo&ÓHh&Õsr

P = P × Q ′ îφ(< p, q′ >) =< φ(p), q′ >

vI . . .

k_a` kCo&ÓHh#ÕsrQ = Q × Q ′ î

ψ(< p, q′ >) = (λ′)∗(q′, ψ(p)) φ′((δ′)∗(q′, ψ(p)))v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

::£7F:ãGP<£7£¦LGPQRGS =ED WJZ98;:9§(?J:9§E:Jû É ¸ `mc p1 =< p1, q

′

1 > Ò p2 =< p2, q′

2 >v

n u co c< p1, α1α2, γ1, ω1 >² p2, α2, γ2, ω1ω2 >

⇔∀ω1∃ω2 î

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


′

1 > Ò p2 =< p2, q′

2 >v


⇔∀ω1∃ω2 îï@v

< p1, α1α2, γ1, ω1 >²< p2, α2, γ2, ω1ω2 >v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


′

1 > Ò p2 =< p2, q′

2 >v



< p1, α1α2, γ1, ω1 >²< p2, α2, γ2, ω1ω2 >v

pv(δ′)∗(q′1, ω2) = q′2

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


′

1 > Ò p2 =< p2, q′

2 >v



< p1, α1α2, γ1, ω1 >²< p2, α2, γ2, ω1ω2 >v

pv(δ′)∗(q′1, ω2) = q′2ðv(λ′)∗(q′1, ω2) = ω2

v⇒k(d irmkªn iÓ£jlk(dacs_c dac

< p1, α1α2, γ1, ω1 >² p2, α2, γ2, ω1ω2 >

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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


′

1 > Ò p2 =< p2, q′

2 >v



< p1, α1α2, γ1, ω1 >²< p2, α2, γ2, ω1ω2 >v

pv(δ′)∗(q′1, ω2) = q′2ðv(λ′)∗(q′1, ω2) = ω2

v⇒k(d irmkªn iÓ£jlk(dacs_c dac

< p1, α1α2, γ1, ω1 >² p2, α2, γ2, ω1ω2 >

⇐k(d irmkªn iÓ£jlk(dacs_c dac

< p1, α1α2, γ1, ω1 >²< p2, α2, γ2, ω1ω2 >v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


ÿ É « ¾BÉ ¸ T =< Σ× Ξ∗, Γ∗,P,Q, s,∆, d , λ, out, φ, ψ >EJ_hacdi;R `h Ò

T ′ =< Ω× Σ∗,Q ′, s ′, δ′, λ′, φ′ >n i;n@&`&iEno cs_a`&`@d_hacdgi;R `h v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


ÿ É « ¾BÉ ¸ T =< Σ× Ξ∗, Γ∗,P,Q, s,∆, d , λ, out, φ, ψ >EJ_hacdi;R `h Ò

T ′ =< Ω× Σ∗,Q ′, s ′, δ′, λ′, φ′ >n i;n@&`&iEno cs_a`&`@d_hacdgi;R `h v⇒

∃T E@ _hacdgi;R `h î

fT= fT fT ′

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

áX?RD(STI

^9k(>rgck dacTo&ÓHh&Õsr k&Õ nsi;ka_a`dac

T ′

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

:MWúDãZW>5 7L8E¦?B<Wù¤C8W4¥;RDFY

Ó>Fnsjldnfì£c o#` `âoVnac×Vm&cs_c Ò h`&i;ki£clF`km&nd&rLk(hac >k@`&kk&Õ nsiλ′(q′, σ)

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

:MWúDãZW>5 7L8E¦?B<Wù¤C8W4¥;RDFY

Ó>Fnsjldnfì£c o#` `âoVnac×Vm&cs_c Ò h`&i;ki£clF`km&nd&rLk(hac >k@`&kk&Õ nsiλ′(q′, σ)

a : abab

a ↓ α b ↓ β1b ↓ β2

p1 p2p3p0

q0

ql

q1

q0

q1 q1

a ↓ α2

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

ÎDªDF<9GFDO<;:ùZ98W4¥;RDFY 9I¤C8W4¥;RDFY

Bc ack( d` n@hac@n_a`dcs_cCÔ@c_n_λ′(q′, σ)

kli£cVknao cF`fhk6d rsji£c v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.



kli£cVknao cF`fhk6d rsji£c v Ó>Fn jldnf`Ti£cl `dac n jlki£c[o#` àOoTnac×Vmcs_cd m&nam#nsÓ_k @` Ti£cln@hac@n_k(

λ′(q′, σ)kan v cmoni£chacoak( Ó>Xsckka_a`Xscnc×Vmcs_c ñ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.



kli£cVknao cF`fhk6d rsji£c v Ó>Fn jldnf`Ti£cl `dac n jlki£c[o#` àOoTnac×Vmcs_cd m&nam#nsÓ_k @` Ti£cln@hac@n_k(

λ′(q′, σ)kan v cmoni£chacoak( Ó>Xsckka_a`Xscnc×Vmcs_c ñ

a : ababbc

a ↓ α b ↓ β1 b ↓ β2c ↓ γ

xyzαβ1α2 yzαβ1α2 zαβ1α2xyz

a ↓ α2b ↓ β3

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

ÎDªDF<9GFDO<;:ùZ98W4¥;RDFYÎI¤C8W4¥;RDFY

Bc ack(âk_&Õ ñ v `Xsc mhac`d hn ñ ` u ksck(×` Ò m#n u cs_an ò&ÓHhdÒoVn@adnoadnfÓ>_ansdk` ñ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


Bc ack(âk_&Õ ñ v `Xsc mhac`d hn ñ ` u ksck(×` Ò m#n u cs_an ò&ÓHhdÒoVn@adnoadnfÓ>_ansdk` ñ Ó>Fn jldnf`Ti£cX_h#o c i£c[knao cF`[naghacdn[_a` ksckk ñ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


