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I out : Q × Γ× 0, 1 → Ω∗ Ò Dom(out) = Dom(d)k&Õsn i;dacbrdmk v
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I out : Q × Γ× 0, 1 → Ω∗ Ò Dom(out) = Dom(d)k&Õsn i;dacbrdmk v
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<;NOG9ܦL8;:9QRGS Ò mÓ£iE`_an
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I T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ > ÒI κ = (p, α, γ, ω)
` WJ<;NOG9ܦL8;:9QRGS Ò mÓ£iE`_anI p ∈ (P ∪ Q)
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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I T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ > ÒI κ = (p, α, γ, ω)
` WJ<;NOG9ܦL8;:9QRGS Ò mÓ£iE`_anI p ∈ (P ∪ Q)
I α ∈ Σ∗
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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I T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ > ÒI κ = (p, α, γ, ω)
` WJ<;NOG9ܦL8;:9QRGS Ò mÓ£iE`_anI p ∈ (P ∪ Q)
I α ∈ Σ∗
I γ ∈ Γ∗v
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I T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ > ÒI κ = (p, α, γ, ω)
` WJ<;NOG9ܦL8;:9QRGS Ò mÓ£iE`_anI p ∈ (P ∪ Q)
I α ∈ Σ∗
I γ ∈ Γ∗v
I ω ∈ Ω∗v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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à m&nκ′ = (p′, α′, γ′, ω′)
kκ′′ = (p′′, α′′, γ′′, ω′′) Ò _an Z98>DߣW? κ′`κ′′
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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à m&nκ′ = (p′, α′, γ′, ω′)
kκ′′ = (p′′, α′′, γ′′, ω′′) Ò _an Z98>DߣW? κ′`κ′′
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$#"&%' |()+* ),-
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à m&nκ′ = (p′, α′, γ′, ω′)
kκ′′ = (p′′, α′′, γ′′, ω′′) Ò _an Z98>DߣW? κ′`κ′′
I p′ ∈ P Ò α′ = σα′′ Ò σ ∈ Σ kp′′ = ∆(p′, σ) Ò γ′′ = γ′λ(p′, σ) Ò
ω′′ = ω′v
I p′ ∈ Q Ò γ′ = bγ′′ Ò b ∈ Γ Ò γ′′ 6= εk
p′′ = d(p′, b, 1) Ò α′′ = α′ Òω′′ = ω out(p′, b, 1)
v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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à m&nκ′ = (p′, α′, γ′, ω′)
kκ′′ = (p′′, α′′, γ′′, ω′′) Ò _an Z98>DߣW? κ′`κ′′
I p′ ∈ P Ò α′ = σα′′ Ò σ ∈ Σ kp′′ = ∆(p′, σ) Ò γ′′ = γ′λ(p′, σ) Ò
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)
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$#"&%' |()+* ),-
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à m&nκ′ = (p′, α′, γ′, ω′)
kκ′′ = (p′′, α′′, γ′′, ω′′) Ò _an Z98>DߣW? κ′`κ′′
I p′ ∈ P Ò α′ = σα′′ Ò σ ∈ Σ kp′′ = ∆(p′, σ) Ò γ′′ = γ′λ(p′, σ) Ò
ω′′ = ω′v
I p′ ∈ Q Ò γ′ = bγ′′ Ò b ∈ Γ Ò γ′′ 6= εk
p′′ = d(p′, b, 1) Ò α′′ = α′ Òω′′ = ω out(p′, b, 1)
vI p′ ∈ Q Ò γ′ = b ∈ Γ
kγ′′ = ε Ò p′′ = d(p′, b, 0) Ò α′′ = α′ Ò
ω′′ = ω′ out(p′, b, 0)v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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:
`Iä ÉJå ¬ ¿ ¬Mæ Éçº « ÀBÍÏÁ迾B¸XÐfÁVÑ ÀÉ· «æ ÉØTÉ « ¼ ´
Á ²2æR«2¬é
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2Ó£jlk(dac dacκ′ ² κ′′
I κ′=κ′′ Ò iÓ£jlk(dac 0 v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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2Ó£jlk(dac dacκ′ ² κ′′
I κ′=κ′′ Ò iÓ£jlk(dac 0 vI κ′`κ ² κ′′ Ò iÓ£jlk(dac 1+ iÓ£jlk(dacs_cCdac
κ ² κ′′v
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! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. ëP\4GP Æ<;:ì02104365 798;:9<>=@?BAC=EDF8
T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >`EJ_hacdi;R `h ví k(m
L(T ) = (α, ω) î
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T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >`EJ_hacdi;R `h ví k(m
L(T ) = (α, ω) îï@v∃f ∈ P, ∃γ ∈ Γ∗, ∃ω1 ∈ Ω
∗ Ò< i , α, ε, ε >²< f , ε, γ, ω1 >
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >`EJ_hacdi;R `h ví k(m
L(T ) = (α, ω) îï@v∃f ∈ P, ∃γ ∈ Γ∗, ∃ω1 ∈ Ω
∗ Ò< i , α, ε, ε >²< f , ε, γ, ω1 >pv
∃q ∈ Q, ∃ω2 ∈ Ω∗ î
< φ(f ), ε, γ, ω1 >²< q, ε, ε, ω1ω2 >
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. ëP\4GP Æ<;:ì02104365 798;:9<>=@?BAC=EDF8
T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >`EJ_hacdi;R `h ví k(m
L(T ) = (α, ω) îï@v∃f ∈ P, ∃γ ∈ Γ∗, ∃ω1 ∈ Ω
∗ Ò< i , α, ε, ε >²< f , ε, γ, ω1 >pv
∃q ∈ Q, ∃ω2 ∈ Ω∗ î
< φ(f ), ε, γ, ω1 >²< q, ε, ε, ω1ω2 >ðvω = ω1ω2ψ(q)
v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >`EJ_hacdi;R `h ví k(m
L(T ) = (α, ω) îï@v∃f ∈ P, ∃γ ∈ Γ∗, ∃ω1 ∈ Ω
∗ Ò< i , α, ε, ε >²< 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 >`EJ_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$#"&%' |()+* ),-
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T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >`EJ_hacdi;R `h v
l = max |λ(p, σ)|,
Ià m&n
q′, q′′ ∈ Q_an o `mk kon i
< q′, α, γ, ω >²< q′′, α, γ′, ω′ >k(4c iÓ£jlk(dac î
|γ′| − |γ|.
I ó d irmkn[hn dgh`&Õ nsi;ka_a` v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >`EJ_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|.
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$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. ò8;:9§DF8>=FGP8;:H<>DO<;:Þ02104365 7L8;:H<>=?BAC=DF8
T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >`EJ_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|.
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$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. ( "#'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$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. ( "#'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
a : a
p0
p1 qa
qb
φ
q$f
a : a, b : b
$ : $
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. áCYl:ÞG÷8;:9QRG;WJ<;:EB<9G÷02104365 7L8;:9<=?BAC=ED;89GJIRôõ¦L<9 FQRGSL7F:
rep0,1
I ö rdmk(_c rep0,1 : a, b, 0, 1∗ → A, a, b∗ Ò
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. áCYl:ÞG÷8;:9QRG;WJ<;:EB<9G÷02104365 7L8;:9<=?BAC=ED;89GJIRôõ¦L<9 FQRGSL7F:
rep0,1
I ö rdmk(_c rep0,1 : a, b, 0, 1∗ → A, a, b∗ Ò
I
rep0,1(ci ) =
ε,cm&n
ci ∈ 0, 1
¬!k(dacÔ`
.
