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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 ,(δa)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✱ δa∶Q → Q ✐s ❛ ❧✐♥❡❛r ♠❛♣ ❚❤❡ ❧❛♥❣✉❛❣❡ ❛❝❝❡♣t❡❞ ❜② A ✐s ❛ ♠❛♣ LA∶A∗ → R ❞❡✜♥❡❞ ❜② w ∈ A∗ ↦ f (δw(q✵))

✹ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡

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

✺ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡

L(u) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ✷∣u∣a ✐❢ ∣u∣b ❡✈❡♥ ✫ ∣u∣c = ✵✱ ✵ ♦t❤❡r✇✐s❡ ⟨R✷,(✶,✵),f ,(δa)a∈A⟩ δa(x,y) = (✷x,✷y) δb(x,y) = (y,x) δc(x,y) = (✵,✵) f (x,y) = x

✻ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡

L(u) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ✷∣u∣a ✐❢ ∣u∣b ❡✈❡♥ ✫ ∣u∣c = ✵✱ ✵ ♦t❤❡r✇✐s❡ ⟨R✷,(✶,✵),f ,(δa)a∈A⟩ δa(x,y) = (✷x,✷y) δb(x,y) = (y,x) δc(x,y) = (✵,✵) f (x,y) = x a

✻ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡

L(u) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ✷∣u∣a ✐❢ ∣u∣b ❡✈❡♥ ✫ ∣u∣c = ✵✱ ✵ ♦t❤❡r✇✐s❡ ⟨R✷,(✶,✵),f ,(δa)a∈A⟩ δa(x,y) = (✷x,✷y) δb(x,y) = (y,x) δc(x,y) = (✵,✵) f (x,y) = x ab

✻ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡

L(u) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ✷∣u∣a ✐❢ ∣u∣b ❡✈❡♥ ✫ ∣u∣c = ✵✱ ✵ ♦t❤❡r✇✐s❡ ⟨R✷,(✶,✵),f ,(δa)a∈A⟩ δa(x,y) = (✷x,✷y) δb(x,y) = (y,x) δc(x,y) = (✵,✵) f (x,y) = x abb

✻ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡

L(u) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ✷∣u∣a ✐❢ ∣u∣b ❡✈❡♥ ✫ ∣u∣c = ✵✱ ✵ ♦t❤❡r✇✐s❡ ⟨R✷,(✶,✵),f ,(δa)a∈A⟩ δa(x,y) = (✷x,✷y) δb(x,y) = (y,x) δc(x,y) = (✵,✵) f (x,y) = x abba

✻ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡

L(u) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ✷∣u∣a ✐❢ ∣u∣b ❡✈❡♥ ✫ ∣u∣c = ✵✱ ✵ ♦t❤❡r✇✐s❡ ⟨R✷,(✶,✵),f ,(δa)a∈A⟩ δa(x,y) = (✷x,✷y) δb(x,y) = (y,x) δc(x,y) = (✵,✵) f (x,y) = x abbaa

✻ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡

L(u) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ✷∣u∣a ✐❢ ∣u∣b ❡✈❡♥ ✫ ∣u∣c = ✵✱ ✵ ♦t❤❡r✇✐s❡ ⟨R✷,(✶,✵),f ,(δa)a∈A⟩ δa(x,y) = (✷x,✷y) δb(x,y) = (y,x) δc(x,y) = (✵,✵) f (x,y) = x abbaa ↦ ✽

✻ ✴ ✸✶

❲❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡s✿ ❛♥ ❡①❛♠♣❧❡

L(u) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ✷∣u∣a ✐❢ ∣u∣b ❡✈❡♥ ✫ ∣u∣c = ✵✱ ✵ ♦t❤❡r✇✐s❡ ⟨R✷,(✶,✵),f ,(δa)a∈A⟩ δa(x,y) = (✷x,✷y) δb(x,y) = (y,x) δc(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

❆✉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❤♠✮ ❚❤❡ ❝♦❛❧❣❡❜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❤♠✮ ❚❤❡ ❝♦❛❧❣❡❜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❤♠✮ ❚❤❡ ❝♦❛❧❣❡❜r❛✐❝ ✈✐❡✇ ❜r✐♥❣s ✐ts ♦✇♥ ❛❞✈❛♥t❛❣❡s✿ ✭❡✳❣✳ ❝❤❡❝❦✐♥❣ ◆❋❆ ❡q✉✐✈❛❧❡♥❝❡s ✉s✐♥❣ ✉♣✲t♦ t❡❝❤♥✐q✉❡s ❢♦r ❜✐s✐♠✉❧❛t✐♦♥s✮

✾ ✴ ✸✶

❚❤♦♠❛s ❈♦❧❝♦♠❜❡t ✏❆❧❣è❜r❡s❄ ❈♦✲❛❧❣è❜r❡s❄ ▼❛✐s ✐❧s ♥❡ s♦♥t ♥✐ ❧✬✉♥ ♥✐ ❧✬❛✉tr❡ ✦✑ ❆♥ ❛✉t♦♠❛t♦♥ ♣r♦❝❡ss❡s ❛♥ ✐♥♣✉t✱ r❡s♣❡❝t✐♥❣ ✐ts str✉❝t✉r❡ ✭✇♦r❞✱ tr❡❡✱ ✐♥✜♥✐t❡ ✇♦r❞ ♦r tr❡❡✱ tr❛❝❡✱ ✳ ✳ ✳ ✮ ♦✉t♣✉ts ❛ q✉❛♥t✐t② ✐♥ s♦♠❡ ✉♥✐✈❡rs❡ ♦❢ ♦✉t♣✉t ✈❛❧✉❡s ✭❇♦♦❧❡❛♥ ✈❛❧✉❡s✱ ♣r♦❜❛❜✐❧✐t✐❡s✱ ✈❡❝t♦r s♣❛❝❡✱ ✇♦r❞s✱ ✳ ✳ ✳ ✮ ❆✉t♦♠❛t❛ ❛r❡ ❢✉♥❝t♦rs✦✦✦