Bc ack(âk_&Õ ñ v `Xsc mhac`d hn ñ ` u ksck(×` Ò m#n u cs_an ò&ÓHhdÒoVn@adnoadnfÓ>_ansdk` ñ Ó>Fn jldnf`Ti£cX_h#o c i£c[knao cF`[naghacdn[_a` ksckk ña : ababbc

a ↓ α b ↓ β1b ↓ β2

c ↓ γ

xyxyαβ1α2

εyαβ1α2

a ↓ α2 b ↓ β3

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

ÎDªDF<9GFDO<;:ùZ98W4¥;RDFY

a : ababbc

a ↓ α b ↓ β1b ↓ β2

c ↓ γ

xyxyαβ1α2

εyαβ1α2

a ↓ α2 b ↓ β3

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


a : ababbc

a ↓ α b ↓ β1b ↓ β2

c ↓ γ

xyxyαβ1α2

εyαβ1α2

a ↓ α2 b ↓ β3

Bclk(>rgk(hacF`X_a` k kon i;k dac ï _ÓHmc P_an@` _mcs_an h`&Õ nsi;k v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


a : ababbc

a ↓ α b ↓ β1b ↓ β2

c ↓ γ

xyxyαβ1α2

εyαβ1α2

a ↓ α2 b ↓ β3

Bclk(>rgk(hacF`X_a` k kon i;k dac ï _ÓHmc P_an@` _mcs_an h`&Õ nsi;k vVh r u k hn@a`Lk ñ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


a : ababbc

a ↓ α b ↓ β1b ↓ β2

c ↓ γ

xyxyαβ1α2

εyαβ1α2

a ↓ α2 b ↓ β3

Bclk(>rgk(hacF`X_a` k kon i;k dac ï _ÓHmc P_an@` _mcs_an h`&Õ nsi;k vVh r u k hn@a`Lk ñ cs_anÔ`k. . .

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


a : ababbc

a ↓ α b ↓ β1b ↓ β2

c ↓ γ

xyxyαβ1α2

εyαβ1α2

a ↓ α2 b ↓ β3

Bclk(>rgk(hacF`X_a` k kon i;k dac ï _ÓHmc P_an@` _mcs_an h`&Õ nsi;k vVh r u k hn@a`Lk ñ cs_anÔ`k. . .

d 4c v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

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walkP

I walkP : P × Σ∗ → P ∪ Q

a : ababbc

a ↓ α b ↓ β1 b ↓ β2c ↓ γ


a ↓ α2b ↓ β3

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


walkP


I walkP(p, ε) = p

a : ababbc

a ↓ α b ↓ β1 b ↓ β2c ↓ γ


a ↓ α2b ↓ β3

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


walkP


I walkP(p, ε) = pI

walkP(p, σα′) =

∆(p, σ) Ò cm&n ∆(p, σ) ∈ Q

walkP(∆(p, σ), α′) Ò cm#n ∆(p, σ) ∈ P

¬!k(dacaÔ` v

a : ababbc

a ↓ α b ↓ β1 b ↓ β2c ↓ γ


a ↓ α2b ↓ β3

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

¤C8W4¥;RDFY[G úGâ5 WJ<>=7L8E¦L 79GP§;<W÷8>D:9§E:9<DLûª=7F:£7£ê ®WF7§ß£W?J: 5

restP

I restP : P × Σ∗ → Σ∗

a : ababbc

a ↓ α b ↓ β1 b ↓ β2c ↓ γ


a ↓ α2b ↓ β3

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


restP


I restP(p, ε) = ε

a : ababbc

a ↓ α b ↓ β1 b ↓ β2c ↓ γ


a ↓ α2b ↓ β3

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


restP


I restP(p, ε) = εI

restP(p, σα′) =

α′ Ò cm#n ∆(p, σ) ∈ Q

restP(∆(p, σ), α′) Ò cm&n ∆(p, σ) ∈ P

¬!k(dacaÔ` v

a : ababbc

a ↓ α b ↓ β1 b ↓ β2c ↓ γ


a ↓ α2b ↓ β3

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

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printP

I printP : P × Σ∗ → Γ∗

a : ababbc

a ↓ α b ↓ β1 b ↓ β2c ↓ γ


a ↓ α2b ↓ β3

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


printP


I printP(p, ε) = ε

a : ababbc

a ↓ α b ↓ β1 b ↓ β2c ↓ γ


a ↓ α2b ↓ β3

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


printP


I printP(p, ε) = ε

I

printP(p, σα′) =

λ(p, σ) Ò cm&n ∆(p, σ) ∈ Q

λ(p, σ) printP(∆(p, σ), α′) Ò cm&n ∆(p, σ) ∈ P

¬!k(dacaÔ` v

a : ababbc

a ↓ α b ↓ β1 b ↓ β2c ↓ γ


a ↓ α2b ↓ β3

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

U:9 F§Wú=FGPY¦@BGP8;:£7walkP

restP

printP

∀p ∈ P Ò ∀α, β ∈ Σ∗ Ò ∀γ ∈ Γ∗ Ò ∀ω ∈ Ξ∗ îï@v!walkP(p, α)⇒

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


restP

printP

∀p ∈ P Ò ∀α, β ∈ Σ∗ Ò ∀γ ∈ Γ∗ Ò ∀ω ∈ Ξ∗ îï@v!walkP(p, α)⇒pvrestP(p, α)

`dac_coam&cα

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


restP

printP


`dac_coam&cαðv

< p, αβ, γ, ω >²< walkP(p, α),restP(p, α)β, γprintP(p, α), ω > .