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. áCYl:ÞG÷8;:9QRG;WJ<;:EB<9G÷02104365 7L8;:9<=?BAC=ED;89GJIRôõ¦L<9 FQRGSL7F:
rep0,1
I ö rdmk(_c rep0,1 : a, b, 0, 1∗ → A, a, b∗ Ò
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$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. ôõ¦L<9 FQRGSL7F:
rep0,1<;:Þ :98£7LGP<9 (:
a ↓ a, b ↓ b
0 ↓ 0
1 ↓ 1
0 : ε
1 : ε
a : a, b : b
a : a, b : b
φp0
q0
q1
f
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. áøD?BGP<ùYl:EB W¡ZW>5GP<£7;DF8>DE=ED;<ù02104365 7L8;:H<>=?BAC=DF8PIPôõ¦L<9 FQRGSL74:ins0,1
I ö rdmk(_c ins0,1 : a, b+ → a, b, 0, 1∗ Ò
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. áøD?BGP<ùYl:EB W¡ZW>5GP<£7;DF8>DE=ED;<ù02104365 7L8;:H<>=?BAC=DF8PIPôõ¦L<9 FQRGSL74:ins0,1
I ö rdmk(_c ins0,1 : a, b+ → a, b, 0, 1∗ Ò
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$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. ( "#'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$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. ( "#'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$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. ( "#'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
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. ( "#'y)+* ),- /. ôõ¦L<9 FQRGSL7F:
ins0,1<;:ù :98£7LGP<9 :
b ↓ bb ↓ b
φ φ
a ↓ a
a ↓ a
p1p0
q1
0
B1
0
B1
1
EA1
0
EA1
1
OA1
1
b : b
a : 1a
a : 0a
b : 1b
b : b
a : aa : a
a : a
b : b
b : b
a : 1a
b : b
a : a
b : b
OA1
0
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
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
ôOWJ89Y¦BGP8WJ§; (:ì<;:Þ8>D\¦@L7F:£7F:
ÿ É « ¾BÉ ¸ T =< Σ× Ω∗, Γ∗,P,Q, s,∆, d , λ, out, φ, ψ >EJ_hacdi;R `h Ò
T ′ =< Ω× Ξ∗,Q ′, s ′, δ′, λ′, φ′ >Bn i;n@&`&iEno cs_a`&`d_hacdgi;R `h v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
ôOWJ89Y¦BGP8WJ§; (:ì<;:Þ8>D\¦@L7F:£7F:
ÿ É « ¾BÉ ¸ T =< Σ× Ω∗, Γ∗,P,Q, s,∆, d , λ, out, φ, ψ >EJ_hacdi;R `h Ò
T ′ =< Ω× Ξ∗,Q ′, s ′, δ′, λ′, φ′ >Bn i;n@&`&iEno cs_a`&`d_hacdgi;R `h v⇒
∃T E@ _hacdgi;R `h î
fT= fT ′ fT
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
áX?RD(STIêT=7>WESB<LG£S
^9k(>rgck dacT ′o&ÓHh#Õsr k&Õsn i;ka_a`dac
T
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
áX?RD(STIêT=7>WESB<LG£S
^9k(>rgck dacT ′o&ÓHh#Õsr k&Õsn i;ka_a`dac
T
⇒`mchs_anonhnko`#iE`dk`[dacÓ>_an dk î
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
áX?RD(STIêT=7>WESB<LG£S
^9k(>rgck dacT ′o&ÓHh#Õsr k&Õsn i;ka_a`dac
T
⇒`mchs_anonhnko`#iE`dk`[dacÓ>_an dk îI P = P × Q ′
vI Q = Q × Q ′
vI i =< s, s ′ >
v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
. . .GPY¦BGP8;:9YXDúG÷Z98>DߣW£?BG>7;D
I ÓHhono&ÓHh#Õsr
P = P × Q ′ î∆(< p, q′ >, σ) =< ∆(p, σ), q′ >λ(< p, q′ >, σ) = λ(p, σ)
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
. . .GPY¦BGP8;:9YXDúG÷Z98>DߣW£?BG>7;D
I ÓHhono&ÓHh#Õsr
P = P × Q ′ î∆(< p, q′ >, σ) =< ∆(p, σ), q′ >λ(< p, q′ >, σ) = λ(p, σ)
I . . .cfn@&`k6o&ÓHh#Õsr
Q = Q × Q ′
d(< p, q′ >, b, j) =< d(p, b, j), (δ′)∗(q′, out(p, b, j)) > Ò scj ∈ 0, 1
vout(< p, q′ >, b, j) = (λ′)∗(q′, out(p, b, j)) Ò sc j ∈ 0, 1
v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
. . .<;:9 F8;:ES GO¦E=(PWJ§;GSL7F:õ\:õ\:9§(ê8 §E:9<>D
I ÓHhon[_a` kCo&ÓHh&Õsr
P = P × Q ′ îφ(< p, q′ >) =< φ(p), q′ >
v
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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
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::£7F:ãGP<£7£¦LGPQRGS =ED WJZ98;:9§(?J:9§E:Jû É ¸ `mc p1 =< p1, q
′
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2 >v
n u co c< p1, α1α2, γ1, ω1 >² p2, α2, γ2, ω1ω2 >
⇔∀ω1∃ω2 î
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′
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n u co c< p1, α1α2, γ1, ω1 >² p2, α2, γ2, ω1ω2 >
⇔∀ω1∃ω2 îï@v
< p1, α1α2, γ1, ω1 >²< p2, α2, γ2, ω1ω2 >v
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::£7F:ãGP<£7£¦LGPQRGS =ED WJZ98;:9§(?J:9§E:Jû É ¸ `mc p1 =< p1, q
′
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2 >v
n u co c< p1, α1α2, γ1, ω1 >² p2, α2, γ2, ω1ω2 >
⇔∀ω1∃ω2 îï@v
< p1, α1α2, γ1, ω1 >²< p2, α2, γ2, ω1ω2 >v
pv(δ′)∗(q′1, ω2) = q′2
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n u co c< p1, α1α2, γ1, ω1 >² p2, α2, γ2, ω1ω2 >
⇔∀ω1∃ω2 îï@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
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n u co c< p1, α1α2, γ1, ω1 >² p2, α2, γ2, ω1ω2 >
⇔∀ω1∃ω2 îï@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
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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
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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 ′
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áX?RD(STI
^9k(>rgck dacTo&ÓHh&Õsr k&Õ nsi;ka_a`dac
T ′
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:MWúDãZW>5 7L8E¦?B<Wù¤C8W4¥;RDFY
Ó>Fnsjldnf`i£c o#` `âoVnac×Vm&cs_c Ò h`&i;ki£clF`km&nd&rLk(hac >k@`&kk&Õ nsiλ′(q′, σ)
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a : abab
a ↓ α b ↓ β1b ↓ β2
p1 p2p3p0
q0
ql
q1
q0
q1 q1
a ↓ α2
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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
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Bc ack( d` n@hac@n_a`dcs_cCÔ@c_n_λ′(q′, σ)
kli£cVknao cF`fhk6d rsji£c v Ó>Fn jldnf`Ti£cl `dac n jlki£c[o#` `aOoTnac×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
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Bc ack( d` n@hac@n_a`dcs_cCÔ@c_n_λ′(q′, σ)
kli£cVknao cF`fhk6d rsji£c v Ó>Fn jldnf`Ti£cl `dac n jlki£c[o#` `aOoTnac×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$#"&%' |()+* ),-
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Bc ack(âk_&Õ ñ v `Xsc mhac`d hn ñ ` u ksck(×` Ò m#n u cs_an `o&ÓHhd`OoVn@adnoadnfÓ>_ansdk` ñ
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Bc ack(âk_&Õ ñ v `Xsc mhac`d hn ñ ` u ksck(×` Ò m#n u cs_an `o&ÓHhd`OoVn@adnoadnfÓ>_ansdk` ñ Ó>Fn jldnf`Ti£cX_h#o c i£c[knao cF`[naghacdn[_a` ksckk ñ
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Bc ack(âk_&Õ ñ v `Xsc mhac`d hn ñ ` u ksck(×` Ò m#n u cs_an `o&ÓHhd`OoVn@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$#"&%' |()+* ),-
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a : ababbc
a ↓ α b ↓ β1b ↓ β2
c ↓ γ
xyxyαβ1α2
εyαβ1α2
a ↓ α2 b ↓ β3
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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$#"&%' |()+* ),-
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a : ababbc
a ↓ α b ↓ β1b ↓ β2
c ↓ γ
xyxyαβ1α2
εyαβ1α2
a ↓ α2 b ↓ β3
Bclk(>rgk(hacF`X_a` k kon i;k dac ï _ÓHmc P_an@` _mcs_an h`&Õ nsi;k vVh r u k hn@a`Lk ñ
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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ÎDªDF<9GFDO<;:ùZ98W4¥;RDFY
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$#"&%' |()+* ),-
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ÎDªDF<9GFDO<;:ùZ98W4¥;RDFY
a : ababbc
a ↓ α b ↓ β1b ↓ β2
c ↓ γ
xyxyαβ1α2
εyαβ1α2
a ↓ α2 b ↓ β3
Bclk(>rgk(hacF`X_a` k kon i;k dac ï _ÓHmc P_an@` _mcs_an h`&Õ nsi;k vVh r u k hn@a`Lk ñ cs_anÔ`k. . .
