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❆✉t♦♠❛t❛ ♠✐♥✐♠✐③❛t✐♦♥ ❛ ❧✐❣❤t✇❡✐❣❤t ❝❛t❡❣♦r✐❝❛❧ ❛♣♣r♦❛❝❤ ❚❤♦♠❛s ❈♦❧❝♦♠❜❡t ❛♥❞ ❉❛♥✐❡❧❛ P❡tr✐➩❛♥ ❈◆❘❙ ✫ ■❘■❋✱ P❛r✐s ✼ ❖P❈❚ ✷✵✶✼✱ ❱✐❡♥♥❛✱ ✷✾ ❏✉❧② ✷✵✶✼

❖✈❡r✈✐❡✇

❼ ▼♦t✐✈❛t✐♦♥✿ ❤②❜r✐❞ s❡t✲✈❡❝t♦r ❛✉t♦♠❛t❛ ❼ ❇②♣r♦❞✉❝t✿ ❛ ❧✐❣❤t✇❡✐❣❤t ❝❛t❡❣♦r②✲t❤❡♦r❡t✐❝ ❛♣♣r♦❛❝❤ ❼ ❆✉t♦♠❛t❛ ❛r❡ ❢✉♥❝t♦rs✦ ▼✐♥✐♠✐③❛t✐♦♥ ✐♥ t❤✐s s❡tt✐♥❣✳ ❼ ❊①❛♠♣❧❡s ❼ ❖♣❡♥ ♣r♦❜❧❡♠s✦ ✷ ✴ ✸✶

▼♦t✐✈❛t✐♦♥

❆✉t♦♠❛t❛ ❢♦r ✇❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s ❖♥❝❡ ✉♣♦♥ ❛ t✐♠❡ ✇❡✐❣❤t❡❞ ❛✉t♦♠❛t❛ ✇❡r❡ ✐♥tr♦❞✉❝❡❞ ❜② ❬▼✳✲P✳ ❙❝❤üt③❡♥❜❡r❣❡r✱ ✶✾✻✶❪ ❖♥ t❤❡ ❞❡✜♥✐t✐♦♥ ♦❢ ❛ ❢❛♠✐❧② ♦❢ ❛✉t♦♠❛t❛ ❆ ♠✐♥✐♠✐③❛t✐♦♥ ❛❧❣♦r✐t❤♠ ✐s ❛❧s♦ ♣r♦✈✐❞❡❞✳ ✸ ✴ ✸✶