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


restP

printP


`dac_coam&cαðv


v cm&nwalkP(p, α) ∈ P ⇒ restP(p, α) = ε

v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


restP

printP


`dac_coam&cαðv



v v

walkP(p, α) ∈ Q Ò ⇒ |restP(p, α)| <|α|v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


restP

printP


`dac_coam&cαðv



v v


v¬!walkP(p, α)⇒ @

h`&Õsn iªn_< p, αβ, γ, ω >

iÓ£jlk(dac"!n_|α| − 1

v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


restP

printP


`dac_coam&cαðv



v v


v¬!walkP(p, α)⇒ @

h`&Õsn iªn_< p, αβ, γ, ω >

iÓ£jlk(dac"!n_|α| − 1

vnmcscs_a`&_onk(d irmkn

|α|v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

¤C8W4¥;RDFYâ5 WJ<>=7L8¦L 7LGP§;<£W÷8>D:9§E:9<>DLûÎôõ¦L<9 FQRGS <;:Z98>Dß£W?BG>7;Dª5

walkQ

walkQ : Q × Σ∗ × Γ∗ → P ∪ Q

I walkQ(q, α, ε) = q

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


walkQ

walkQ : Q × Σ∗ × Γ∗ → P ∪ Q

I walkQ(q, α, ε) = qI

walkQ(q, α, b) =

d(q, b, 0)cm&n

d(q, b, 0) ∈ Q

walkP(d(q, b, 0), α) Ò cm#n d(q, b, 0) ∈ Pk

walkP(d(q, b, 0), α) ∈ P

walkQ(walkP(d(q, b, 0), α), restP(d(q, b, 0), α),

printP(d(q, b, 0), α))cm&nd(q, b, 0) ∈ P

kwalkP(d(q, b, 0), α) ∈ Qw a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


walkQ

§ W4¥# GS÷=(9¦%$4:&

walkQ(q, α, bγ′) =

walkQ(d(q, b, 1), α, γ′)cm#n

d(q, b, 1) ∈ Q

walkP(d(q, b, 1), α) Ò cm&n d(q, b, 1) ∈ Pk


walkQ(walkP(d(q, b, 1), α), restP(d(q, b, 1), α),

γ′ printP(d(q, b, 1), α))cm#nd(q, b, 1) ∈ P

kwalkP(d(q, b, 1), α) ∈ Q

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

¤C8W4¥;RDFYâ5 WJ<>=7L8¦L 7LGP§;<£W÷8>D:9§E:9<>DLû'ª=7F:£7£ê ®WF7Þ§ß£W?J:5restQ

I restQ : Q × Σ∗ × Γ∗ → Σ∗

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


I restQ : Q × Σ∗ × Γ∗ → Σ∗

I restQ(q, α, ε) = ε

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


I restQ : Q × Σ∗ × Γ∗ → Σ∗

I restQ(q, α, ε) = εI

restQ(q, α, b) =

αcm&n

d(q, b, 0) ∈ Q

restP(d(q, b, 0), α) Ò cm&n d(q, b, 0) ∈ Pk


restQ(walkP(d(q, b, 0), α), restP(d(q, b, 0), α),

printP(d(q, b, 0), α))cm&nd(q, b, 0) ∈ P


! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


§ W4¥# GS =(9¦%$4:&

restQ(q, α, bγ′) =

restQ(d(q, b, 1), α, γ′)cm&n

d(q, b, 1) ∈ Q

restP(d(q, b, 1), α) Ò cm#n d(q, b, 1) ∈ Pk


restQ(walkP(d(q, b, 1), α), restP(d(q, b, 1), α),

γ′ printP(d(q, b, 1), α))cm#nd(q, b, 1) ∈ P


w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

¤C8W4¥;RDFYâ5 WJ<>=7L8¦L 7LGP§;<£W÷8>D:9§E:9<>DLû :9Z9GF=Æ§ WJZ;: :£7;:5printQ

printQ : Q × Σ∗ × Γ∗ → Γ∗

I printQ(q, α, ε) = ε

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


printQ : Q × Σ∗ × Γ∗ → Γ∗

I printQ(q, α, ε) = εI

printQ(q, α, b) =

εcm&n

d(q, b, 0) ∈ Q

printP(d(q, b, 0), α) Ò cm#n d(q, b, 0) ∈ Pk


printQ(walkP(d(q, b, 0), α), restP(d(q, b, 0), α),

printP(d(q, b, 0), α))cm#nd(q, b, 0) ∈ P


! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


§®W4¥# GS =(9¦%$4:&

printQ(q, α, bγ′) =

printQ(d(q, b, 1), α, γ′)cm&n

d(q, b, 1) ∈ Q

γ′ printP(d(q, b, 1), α) Ò cm#n d(q, b, 1) ∈ Pk


printQ(walkP(d(q, b, 1), α), restP(d(q, b, 1), α),

γ′ printP(d(q, b, 1), α))cm&nd(q, b, 1) ∈ P


w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

¤C8W4¥;RDFYâ5 WJ<>=7L8¦L 7LGP§;<£W÷8>D:9§E:9<>DLûÎôõ¦L<9 FQRGS <;:G;\(ß£W?J: 5

outputQ

outputQ : Q × Σ∗ × Γ∗ → Ξ∗

I outputQ(q, α, ε) = ε

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

¤C8W4¥;RDFYâ5 WJ<>=7L8¦L 7LGP§;<£W÷8>D:9§E:9<>DLûÎôõ¦L<9 FQRGS <;:G;\(ß£W?J: 5

outputQ

outputQ : Q × Σ∗ × Γ∗ → Ξ∗

I outputQ(q, α, ε) = εI

outputQ(q, α, b) =

εcm&n

out(q, b, 0) ∈ Q

out(q, b, 0) Ò cm#n d(q, b, 0) ∈ Pk


out(q, b, 0) outputQ(walkP(d(q, b, 0), α),

restP(d(q, b, 0), α), printP(d(q, b, 0), α))cm#nd(q, b, 0) ∈ P


! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

¤C8W4¥;RDFYâ5 WJ<>=7L8¦L 7LGP§;<£W÷8>D:9§E:9<>DLûTál\(ß£W?B<;:âN¦L<9 FQRGS5outputQ