d 4c v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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walkP
I walkP : P × Σ∗ → P ∪ Q
a : ababbc
a ↓ α b ↓ β1 b ↓ β2c ↓ γ
xyzαβ1α2 yzαβ1α2 zαβ1α2xyz
a ↓ α2b ↓ β3
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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walkP
I walkP : P × Σ∗ → P ∪ Q
I walkP(p, ε) = p
a : ababbc
a ↓ α b ↓ β1 b ↓ β2c ↓ γ
xyzαβ1α2 yzαβ1α2 zαβ1α2xyz
a ↓ α2b ↓ β3
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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walkP
I walkP : P × Σ∗ → P ∪ Q
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
xyzαβ1α2 yzαβ1α2 zαβ1α2xyz
a ↓ α2b ↓ β3
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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restP
I restP : P × Σ∗ → Σ∗
a : ababbc
a ↓ α b ↓ β1 b ↓ β2c ↓ γ
xyzαβ1α2 yzαβ1α2 zαβ1α2xyz
a ↓ α2b ↓ β3
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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restP
I restP : P × Σ∗ → Σ∗
I restP(p, ε) = ε
a : ababbc
a ↓ α b ↓ β1 b ↓ β2c ↓ γ
xyzαβ1α2 yzαβ1α2 zαβ1α2xyz
a ↓ α2b ↓ β3
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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restP
I restP : P × Σ∗ → Σ∗
I restP(p, ε) = εI
restP(p, σα′) =
α′ Ò cm#n ∆(p, σ) ∈ Q
restP(∆(p, σ), α′) Ò cm&n ∆(p, σ) ∈ P
¬!k(dacaÔ` v
a : ababbc
a ↓ α b ↓ β1 b ↓ β2c ↓ γ
xyzαβ1α2 yzαβ1α2 zαβ1α2xyz
a ↓ α2b ↓ β3
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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printP
I printP : P × Σ∗ → Γ∗
a : ababbc
a ↓ α b ↓ β1 b ↓ β2c ↓ γ
xyzαβ1α2 yzαβ1α2 zαβ1α2xyz
a ↓ α2b ↓ β3
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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printP
I printP : P × Σ∗ → Γ∗
I printP(p, ε) = ε
a : ababbc
a ↓ α b ↓ β1 b ↓ β2c ↓ γ
xyzαβ1α2 yzαβ1α2 zαβ1α2xyz
a ↓ α2b ↓ β3
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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printP
I printP : P × Σ∗ → Γ∗
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
xyzαβ1α2 yzαβ1α2 zαβ1α2xyz
a ↓ α2b ↓ β3
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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restP
printP
∀p ∈ P Ò ∀α, β ∈ Σ∗ Ò ∀γ ∈ Γ∗ Ò ∀ω ∈ Ξ∗ îï@v!walkP(p, α)⇒
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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
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restP
printP
∀p ∈ P Ò ∀α, β ∈ Σ∗ Ò ∀γ ∈ Γ∗ Ò ∀ω ∈ Ξ∗ îï@v!walkP(p, α)⇒pvrestP(p, α)
`dac_coam&cαðv
< p, αβ, γ, ω >²< walkP(p, α),restP(p, α)β, γprintP(p, α), ω > .
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restP
printP
∀p ∈ P Ò ∀α, β ∈ Σ∗ Ò ∀γ ∈ Γ∗ Ò ∀ω ∈ Ξ∗ îï@v!walkP(p, α)⇒pvrestP(p, α)
`dac_coam&cαðv
< p, αβ, γ, ω >²< walkP(p, α),restP(p, α)β, γprintP(p, α), ω > .
v cm&nwalkP(p, α) ∈ P ⇒ restP(p, α) = ε
v
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restP
printP
∀p ∈ P Ò ∀α, β ∈ Σ∗ Ò ∀γ ∈ Γ∗ Ò ∀ω ∈ Ξ∗ îï@v!walkP(p, α)⇒pvrestP(p, α)
`dac_coam&cαðv
< p, αβ, γ, ω >²< walkP(p, α),restP(p, α)β, γprintP(p, α), ω > .
v cm&nwalkP(p, α) ∈ P ⇒ restP(p, α) = ε
v v
walkP(p, α) ∈ Q Ò ⇒ |restP(p, α)| <|α|v
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restP
printP
∀p ∈ P Ò ∀α, β ∈ Σ∗ Ò ∀γ ∈ Γ∗ Ò ∀ω ∈ Ξ∗ îï@v!walkP(p, α)⇒pvrestP(p, α)
`dac_coam&cαðv
< p, αβ, γ, ω >²< walkP(p, α),restP(p, α)β, γprintP(p, α), ω > .
v cm&nwalkP(p, α) ∈ P ⇒ restP(p, α) = ε
v v
walkP(p, α) ∈ Q Ò ⇒ |restP(p, α)| <|α|v
v¬!walkP(p, α)⇒ @
h`&Õsn iªn_< p, αβ, γ, ω >
iÓ£jlk(dac"!n_|α| − 1
v
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restP
printP
∀p ∈ P Ò ∀α, β ∈ Σ∗ Ò ∀γ ∈ Γ∗ Ò ∀ω ∈ Ξ∗ îï@v!walkP(p, α)⇒pvrestP(p, α)
`dac_coam&cαðv
< p, αβ, γ, ω >²< walkP(p, α),restP(p, α)β, γprintP(p, α), ω > .