✶✵ ✴ ✸✶

❚❤♦♠❛s ❈♦❧❝♦♠❜❡t ✏❆❧❣è❜r❡s❄ ❈♦✲❛❧❣è❜r❡s❄ ▼❛✐s ✐❧s ♥❡ s♦♥t ♥✐ ❧✬✉♥ ♥✐ ❧✬❛✉tr❡ ✦✑ ❆♥ ❛✉t♦♠❛t♦♥ ♣r♦❝❡ss❡s ❛♥ ✐♥♣✉t✱ r❡s♣❡❝t✐♥❣ ✐ts str✉❝t✉r❡ ✭✇♦r❞✱ tr❡❡✱ ✐♥✜♥✐t❡ ✇♦r❞ ♦r tr❡❡✱ tr❛❝❡✱ ✳ ✳ ✳ ✮ ♦✉t♣✉ts ❛ q✉❛♥t✐t② ✐♥ s♦♠❡ ✉♥✐✈❡rs❡ ♦❢ ♦✉t♣✉t ✈❛❧✉❡s ✭❇♦♦❧❡❛♥ ✈❛❧✉❡s✱ ♣r♦❜❛❜✐❧✐t✐❡s✱ ✈❡❝t♦r s♣❛❝❡✱ ✇♦r❞s✱ ✳ ✳ ✳ ✮ ❆✉t♦♠❛t❛ ❛r❡ ❢✉♥❝t♦rs✦✦✦

✶✵ ✴ ✸✶

❲♦r❞ ❛✉t♦♠❛t❛

❞❡t❡r♠✐♥✐st✐❝ ❛✉t♦♠❛t❛ ✶ Q ✷ ✐♥ ❙❡t ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❛✉t♦♠❛t❛ ✶ Q ✶ ✐♥ ❘❡❧ ✇❡✐❣❤t❡❞ ❛✉t♦♠❛t❛ S Q S ✐♥ ▼♦❞S ❙✉❜s❡q✳ tr❛♥s❞✉❝❡rs ✶ Q ✶ ✐♥ ❑❧(T )

a a a a

❲❡ s❡❡ ❛ ♣❛tt❡r♥ ❡♠❡r❣✐♥❣✦

✶✶ ✴ ✸✶

❲♦r❞ ❛✉t♦♠❛t❛

❞❡t❡r♠✐♥✐st✐❝ ❛✉t♦♠❛t❛ ✶ Q ✷ ✐♥ ❙❡t ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❛✉t♦♠❛t❛ ✶ Q ✶ ✐♥ ❘❡❧ ✇❡✐❣❤t❡❞ ❛✉t♦♠❛t❛ S Q S ✐♥ ▼♦❞S ❙✉❜s❡q✳ tr❛♥s❞✉❝❡rs ✶ Q ✶ ✐♥ ❑❧(T )

a a a a

❲❡ s❡❡ ❛ ♣❛tt❡r♥ ❡♠❡r❣✐♥❣✦

✶✶ ✴ ✸✶

❲♦r❞ ❛✉t♦♠❛t❛ ❛r❡ ❢✉♥❝t♦rs A∶I → C , ✇❤❡r❡ t❤❡ ✐♥♣✉t ❝❛t❡❣♦r② I ✐s ❢r❡❡❧② ❣❡♥❡r❛t❡❞ ❜② ✐♥ st❛t❡s ♦✉t

▷ a ◁ ✶✷ ✴ ✸✶

❲♦r❞ ❛✉t♦♠❛t❛ ❛r❡ ❢✉♥❝t♦rs A∶I → C , ✇❤❡r❡ t❤❡ ✐♥♣✉t ❝❛t❡❣♦r② I ✐s ❢r❡❡❧② ❣❡♥❡r❛t❡❞ ❜② ✐♥ st❛t❡s ♦✉t

▷ a ◁

❞❡t❡r♠✐♥✐st✐❝ ❛✉t♦♠❛t❛ A∶I → ❙❡t ✐♥ ↦ ✶ ❛♥❞ ♦✉t ↦ ✷

✶✷ ✴ ✸✶

❲♦r❞ ❛✉t♦♠❛t❛ ❛r❡ ❢✉♥❝t♦rs A∶I → C , ✇❤❡r❡ t❤❡ ✐♥♣✉t ❝❛t❡❣♦r② I ✐s ❢r❡❡❧② ❣❡♥❡r❛t❡❞ ❜② ✐♥ st❛t❡s ♦✉t

▷ a ◁

❞❡t❡r♠✐♥✐st✐❝ ❛✉t♦♠❛t❛ A∶I → ❙❡t ✐♥ ↦ ✶ ❛♥❞ ♦✉t ↦ ✷ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❛✉t♦♠❛t❛ A∶I → ❘❡❧ ✐♥ ↦ ✶ ❛♥❞ ♦✉t ↦ ✶

✶✷ ✴ ✸✶

❲♦r❞ ❛✉t♦♠❛t❛ ❛r❡ ❢✉♥❝t♦rs A∶I → C , ✇❤❡r❡ t❤❡ ✐♥♣✉t ❝❛t❡❣♦r② I ✐s ❢r❡❡❧② ❣❡♥❡r❛t❡❞ ❜② ✐♥ st❛t❡s ♦✉t

▷ a ◁

❞❡t❡r♠✐♥✐st✐❝ ❛✉t♦♠❛t❛ A∶I → ❙❡t ✐♥ ↦ ✶ ❛♥❞ ♦✉t ↦ ✷ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❛✉t♦♠❛t❛ A∶I → ❘❡❧ ✐♥ ↦ ✶ ❛♥❞ ♦✉t ↦ ✶ ✇❡✐❣❤t❡❞ ❛✉t♦♠❛t❛ A∶I → ▼♦❞S ✐♥ ↦ S ❛♥❞ ♦✉t ↦ S

✶✷ ✴ ✸✶

❲♦r❞ ❛✉t♦♠❛t❛ ❛r❡ ❢✉♥❝t♦rs A∶I → C , ✇❤❡r❡ t❤❡ ✐♥♣✉t ❝❛t❡❣♦r② I ✐s ❢r❡❡❧② ❣❡♥❡r❛t❡❞ ❜② ✐♥ st❛t❡s ♦✉t

▷ a ◁

❞❡t❡r♠✐♥✐st✐❝ ❛✉t♦♠❛t❛ A∶I → ❙❡t ✐♥ ↦ ✶ ❛♥❞ ♦✉t ↦ ✷ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ❛✉t♦♠❛t❛ A∶I → ❘❡❧ ✐♥ ↦ ✶ ❛♥❞ ♦✉t ↦ ✶ ✇❡✐❣❤t❡❞ ❛✉t♦♠❛t❛ A∶I → ▼♦❞S ✐♥ ↦ S ❛♥❞ ♦✉t ↦ S s✉❜s❡q✳ tr❛♥s❞✉❝❡rs A∶I → ❑❧(T ) ✐♥ ↦ ✶ ❛♥❞ ♦✉t ↦ ✶