§®W4¥# GS =(9¦%$4:&

outputQ(q, α, bγ′) =

out(q, b, 1) outputQ(d(q, b, 1), α, γ′)cm&n

d(q, b, 1) ∈ Q

γ′ out(q, b, 1) Ò cm#n d(q, b, 1) ∈ Pk


out(q, b, 1) outputQ(walkP(d(q, b, 1), α),

restP(d(q, b, 1), α), γ′ printP(d(q, b, 1), α))cm&n

d(q, b, 1) ∈ Pk

walkP(d(q, b, 1), α) ∈ Q

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

U:9 F§W <9G÷ :\4§E:£7walkQ

restQ

printQ

outputQ

É ¸ ∀q ∈ Q Ò ∀α, β ∈ Σ∗ Ò ∀γ ∈ Γ∗ k ∀ω ∈ Ω∗ï@v!walkQ(q, α, γ) ⇒

< q, αβ, γ, ω >²

< walkQ(q, α, γ), restQ(q, α, γ)β, printQ(q, α, γ), ωω′ >

ω′ = outputQ(q, α, γ).

restQ(q, α, γ) 6= ε ⇒dc(iÓ£ u k(_Co&Ó>Fn j`dCkonsi

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

U:9 F§W <9G÷ :\4§E:£7walkQ

restQ

printQ

outputQ

É ¸ ∀q ∈ Q Ò ∀α, β ∈ Σ∗ Ò ∀γ ∈ Γ∗ k ∀ω ∈ Ω∗ï@v!walkQ(q, α, γ) ⇒

< q, αβ, γ, ω >²

< walkQ(q, α, γ), restQ(q, α, γ)β, printQ(q, α, γ), ωω′ >

ω′ = outputQ(q, α, γ).

restQ(q, α, γ) 6= ε ⇒dc(iÓ£ u k(_Co&Ó>Fn j`dCkonsi

pv¬!walkQ(q, α, γ) ⇒ @

< q, αβ, γ, ω >²< p, β′, γ′, ω >,

scfm#na_an|β′| < |β|

kkβ′ = β

kγ′ = ε

vw a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

U:9 F§W <9G÷ :\4§E:£7walkQ

restQ

printQ

outputQ

*)@Z98W?Lê2L¢ªDF<9GFD,+

Ë « ºH¸ ² ¸2¼JÉ.-fÇ9¼ æR«` bk(dk(hacF` î(α′, γ′) ≺ (α′′, γ′′)⇔

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

U:9 F§W <9G÷ :\4§E:£7walkQ

restQ

printQ

outputQ


Ë « ºH¸ ² ¸2¼JÉ.-fÇ9¼ æR«` bk(dk(hacF` î(α′, γ′) ≺ (α′′, γ′′)⇔|α′| < |α′′|

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

U:9 F§W <9G÷ :\4§E:£7walkQ

restQ

printQ

outputQ


Ë « ºH¸ ² ¸2¼JÉ.-fÇ9¼ æR«` bk(dk(hacF` î(α′, γ′) ≺ (α′′, γ′′)⇔|α′| < |α′′| ∨|α′| = |α′′|&|γ′| < |γ′′|

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

U:9 F§W <9G÷ :\4§E:£7walkQ

restQ

printQ

outputQ


Ë « ºH¸ ² ¸2¼JÉ.-fÇ9¼ æR«` bk(dk(hacF` î(α′, γ′) ≺ (α′′, γ′′)⇔|α′| < |α′′| ∨|α′| = |α′′|&|γ′| < |γ′′|(Σ∗ × Γ∗,≺)

brd i;k(hacdn

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

U:9 F§W <9G÷ :\4§E:£7walkQ

restQ

printQ

outputQ


Ë « ºH¸ ² ¸2¼JÉ.-fÇ9¼ æR«` bk(dk(hacF` î(α′, γ′) ≺ (α′′, γ′′)⇔|α′| < |α′′| ∨|α′| = |α′′|&|γ′| < |γ′′|(Σ∗ × Γ∗,≺)

brd i;k(hacdn^>_h rm_srhdacfk(d irmk nLdn j` _on_an(Σ∗ × Γ∗,≺) ñ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

¤lWH=(RD?B<9Gâ?RD7F:&BG÷Z98>D?BGì WJ<>=7L8E¦L FQRG£SL7;:ú<;:T = T T ′

I

Out(λ′) = λ′(q′, a) | q′ ∈ Q ′, a ∈ Ω ∪ φ′(q′) | q′ ∈ Q ′ ∪

λ′(q′, a)φ′(δ′(q′, a)) | q′ ∈ Q ′, a ∈ Ω.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


I


λ′(q′, a)φ′(δ′(q′, a)) | q′ ∈ Q ′, a ∈ Ω.

I

Rest(λ′) =(x , z) | z =

∏ki=1

printP(pi , ti ), x = rest(pk , tk),mÓi`_ant0 ∈ Out(λ′), pi ∈ P

rgiEno#`_onh#o cs_r &noak`_anti+1 = restP(pi , ti )

sc0 ≤ i < k

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


I


λ′(q′, a)φ′(δ′(q′, a)) | q′ ∈ Q ′, a ∈ Ω.