v cm&nwalkP(p, α) ∈ P ⇒ restP(p, α) = ε
v v
walkP(p, α) ∈ Q Ò ⇒ |restP(p, α)| <|α|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
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walkQ
walkQ : Q × Σ∗ × Γ∗ → P ∪ Q
I walkQ(q, α, ε) = q
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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walkQ
walkQ : Q × Σ∗ × Γ∗ → P ∪ Q
I walkQ(q, α, ε) = qI
walkQ(q, α, b) =
d(q, b, 0)cm&n
d(q, b, 0) ∈ Q
walkP(d(q, b, 0), α) Ò cm#n d(q, b, 0) ∈ Pk
walkP(d(q, b, 0), α) ∈ P
walkQ(walkP(d(q, b, 0), α), restP(d(q, b, 0), α),
printP(d(q, b, 0), α))cm&nd(q, b, 0) ∈ P
kwalkP(d(q, b, 0), α) ∈ Qw a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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walkQ
§ W4¥# GS÷=(9¦%$4:&
walkQ(q, α, bγ′) =
walkQ(d(q, b, 1), α, γ′)cm#n
d(q, b, 1) ∈ Q
walkP(d(q, b, 1), α) Ò cm&n d(q, b, 1) ∈ Pk
walkP(d(q, b, 1), α) ∈ P
walkQ(walkP(d(q, b, 1), α), restP(d(q, b, 1), α),
γ′ printP(d(q, b, 1), α))cm#nd(q, b, 1) ∈ P
kwalkP(d(q, b, 1), α) ∈ Q
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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I restQ : Q × Σ∗ × Γ∗ → Σ∗
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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I restQ : Q × Σ∗ × Γ∗ → Σ∗
I restQ(q, α, ε) = ε
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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
walkP(d(q, b, 0), α) ∈ P
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
kwalkP(d(q, b, 0), α) ∈ Qw a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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§ W4¥# GS =(9¦%$4:&
restQ(q, α, bγ′) =
restQ(d(q, b, 1), α, γ′)cm&n
d(q, b, 1) ∈ Q
restP(d(q, b, 1), α) Ò cm#n d(q, b, 1) ∈ Pk
walkP(d(q, b, 1), α) ∈ P
restQ(walkP(d(q, b, 1), α), restP(d(q, b, 1), α),
γ′ printP(d(q, b, 1), α))cm#nd(q, b, 1) ∈ P
kwalkP(d(q, b, 1), α) ∈ Q
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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printQ : Q × Σ∗ × Γ∗ → Γ∗
I printQ(q, α, ε) = ε
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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printQ : Q × Σ∗ × Γ∗ → Γ∗
I printQ(q, α, ε) = εI
printQ(q, α, b) =
εcm&n
d(q, b, 0) ∈ Q
printP(d(q, b, 0), α) Ò cm#n d(q, b, 0) ∈ Pk
walkP(d(q, b, 0), α) ∈ P
printQ(walkP(d(q, b, 0), α), restP(d(q, b, 0), α),
printP(d(q, b, 0), α))cm#nd(q, b, 0) ∈ P
kwalkP(d(q, b, 0), α) ∈ Qw a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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§®W4¥# GS =(9¦%$4:&
printQ(q, α, bγ′) =
printQ(d(q, b, 1), α, γ′)cm&n
d(q, b, 1) ∈ Q
γ′ printP(d(q, b, 1), α) Ò cm#n d(q, b, 1) ∈ Pk
walkP(d(q, b, 1), α) ∈ P
printQ(walkP(d(q, b, 1), α), restP(d(q, b, 1), α),
γ′ printP(d(q, b, 1), α))cm&nd(q, b, 1) ∈ P
kwalkP(d(q, b, 1), α) ∈ Q
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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outputQ
outputQ : Q × Σ∗ × Γ∗ → Ξ∗
I outputQ(q, α, ε) = ε
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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outputQ
outputQ : Q × Σ∗ × Γ∗ → Ξ∗
I outputQ(q, α, ε) = εI
outputQ(q, α, b) =
εcm&n
out(q, b, 0) ∈ Q
out(q, b, 0) Ò cm#n d(q, b, 0) ∈ Pk
walkP(d(q, b, 0), α) ∈ P
out(q, b, 0) outputQ(walkP(d(q, b, 0), α),
restP(d(q, b, 0), α), printP(d(q, b, 0), α))cm#nd(q, b, 0) ∈ P
kwalkP(d(q, b, 0), α) ∈ Qw a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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§®W4¥# GS =(9¦%$4:&
outputQ(q, α, bγ′) =
out(q, b, 1) outputQ(d(q, b, 1), α, γ′)cm&n
d(q, b, 1) ∈ Q
γ′ out(q, b, 1) Ò cm#n d(q, b, 1) ∈ Pk
walkP(d(q, b, 1), α) ∈ P
out(q, b, 1) outputQ(walkP(d(q, b, 1), α),
restP(d(q, b, 1), α), γ′ printP(d(q, b, 1), α))cm&n
d(q, b, 1) ∈ Pk
walkP(d(q, b, 1), α) ∈ Q
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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$#"&%' |()+* ),-
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U:9 F§W <9G÷ :\4§E:£7walkQ
restQ
printQ
outputQ
É ¸ ∀q ∈ Q Ò ∀α, β ∈ Σ∗ Ò ∀γ ∈ Γ∗ k ∀ω ∈ Ω∗ï@v!walkQ(q, α, γ) ⇒
< q, αβ, γ, ω >²
< walkQ(q, α, γ), restQ(q, α, γ)β, printQ(q, α, γ), ωω′ >
ω′ = outputQ(q, α, γ).
restQ(q, α, γ) 6= ε ⇒dc(iÓ£ u k(_Co&Ó>Fn j`dCkonsi
pv¬!walkQ(q, α, γ) ⇒ @
< q, αβ, γ, ω >²< p, β′, γ′, ω >,
scfm#na_an|β′| < |β|
kkβ′ = β
kγ′ = ε
vw a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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$#"&%' |()+* ),-
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restQ
printQ
outputQ
*)@Z98W?Lê2L¢ªDF<9GFD,+
Ë « ºH¸ ² ¸2¼JÉ.-fÇ9¼ æR«` bk(dk(hacF` î(α′, γ′) ≺ (α′′, γ′′)⇔|α′| < |α′′|
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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restQ
printQ
outputQ
*)@Z98W?Lê2L¢ªDF<9GFD,+
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$#"&%' |()+* ),-
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restQ
printQ
outputQ
*)@Z98W?Lê2L¢ªDF<9GFD,+
Ë « ºH¸ ² ¸2¼JÉ.-fÇ9¼ æR«` bk(dk(hacF` î(α′, γ′) ≺ (α′′, γ′′)⇔|α′| < |α′′| ∨|α′| = |α′′|&|γ′| < |γ′′|(Σ∗ × Γ∗,≺)
brd i;k(hacdn
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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′′)⇔|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$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
¤lWH=(RD?B<9Gâ?RD7F:&BG÷Z98>D?BGì WJ<>=7L8E¦L FQRG£SL7;:ú<;:T = T T ′
I
Out(λ′) = λ′(q′, a) | q′ ∈ Q ′, a ∈ Ω ∪ φ′(q′) | q′ ∈ Q ′ ∪
λ′(q′, a)φ′(δ′(q′, a)) | q′ ∈ Q ′, a ∈ Ω.
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
¤lWH=(RD?B<9Gâ?RD7F:&BG÷Z98>D?BGì WJ<>=7L8E¦L FQRG£SL7;:ú<;:T = T T ′
I
Out(λ′) = λ′(q′, a) | q′ ∈ Q ′, a ∈ Ω ∪ φ′(q′) | q′ ∈ Q ′ ∪
λ′(q′, a)φ′(δ′(q′, a)) | q′ ∈ Q ′, a ∈ Ω.
I
Rest(λ′) =(x , z) | z =
∏ki=1
printP(pi , ti ), x = rest(pk , tk),mÓi`_ant0 ∈ Out(λ′), pi ∈ P
rgiEno#`_onh#o cs_r &noak`_anti+1 = restP(pi , ti )
sc0 ≤ i < k
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
¤lWH=(RD?B<9Gâ?RD7F:&BG÷Z98>D?BGì WJ<>=7L8E¦L FQRG£SL7;:ú<;:T = T T ′
I
Out(λ′) = λ′(q′, a) | q′ ∈ Q ′, a ∈ Ω ∪ φ′(q′) | q′ ∈ Q ′ ∪
λ′(q′, a)φ′(δ′(q′, a)) | q′ ∈ Q ′, a ∈ Ω.
