r rs rsrs t r - - PowerPoint PPT Presentation

❚r❡❡ ❙❡r✐❡s ❚r❛♥s❞✉❝❡rs ❛♥❞ ❲❡✐❣❤t❡❞ ❚r❡❡ ❆✉t♦♠❛t❛

❆♥❞r❡❛s ▼❛❧❡tt✐✱ ❍❡✐❦♦ ❱♦❣❧❡r ❉❡❝❡♠❜❡r ✷✱ ✷✵✵✸ ✶✳ ❇❛s✐❝ ❉❡✜♥✐t✐♦♥s ✷✳ ❆ ❙tr❛♥❣❡ ❙❡♠✐r✐♥❣ ✸✳ ❊q✉✐✈❛❧❡♥❝❡ ❘❡s✉❧t ✹✳ ❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❖✉t❧♦♦❦

✶ ❉❡❝❡♠❜❡r ✷✱ ✷✵✵✸

• ❡♥❡r❛❧✐③❛t✐♦♥ ❍✐❡r❛r❝❤②

weighted tree automaton L ∈ A TΣ

• tree series

transducer τ : TΣ − → A T∆

• weighted automaton

L ∈ A Σ∗

• weighted transducer

τ : Σ∗ − → A ∆∗

• tree automaton

L ⊆ TΣ tree transducer τ : TΣ − → P(T∆) string automaton L ⊆ Σ∗ generalized sequential machine τ : Σ∗ − → P(∆∗)

❇❛s✐❝ ❉❡✜♥✐t✐♦♥s ✷ ❉❡❝❡♠❜❡r ✷✱ ✷✵✵✸

❇♦tt♦♠✲❯♣ ❚r❡❡ ❙❡r✐❡s ❚r❛♥s❞✉❝❡rs

M = (Q, Σ, ∆, A, F, µ)

• ✐♥♣✉t ❛♥❞ ♦✉t♣✉t r❛♥❦❡❞ ❛❧♣❤❛❜❡t Σ = ∆ = {σ(2), α(0), β(0)}✱
• st❛t❡s ❛♥❞ ✜♥❛❧ st❛t❡s Q = F = {p, q}✱
• s❡♠✐r✐♥❣ A = P = (P(N∗

+), ∪, ◦, ∅, {ε}) ✇✐t❤ P1 ◦ P2 = { ab | a ∈ P1, b ∈ P2 }✱ ❛♥❞

• tr❡❡ r❡♣r❡s❡♥t❛t✐♦♥ µ

α − → q {ε} α β − → q {ε} β σ q x1 q x2 − → q {ε} σ x1 x2 σ r1 x1 r2 x2 − → p {i} xi

❇❛s✐❝ ❉❡✜♥✐t✐♦♥s ✸ ❉❡❝❡♠❜❡r ✷✱ ✷✵✵✸

q {ε} σ σ α β α σ q {ε} σ α β q {ε} α p {1} σ α β σ σ α β α σ σ q {ε} α q {ε} β q {ε} α σ p {1} α q {ε} α p {2} α p {1.1} α σ p {2} β q {ε} α p {1.2} β

3

❇❛s✐❝ ❉❡✜♥✐t✐♦♥s ✹ ❉❡❝❡♠❜❡r ✷✱ ✷✵✵✸

❚r❡❡ ❙❡r✐❡s

• ❛ tr❡❡ s❡r✐❡s ϕ ✐s ❛ ♠❛♣♣✐♥❣ ♦❢ t②♣❡ T∆(V ) −

→ A❀ (ϕ, t) ✐s ✉s❡❞ t♦ ❞❡♥♦t❡ ϕ(t)

• t❤❡ ❝❧❛ss ♦❢ ❛❧❧ tr❡❡ s❡r✐❡s ✐s ❞❡♥♦t❡❞ A

T∆(V )

• t❤❡ s✉♣♣♦rt ♦❢ ❛ tr❡❡ s❡r✐❡s ϕ ✐s ❞❡✜♥❡❞ t♦ ❜❡ supp(ϕ) = { t ∈ T∆(V ) | (ϕ, t) = 0 }
• ϕ ✐s ♣♦❧②♥♦♠✐❛❧ ✐✛ ✐ts s✉♣♣♦rt ✐s ✜♥✐t❡❀ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❧❛ss ✐s AT∆(V )
• ▲❡t ϕ ∈ A

T∆(Xk) ✱ (ψ1, . . . , ψk) ∈ A T∆(V ) k✳ ❙✉❜st✐t✉t✐♦♥ ♦❢ (ψ1, . . . , ψk) ✐♥t♦ ϕ ✐s ϕ ← − (ψ1, . . . , ψk) =

• t∈supp(ϕ)

(∀i∈[k]): ti∈supp(ψi)

• (ϕ, t)⊙(ψ1, t1)⊙· · ·⊙(ψk, tk)
• t[t1, . . . , tk].