I

Rest(λ′) =(x , z) | z =

∏ki=1

printP(pi , ti ), x = rest(pk , tk),mÓi`_ant0 ∈ Out(λ′), pi ∈ P

rgiEno#`_onh#o cs_r &noak`_anti+1 = restP(pi , ti )

sc0 ≤ i < k

I

Print(λ′) = printQ(q, x , bz) | q ∈ Q, (x , z) ∈ Out(λ′), b ∈ Γ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′.

êT=7>WESB<LG£S

I P = Q ′ × P × Print(λ′)mhac((dnLdn j` _on ñ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


êT=7>WESB<LG£S


I Q0 = Q ′ × Q × Rest(λ′)× 0mhac((dnLdn j` _on ñ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


êT=7>WESB<LG£S




w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


êT=7>WESB<LG£S




I Q1 = Q0 ∪ Q1

Bmhac((dnLdnsj2` _on ñ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


êT=7>WESB<LG£S




I Q1 = Q0 ∪ Q1

Bmhac((dnLdnsj2` _on ñI s =< s ′, s, ε >

dacÔ@c dnfÓ>_ansdk` ñ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′ I

∆Gλ

I

∆(< q′, p, γ >, a) =

< δ′(q′, a),walkP(p, λ′(q′, a)), ε > Ò cm#n

walkP(p, λ′(q′, a)) ∈ P

< δ′(q′, a),walkP(p, λ(q′, a)),

restP(p, λ′(q′, a)), ε >

k(dacÔa` v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′ I

∆Gλ

I

∆(< q′, p, γ >, a) =

< δ′(q′, a),walkP(p, λ′(q′, a)), ε > Ò cm#n

walkP(p, λ′(q′, a)) ∈ P

< δ′(q′, a),walkP(p, λ(q′, a)),

restP(p, λ′(q′, a)), ε >

k(dacÔa` vI

λ(< q′, p, γ >, a) = γ printP(p, λ′(q′, a)).

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′ I d §(ê8ß¦

Q0I

d(< q′, q, α, γ, 0 >, b, 1) =

< q′, d(q, b, 1), α, γ, 0 >cm&n

d(q, b, 1) ∈ Q

< q′,walkP(q1, α), restP(q1, α),

γ printP(q1, α), 0 >cm&nq1 = d(q, b, 1) ∈ P

kwalkP(q1, α) ∈ Q

< q′,walkP(q1, α), γ printP(q1, α) >cm&nq1 = d(q, b, 1)

kwalkP(q1, α) ∈ P.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


Q0I

d(< q′, q, α, γ, 0 >, b, 0) =

< q′,walkQ(q, α, bγ), printQ(q, α, bγ) >cm&nwalkQ(q, α, bγ) ∈ P

< q′,walkQ(q, α, bγ),

restQ(q, α, bγ), printQ(q, α, bγ), 0 >cm&nwalkQ(q, α, bγ) ∈ Q.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


Q1I

d(< q′, q, α, γ, 1 >, b, 1) =

< q′, d(q, b, 1), α, γ, 1 >cm&n

d(q, b, 1) ∈ Q

< q′,walkP(q1, α), restP(q1, α),

γ printP(q1, α), 1 >cm&nq1 = d(q, b, 1) ∈ P

kwalkP(q1, α) ∈ Q

< q′, φ(walkP(q1, α)), ε,

γ printP(q1, α), 0 >cm&nq1 = d(q, b, 1)

kwalkP(q1, α) ∈ P.w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


Q1I

d(< q′, q, γ, α, 1 >, b, 0) =

< q′, φ(walkQ(q, α, bγ)), ε,

printQ(q, α, bγ), 0 >cm&nwalkQ(q, α, bγ) ∈ P

< q′,walkQ(q, α, bγ), restQ(q, α, bγ),

printQ(q, α, bγ), 1 >cm&nwalkQ(q, α, bγ) ∈ Q.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′ I λ §(ê8ß¦

QI

out(< q′, q, α, γ, j >, b, 1) = out(q, b, 1)

out(< q′, q, α, γ, j >, b, 0) = outputQ(q, α, bγ).

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′ I%/[=(PWJ§;GSì\:õ\:9§(ê8 §E:9<>D

φI

φ(< q′, p, γ >) =

< q′,walkP(p, φ′(q′)), restP(p, φ

′(q′)),

γ printP(p, φ′(q′), 1) >cm#n

walkP(p, φ(q′)) ∈ Q

< q′, φ(p1), ε, γ printP(p, φ′(q′)), 0 >cm#n

p1 = walkP(p, φ(q′)) ∈ P

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′ I%/[=(PWJ§;GSì\:õ\:9§(ê8 §E:9<>D