I
Rest(λ′) =(x , z) | z =
∏ki=1
printP(pi , ti ), x = rest(pk , tk),mÓi`_ant0 ∈ Out(λ′), pi ∈ P
rgiEno#`_onh#o cs_r &noak`_anti+1 = restP(pi , ti )
sc0 ≤ i < k
I
Print(λ′) = printQ(q, x , bz) | q ∈ Q, (x , z) ∈ Out(λ′), b ∈ Γ
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′.
êT=7>WESB<LG£S
I P = Q ′ × P × Print(λ′)mhac((dnLdn j` _on ñ
I Q0 = Q ′ × Q × Rest(λ′)× 0mhac((dnLdn j` _on ñ
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′.
êT=7>WESB<LG£S
I P = Q ′ × P × Print(λ′)mhac((dnLdn j` _on ñ
I Q0 = Q ′ × Q × Rest(λ′)× 0mhac((dnLdn j` _on ñ
I Q1 = Q ′ × Q × Rest(λ′)× 1mhac((dnLdn j` _on ñ
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′.
êT=7>WESB<LG£S
I P = Q ′ × P × Print(λ′)mhac((dnLdn j` _on ñ
I Q0 = Q ′ × Q × Rest(λ′)× 0mhac((dnLdn j` _on ñ
I Q1 = Q ′ × Q × Rest(λ′)× 1mhac((dnLdn j` _on ñ
I Q1 = Q0 ∪ Q1
Bmhac((dnLdnsj2` _on ñ
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′.
êT=7>WESB<LG£S
I P = Q ′ × P × Print(λ′)mhac((dnLdn j` _on ñ
I Q0 = Q ′ × Q × Rest(λ′)× 0mhac((dnLdn j` _on ñ
I Q1 = Q ′ × Q × Rest(λ′)× 1mhac((dnLdn j` _on ñ
I Q1 = Q0 ∪ Q1
Bmhac((dnLdnsj2` _on ñI s =< s ′, s, ε >
dacÔ@c dnfÓ>_ansdk` ñ
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′ I
∆Gλ
I
∆(< q′, p, γ >, a) =
< δ′(q′, a),walkP(p, λ′(q′, a)), ε > Ò cm#n
walkP(p, λ′(q′, a)) ∈ P
< δ′(q′, a),walkP(p, λ(q′, a)),
restP(p, λ′(q′, a)), ε >
k(dacÔa` v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′ I
∆Gλ
I
∆(< q′, p, γ >, a) =
< δ′(q′, a),walkP(p, λ′(q′, a)), ε > Ò cm#n
walkP(p, λ′(q′, a)) ∈ P
< δ′(q′, a),walkP(p, λ(q′, a)),
restP(p, λ′(q′, a)), ε >
k(dacÔa` vI
λ(< q′, p, γ >, a) = γ printP(p, λ′(q′, a)).
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′ I d §(ê8ߦ
Q0I
d(< q′, q, α, γ, 0 >, b, 1) =
< q′, d(q, b, 1), α, γ, 0 >cm&n
d(q, b, 1) ∈ Q
< q′,walkP(q1, α), restP(q1, α),
γ printP(q1, α), 0 >cm&nq1 = d(q, b, 1) ∈ P
kwalkP(q1, α) ∈ Q
< q′,walkP(q1, α), γ printP(q1, α) >cm&nq1 = d(q, b, 1)
kwalkP(q1, α) ∈ P.
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′ I d §(ê8ߦ
Q0I
d(< q′, q, α, γ, 0 >, b, 0) =
< q′,walkQ(q, α, bγ), printQ(q, α, bγ) >cm&nwalkQ(q, α, bγ) ∈ P
< q′,walkQ(q, α, bγ),
restQ(q, α, bγ), printQ(q, α, bγ), 0 >cm&nwalkQ(q, α, bγ) ∈ Q.
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′ I d §(ê8ߦ
Q1I
d(< q′, q, α, γ, 1 >, b, 1) =
< q′, d(q, b, 1), α, γ, 1 >cm&n
d(q, b, 1) ∈ Q
< q′,walkP(q1, α), restP(q1, α),
γ printP(q1, α), 1 >cm&nq1 = d(q, b, 1) ∈ P
kwalkP(q1, α) ∈ Q
< q′, φ(walkP(q1, α)), ε,
γ printP(q1, α), 0 >cm&nq1 = d(q, b, 1)
kwalkP(q1, α) ∈ P.w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′ I d §(ê8ߦ
Q1I
d(< q′, q, γ, α, 1 >, b, 0) =
< q′, φ(walkQ(q, α, bγ)), ε,
printQ(q, α, bγ), 0 >cm&nwalkQ(q, α, bγ) ∈ P
< q′,walkQ(q, α, bγ), restQ(q, α, bγ),
printQ(q, α, bγ), 1 >cm&nwalkQ(q, α, bγ) ∈ Q.
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′ I λ §(ê8ߦ
QI
out(< q′, q, α, γ, j >, b, 1) = out(q, b, 1)
out(< q′, q, α, γ, j >, b, 0) = outputQ(q, α, bγ).
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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$#"&%' |()+* ),-
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UXWJ<>=7L8E¦L FQRGSL7;:ú<;:T = T T ′ I%/[=(PWJ§;GSì\:õ\:9§(ê8 §E:9<>D
ψI
ψ(< q′, q, α, γ, j >) =
outputQ(q, α, γ) ψ(q1)mÓ£iE`_anq1 = walkQ(q, α, γ) ∈ Q, j = 0,
α = εk
printQ(q, α, γ) = ε
¬!k(dacaÔ`
.
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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T = T T ′ I
ÿ É « ¾BÉ ¸ fT= fT fT ′
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W É ¸
I p1, p2 ∈ P ∪ Q0
v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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$#"&%' |()+* ),-
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T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W É ¸
I p1, p2 ∈ P ∪ Q0
vI q′i Ò pi Ò xi
kzi
sci ∈ 1, 2 î
pi = < q′i , pi , xi , zi , 0 >cm&n
pi ∈ Q0
pi = < q′i , pi , zi >k
xi = εm&n
pi ∈ Pv
I ∀β1, β2 ∈ Ω∗ Ò ∀γ1, γ2 ∈ Γ
∗ Ò ω1, ω2 ∈ Ξ∗ î
< p1, β1β2, γ1, ω1 >²< p2, β2, γ2, ω2 >⇒
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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
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$#"&%' |()+* ),-
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T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W É ¸
I p1, p2 ∈ P ∪ Q0
vI q′i Ò pi Ò xi
kzi
sci ∈ 1, 2 î
pi = < q′i , pi , xi , zi , 0 >cm&n
pi ∈ Q0
pi = < q′i , pi , zi >k
xi = εm&n
pi ∈ Pv
I ∀β1, β2 ∈ Ω∗ Ò ∀γ1, γ2 ∈ Γ
∗ Ò ω1, ω2 ∈ Ξ∗ î
< p1, β1β2, γ1, ω1 >²< p2, β2, γ2, ω2 >⇒
∀α :
(δ′)∗(q1, β1) = q2
k< p1, x1(λ
′)∗(q1, β1)α, γ1z1, ω1 >²< p2, x2α, γ2z2, ω2 > .
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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$#"&%' |()+* ),-
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T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W
É ¸I p1 =< q′1, p1, x1, z1, 1 >
kq2 =< q′2, p2, x2, z2, 1 >
scn_Q1
vI
< p1, ε, γ1, ω1 >²< p2, ε, γ2, ω2 >⇒
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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$#"&%' |()+* ),-
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T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W
É ¸I p1 =< q′1, p1, x1, z1, 1 >
kq2 =< q′2, p2, x2, z2, 1 >
scn_Q1
vI
< p1, ε, γ1, ω1 >²< p2, ε, γ2, ω2 >⇒
q′
1=q′
2
< p1, x1, γ1z1, ω1 >²< p2, x2, γ2z2, ω2 > .