✶✷ ✴ ✸✶

▲❛♥❣✉❛❣❡s ❛r❡ ❢✉♥❝t♦rs L∶O → C , ✇❤❡r❡ O ✐s t❤❡ ❢✉❧❧ s✉❜❝❛t❡❣♦r② ♦❢ I ♦♥ ♦❜❥❡❝ts ✐♥ ❛♥❞ ♦✉t ✐♥ ♦✉t

▷w◁ ∶ w∈A∗

❆ ❧❛♥❣✉❛❣❡ ❝❛♥ ❜❡ ♠♦❞❡❧❧❡❞ ❛s ❛ ❢✉♥❝t♦r

❙❡t

❙❡t s♦ t❤❛t

❙❡t ✐♥

✶ ❛♥❞

❙❡t ♦✉t

✷ ❋♦r ❛❧❧ ✇❡ ❤❛✈❡

❙❡t

✶ ✷ ✐♥ ❙❡t✳ ❆❧t❡r♥❛t✐✈❡❧②✱ ❝❛♥ ❜❡ ♠♦❞❡❧❧❡❞ ❛s ❛ ❢✉♥❝t♦r

❘❡❧

❘❡❧ s♦ t❤❛t

❘❡❧ ✐♥

✶ ❛♥❞

❘❡❧ ♦✉t

✶ ❋♦r ❛❧❧ ✇❡ ❤❛✈❡

❘❡❧

✶ ✶ ✐♥ ❘❡❧✳

✶✸ ✴ ✸✶

▲❛♥❣✉❛❣❡s ❛r❡ ❢✉♥❝t♦rs L∶O → C , ✇❤❡r❡ O ✐s t❤❡ ❢✉❧❧ s✉❜❝❛t❡❣♦r② ♦❢ I ♦♥ ♦❜❥❡❝ts ✐♥ ❛♥❞ ♦✉t ✐♥ ♦✉t

▷w◁ ∶ w∈A∗

❆ ❧❛♥❣✉❛❣❡ L ⊆ A∗ ❝❛♥ ❜❡ ♠♦❞❡❧❧❡❞ ❛s ❛ ❢✉♥❝t♦r L❙❡t∶O → ❙❡t s♦ t❤❛t L❙❡t(✐♥) = ✶ ❛♥❞ L❙❡t(♦✉t) = ✷, ❋♦r ❛❧❧ w ∈ A∗ ✇❡ ❤❛✈❡ L❙❡t(▷w◁)∶✶ → ✷ ✐♥ ❙❡t✳ ❆❧t❡r♥❛t✐✈❡❧②✱ ❝❛♥ ❜❡ ♠♦❞❡❧❧❡❞ ❛s ❛ ❢✉♥❝t♦r

❘❡❧

❘❡❧ s♦ t❤❛t

❘❡❧ ✐♥

✶ ❛♥❞

❘❡❧ ♦✉t

✶ ❋♦r ❛❧❧ ✇❡ ❤❛✈❡

❘❡❧

✶ ✶ ✐♥ ❘❡❧✳

✶✸ ✴ ✸✶

▲❛♥❣✉❛❣❡s ❛r❡ ❢✉♥❝t♦rs L∶O → C , ✇❤❡r❡ O ✐s t❤❡ ❢✉❧❧ s✉❜❝❛t❡❣♦r② ♦❢ I ♦♥ ♦❜❥❡❝ts ✐♥ ❛♥❞ ♦✉t ✐♥ ♦✉t

▷w◁ ∶ w∈A∗

❆ ❧❛♥❣✉❛❣❡ L ⊆ A∗ ❝❛♥ ❜❡ ♠♦❞❡❧❧❡❞ ❛s ❛ ❢✉♥❝t♦r L❙❡t∶O → ❙❡t s♦ t❤❛t L❙❡t(✐♥) = ✶ ❛♥❞ L❙❡t(♦✉t) = ✷, ❋♦r ❛❧❧ w ∈ A∗ ✇❡ ❤❛✈❡ L❙❡t(▷w◁)∶✶ → ✷ ✐♥ ❙❡t✳ ❆❧t❡r♥❛t✐✈❡❧②✱ L ⊆ A∗ ❝❛♥ ❜❡ ♠♦❞❡❧❧❡❞ ❛s ❛ ❢✉♥❝t♦r L❘❡❧∶O → ❘❡❧ s♦ t❤❛t L❘❡❧(✐♥) = ✶ ❛♥❞ L❘❡❧(♦✉t) = ✶. ❋♦r ❛❧❧ w ∈ A∗ ✇❡ ❤❛✈❡ L❘❡❧(▷w◁)∶✶ → ✶ ✐♥ ❘❡❧✳

✶✸ ✴ ✸✶

❆❝❝❡♣t✐♥❣ ❛ ❧❛♥❣✉❛❣❡ ✭t❤❡ ❢✉♥❝t♦r ✈❡rs✐♦♥✮

❆♥ ❛✉t♦♠❛t♦♥ A ❛❝❝❡♣ts ❛ ❧❛♥❣✉❛❣❡ L ✇❤❡♥ t❤❡ ♥❡①t ❞✐❛❣r❛♠ ❝♦♠♠✉t❡s ✐♥ ♦✉t O C ✐♥ st❛t❡s ♦✉t I

▷w◁ ∶ w∈A∗ L ▷ a ◁ A

❋♦r ❡✈❡r② ❧❛♥❣✉❛❣❡ ✇❡ ❤❛✈❡ ❛ ❝❛t❡❣♦r② ❆✉t♦ ♦❢ ❛✉t♦♠❛t❛ ❛❝❝❡♣t✐♥❣ ✳

✶✹ ✴ ✸✶

❆❝❝❡♣t✐♥❣ ❛ ❧❛♥❣✉❛❣❡ ✭t❤❡ ❢✉♥❝t♦r ✈❡rs✐♦♥✮

❆♥ ❛✉t♦♠❛t♦♥ A ❛❝❝❡♣ts ❛ ❧❛♥❣✉❛❣❡ L ✇❤❡♥ t❤❡ ♥❡①t ❞✐❛❣r❛♠ ❝♦♠♠✉t❡s ✐♥ ♦✉t O C ✐♥ st❛t❡s ♦✉t I