ψI

ψ(< q′, q, α, γ, j >) =

outputQ(q, α, γ) ψ(q1)mÓ£iE`_anq1 = walkQ(q, α, γ) ∈ Q, j = 0,

α = εk

printQ(q, α, γ) = ε

¬!k(dacaÔ`

.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ I

ÿ É « ¾BÉ ¸ fT= fT fT ′

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W É ¸

I p1, p2 ∈ P ∪ Q0

v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W É ¸

I p1, p2 ∈ P ∪ Q0

vI q′i Ò pi Ò xi

kzi

sci ∈ 1, 2 î

pi = < q′i , pi , xi , zi , 0 >cm&n

pi ∈ Q0

pi = < q′i , pi , zi >k

xi = εm&n

pi ∈ Pv

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W É ¸

I p1, p2 ∈ P ∪ Q0


kzi

sci ∈ 1, 2 î

pi = < q′i , pi , xi , zi , 0 >cm&n

pi ∈ Q0

pi = < q′i , pi , zi >k

xi = εm&n

pi ∈ Pv

I ∀β1, β2 ∈ Ω∗ Ò ∀γ1, γ2 ∈ Γ

∗ Ò ω1, ω2 ∈ Ξ∗ î

< p1, β1β2, γ1, ω1 >²< p2, β2, γ2, ω2 >⇒

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W É ¸

I p1, p2 ∈ P ∪ Q0


kzi

sci ∈ 1, 2 î

pi = < q′i , pi , xi , zi , 0 >cm&n

pi ∈ Q0

pi = < q′i , pi , zi >k

xi = εm&n

pi ∈ Pv

I ∀β1, β2 ∈ Ω∗ Ò ∀γ1, γ2 ∈ Γ

∗ Ò ω1, ω2 ∈ Ξ∗ î

< p1, β1β2, γ1, ω1 >²< p2, β2, γ2, ω2 >⇒

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W É ¸

I p1, p2 ∈ P ∪ Q0


kzi

sci ∈ 1, 2 î

pi = < q′i , pi , xi , zi , 0 >cm&n

pi ∈ Q0

pi = < q′i , pi , zi >k

xi = εm&n

pi ∈ Pv

I ∀β1, β2 ∈ Ω∗ Ò ∀γ1, γ2 ∈ Γ

∗ Ò ω1, ω2 ∈ Ξ∗ î

< p1, β1β2, γ1, ω1 >²< p2, β2, γ2, ω2 >⇒

∀α :

(δ′)∗(q1, β1) = q2

k< p1, x1(λ

′)∗(q1, β1)α, γ1z1, ω1 >²< p2, x2α, γ2z2, ω2 > .

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W

É ¸I p1 =< q′1, p1, x1, z1, 1 >

kq2 =< q′2, p2, x2, z2, 1 >

scn_Q1

v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W

É ¸I p1 =< q′1, p1, x1, z1, 1 >

kq2 =< q′2, p2, x2, z2, 1 >

scn_Q1

vI

< p1, ε, γ1, ω1 >²< p2, ε, γ2, ω2 >⇒

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W

É ¸I p1 =< q′1, p1, x1, z1, 1 >

kq2 =< q′2, p2, x2, z2, 1 >

scn_Q1

vI

< p1, ε, γ1, ω1 >²< p2, ε, γ2, ω2 >⇒

q′

1=q′

2

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W

É ¸I p1 =< q′1, p1, x1, z1, 1 >

kq2 =< q′2, p2, x2, z2, 1 >

scn_Q1

vI

< p1, ε, γ1, ω1 >²< p2, ε, γ2, ω2 >⇒

q′

1=q′

2

< p1, x1, γ1z1, ω1 >²< p2, x2, γ2z2, ω2 > .

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W É ¸

I q′1, q′

2 ∈ Q ′ Ò β1 ∈ Ω∗ Ò p1, p2 ∈ P

v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W É ¸

I q′1, q′

2 ∈ Q ′ Ò β1 ∈ Ω∗ Ò p1, p2 ∈ P

vI

(δ′)∗(q′1, β) = q′2k

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W É ¸

I q′1, q′

2 ∈ Q ′ Ò β1 ∈ Ω∗ Ò p1, p2 ∈ P

vI

(δ′)∗(q′1, β) = q′2kI

< p1, (λ′)∗(q′1, β1), ε, ε >²< p2, ε, γ1, ω1 >⇒

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W É ¸

I q′1, q′

2 ∈ Q ′ Ò β1 ∈ Ω∗ Ò p1, p2 ∈ P

vI

(δ′)∗(q′1, β) = q′2kI

< p1, (λ′)∗(q′1, β1), ε, ε >²< p2, ε, γ1, ω1 >⇒

I

< p1, β1, ε, ε >²< p2, ε, γ2, ω1 >,scfm#na_anp1 =< q′1, p1, ε >

kp2 =< q′2, p2, z >

scfd m#n@`zv

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W

ï@vfT(α) = ω ⇒

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W

ï@vfT(α) = ω ⇒pv

fT (fT ′(α)) = ω

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W


fT (fT ′(α)) = ωðv!fT ′(α)&!fT (fT ′(α)) ⇒

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W


fT (fT ′(α)) = ωðv!fT ′(α)&!fT (fT ′(α)) ⇒ v!f

T(α)

v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

ôOWJ89Y¦@BGR8WJ§; (:ì<;:Þ8>D\¦@L7F:£7F:PI

ÿ É « ¾BÉ ¸ ^>ÓHÖB` _o#ro cs_ h`&iE_coak(Lk[ EJ_hac@dgi;R `hkbrdmk(k Ò Ô(k(_an m#nLn@k(kd` `h`&iE_coak(4cEJ_hacdi;R `h ñ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

òTDßP<9G.$FDE=F : RDFYl: É ¸

∀ T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >,

∃ T =< Σ× Ω∗, (Σ× P), P, Q, i , ∆, d , λ, out, φ, ψ >:

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


∀ T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >,


ï@v∀ p ∈ P∀σ ∈ Σ

λ(p, σ) =< p, σ >

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


∀ T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >,


ï@v∀ p ∈ P∀σ ∈ Σ

λ(p, σ) =< p, σ >

pvfT = f

T.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

áX?RD(S

vvv n_dnon ^ ó#0 ~21à3 ó4 ñ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

áX?RD(S

vvv n_dnon ^ ó#0 ~21à3 ó4 ña ↓ aba b ↓ cdd

p0 p1p2

p0 p1 p2

a ↓ (p0, a) b ↓ (p1, b)

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

¤C8W4¥;RDFY[Gï@v hn@a` ï

λ(q, a) = ε⇒

d(., < p, a >, 1) =

d∗(., λ(p, a), 1)?

d∗(., λ(p, a), 0)?