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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$#"&%' |()+* ),-
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T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W É ¸
I q′1, q′
2 ∈ Q ′ Ò β1 ∈ Ω∗ Ò p1, p2 ∈ P
vI
(δ′)∗(q′1, β) = q′2k
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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$#"&%' |()+* ),-
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T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W É ¸
I q′1, q′
2 ∈ Q ′ Ò β1 ∈ Ω∗ Ò p1, p2 ∈ P
vI
(δ′)∗(q′1, β) = q′2kI
< p1, (λ′)∗(q′1, β1), ε, ε >²< p2, ε, γ1, ω1 >⇒
I
< p1, β1, ε, ε >²< p2, ε, γ2, ω1 >,scfm#na_anp1 =< q′1, p1, ε >
kp2 =< q′2, p2, z >
scfd m#n@`zv
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W
ï@vfT(α) = ω ⇒
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W
ï@vfT(α) = ω ⇒pv
fT (fT ′(α)) = ω
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W
ï@vfT(α) = ω ⇒pv
fT (fT ′(α)) = ωðv!fT ′(α)&!fT (fT ′(α)) ⇒
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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T = T T ′ IPKWJ :\:£7;D(R=79§W;7>W
ï@vfT(α) = ω ⇒pv
fT (fT ′(α)) = ωðv!fT ′(α)&!fT (fT ′(α)) ⇒ v!f
T(α)
v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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
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òTDßP<9G.$FDE=F : RDFYl: É ¸
∀ T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >,
∃ T =< Σ× Ω∗, (Σ× P), P, Q, i , ∆, d , λ, out, φ, ψ >:
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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òTDßP<9G.$FDE=F : RDFYl: É ¸
∀ T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >,
∃ T =< Σ× Ω∗, (Σ× P), P, Q, i , ∆, d , λ, out, φ, ψ >:
ï@v∀ p ∈ P∀σ ∈ Σ
λ(p, σ) =< p, σ >
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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òTDßP<9G.$FDE=F : RDFYl: É ¸
∀ T =< Σ× Ω∗, Γ∗,P,Q, i ,∆, d , λ, out, φ, ψ >,
∃ T =< Σ× Ω∗, (Σ× P), P, Q, i , ∆, d , λ, out, φ, ψ >:
ï@v∀ p ∈ P∀σ ∈ Σ
λ(p, σ) =< p, σ >
pvfT = f
T.
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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áX?RD(S
vvv n_dnon ^ ó#0 ~21à3 ó4 ñ
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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$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
¤C8W4¥;RDFY[Gï@v hn@a` ï
λ(q, a) = ε⇒
d(., < p, a >, 1) =
d∗(., λ(p, a), 1)?
d∗(., λ(p, a), 0)?
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
¤C8W4¥;RDFY[Gï@v hn@a` ï
λ(q, a) = ε⇒
d(., < p, a >, 1) =
d∗(., λ(p, a), 1)?
d∗(., λ(p, a), 0)?pv 5 `×2`dk`ìPi£c[ck( n@&`&i;dkk(Lona dacλ(p, a)
v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
¤C8W4¥;RDFY[Gï@v hn@a` ï
λ(q, a) = ε⇒
d(., < p, a >, 1) =
d∗(., λ(p, a), 1)?
d∗(., λ(p, a), 0)?pv 5 `×2`dk`ìPi£c[ck( n@&`&i;dkk(Lona dacλ(p, a)
vðv hn@a` p î
(p0, b0)(p1, b1) . . . (pn, bn)
γ0γ1 . . . γn γ′
0γ1 . . . γn
(p1, b1) . . . (pn, bn)
a : b
c : ab a : ε
(p0, b0) : abεb
q1 q2q3
q
p = (p, γ′
0)
p
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
¤C8W4¥;RDFY[G%)@Z98W?Lê9¢DF<9GFD,+
hn@a` ð îp1
q1 q2 q3
p2
γ′
0γ1 . . . γn
γ′′
0 γ1 . . . γn+1
a : ω1 b : ω2
c : ω3
(p1, γ′
0, ω)
bn+1 ↓ γn+1
bn+1 ↓ (p1, bn+1)
(p2, γ′′
0 , ωω1ω2ω3)
γ′
0γ1 . . . γn+1
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L 79GP§;<W÷8>DªDF<9GFDHIRôõ¦L<9 FQRGS÷\:ùZ98>DߣW?BG>7;D5walk
walk : Q × Γ∗ → P ∪ Q
I |γ| ≤ 1 Òwalk(q, γ) = q
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L 79GP§;<W÷8>DªDF<9GFDHIRôõ¦L<9 FQRGS÷\:ùZ98>DߣW?BG>7;D5walk
walk : Q × Γ∗ → P ∪ Q
I |γ| ≤ 1 Òwalk(q, γ) = q
I
cm&nγ = bγ′
kd(q, b, 1) ∈ P Ò _an
walk(q, γ) = d(q, b, 1).
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L 79GP§;<W÷8>DªDF<9GFDHIRôõ¦L<9 FQRGS÷\:ùZ98>DߣW?BG>7;D5walk
walk : Q × Γ∗ → P ∪ Q
I |γ| ≤ 1 Òwalk(q, γ) = q
I
cm&nγ = bγ′
kd(q, b, 1) ∈ P Ò _an
walk(q, γ) = d(q, b, 1).
I
cm&nγ = bγ′
kd(q, b, 1) ∈ Q Ò _an
walk(q, bγ′) = walk(d(q, b, 1), γ′).
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L 79GP§;<W÷8>DªDF<9GFDHIRôõ¦L<9 FQRGS÷\:ùZ98>DߣW?BG>7;D5rest
rest : Q × Γ∗ → Γ∗
I |γ| ≤ 1 Òrest(q, γ) = γ
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L 79GP§;<W÷8>DªDF<9GFDHIRôõ¦L<9 FQRGS÷\:ùZ98>DߣW?BG>7;D5rest
rest : Q × Γ∗ → Γ∗
I |γ| ≤ 1 Òrest(q, γ) = γ
I
cm&nγ = bγ′
kd(q, b, 1) ∈ P Ò _an
rest(q, γ) = γ′.
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L 79GP§;<W÷8>DªDF<9GFDHIRôõ¦L<9 FQRGS÷\:ùZ98>DߣW?BG>7;D5rest
rest : Q × Γ∗ → Γ∗
I |γ| ≤ 1 Òrest(q, γ) = γ
I
cm&nγ = bγ′
kd(q, b, 1) ∈ P Ò _an
rest(q, γ) = γ′.
I
cm&nγ = bγ′
kd(q, b, 1) ∈ Q Ò _anrest(q, bγ′) = rest(d(q, b, 1), γ ′)
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L 79GP§;<W÷8>DªDF<9GFDHIRôõ¦L<9 FQRGS÷\:ùZ98>DߣW?BG>7;D5output
output : Q × Γ∗ → Ω∗
I |γ| ≤ 1 Òoutput(q, γ) = ε
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L 79GP§;<W÷8>DªDF<9GFDHIRôõ¦L<9 FQRGS÷\:ùZ98>DߣW?BG>7;D5output
output : Q × Γ∗ → Ω∗
I |γ| ≤ 1 Òoutput(q, γ) = ε
I
cm&nγ = bγ′
kd(q, b, 1) ∈ P Ò _an
output(q, γ) = out(q, b, 1).
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L 79GP§;<W÷8>DªDF<9GFDHIRôõ¦L<9 FQRGS÷\:ùZ98>DߣW?BG>7;D5output
output : Q × Γ∗ → Ω∗
I |γ| ≤ 1 Òoutput(q, γ) = ε
I
cm&nγ = bγ′
kd(q, b, 1) ∈ P Ò _an
output(q, γ) = out(q, b, 1).