▷w◁ ∶ w∈A∗ L ▷ a ◁ A

❋♦r ❡✈❡r② ❧❛♥❣✉❛❣❡ L∶O → C ✇❡ ❤❛✈❡ ❛ ❝❛t❡❣♦r② ❆✉t♦L ♦❢ ❛✉t♦♠❛t❛ ❛❝❝❡♣t✐♥❣ L✳

✶✹ ✴ ✸✶

❆✉t♦♠❛t❛ ❛s ❢✉♥❝t♦rs✿ ♠✐♥✐♠✐③❛t✐♦♥

▼✐♥✐♠✐❛❧ ❆✉t♦♠❛t♦♥ ▼✐♥(L) ❢♦r ❛ ▲❛♥❣✉❛❣❡

❲❤❡♥ ❞♦❡s ❛ ❵♠✐♥✐♠❛❧✬ ❛✉t♦♠❛t♦♥ ❛❝❝❡♣t✐♥❣ ❛ ❧❛♥❣✉❛❣❡ L ❡①✐st❄ O C I

L ▼✐♥(L)? ▼✐♥(L) A✐♥✐t(L) A❢✐♥❛❧(L)

■❢ t❤❡ ❝❛t❡❣♦r② ♦❢ ❛✉t♦♠❛t❛ ❛❝❝❡♣t✐♥❣ ❤❛s ❼ ❛♥ ✐♥✐t✐❛❧ ♦❜❥❡❝t

✐♥✐t

✱ ❼ ❛ ✜♥❛❧ ♦❜❥❡❝t

❢✐♥❛❧

✱ ❛♥❞✱ ❼ ❛ ❢❛❝t♦r✐③❛t✐♦♥ s②st❡♠ t❤❡♥ ▼✐♥ ✐s ♦❜t❛✐♥❡❞ ❛s t❤❡ ❢❛❝t♦r✐③❛t✐♦♥

✐♥✐t

▼✐♥

❢✐♥❛❧ ✶✺ ✴ ✸✶

▼✐♥✐♠✐❛❧ ❆✉t♦♠❛t♦♥ ▼✐♥(L) ❢♦r ❛ ▲❛♥❣✉❛❣❡

❲❤❡♥ ❞♦❡s ❛ ❵♠✐♥✐♠❛❧✬ ❛✉t♦♠❛t♦♥ ❛❝❝❡♣t✐♥❣ ❛ ❧❛♥❣✉❛❣❡ L ❡①✐st❄ O C I

L ▼✐♥(L)? ▼✐♥(L) A✐♥✐t(L) A❢✐♥❛❧(L)

■❢ t❤❡ ❝❛t❡❣♦r② ♦❢ ❛✉t♦♠❛t❛ ❛❝❝❡♣t✐♥❣ L ❤❛s ❼ ❛♥ ✐♥✐t✐❛❧ ♦❜❥❡❝t A✐♥✐t(L)✱ ❼ ❛ ✜♥❛❧ ♦❜❥❡❝t

❢✐♥❛❧

✱ ❛♥❞✱ ❼ ❛ ❢❛❝t♦r✐③❛t✐♦♥ s②st❡♠ t❤❡♥ ▼✐♥ ✐s ♦❜t❛✐♥❡❞ ❛s t❤❡ ❢❛❝t♦r✐③❛t✐♦♥

✐♥✐t

▼✐♥

❢✐♥❛❧ ✶✺ ✴ ✸✶

▼✐♥✐♠✐❛❧ ❆✉t♦♠❛t♦♥ ▼✐♥(L) ❢♦r ❛ ▲❛♥❣✉❛❣❡

❲❤❡♥ ❞♦❡s ❛ ❵♠✐♥✐♠❛❧✬ ❛✉t♦♠❛t♦♥ ❛❝❝❡♣t✐♥❣ ❛ ❧❛♥❣✉❛❣❡ L ❡①✐st❄ O C I

L ▼✐♥(L)? ▼✐♥(L) A✐♥✐t(L) A❢✐♥❛❧(L)

■❢ t❤❡ ❝❛t❡❣♦r② ♦❢ ❛✉t♦♠❛t❛ ❛❝❝❡♣t✐♥❣ L ❤❛s ❼ ❛♥ ✐♥✐t✐❛❧ ♦❜❥❡❝t A✐♥✐t(L)✱ ❼ ❛ ✜♥❛❧ ♦❜❥❡❝t A❢✐♥❛❧(L)✱ ❛♥❞✱ ❼ ❛ ❢❛❝t♦r✐③❛t✐♦♥ s②st❡♠ t❤❡♥ ▼✐♥ ✐s ♦❜t❛✐♥❡❞ ❛s t❤❡ ❢❛❝t♦r✐③❛t✐♦♥

✐♥✐t

▼✐♥

❢✐♥❛❧ ✶✺ ✴ ✸✶

▼✐♥✐♠✐❛❧ ❆✉t♦♠❛t♦♥ ▼✐♥(L) ❢♦r ❛ ▲❛♥❣✉❛❣❡

❲❤❡♥ ❞♦❡s ❛ ❵♠✐♥✐♠❛❧✬ ❛✉t♦♠❛t♦♥ ❛❝❝❡♣t✐♥❣ ❛ ❧❛♥❣✉❛❣❡ L ❡①✐st❄ O C I

L ▼✐♥(L)? ▼✐♥(L) A✐♥✐t(L) A❢✐♥❛❧(L)

■❢ t❤❡ ❝❛t❡❣♦r② ♦❢ ❛✉t♦♠❛t❛ ❛❝❝❡♣t✐♥❣ L ❤❛s ❼ ❛♥ ✐♥✐t✐❛❧ ♦❜❥❡❝t A✐♥✐t(L)✱ ❼ ❛ ✜♥❛❧ ♦❜❥❡❝t A❢✐♥❛❧(L)✱ ❛♥❞✱ ❼ ❛ ❢❛❝t♦r✐③❛t✐♦♥ s②st❡♠ t❤❡♥ ▼✐♥(L) ✐s ♦❜t❛✐♥❡❞ ❛s t❤❡ ❢❛❝t♦r✐③❛t✐♦♥ A✐♥✐t(L) ↠ ▼✐♥(L) ↣ A❢✐♥❛❧(L).