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


λ(q, a) = ε⇒

d(., < p, a >, 1) =

d∗(., λ(p, a), 1)?

d∗(., λ(p, a), 0)?pv 5 `×2`dk`ìPi£c[ck( n@&`&i;dkk(Lona dacλ(p, a)

v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


λ(q, a) = ε⇒

d(., < p, a >, 1) =

d∗(., λ(p, a), 1)?

d∗(., λ(p, a), 0)?pv 5 `×2`dk`ìPi£c[ck( n@&`&i;dkk(Lona dacλ(p, a)

vðv hn@a` p î

(p0, b0)(p1, b1) . . . (pn, bn)

γ0γ1 . . . γn γ′

0γ1 . . . γn

(p1, b1) . . . (pn, bn)

a : b

c : ab a : ε

(p0, b0) : abεb

q1 q2q3

q

p = (p, γ′

0)

p

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

¤C8W4¥;RDFY[G%)@Z98W?Lê9¢DF<9GFD,+

hn@a` ð îp1

q1 q2 q3

p2

γ′

0γ1 . . . γn

γ′′

0 γ1 . . . γn+1

a : ω1 b : ω2

c : ω3

(p1, γ′

0, ω)

bn+1 ↓ γn+1

bn+1 ↓ (p1, bn+1)

(p2, γ′′

0 , ωω1ω2ω3)

γ′

0γ1 . . . γn+1

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

UXWJ<>=7L8E¦L 79GP§;<W÷8>DªDF<9GFDHIRôõ¦L<9 FQRGS÷\:ùZ98>Dß£W?BG>7;D5walk

walk : Q × Γ∗ → P ∪ Q

I |γ| ≤ 1 Òwalk(q, γ) = q

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


walk : Q × Γ∗ → P ∪ Q


I

cm&nγ = bγ′

kd(q, b, 1) ∈ P Ò _an

walk(q, γ) = d(q, b, 1).

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


walk : Q × Γ∗ → P ∪ Q


I

cm&nγ = bγ′

kd(q, b, 1) ∈ P Ò _an

walk(q, γ) = d(q, b, 1).

I

cm&nγ = bγ′

kd(q, b, 1) ∈ Q Ò _an

walk(q, bγ′) = walk(d(q, b, 1), γ′).

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

UXWJ<>=7L8E¦L 79GP§;<W÷8>DªDF<9GFDHIRôõ¦L<9 FQRGS÷\:ùZ98>Dß£W?BG>7;D5rest

rest : Q × Γ∗ → Γ∗

I |γ| ≤ 1 Òrest(q, γ) = γ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


rest : Q × Γ∗ → Γ∗


I

cm&nγ = bγ′

kd(q, b, 1) ∈ P Ò _an

rest(q, γ) = γ′.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


rest : Q × Γ∗ → Γ∗


I

cm&nγ = bγ′

kd(q, b, 1) ∈ P Ò _an

rest(q, γ) = γ′.

I

cm&nγ = bγ′

kd(q, b, 1) ∈ Q Ò _anrest(q, bγ′) = rest(d(q, b, 1), γ ′)

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

UXWJ<>=7L8E¦L 79GP§;<W÷8>DªDF<9GFDHIRôõ¦L<9 FQRGS÷\:ùZ98>Dß£W?BG>7;D5output

output : Q × Γ∗ → Ω∗

I |γ| ≤ 1 Òoutput(q, γ) = ε

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.




I

cm&nγ = bγ′

kd(q, b, 1) ∈ P Ò _an

output(q, γ) = out(q, b, 1).

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.




I

cm&nγ = bγ′

kd(q, b, 1) ∈ P Ò _an

output(q, γ) = out(q, b, 1).

I

cm&nγ = bγ′

kd(q, b, 1) ∈ Q Ò _an

output(q, bγ′) = out(q, b, 1)output(d(q, b, 1), γ ′)

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

UXWJ<>=7L8E¦L FQRGS <;:T

nsiEn@dndac nLn@k(k(_cn_idn ñI

Suf (λ) = γ ∈ Γ∗ | ∃p ∈ P, σ ∈ Σ, β ∈ Γ∗,scfm#nka_an

λ(p, σ) = β γ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

UXWJ<>=7L8E¦L FQRGS <;:T

nsiEn@dndac nLn@k(k(_cn_idn ñI

Suf (λ) = γ ∈ Γ∗ | ∃p ∈ P, σ ∈ Σ, β ∈ Γ∗,scfm#nka_an

λ(p, σ) = β γ

I

Pref (out) = print(q, γ) | q ∈ Q, γ ∈ Suf (λ).

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

UXWJ<>=7L8E¦L FQRGS <;:TI6êT=7>W£SB<9GS

I P = P × Suf (λ)× Pref (out)v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.



I Q = Q × Γ ∪ ε × Pref (out)v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.



I Q = Q × Γ ∪ ε × Pref (out)v

I (i) =v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

UXWJ<>=7L8E¦L FQRGS <;:T .∆

∆(< p, γ, ω >, σ) =

< ∆(p, σ), γ, ω >cm#n

∆(p, σ) ∈ P

< walk(q, γ), rest(q, γ), ω print(q, γ) >,mÓ£iE`_anq = ∆(p, σ) ∈ Q.

¬!k(dacÔ` v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

ôõ¦L<9 FQRGPG>7;Dins0,1

rep0,1

Grep0,1 ins0,1


w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


rep0,1

Grep0,1 ins0,1


I

ins0,1

(k∏

i=0

(ami bni )

)=

k∏

i=0

(ciami bni )0,

ci ≡k∑

j=i

mj( mod 2).

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


rep0,1

Grep0,1 ins0,1


w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


rep0,1

Grep0,1 ins0,1


I

rep0,1(ci ) =

ε,cm&n

ci ∈ 0, 1

¬!k(dacÔ`

.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


rep0,1

Grep0,1 ins0,1


I

rep0,1(ci ) =

ε,cm&n

ci ∈ 0, 1

¬!k(dacÔ`

.

