❚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 tree series automaton transducer L ∈ A � � T Σ � � τ : T Σ − → A � � T ∆ � � weighted automaton weighted transducer tree transducer tree automaton τ : Σ ∗ − � Σ ∗ � � ∆ ∗ � L ⊆ T Σ τ : T Σ − → P ( T ∆ ) L ∈ A � � → A � � generalized string automaton sequential machine L ⊆ Σ ∗ 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❤ P 1 ◦ P 2 = { ab | a ∈ P 1 , b ∈ P 2 } ✱ ❛♥❞ • tr❡❡ r❡♣r❡s❡♥t❛t✐♦♥ µ q q − → − → α β { ε } α { ε } β q σ σ p q q r 1 r 2 − → − → { ε } σ { i } x i x 1 x 2 x 1 x 2 x 1 x 2 ❇❛s✐❝ ❉❡✜♥✐t✐♦♥s ✸ ❉❡❝❡♠❜❡r ✷✱ ✷✵✵✸
q σ σ α { ε } α β σ p q q σ σ { 1 } { ε } { ε } α α β α β σ p σ σ q σ { 2 } α p q σ α 3 q q { ε } α p α β { 1 } α { ε } α { ε } α { ε } β { 1 . 1 } α σ p p q { 1 . 2 } β { ε } α { 2 } β ❇❛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 A � T ∆ ( V ) � � k ✳ ❙✉❜st✐t✉t✐♦♥ ♦❢ ( ψ 1 , . . . , ψ k ) • ▲❡t ϕ ∈ A � � T ∆ ( X k ) � � ✱ ( ψ 1 , . . . , ψ k ) ∈ A � � T ∆ ( V ) � ✐♥t♦ ϕ ✐s � � � ϕ ← − ( ψ 1 , . . . , ψ k ) = ( ϕ, t ) ⊙ ( ψ 1 , t 1 ) ⊙· · ·⊙ ( ψ k , t k ) t [ t 1 , . . . , t k ] . t ∈ supp( ϕ ) ( ∀ i ∈ [ k ]): t i ∈ supp( ψ i ) ❇❛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 ( µ k ) k ∈ N ♦❢ t②♣❡ µ k : Σ ( k ) − � Q × Q k . → A � � T ∆ ( X k ) � ❇❛s✐❝ ❉❡✜♥✐t✐♦♥s ✻ ❉❡❝❡♠❜❡r ✷✱ ✷✵✵✸
❙❡♠❛♥t✐❝s ♦❢ ❇♦tt♦♠✲✉♣ ❚r❡❡ ❙❡r✐❡s ❚r❛♥s❞✉❝❡rs � Q � k − � Q � µ k ( σ ) : A � � T ∆ � → A � � T ∆ � � � � µ k ( σ )( R 1 , . . . , R k ) q = µ k ( σ ) q, ( q 1 ,...,q k ) ← − ( R 1 ) q 1 , . . . , ( R k ) q k . ( q 1 ,...,q k ) ∈ Q k � Q ■♥✐t✐❛❧ ❤♦♠♦♠♦r♣❤✐s♠✿ h µ : T Σ − → A � � T ∆ � h µ ( σ ( s 1 , . . . , s k )) = µ k ( σ )( h µ ( s 1 ) , . . . , h µ ( s k )) tr❡❡✲t♦✲tr❡❡✲s❡r✐❡s tr❛♥s❢♦r♠❛t✐♦♥ ❝♦♠♣✉t❡❞ ❜② M ✐s τ M : T Σ − → A � � T ∆ � � � τ M ( s ) = h µ ( s ) q q ∈ F ❇❛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 ( µ k ) k ∈ N ♦❢ t②♣❡ µ k : Σ ( k ) − → A Q × Q k ✳ ❙❡♠❛♥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❡✳ � ∗ ◦ ( { ε } ∪ { ( n, ϕ ) | n ∈ N S = N B B = A � � T ∆ � + , ϕ ∈ A � � T ∆ ( X n ) � � } ) S = ( S, ∪ , ◦ , ∅ , { ε } ) ✇✐t❤ ❛❞❞✐t✐♦♥ ❜❡✐♥❣ ❞❡✜♥❡❞ ❢♦r ❡✈❡r② ❡❧❡♠❡♥t b ∈ B ❛♥❞ ❡✈❡r② t✇♦ s❡♠✐r✐♥❣ ❡❧❡♠❡♥ts S 1 , S 2 ∈ S ❜② ( S 1 ∪ S 2 )( b ) = S 1 ( b ) + S 2 ( 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 S 1 , S 2 ∈ S ❜② � ( S 1 ◦ S 2 )( b ) = S 1 ( b 1 ) · S 2 ( b 2 ) . ✭✷✮ b 1 ,b 2 ∈ B, b = b 1 ← b 2 ❆ ❙tr❛♥❣❡ ❙❡♠✐r✐♥❣ ✾ ❉❡❝❡♠❜❡r ✷✱ ✷✵✵✸
❲r❛♣♣✐♥❣ ❙✉❜st✐t✉t✐♦♥ − : B 2 − ❖♥ B ✇❡ ❞❡✜♥❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♦♣❡r❛t✐♦♥ ← → B ✿ � ∗ ♦r b = ε, a ← − b ✱ ✐❢ a ∈ A � � T ∆ � = a.b 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 ∆ ( X k ) � � − → A � � T ∆ ( X ) � � | k ∈ N ) ❛r❡ ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s✳ a ← − k b = a [ x i /x i + 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 ✷✱ ✷✵✵✸
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