✶✺ ✴ ✸✶

❚r✐✈✐❛❧ ❡①❛♠♣❧❡

❞❡t❡r♠✐♥✐st✐❝ ❛✉t♦♠❛t❛✱ ✐✳❡✳ (❙❡t,✶,✷)✲❛✉t♦♠❛t❛ ❛❝❝❡♣t✐♥❣ ❛ (❙❡t,✶,✷)✲❧❛♥❣✉❛❣❡ A∗ ✶ Q ✷ ✷A∗

L? r❡❛❝❤❡❞❙t❛t❡ L ε i f ❛❝❝❡♣t❡❞▲❛♥❣✉❛❣❡ ε? ✶✻ ✴ ✸✶

❚r✐✈✐❛❧ ❡①❛♠♣❧❡

❞❡t❡r♠✐♥✐st✐❝ ❛✉t♦♠❛t❛✱ ✐✳❡✳ (❙❡t,✶,✷)✲❛✉t♦♠❛t❛ ❛❝❝❡♣t✐♥❣ ❛ (❙❡t,✶,✷)✲❧❛♥❣✉❛❣❡ A∗ ✶ ▼✐♥(L) ✷ ✷A∗

L? r❡❛❝❤❡❞❙t❛t❡ L ε i f ❛❝❝❡♣t❡❞▲❛♥❣✉❛❣❡ ε? ✶✻ ✴ ✸✶

▼✐♥✐♠✐③❛t✐♦♥ ✈✐❛ ❡♣✐✲♠♦♥♦ ❢❛❝t♦r✐s❛t✐♦♥s

■❢ t❤❡ ♦✉t♣✉t ❝❛t❡❣♦r② C ❤❛s ❝♦✉♥t❛❜❧❡ ♣♦✇❡rs ❛♥❞ ❝♦♣♦✇❡rs✱ ❛♥❞✱ ❛♥❞ ❡♣✐✲♠♦♥♦ ❢❛❝t♦r✐s❛t✐♦♥ s②st❡♠✱ t❤❡♥ t❤❡ ♠✐♥✐♠✐❛❧ ❛✉t♦♠❛t♦♥ ❢♦r L ✐s ❝♦♠♣✉t❡❞ ❛s ❢♦❧❧♦✇s ∐

u∈A∗ I

I ▼✐♥(L) F ∏

u∈A∗ F L? L ε i f ε? ✶✼ ✴ ✸✶

❚❤✉s ❢❛r ✇❡ ❤❛✈❡ r❡✐♥✈❡♥t❡❞ t❤❡ ✇❤❡❡❧ ✳✳✳ ❍♦✇❡✈❡r✱ t❤❡ ✇❤❡❡❧ ✇❛s ❛ ♣r❡tt② ❛✇❡s♦♠❡ ✐♥✈❡♥t✐♦♥✦

✶✽ ✴ ✸✶

❚❤✉s ❢❛r ✇❡ ❤❛✈❡ r❡✐♥✈❡♥t❡❞ t❤❡ ✇❤❡❡❧ ✳✳✳ ❍♦✇❡✈❡r✱ t❤❡ ✇❤❡❡❧ ✇❛s ❛ ♣r❡tt② ❛✇❡s♦♠❡ ✐♥✈❡♥t✐♦♥✦

✶✽ ✴ ✸✶

❲❤❛t ✐❢ t❤❡ ♦✉t♣✉t ❝❛t❡❣♦r② ✐s ♥♦t ♥✐❝❡❄

❚✇♦ ❛♣♣❧✐❝❛t✐♦♥s

❙✉❜s❡q✉❡♥t✐❛❧ tr❛♥s❞✉❝❡rs t❤❡ ♦✉t♣✉t ❝❛t❡❣♦r② ❤❛s ❝♦♣♦✇❡rs✱ ❢❛❝t♦r✐③❛t✐♦♥ s②st❡♠✱ ❜✉t ❞♦❡s ♥♦t ❤❛✈❡ ♣r♦❞✉❝ts✳ ❍②❜r✐❞ s❡t✲✈❡❝t♦r ❛✉t♦♠❛t❛ ❛ ❝♦st✉♠✲♠❛❞❡ ♦✉t♣✉t ❝❛t❡❣♦r② t❤❛t ❤❛s ❛❧❧ ♣♦✇❡rs ❛♥❞ ❝♦♣♦✇❡rs✱ ❜✉t ✇❤❡r❡ t❤❡ ❢❛❝t♦r✐s❛t✐♦♥ s②st❡♠ ✐s ♥♦t ✏♥✐❝❡✑ ❡♥♦✉❣❤ t♦ ❣✐✈❡ ❛ ♠❡❛♥✐♥❣❢✉❧ ♥♦t✐♦♥ ♦❢ ♠✐♥✐♠✐③❛t✐♦♥✳

✶✾ ✴ ✸✶

❚✇♦ ❛♣♣❧✐❝❛t✐♦♥s

❙✉❜s❡q✉❡♥t✐❛❧ tr❛♥s❞✉❝❡rs t❤❡ ♦✉t♣✉t ❝❛t❡❣♦r② ❤❛s ❝♦♣♦✇❡rs✱ ❢❛❝t♦r✐③❛t✐♦♥ s②st❡♠✱ ❜✉t ❞♦❡s ♥♦t ❤❛✈❡ ♣r♦❞✉❝ts✳ ❍②❜r✐❞ s❡t✲✈❡❝t♦r ❛✉t♦♠❛t❛ ❛ ❝♦st✉♠✲♠❛❞❡ ♦✉t♣✉t ❝❛t❡❣♦r② t❤❛t ❤❛s ❛❧❧ ♣♦✇❡rs ❛♥❞ ❝♦♣♦✇❡rs✱ ❜✉t ✇❤❡r❡ t❤❡ ❢❛❝t♦r✐s❛t✐♦♥ s②st❡♠ ✐s ♥♦t ✏♥✐❝❡✑ ❡♥♦✉❣❤ t♦ ❣✐✈❡ ❛ ♠❡❛♥✐♥❣❢✉❧ ♥♦t✐♦♥ ♦❢ ♠✐♥✐♠✐③❛t✐♦♥✳