❇❛s✐❝ ❉❡✜♥✐t✐♦♥s ✺ ❉❡❝❡♠❜❡r ✷✱ ✷✵✵✸

❇♦tt♦♠✲✉♣ ❚r❡❡ ❙❡r✐❡s ❚r❛♥s❞✉❝❡rs

M = (Q, Σ, ∆, A, F, µ)✱ ✇❤❡r❡

• Q ❛♥❞ F ⊆ Q ❛r❡ ✜♥✐t❡ s❡ts ♦❢ st❛t❡s ❛♥❞ ✜♥❛❧ st❛t❡s✱ r❡s♣✳✱
• Σ ❛♥❞ ∆ ❛r❡ t❤❡ ✐♥♣✉t ❛♥❞ ♦✉t♣✉t r❛♥❦❡❞ ❛❧♣❤❛❜❡ts✱ r❡s♣✳✱
• A = (A, ⊕, ⊙, 0, 1) ✐s ❛ s❡♠✐r✐♥❣
• µ ✐s ❛ ❢❛♠✐❧② ♦❢ ♠❛♣♣✐♥❣s (µk)k∈N ♦❢ t②♣❡

µk : Σ(k) − → A T∆(Xk) Q×Qk.

❇❛s✐❝ ❉❡✜♥✐t✐♦♥s ✻ ❉❡❝❡♠❜❡r ✷✱ ✷✵✵✸

❙❡♠❛♥t✐❝s ♦❢ ❇♦tt♦♠✲✉♣ ❚r❡❡ ❙❡r✐❡s ❚r❛♥s❞✉❝❡rs

µk(σ) :

• A

T∆ Qk − → A T∆ Q µk(σ)(R1, . . . , Rk)q =

• (q1,...,qk)∈Qk

µk(σ)q,(q1,...,qk) ← −

• (R1)q1, . . . , (Rk)qk
• .

■♥✐t✐❛❧ ❤♦♠♦♠♦r♣❤✐s♠✿ hµ : TΣ − → A T∆ Q hµ(σ(s1, . . . , sk)) = µk(σ)(hµ(s1), . . . , hµ(sk)) tr❡❡✲t♦✲tr❡❡✲s❡r✐❡s tr❛♥s❢♦r♠❛t✐♦♥ ❝♦♠♣✉t❡❞ ❜② M ✐s τM : TΣ − → A T∆

• τM(s) =
• q∈F

hµ(s)q

❇❛s✐❝ ❉❡✜♥✐t✐♦♥s ✼ ❉❡❝❡♠❜❡r ✷✱ ✷✵✵✸

❇♦tt♦♠✲✉♣ ❲❡✐❣❤t❡❞ ❚r❡❡ ❆✉t♦♠❛t❛

M = (Q, Σ, A, F, µ)✱ ✇❤❡r❡

• Q ❛♥❞ F ⊆ Q ❛r❡ ✜♥✐t❡ s❡ts ♦❢ st❛t❡s ❛♥❞ ✜♥❛❧ st❛t❡s✱ r❡s♣✳✱
• Σ ✐s t❤❡ ✐♥♣✉t r❛♥❦❡❞ ❛❧♣❤❛❜❡t✱ r❡s♣✳✱
• A = (A, ⊕, ⊙, 0, 1) ✐s ❛ s❡♠✐r✐♥❣
• µ ✐s ❛ ❢❛♠✐❧② ♦❢ ♠❛♣♣✐♥❣s (µk)k∈N ♦❢ t②♣❡ µk : Σ(k) −

→ AQ×Qk✳ ❙❡♠❛♥t✐❝s ✐s s✐♠✐❧❛r❧② ❞❡✜♥❡❞ ❛s ✐t ✐s ❢♦r ❜♦tt♦♠✲✉♣ tr❡❡ s❡r✐❡s tr❛♥s❞✉❝❡rs✳

❇❛s✐❝ ❉❡✜♥✐t✐♦♥s ✽ ❉❡❝❡♠❜❡r ✷✱ ✷✵✵✸

❆ ❙❡♠✐r✐♥❣❄

▲❡t A = (A, ⊕, ⊙, 0, 1) ❜❡ ❛ s❡♠✐r✐♥❣✳ ❲❡ ❞❡✜♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❛❧❣❡❜r❛✐❝ str✉❝t✉r❡✳ B = A T∆ ∗ ◦ ({ε} ∪ { (n, ϕ) | n ∈ N