❱❡❝t♦r ❛✉t♦♠❛t❛ ❆♥ ✈❡❝t♦r ❛✉t♦♠❛t♦♥ ✐s ❛ t✉♣❧❡ A = ⟨ Q , q ✵ , f , ( δ a ) a ∈ A ⟩ ❼ Q ✐s ❛♥ R ✲✈❡❝t♦r s♣❛❝❡ ❼ q ✵ ✐s ❛♥ ✐♥✐t✐❛❧ ✈❡❝t♦r ✐♥ Q ❼ f ∶ Q → R ❛ss♦❝✐❛t❡s t♦ ❡❛❝❤ st❛t❡ ❛♥ ♦✉t♣✉t ✈❛❧✉❡ ❼ ❢♦r ❡❛❝❤ a ∈ A ✱ δ a ∶ Q → Q ✐s ❛ ❧✐♥❡❛r ♠❛♣ ❚❤❡ ❧❛♥❣✉❛❣❡ ❛❝❝❡♣t❡❞ ❜② A ✐s ❛ ♠❛♣ L A ∶ A ∗ → R ❞❡✜♥❡❞ ❜② w ∈ A ∗ ↦ f ( δ w ( q ✵ )) ✹ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡ ❈♦♥s✐❞❡r t❤❡ ❛❧♣❤❛❜❡t A = { a , b , c } ❛♥❞ t❤❡ ❧❛♥❣✉❛❣❡ L ∶ A ∗ → R ⎧ ⎪ ⎪ ✷ ∣ u ∣ a ✐❢ ∣ u ∣ b ✐s ❡✈❡♥ ❛♥❞ ∣ u ∣ c = ✵✱ ⎪ L ( u ) = ⎨ ⎪ ⎪ ⎪ ✵ ♦t❤❡r✇✐s❡ ⎩ ❆♥ ❛✉t♦♠❛t♦♥ ❛❝❝❡♣t✐♥❣ t❤✐s ❧❛♥❣✉❛❣❡ ✐s ⟨ R ✷ , ( ✶ , ✵ ) , f , ( δ a ) a ∈ A ⟩ ✺ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡ ⎧ ⎪ ✷ ∣ u ∣ a ⎪ ✐❢ ∣ u ∣ b ❡✈❡♥ ✫ ∣ u ∣ c = ✵✱ ⎪ L ( u ) = ⎨ ⎪ ⎪ ✵ ♦t❤❡r✇✐s❡ ⎪ ⎩ ⟨ R ✷ , ( ✶ , ✵ ) , f , ( δ a ) a ∈ A ⟩ δ a ( x , y ) = ( ✷ x , ✷ y ) δ b ( x , y ) = ( y , x ) δ c ( x , y ) = ( ✵ , ✵ ) f ( x , y ) = x ✻ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡ ⎧ ⎪ ✷ ∣ u ∣ a ⎪ ✐❢ ∣ u ∣ b ❡✈❡♥ ✫ ∣ u ∣ c = ✵✱ ⎪ L ( u ) = ⎨ ⎪ ⎪ ✵ ♦t❤❡r✇✐s❡ ⎪ ⎩ ⟨ R ✷ , ( ✶ , ✵ ) , f , ( δ a ) a ∈ A ⟩ δ a ( x , y ) = ( ✷ x , ✷ y ) δ b ( x , y ) = ( y , x ) δ c ( x , y ) = ( ✵ , ✵ ) f ( x , y ) = x a ✻ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡ ⎧ ⎪ ✷ ∣ u ∣ a ⎪ ✐❢ ∣ u ∣ b ❡✈❡♥ ✫ ∣ u ∣ c = ✵✱ ⎪ L ( u ) = ⎨ ⎪ ⎪ ✵ ♦t❤❡r✇✐s❡ ⎪ ⎩ ⟨ R ✷ , ( ✶ , ✵ ) , f , ( δ a ) a ∈ A ⟩ δ a ( x , y ) = ( ✷ x , ✷ y ) δ b ( x , y ) = ( y , x ) δ c ( x , y ) = ( ✵ , ✵ ) f ( x , y ) = x ab ✻ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡ ⎧ ⎪ ✷ ∣ u ∣ a ⎪ ✐❢ ∣ u ∣ b ❡✈❡♥ ✫ ∣ u ∣ c = ✵✱ ⎪ L ( u ) = ⎨ ⎪ ⎪ ✵ ♦t❤❡r✇✐s❡ ⎪ ⎩ ⟨ R ✷ , ( ✶ , ✵ ) , f , ( δ a ) a ∈ A ⟩ δ a ( x , y ) = ( ✷ x , ✷ y ) δ b ( x , y ) = ( y , x ) δ c ( x , y ) = ( ✵ , ✵ ) f ( x , y ) = x abb ✻ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡ ⎧ ⎪ ✷ ∣ u ∣ a ⎪ ✐❢ ∣ u ∣ b ❡✈❡♥ ✫ ∣ u ∣ c = ✵✱ ⎪ L ( u ) = ⎨ ⎪ ⎪ ✵ ♦t❤❡r✇✐s❡ ⎪ ⎩ ⟨ R ✷ , ( ✶ , ✵ ) , f , ( δ a ) a ∈ A ⟩ δ a ( x , y ) = ( ✷ x , ✷ y ) δ b ( x , y ) = ( y , x ) δ c ( x , y ) = ( ✵ , ✵ ) f ( x , y ) = x abba ✻ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡ ⎧ ⎪ ✷ ∣ u ∣ a ⎪ ✐❢ ∣ u ∣ b ❡✈❡♥ ✫ ∣ u ∣ c = ✵✱ ⎪ L ( u ) = ⎨ ⎪ ⎪ ✵ ♦t❤❡r✇✐s❡ ⎪ ⎩ ⟨ R ✷ , ( ✶ , ✵ ) , f , ( δ a ) a ∈ A ⟩ δ a ( x , y ) = ( ✷ x , ✷ y ) δ b ( x , y ) = ( y , x ) δ c ( x , y ) = ( ✵ , ✵ ) f ( x , y ) = x abbaa ✻ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡ ⎧ ⎪ ✷ ∣ u ∣ a ⎪ ✐❢ ∣ u ∣ b ❡✈❡♥ ✫ ∣ u ∣ c = ✵✱ ⎪ L ( u ) = ⎨ ⎪ ⎪ ✵ ♦t❤❡r✇✐s❡ ⎪ ⎩ ⟨ R ✷ , ( ✶ , ✵ ) , f , ( δ a ) a ∈ A ⟩ δ a ( x , y ) = ( ✷ x , ✷ y ) δ b ( x , y ) = ( y , x ) δ c ( x , y ) = ( ✵ , ✵ ) f ( x , y ) = x abbaa ↦ ✽ ✻ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡ ⎧ ⎪ ✷ ∣ u ∣ a ⎪ ✐❢ ∣ u ∣ b ❡✈❡♥ ✫ ∣ u ∣ c = ✵✱ ⎪ L ( u ) = ⎨ ⎪ ⎪ ✵ ♦t❤❡r✇✐s❡ ⎪ ⎩ ⟨ R ✷ , ( ✶ , ✵ ) , f , ( δ a ) a ∈ A ⟩ δ a ( x , y ) = ( ✷ x , ✷ y ) δ b ( x , y ) = ( y , x ) δ c ( x , y ) = ( ✵ , ✵ ) f ( x , y ) = x abbaac ↦ ✵ ✻ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡ ❚❤❡ ✏r❡❛❝❤❛❜❧❡✑ ✈❡❝t♦rs ❛r❡ ♦♥ t❤❡ ✏✉♥✐♦♥✑ ♦❢ t✇♦ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✳ ♠❛✐♥t❛✐♥✐♥❣ ♦♥❡ ❜✐t ❛♥❞ ♦♥❡ r❡❛❧ ✐s ❜❡tt❡r t❤❛♥ t✇♦ r❡❛❧s ✼ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡ ❚❤❡ ✏r❡❛❝❤❛❜❧❡✑ ✈❡❝t♦rs ❛r❡ ♦♥ t❤❡ ✏✉♥✐♦♥✑ ♦❢ t✇♦ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✳ ♠❛✐♥t❛✐♥✐♥❣ ♦♥❡ ❜✐t ❛♥❞ ♦♥❡ r❡❛❧ ✐s ❜❡tt❡r t❤❛♥ t✇♦ r❡❛❧s ✼ ✴ ✸✶