✶✾ ✴ ✸✶

❙✉❜s❡q✉❡♥t✐❛❧ tr❛♥s❞✉❝❡rs à ❧❛ ❈❤♦✛r✉t

▼✐♥✐♠✐③❛t✐♦♥ ♦❢ s✉❜s❡q✉❡♥t✐❛❧ tr❛♥s❞✉❝❡rs à ❧❛ ❈❤♦✛r✉t

✵ ✶ ✷ ✸ ε a a ε ba ba ε a∣ab a∣abba a∣abba a∣bab a∣bab a∣b a∣abb a∣abba a∣abba a∣ab a∣ab a∣b b∣b b∣ba b∣ba b∣ba b∣ba b∣b b∣ab b∣ab b∣ba b∣ba

✷✵ ✴ ✸✶

▼✐♥✐♠✐③❛t✐♦♥ ♦❢ s✉❜s❡q✉❡♥t✐❛❧ tr❛♥s❞✉❝❡rs à ❧❛ ❈❤♦✛r✉t

✵ ✶ ✷ ✸ ε a a ε ba ba ε a∣ab a∣abba a∣abba a∣bab a∣bab a∣b a∣abb a∣abba a∣abba a∣ab a∣ab a∣b b∣b b∣ba b∣ba b∣ba b∣ba b∣b b∣ab b∣ab b∣ba b∣ba

✷✵ ✴ ✸✶

▼✐♥✐♠✐③❛t✐♦♥ ♦❢ s✉❜s❡q✉❡♥t✐❛❧ tr❛♥s❞✉❝❡rs à ❧❛ ❈❤♦✛r✉t

✵ ✶ ✷ ✸ ε a a ε ba ba ε a∣ab a∣abba a∣abba a∣bab a∣bab a∣b a∣abb a∣abba a∣abba a∣ab a∣ab a∣b b∣b b∣ba b∣ba b∣ba b∣ba b∣b b∣ab b∣ab b∣ba b∣ba

✷✵ ✴ ✸✶

▼✐♥✐♠✐③❛t✐♦♥ ♦❢ s✉❜s❡q✉❡♥t✐❛❧ tr❛♥s❞✉❝❡rs à ❧❛ ❈❤♦✛r✉t

✵ ✶ ✷ ✸ ε a a ε ba ba ε a∣ab a∣abba a∣abba a∣bab a∣bab a∣b a∣abb a∣abba a∣abba a∣ab a∣ab a∣b b∣b b∣ba b∣ba b∣ba b∣ba b∣b b∣ab b∣ab b∣ba b∣ba

✷✵ ✴ ✸✶

▼✐♥✐♠✐③❛t✐♦♥ ♦❢ s✉❜s❡q✉❡♥t✐❛❧ tr❛♥s❞✉❝❡rs à ❧❛ ❈❤♦✛r✉t

✵ ✶ ✷ ✸ ε a ε ba ba ε a∣ab a∣abba a∣abba a∣bab a∣bab a∣b a∣abb a∣abba a∣abba a∣ab a∣ab a∣b b∣b b∣ba b∣ba b∣ba b∣ba b∣b b∣ab b∣ab b∣ba b∣ba

✷✵ ✴ ✸✶

▼✐♥✐♠✐③❛t✐♦♥ ♦❢ s✉❜s❡q✉❡♥t✐❛❧ tr❛♥s❞✉❝❡rs à ❧❛ ❈❤♦✛r✉t

✵ ✶ ✷ ✸ ε a ε ba ba ε a∣ab a∣abba a∣abba a∣bab a∣bab a∣b a∣abb a∣abba a∣abba a∣ab a∣ab a∣b b∣b b∣ba b∣ba b∣ba b∣ba b∣b b∣ab b∣ab b∣ba b∣ba

✷✵ ✴ ✸✶

▼✐♥✐♠✐③❛t✐♦♥ ♦❢ s✉❜s❡q✉❡♥t✐❛❧ tr❛♥s❞✉❝❡rs à ❧❛ ❈❤♦✛r✉t

✵ ✶ ✷ ✸ ε a ε ba ba ε a∣ab a∣abba a∣abba a∣bab a∣bab a∣b a∣abb a∣abba a∣abba a∣ab a∣ab a∣b b∣b b∣ba b∣ba b∣ba b∣ba b∣b b∣ab b∣ab b∣ba b∣ba

✷✵ ✴ ✸✶

▼✐♥✐♠✐③❛t✐♦♥ ♦❢ s✉❜s❡q✉❡♥t✐❛❧ tr❛♥s❞✉❝❡rs à ❧❛ ❈❤♦✛r✉t

✵ ✶ ✷ ✸ ✵′ ✶′ ε ε a ε ba a∣ab a∣abba a∣bab a∣b a∣abb a∣ab b∣b b∣ba b∣b b∣b b∣ab b∣ba

✷✶ ✴ ✸✶

❙✉❜s❡q✉❡♥t✐❛❧ tr❛♥s❞✉❝❡rs ❛s ❢✉♥❝t♦rs

❆ s✉❜s❡q✉❡♥t✐❛❧ tr❛♥s❞✉❝❡rs ✇✐t❤ ♦✉t♣✉t ❛❧♣❤❛❜❡t B ✐s ❡ss❡♥t✐❛❧❧② ❛ ❢✉♥❝t♦r A∶I → ❑❧(T ) ❢♦r t❤❡ ♠♦♥❛❞ T ∶❙❡t → ❙❡t ❞❡✜♥❡❞ ❜② T (X) = B∗ × X + ✶. ❚❤❛t ✐s✱ ✇❡ ❤❛✈❡ t❤❡ ❞❛t❛ ✶ Q ✶ ✐♥ ❑❧(T )

a

❚❤❡ ❝❛t❡❣♦r② ❑❧ ❞♦❡s ♥♦t ❤❛✈❡ ♣♦✇❡rs ♦r ♣r♦❞✉❝ts✦✦ ❚❤✐s ✐s ✇❤② ✇❡ ❝❛♥♥♦t ❥✉st ✉s❡ ❝♦❛❧❣❡❜r❛s ❢♦r ✶ ✶ ✱ s❡❡ ❬❍❛♥s❡♥✱ ✷✵✶✵❪