I
cm&nγ = bγ′
kd(q, b, 1) ∈ Q Ò _an
output(q, bγ′) = out(q, b, 1)output(d(q, b, 1), γ ′)
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L FQRGS <;:T
nsiEn@dndac nLn@k(k(_cn_idn ñI
Suf (λ) = γ ∈ Γ∗ | ∃p ∈ P, σ ∈ Σ, β ∈ Γ∗,scfm#nka_an
λ(p, σ) = β γ
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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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$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L FQRGS <;:TI6êT=7>W£SB<9GS
I P = P × Suf (λ)× Pref (out)v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L FQRGS <;:TI6êT=7>W£SB<9GS
I P = P × Suf (λ)× Pref (out)v
I Q = Q × Γ ∪ ε × Pref (out)v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L FQRGS <;:TI6êT=7>W£SB<9GS
I P = P × Suf (λ)× Pref (out)v
I Q = Q × Γ ∪ ε × Pref (out)v
I (i) =< i , ε, ε >v
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
UXWJ<>=7L8E¦L FQRGS <;:T .∆
∆(< p, γ, ω >, σ) =
< ∆(p, σ), γ, ω >cm#n
∆(p, σ) ∈ P
< walk(q, γ), rest(q, γ), ω print(q, γ) >,mÓ£iE`_anq = ∆(p, σ) ∈ Q.
¬!k(dacÔ` v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
ôõ¦L<9 FQRGPG>7;Dins0,1
rep0,1
Grep0,1 ins0,1
I ö rdmk(_c ins0,1 : a, b+ → a, b, 0, 1∗ Ò
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
ôõ¦L<9 FQRGPG>7;Dins0,1
rep0,1
Grep0,1 ins0,1
I ö rdmk(_c ins0,1 : a, b+ → a, b, 0, 1∗ Ò
I
ins0,1
(k∏
i=0
(ami bni )
)=
k∏
i=0
(ciami bni )0,
ci ≡k∑
j=i
mj( mod 2).
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ôõ¦L<9 FQRGPG>7;Dins0,1
rep0,1
Grep0,1 ins0,1
I ö rdmk(_c rep0,1 : a, b, 0, 1∗ → A, a, b∗ Ò
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
ôõ¦L<9 FQRGPG>7;Dins0,1
rep0,1
Grep0,1 ins0,1
I ö rdmk(_c rep0,1 : a, b, 0, 1∗ → A, a, b∗ Ò
I
rep0,1(ci ) =
ε,cm&n
ci ∈ 0, 1
¬!k(dacÔ`
.
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
![Page 187: logic.fmi.uni-sofia.bglogic.fmi.uni-sofia.bg/theses/Gerdjikov-talk1.pdf · ! "# $# "&% ' ( )+*) ,- /. $# "&% ' | % /% + { { % ü 'yý { {@ {| $# "&% ' | % /% + { { % ü 'yý { {@](https://reader034.fdocuments.net/reader034/viewer/2022042803/5f47626f7470a54705624c55/html5/thumbnails/187.jpg)
! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
ôõ¦L<9 FQRGPG>7;Dins0,1
rep0,1
Grep0,1 ins0,1
I ö rdmk(_c rep0,1 : a, b, 0, 1∗ → A, a, b∗ Ò
I
rep0,1(ci ) =
ε,cm&n
ci ∈ 0, 1
¬!k(dacÔ`
.
I
rep0,1(αcβ) =
αA rep0,1(β)cm&n
α ∈ a, b∗, c = 1
α rep0,1(β)cm&n
α ∈ a, b∗, c = 0
¬!k(dacaÔ`
,
αA
a → A
vw a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
![Page 188: logic.fmi.uni-sofia.bglogic.fmi.uni-sofia.bg/theses/Gerdjikov-talk1.pdf · ! "# $# "&% ' ( )+*) ,- /. $# "&% ' | % /% + { { % ü 'yý { {@ {| $# "&% ' | % /% + { { % ü 'yý { {@](https://reader034.fdocuments.net/reader034/viewer/2022042803/5f47626f7470a54705624c55/html5/thumbnails/188.jpg)
! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
ôõ¦L<9 FQRGPG>7;Dins0,1
rep0,1
Grep0,1 ins0,1
rep0,1(ins0,1
(k∏
i=0
(ami bni )
)) =
k∏
i=0
(cmi
i bni ),
mÓ£iE`_an
ci = A ⇔
k∑
j=i+1
mj ≡ 1( mod 2)
ci = a ⇔k∑
j=i+1
mj ≡ 0( mod 2).
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
![Page 189: logic.fmi.uni-sofia.bglogic.fmi.uni-sofia.bg/theses/Gerdjikov-talk1.pdf · ! "# $# "&% ' ( )+*) ,- /. $# "&% ' | % /% + { { % ü 'yý { {@ {| $# "&% ' | % /% + { { % ü 'yý { {@](https://reader034.fdocuments.net/reader034/viewer/2022042803/5f47626f7470a54705624c55/html5/thumbnails/189.jpg)
! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
ôõ¦L<9 FQRGSL7F:rep0,1 ins0,1
ÿ É « ¾BÉ ¸ @E@J_hacdgi;R `h
T îfT = rep0,1 ins0,1.
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
![Page 190: logic.fmi.uni-sofia.bglogic.fmi.uni-sofia.bg/theses/Gerdjikov-talk1.pdf · ! "# $# "&% ' ( )+*) ,- /. $# "&% ' | % /% + { { % ü 'yý { {@ {| $# "&% ' | % /% + { { % ü 'yý { {@](https://reader034.fdocuments.net/reader034/viewer/2022042803/5f47626f7470a54705624c55/html5/thumbnails/190.jpg)
! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
áX?RD(STI¤Xê89§;G÷<;:¥;BAl?RDF<LG£SI
n rsm&cF` Ò Ô` ∃ E@J_hacdi;R `h T1 îfT1
= rep0,1 ins0,1.
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
![Page 191: logic.fmi.uni-sofia.bglogic.fmi.uni-sofia.bg/theses/Gerdjikov-talk1.pdf · ! "# $# "&% ' ( )+*) ,- /. $# "&% ' | % /% + { { % ü 'yý { {@ {| $# "&% ' | % /% + { { % ü 'yý { {@](https://reader034.fdocuments.net/reader034/viewer/2022042803/5f47626f7470a54705624c55/html5/thumbnails/191.jpg)
! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
áX?RD(STI¤Xê89§;G÷<;:¥;BAl?RDF<LG£SI
n rsm&cF` Ò Ô` ∃ E@J_hacdi;R `h T1 îfT1
= rep0,1 ins0,1.
I
⇒ ∃ T =< Σ× Ω∗, (Σ× P),P,Q, s,∆, d , λ, out, φ, ψ >:
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
![Page 192: logic.fmi.uni-sofia.bglogic.fmi.uni-sofia.bg/theses/Gerdjikov-talk1.pdf · ! "# $# "&% ' ( )+*) ,- /. $# "&% ' | % /% + { { % ü 'yý { {@ {| $# "&% ' | % /% + { { % ü 'yý { {@](https://reader034.fdocuments.net/reader034/viewer/2022042803/5f47626f7470a54705624c55/html5/thumbnails/192.jpg)
! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
áX?RD(STI¤Xê89§;G÷<;:¥;BAl?RDF<LG£SI
n rsm&cF` Ò Ô` ∃ E@J_hacdi;R `h T1 îfT1
= rep0,1 ins0,1.
I
⇒ ∃ T =< Σ× Ω∗, (Σ× P),P,Q, s,∆, d , λ, out, φ, ψ >:
I
∀ p ∈ P, ∀ x ∈ a, b
λ(p, x) =< p, x >
fT = rep0,1 ins0,1.