I

rep0,1(αcβ) =

αA rep0,1(β)cm&n

α ∈ a, b∗, c = 1

α rep0,1(β)cm&n

α ∈ a, b∗, c = 0

¬!k(dacaÔ`

,

αA

a → A

vw a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


rep0,1

Grep0,1 ins0,1

rep0,1(ins0,1

(k∏

i=0

(ami bni )

)) =

k∏

i=0

(cmi

i bni ),

mÓ£iE`_an

ci = A ⇔

k∑

j=i+1

mj ≡ 1( mod 2)

ci = a ⇔k∑

j=i+1

mj ≡ 0( mod 2).

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

ôõ¦L<9 FQRGSL7F:rep0,1 ins0,1

ÿ É « ¾BÉ ¸ @E@J_hacdgi;R `h

T îfT = rep0,1 ins0,1.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

áX?RD(STI¤Xê89§;G÷<;:¥;BAl?RDF<LG£SI

n rsm&cF` Ò Ô` ∃ E@J_hacdi;R `h T1 îfT1

= rep0,1 ins0,1.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.



= rep0,1 ins0,1.

I

⇒ ∃ T =< Σ× Ω∗, (Σ× P),P,Q, s,∆, d , λ, out, φ, ψ >:

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.



= rep0,1 ins0,1.

I

⇒ ∃ T =< Σ× Ω∗, (Σ× P),P,Q, s,∆, d , λ, out, φ, ψ >:

I

∀ p ∈ P, ∀ x ∈ a, b

λ(p, x) =< p, x >

fT = rep0,1 ins0,1.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

áX?RD(STI¤Xê89§;G÷<;:¥;BAl?RDF<LG£S É ¸ `mc α ∈ a+a, b∗

kl = max|out(q, < p, σ >)|, |ψ(q)|

n u co c îI

(s, α, ε, ε) ² (p, β, γ, ω)

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


kl = max|out(q, < p, σ >)|, |ψ(q)|

n u co c îI

(s, α, ε, ε) ² (p, β, γ, ω)

I ⇒ ω = εv

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


kl = max|out(q, < p, σ >)|, |ψ(q)|

n u co c îI

(s, α, ε, ε) ² (p, β, γ, ω)

I ⇒ ω = εv

I

(s, α, ε, ε) ² (p0, ε, γ0, ω0)

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


kl = max|out(q, < p, σ >)|, |ψ(q)|

n u co c îI

(s, α, ε, ε) ² (p, β, γ, ω)

I ⇒ ω = εv

I

(s, α, ε, ε) ² (p0, ε, γ0, ω0)

I ⇒ l(|γ0|+ 1) ≥ |α|v

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

KWJ :\:£7>D(R=79§W2IBáX?RD(S nac u cF` î

I

l = max|out(q, < p, σ >)|, |ψ(q)|

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


I

l = max|out(q, < p, σ >)|, |ψ(q)|

I

M = (l + 1)|Q|+ 1.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.


I

l = max|out(q, < p, σ >)|, |ψ(q)|

I

M = (l + 1)|Q|+ 1.

I

` bk(dk(hacF`mi

Mi=0

îmM = 1

mi−1 = l(∑M

j=i (mj + 1) + 1) scfom&n

M ≥ i ≥ 1.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

KWJ :\:£7>D(R=79§W2IBáX?RD(S

nac u cF` îI αj

Mj=1îαj =

(j−1∏

i=0

a2mi b

)a2mj+1

M∏

i=j+1

ba2mi

.

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

KWJ :\:£7>D(R=79§W2IBáX?RD(S cmhac

I γj Ò pj ∈ P î< s, αj , ε, ε >²< pj , ε, γj , ε >

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.



I ∀r γj ,r

γj ,r

dac_coam&c dacγj

|γj ,r | = r2m0

l

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.



I ∀r γj ,r

γj ,r

dac_coam&c dacγj

|γj ,r | = r2m0

l

I ∀r qj ,r ∈ Q Ò ωj ,r ∈ Ω∗ î

< φ(pj), ε, γj , ε >²< qj ,r , ε, γj ,r , ωj ,r >

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

:¥;BA?RDF<9GFD

I

i < j !γj ,r & !γi ,r ⇒

γj ,r

kγi ,r

ÓHoac i£cs_6osÓHhoaka_a`l `

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

:¥;BA?RDF<9GFD

I

i < j !γj ,r & !γi ,r ⇒

γj ,r

kγi ,r

ÓHoac i£cs_6osÓHhoaka_a`l Ì

r2m0

l−

M∑

k=i+1

(2mk + 1)

k(Lonac ñ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

7êZ98WH=FG*

ÓHhn@k

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

ª<WF7F:9QRGS

I cXn_fhackndac >dkfbrdmk(k Ò scfm#nka_anÓHÖB` _o#ro cEJ_hcdgi;R `h

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

ª<WF7F:9QRGS


I5 ` u rghdk6hacoakcsclscF` _o cd` v n@_cs_ÓÔ(dkrs&noak Ò sci£cl_hnk( E@J_hacdgi>R `hk cl_Õ ñ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.

ª<WF7F:9QRGS


I5 ` u rghdk6hacoakcsclscF` _o cd` v n@_cs_ÓÔ(dkrs&noak Ò sci£cl_hnk( E@J_hacdgi>R `hk cl_Õ ñ

I

n@_cs_ÓEÔ(dklrs&noaksc m#nLn@k(k dacl_cmk(o cfhacoakc ñ

w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-

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logic.fmi.uni-sofia.bglogic.fmi.uni-sofia.bg/theses/Gerdjikov-talk1.pdf · ! "# $# "&% ' ( )+*) ,- /. $# "&% ' | % /% + { { % ü 'yý { {@ {| $# "&% ' | % /% + { { % ü 'yý { {@