+, ϕ ∈ A

T∆(Xn) }) S = NB S = (S, ∪, ◦, ∅, {ε}) ✇✐t❤ ❛❞❞✐t✐♦♥ ❜❡✐♥❣ ❞❡✜♥❡❞ ❢♦r ❡✈❡r② ❡❧❡♠❡♥t b ∈ B ❛♥❞ ❡✈❡r② t✇♦ s❡♠✐r✐♥❣ ❡❧❡♠❡♥ts S1, S2 ∈ S ❜② (S1 ∪ S2)(b) = S1(b) + S2(b). ✭✶✮ ❚❤✐s ❛❞❞✐t✐♦♥ ✐s tr✐✈✐❛❧❧② ❛ss♦❝✐❛t✐✈❡✱ ❝♦♠♠✉t❛t✐✈❡✱ ❛♥❞ ❤❛s ✉♥✐t ❡❧❡♠❡♥t ∅ : B − → N ✇❤✐❝❤ ✐s ❞❡✜♥❡❞ ❢♦r ❡✈❡r② b ∈ B t♦ ❜❡ ∅(b) = 0✳ ❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐s ❞❡✜♥❡❞ ❢♦r ❡✈❡r② ❡❧❡♠❡♥t b ∈ B ❛♥❞ ❡✈❡r② t✇♦ s❡♠✐r✐♥❣ ❡❧❡♠❡♥ts S1, S2 ∈ S ❜② (S1 ◦ S2)(b) =

• b1,b2∈B, b=b1←b2

S1(b1) · S2(b2). ✭✷✮

❆ ❙tr❛♥❣❡ ❙❡♠✐r✐♥❣ ✾ ❉❡❝❡♠❜❡r ✷✱ ✷✵✵✸

❲r❛♣♣✐♥❣ ❙✉❜st✐t✉t✐♦♥

❖♥ B ✇❡ ❞❡✜♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣❡r❛t✐♦♥ ← − : B2 − → B✿ a ← − b = a.b ✱ ✐❢ a ∈ A T∆ ∗ ♦r b = ε, a.(1, ϕ) ← − ψ.b = a.(ϕ ← −0 ψ).b, a.(n, ϕ) ← − ψ.b = a.(n − 1, ϕ ← −0 ψ) ← − b ✱ ✐❢ n > 1, a.(n, ϕ) ← − (m, ψ) = a.(n − 1 + m, ϕ ← −m ψ). ❚❤❡ s✉❜st✐t✉t✐♦♥s ( ← −k : A T∆(X) × A T∆(Xk) − → A T∆(X) | k ∈ N ) ❛r❡ ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✳ a ← −k b = a[xi/xi+k−1 | i > 1] ← − (b) ▲❡♠♠❛✿ (a ← − b) ← − c = a ← − (b ← − c)✳ ▲❡♠♠❛✿ (ϕ ← −m ψ) ← −k τ = ϕ ← −m−1+k (ψ ← −k τ) ✇✐t❤ m = 0✳

❆ ❙tr❛♥❣❡ ❙❡♠✐r✐♥❣ ✶✵ ❉❡❝❡♠❜❡r ✷✱ ✷✵✵✸

❘❡♠❛✐♥✐♥❣ ◗✉❡st✐♦♥s ❛♥❞ ▲✐t❡r❛t✉r❡

• ❉❡t❡r♠✐♥✐st✐❝ tr❡❡ s❡r✐❡s tr❛♥s❞✉❝❡rs❄
• ❚r❡❡ tr❛♥s❞✉❝❡rs✱ ✐✳❡✳✱ ♣♦❧②♥♦♠✐❛❧ tr❡❡ s❡r✐❡s tr❛♥s❞✉❝❡rs ♦✈❡r B
• ❚♦♣✲❞♦✇♥ tr❡❡ s❡r✐❡s tr❛♥s❞✉❝❡rs❄
• o✲s✉❜st✐t✉t✐♦♥❄

❙♦♠❡ ❘❡❢❡r❡♥❝❡s✿ ❙❡✐❞❧✿ ❋✐♥✐t❡ ❚r❡❡ ❆✉t♦♠❛t❛ ✇✐t❤ ❈♦st ❋✉♥❝t✐♦♥s✱ ✶✾✾✹ ❑✉✐❝❤✿ ❋♦r♠❛❧ P♦✇❡r ❙❡r✐❡s ♦✈❡r ❚r❡❡s✱ ✶✾✾✼ ❊♥❣❡❧❢r✐❡t✱ ❋ü❧ö♣✱ ❱♦❣❧❡r✿ ❇♦tt♦♠✲✉♣ ❛♥❞ ❚♦♣✲❞♦✇♥ ❚r❡❡ ❙❡r✐❡s ❚r❛♥s❢♦r♠❛t✐♦♥s✱ ✷✵✵✷ ❋ü❧ö♣✱ ❱♦❣❧❡r✿ ❚r❡❡ ❙❡r✐❡s ❚r❛♥s❢♦r♠❛t✐♦♥s t❤❛t ❘❡s♣❡❝t ❈♦♣②✐♥❣✱ ✷✵✵✸ ❋ü❧ö♣✱ ❱♦❣❧❡r✿ ❲❡✐❣❤t❡❞ ❚r❡❡ ❚r❛♥s❞✉❝❡rs✱ ✷✵✵✸

❈♦♥❝❧✉s✐♦♥s ❛♥❞ ❖✉t❧♦♦❦ ✶✶ ❉❡❝❡♠❜❡r ✷✱ ✷✵✵✸