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
![Page 193: logic.fmi.uni-sofia.bglogic.fmi.uni-sofia.bg/theses/Gerdjikov-talk1.pdf · ! "# $# "&% ' ( )+*) ,- /. $# "&% ' | % /% + { { % ü 'yý { {@ {| $# "&% ' | % /% + { { % ü 'yý { {@](https://reader034.fdocuments.net/reader034/viewer/2022042803/5f47626f7470a54705624c55/html5/thumbnails/193.jpg)
! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
áX?RD(STI¤Xê89§;G÷<;:¥;BAl?RDF<LG£S É ¸ `mc α ∈ a+a, b∗
kl = max|out(q, < p, σ >)|, |ψ(q)|
n u co c îI
(s, α, ε, ε) ² (p, β, γ, ω)
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
![Page 194: logic.fmi.uni-sofia.bglogic.fmi.uni-sofia.bg/theses/Gerdjikov-talk1.pdf · ! "# $# "&% ' ( )+*) ,- /. $# "&% ' | % /% + { { % ü 'yý { {@ {| $# "&% ' | % /% + { { % ü 'yý { {@](https://reader034.fdocuments.net/reader034/viewer/2022042803/5f47626f7470a54705624c55/html5/thumbnails/194.jpg)
! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
áX?RD(STI¤Xê89§;G÷<;:¥;BAl?RDF<LG£S É ¸ `mc α ∈ a+a, b∗
kl = max|out(q, < p, σ >)|, |ψ(q)|
n u co c îI
(s, α, ε, ε) ² (p, β, γ, ω)
I ⇒ ω = εv
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
![Page 195: logic.fmi.uni-sofia.bglogic.fmi.uni-sofia.bg/theses/Gerdjikov-talk1.pdf · ! "# $# "&% ' ( )+*) ,- /. $# "&% ' | % /% + { { % ü 'yý { {@ {| $# "&% ' | % /% + { { % ü 'yý { {@](https://reader034.fdocuments.net/reader034/viewer/2022042803/5f47626f7470a54705624c55/html5/thumbnails/195.jpg)
! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
áX?RD(STI¤Xê89§;G÷<;:¥;BAl?RDF<LG£S É ¸ `mc α ∈ a+a, b∗
kl = max|out(q, < p, σ >)|, |ψ(q)|
n u co c îI
(s, α, ε, ε) ² (p, β, γ, ω)
I ⇒ ω = εv
I
(s, α, ε, ε) ² (p0, ε, γ0, ω0)
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
áX?RD(STI¤Xê89§;G÷<;:¥;BAl?RDF<LG£S É ¸ `mc α ∈ a+a, b∗
kl = max|out(q, < p, σ >)|, |ψ(q)|
n u co c îI
(s, α, ε, ε) ² (p, β, γ, ω)
I ⇒ ω = εv
I
(s, α, ε, ε) ² (p0, ε, γ0, ω0)
I ⇒ l(|γ0|+ 1) ≥ |α|v
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
![Page 197: logic.fmi.uni-sofia.bglogic.fmi.uni-sofia.bg/theses/Gerdjikov-talk1.pdf · ! "# $# "&% ' ( )+*) ,- /. $# "&% ' | % /% + { { % ü 'yý { {@ {| $# "&% ' | % /% + { { % ü 'yý { {@](https://reader034.fdocuments.net/reader034/viewer/2022042803/5f47626f7470a54705624c55/html5/thumbnails/197.jpg)
! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
KWJ :\:£7>D(R=79§W2IBáX?RD(S nac u cF` î
I
l = max|out(q, < p, σ >)|, |ψ(q)|
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
![Page 198: logic.fmi.uni-sofia.bglogic.fmi.uni-sofia.bg/theses/Gerdjikov-talk1.pdf · ! "# $# "&% ' ( )+*) ,- /. $# "&% ' | % /% + { { % ü 'yý { {@ {| $# "&% ' | % /% + { { % ü 'yý { {@](https://reader034.fdocuments.net/reader034/viewer/2022042803/5f47626f7470a54705624c55/html5/thumbnails/198.jpg)
! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
KWJ :\:£7>D(R=79§W2IBáX?RD(S nac u cF` î
I
l = max|out(q, < p, σ >)|, |ψ(q)|
I
M = (l + 1)|Q|+ 1.
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
![Page 199: logic.fmi.uni-sofia.bglogic.fmi.uni-sofia.bg/theses/Gerdjikov-talk1.pdf · ! "# $# "&% ' ( )+*) ,- /. $# "&% ' | % /% + { { % ü 'yý { {@ {| $# "&% ' | % /% + { { % ü 'yý { {@](https://reader034.fdocuments.net/reader034/viewer/2022042803/5f47626f7470a54705624c55/html5/thumbnails/199.jpg)
! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
KWJ :\:£7>D(R=79§W2IBáX?RD(S nac u cF` î
I
l = max|out(q, < p, σ >)|, |ψ(q)|
I
M = (l + 1)|Q|+ 1.
I
` bk(dk(hacF`mi
Mi=0
îmM = 1
mi−1 = l(∑M
j=i (mj + 1) + 1) scfom&n
M ≥ i ≥ 1.
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
KWJ :\:£7>D(R=79§W2IBáX?RD(S
nac u cF` îI αj
Mj=1îαj =
(j−1∏
i=0
a2mi b
)a2mj+1
M∏
i=j+1
ba2mi
.
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
KWJ :\:£7>D(R=79§W2IBáX?RD(S cmhac
I γj Ò pj ∈ P î< s, αj , ε, ε >²< pj , ε, γj , ε >
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
![Page 202: logic.fmi.uni-sofia.bglogic.fmi.uni-sofia.bg/theses/Gerdjikov-talk1.pdf · ! "# $# "&% ' ( )+*) ,- /. $# "&% ' | % /% + { { % ü 'yý { {@ {| $# "&% ' | % /% + { { % ü 'yý { {@](https://reader034.fdocuments.net/reader034/viewer/2022042803/5f47626f7470a54705624c55/html5/thumbnails/202.jpg)
! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
KWJ :\:£7>D(R=79§W2IBáX?RD(S cmhac
I γj Ò pj ∈ P î< s, αj , ε, ε >²< pj , ε, γj , ε >
I ∀r γj ,r
γj ,r
dac_coam&c dacγj
|γj ,r | = r2m0
l
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
KWJ :\:£7>D(R=79§W2IBáX?RD(S cmhac
I γj Ò pj ∈ P î< s, αj , ε, ε >²< pj , ε, γj , ε >
I ∀r γj ,r
γj ,r
dac_coam&c dacγj
|γj ,r | = r2m0
l
I ∀r qj ,r ∈ Q Ò ωj ,r ∈ Ω∗ î
< φ(pj), ε, γj , ε >²< qj ,r , ε, γj ,r , ωj ,r >
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
:¥;BA?RDF<9GFD
I
i < j !γj ,r & !γi ,r ⇒
γj ,r
kγi ,r
ÓHoac i£cs_6osÓHhoaka_a`l `
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
:¥;BA?RDF<9GFD
I
i < j !γj ,r & !γi ,r ⇒
γj ,r
kγi ,r
ÓHoac i£cs_6osÓHhoaka_a`l `I
r2m0
l−
M∑
k=i+1
(2mk + 1)
k(Lonac ñ
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
7êZ98WH=FG*
ÓHhn@k
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
ª<WF7F:9QRGS
I cXn_fhackndac >dkfbrdmk(k Ò scfm#nka_anÓHÖB` _o#ro cEJ_hcdgi;R `h
w a(xy+za )+* ),- /. ! |!s$#"&%' |()+* ),-
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! "#$#"&%' ()+* ),- /. $#"&%' | %/% +%ü'yý@|$#"&%' | %/% +%ü'yý@+| $#"&%' |(þ%'+)+* ),- /.
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