✷✷ ✴ ✸✶

❙✉❜s❡q✉❡♥t✐❛❧ tr❛♥s❞✉❝❡rs ❛s ❢✉♥❝t♦rs

❆ s✉❜s❡q✉❡♥t✐❛❧ tr❛♥s❞✉❝❡rs ✇✐t❤ ♦✉t♣✉t ❛❧♣❤❛❜❡t B ✐s ❡ss❡♥t✐❛❧❧② ❛ ❢✉♥❝t♦r A∶I → ❑❧(T ) ❢♦r t❤❡ ♠♦♥❛❞ T ∶❙❡t → ❙❡t ❞❡✜♥❡❞ ❜② T (X) = B∗ × X + ✶. ❚❤❛t ✐s✱ ✇❡ ❤❛✈❡ t❤❡ ❞❛t❛ ✶ Q ✶ ✐♥ ❑❧(T )

a

❚❤❡ ❝❛t❡❣♦r② ❑❧(T ) ❞♦❡s ♥♦t ❤❛✈❡ ♣♦✇❡rs ♦r ♣r♦❞✉❝ts✦✦ ❚❤✐s ✐s ✇❤② ✇❡ ❝❛♥♥♦t ❥✉st ✉s❡ ❝♦❛❧❣❡❜r❛s ❢♦r SX = (✶ + B∗ × X)A∗ × (✶ + B∗)✱ s❡❡ ❬❍❛♥s❡♥✱ ✷✵✶✵❪

✷✷ ✴ ✸✶

✏●❧✉❡✐♥❣s✑ ♦❢ ✈❡❝t♦r s♣❛❝❡s

▲❡t✬s ❜❛❝❦tr❛❝❦ t♦ t❤❡ ✏❤②❜r✐❞ s❡t✲✈❡❝t♦r✑ ❛✉t♦♠❛t♦♥

❚❤❡ ✏r❡❛❝❤❛❜❧❡✑ ✈❡❝t♦rs ❛r❡ ♦♥ t❤❡ ✏✉♥✐♦♥✑ ♦❢ t✇♦ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s✉❜s♣❛❝❡s✳ ♠❛✐♥t❛✐♥✐♥❣ ♦♥❡ ❜✐t ❛♥❞ ♦♥❡ r❡❛❧ ✐s ❜❡tt❡r t❤❛♥ t✇♦ r❡❛❧s

✷✸ ✴ ✸✶

❲❤❛t ✐s t❤❡ ❣♦♦❞ ❝❛t❡❣♦r② t♦ ❛❝❝♦♠♠♦❞❛t❡ t❤❡ ♥❡✇ ♠♦❞❡❧❄

❆♥ ❡①❛♠♣❧❡ ♦❢ ✏❣❧✉✐♥❣s✑ ♦❢ ✈❡❝t♦r s♣❛❝❡s ✐✳❡✳ ❛ ♠♦♥♦✲❝♦❧✐♠✐t ✐♥ ❱❡❝

p q r

R✷ R✷ R✷ R R R

✷✹ ✴ ✸✶

❚❤❡ ❝❛t❡❣♦r② ●❧✉❡(C)

❆ ❞✐❛❣r❛♠ F∶D → C ✐s ❝❛❧❧❡❞ ❛ ♠♦♥♦✲❝♦❧✐♠✐t ✐❢ ✐t ❤❛s ❛ ♠♦♥♦✲❝♦❝♦♥❡ ✐♥ C✱ t❤❛t ✐s✱ ❛ ❝♦❝♦♥❡ ✇❤❡r❡ ❛❧❧ t❤❡ ✐♥❥❡❝t✐♦♥s ❛r❡ ♠♦♥♦s✳ ❉❡✜♥✐t✐♦♥ ❲❡ ❞❡✜♥❡ ●❧✉❡(C) ❛s t❤❡ ❢r❡❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ C ✉♥❞❡r ♠♦♥♦✲❝♦❧✐♠✐ts✳ ▲❡♠♠❛ ❚❤❡ ❝❛t❡❣♦r② ●❧✉❡ ✐s ❝♦♠♣❧❡t❡ ❛♥❞ ❝♦❝♦♠♣❧❡t❡ ✇❤❡♥❡✈❡r ✐s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ●❧✉❡ ❱❡❝ ❤❛s ❛❧❧ t❤❡ r❡q✉✐r❡❞ ♣r♦♣❡rt✐❡s s♦ t❤❛t ♠✐♥✐♠✐s❛t✐♦♥ ✇♦r❦s s♠♦♦t❤❧②✳

✷✺ ✴ ✸✶

❚❤❡ ❝❛t❡❣♦r② ●❧✉❡(C)

❆ ❞✐❛❣r❛♠ F∶D → C ✐s ❝❛❧❧❡❞ ❛ ♠♦♥♦✲❝♦❧✐♠✐t ✐❢ ✐t ❤❛s ❛ ♠♦♥♦✲❝♦❝♦♥❡ ✐♥ C✱ t❤❛t ✐s✱ ❛ ❝♦❝♦♥❡ ✇❤❡r❡ ❛❧❧ t❤❡ ✐♥❥❡❝t✐♦♥s ❛r❡ ♠♦♥♦s✳ ❉❡✜♥✐t✐♦♥ ❲❡ ❞❡✜♥❡ ●❧✉❡(C) ❛s t❤❡ ❢r❡❡ ❝♦♠♣❧❡t✐♦♥ ♦❢ C ✉♥❞❡r ♠♦♥♦✲❝♦❧✐♠✐ts✳ ▲❡♠♠❛ ❚❤❡ ❝❛t❡❣♦r② ●❧✉❡(C) ✐s ❝♦♠♣❧❡t❡ ❛♥❞ ❝♦❝♦♠♣❧❡t❡ ✇❤❡♥❡✈❡r C ✐s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ●❧✉❡(❱❡❝) ❤❛s ❛❧❧ t❤❡ r❡q✉✐r❡❞ ♣r♦♣❡rt✐❡s s♦ t❤❛t ♠✐♥✐♠✐s❛t✐♦♥ ✇♦r❦s s♠♦♦t❤❧②✳

✷✺ ✴ ✸✶

❙t✐❧❧✱ t❤❡r❡ ✐s ❛ ❝❛t❝❤ ✳✳✳

❲❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ❡✛❡❝t✐✈❡ ♠✐♥✐♠❛❧ ❛✉t♦♠❛t❛✦ ❞❡t❡r♠✐♥✐st✐❝ ✜♥✐t❡ ❛✉t♦♠❛t❛ ❙❡t✜♥ ✜♥✐t❡✲❞✐♠✳ ✈❡❝t♦r ❛✉t♦♠❛t❛ ❱❡❝✜♥ ❡✛❡❝t✐✈❡ ❤②❜r✐❞✲s❡t✲✈❡❝t♦r ❛✉t♦♠❛t❛