❚❤❡ st❛rt✐♥❣ ♣♦✐♥t ❍②❜r✐❞ s❡t✲✈❡❝t♦r ❛✉t♦♠❛t❛ ✏❤❛✈❡✑ ❼ ❛ ✜♥✐t❡ s❡t ♦❢ ❝♦♥tr♦❧ st❛t❡s t❤❛t ❡✈♦❧✈❡ ❧✐❦❡ ❉❋❆s ❼ ❛ ✜♥✐t❡ ✈❡❝t♦r s♣❛❝❡ ❢♦r ❡❛❝❤ ❝♦♥tr♦❧ st❛t❡ ◗✉❡st✐♦♥✳ ❲❤❛t ✐s ❛ s✉✐t❛❜❧❡ ❛✉t♦♠❛t❛ ♠♦❞❡❧ s♦ t❤❛t ♠✐♥✐♠✐s❛t✐♦♥ ✐s ♣♦ss✐❜❧❡ ❛♥❞ ✇❡ r❡tr✐❡✈❡ t❤✐s ✏❤②❜r✐❞✑ ❜❡❤❛✈✐♦✉r❄ ✽ ✴ ✸✶

❆✉t♦♠❛t❛ ❛s ❢✉♥❝t♦rs

▼✐♥✐♠✐③❛t✐♦♥ ❝❛♥ ❜❡ ❡①♣❧❛✐♥❡❞ ✈✐❛ t❤❡ ❞✉❛❧✐t② ❜❡t✇❡❡♥ t❤❡ ❛❧❣❡❜r❛✐❝✲❝♦❛❧❣❡❜r❛✐❝ ✈✐❡✇ ✭❡✳❣✳ ❇r③♦③♦✇s❦✐✬s ❛❧❣♦r✐t❤♠✮ ❚❤❡ ❝♦❛❧❣❡❜r❛✐❝ ✈✐❡✇ ❜r✐♥❣s ✐ts ♦✇♥ ❛❞✈❛♥t❛❣❡s✿ ✭❡✳❣✳ ❝❤❡❝❦✐♥❣ ◆❋❆ ❡q✉✐✈❛❧❡♥❝❡s ✉s✐♥❣ ✉♣✲t♦ t❡❝❤♥✐q✉❡s ❢♦r ❜✐s✐♠✉❧❛t✐♦♥s✮ ❆✉t♦♠❛t❛ ✐♥ ❈❛t❡❣♦r✐❡s✿ ✇❤❛t ✇❡ ❛❧r❡❛❞② ❦♥❡✇ ❆✉t♦♠❛t❛ ❛r❡ ❜♦t❤ ❛❧❣❡❜r❛s ❢♦r ❛ ❢✉♥❝t♦r + ✜♥❛❧ ♠❛♣ ❛♥❞ ❝♦❛❧❣❡❜r❛s ❢♦r ❛ ❢✉♥❝t♦r + ✐♥✐t✐❛❧ ♠❛♣ ✾ ✴ ✸✶

❚❤❡ ❝♦❛❧❣❡❜r❛✐❝ ✈✐❡✇ ❜r✐♥❣s ✐ts ♦✇♥ ❛❞✈❛♥t❛❣❡s✿ ✭❡✳❣✳ ❝❤❡❝❦✐♥❣ ◆❋❆ ❡q✉✐✈❛❧❡♥❝❡s ✉s✐♥❣ ✉♣✲t♦ t❡❝❤♥✐q✉❡s ❢♦r ❜✐s✐♠✉❧❛t✐♦♥s✮ ❆✉t♦♠❛t❛ ✐♥ ❈❛t❡❣♦r✐❡s✿ ✇❤❛t ✇❡ ❛❧r❡❛❞② ❦♥❡✇ ❆✉t♦♠❛t❛ ❛r❡ ❜♦t❤ ❛❧❣❡❜r❛s ❢♦r ❛ ❢✉♥❝t♦r + ✜♥❛❧ ♠❛♣ ❛♥❞ ❝♦❛❧❣❡❜r❛s ❢♦r ❛ ❢✉♥❝t♦r + ✐♥✐t✐❛❧ ♠❛♣ ▼✐♥✐♠✐③❛t✐♦♥ ❝❛♥ ❜❡ ❡①♣❧❛✐♥❡❞ ✈✐❛ t❤❡ ❞✉❛❧✐t② ❜❡t✇❡❡♥ t❤❡ ❛❧❣❡❜r❛✐❝✲❝♦❛❧❣❡❜r❛✐❝ ✈✐❡✇ ✭❡✳❣✳ ❇r③♦③♦✇s❦✐✬s ❛❧❣♦r✐t❤♠✮ ✾ ✴ ✸✶