• ❧✉❡✜♥(❱❡❝✜♥)

✇❤❡r❡ ●❧✉❡✜♥(❱❡❝✜♥) ✐s t❤❡ ❢r❡❡ ❝♦❝♦♠♣❧❡t✐♦♥ ♦❢ ❱❡❝✜♥ ✉♥❞❡r ✜♥✐t❡ ♠♦♥♦✲❝♦❧✐♠✐ts✳

✷✻ ✴ ✸✶

❊①❛♠♣❧❡

❈♦♥s✐❞❡r t❤❡ ✇❡✐❣❤t❡❞ ❧❛♥❣✉❛❣❡ L∶A∗ → R ❣✐✈❡♥ ❜② L(u) = cos(α∣u∣) ❢♦r s♦♠❡ α ✇❤✐❝❤ ✐s ♥♦t ❛ r❛t✐♦♥❛❧ ♠✉❧t✐♣❧❡ ♦❢ π✳ ❚❤❡ ♠✐♥✐♠❛❧ ❛✉t♦♠❛t♦♥ ✐♥ ●❧✉❡(❱❡❝) ✐s ❛ ❝♦✉♥t❛❜❧❡ ❝♦❧✐♠✐t ♦❢ ♦♥❡✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡s✳

α α α

✷✼ ✴ ✸✶

■t s❡❡♠s ✇❡ ❤❛✈❡ ✏❜r♦❦❡♥✑ t❤❡ ♠✐♥✐♠✐s❛t✐♦♥ ✇❤❡❡❧ ✳✳✳ ❚❤❡ ✜①✿ ❛ ❢❛❝t♦r✐s❛t✐♦♥ t❤r♦✉❣❤ s②st❡♠

✷✽ ✴ ✸✶

■t s❡❡♠s ✇❡ ❤❛✈❡ ✏❜r♦❦❡♥✑ t❤❡ ♠✐♥✐♠✐s❛t✐♦♥ ✇❤❡❡❧ ✳✳✳ ❚❤❡ ✜①✿ ❛ ❢❛❝t♦r✐s❛t✐♦♥ t❤r♦✉❣❤ s②st❡♠

✷✽ ✴ ✸✶

❈♦♥❝❧✉s✐♦♥s

❈♦♥❝❧✉s✐♦♥s

❖✉r ❝♦♥tr✐❜✉t✐♦♥✿ ❛ ♥❡✇ ❛✉t♦♠❛t❛ ♠♦❞❡❧✦ ❚❤❡ ❝❛t❡❣♦r②✲t❤❡♦r❡t✐❝ ♣❡rs♣❡❝t✐✈❡ ❤❡❧♣s ✇✐t❤ t❤❡ ❛❝❝✉r❛t❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❤②❜r✐❞ s❡t✲✈❡❝t♦r ❛✉t♦♠❛t❛ ♠♦❞❡❧✳ ◗✉✐t❡ ❛ ❢❡✇ q✉❡st✐♦♥s r❡♠❛✐♥ t♦ ❜❡ ❛♥s✇❡r❡❞✳✳✳ ❈❛♥ ✇❡ ❝❤❛r❛❝t❡r✐s❡ t❤❡ ♣r❡s❤❡❛✈❡s t❤❛t ❛r❡ ♠♦♥♦✲❝♦❧✐♠✐ts ♦❢ r❡♣r❡s❡♥t❛❜❧❡s❄ ✭s♦♠❡ ♣❛rt✐❛❧ r❡s✉❧ts✱ ❡✳❣✳ ✇❡ ♣r♦✈❡❞ t❤❛t t❤❡② ♣r❡s❡r✈❡ ❡q✉❛❧✐s❡rs✱ ❜✉t t❤❛t ✐s ♥♦t s✉✣❝✐❡♥t✮ ❍♦✇ ❞♦ ✇❡ ❡✛❡❝t✐✈❡❧② ♠✐♥✐♠✐s❡ ❤②❜r✐❞✲s❡t✲✈❡❝t♦r ❛✉t♦♠❛t❛❄

✷✾ ✴ ✸✶

❈♦♥❝❧✉s✐♦♥s

❖✉r ❝♦♥tr✐❜✉t✐♦♥✿ ❛ ♥❡✇ ❛✉t♦♠❛t❛ ♠♦❞❡❧✦ ❚❤❡ ❝❛t❡❣♦r②✲t❤❡♦r❡t✐❝ ♣❡rs♣❡❝t✐✈❡ ❤❡❧♣s ✇✐t❤ t❤❡ ❛❝❝✉r❛t❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❤②❜r✐❞ s❡t✲✈❡❝t♦r ❛✉t♦♠❛t❛ ♠♦❞❡❧✳ ◗✉✐t❡ ❛ ❢❡✇ q✉❡st✐♦♥s r❡♠❛✐♥ t♦ ❜❡ ❛♥s✇❡r❡❞✳✳✳ ❈❛♥ ✇❡ ❝❤❛r❛❝t❡r✐s❡ t❤❡ ♣r❡s❤❡❛✈❡s t❤❛t ❛r❡ ♠♦♥♦✲❝♦❧✐♠✐ts ♦❢ r❡♣r❡s❡♥t❛❜❧❡s❄ ✭s♦♠❡ ♣❛rt✐❛❧ r❡s✉❧ts✱ ❡✳❣✳ ✇❡ ♣r♦✈❡❞ t❤❛t t❤❡② ♣r❡s❡r✈❡ ❡q✉❛❧✐s❡rs✱ ❜✉t t❤❛t ✐s ♥♦t s✉✣❝✐❡♥t✮ ❍♦✇ ❞♦ ✇❡ ❡✛❡❝t✐✈❡❧② ♠✐♥✐♠✐s❡ ❤②❜r✐❞✲s❡t✲✈❡❝t♦r ❛✉t♦♠❛t❛❄

✷✾ ✴ ✸✶

❈♦♥❝❧✉s✐♦♥s

❖✉r ❝♦♥tr✐❜✉t✐♦♥✿ ❛ ♥❡✇ ❛✉t♦♠❛t❛ ♠♦❞❡❧✦ ❚❤❡ ❝❛t❡❣♦r②✲t❤❡♦r❡t✐❝ ♣❡rs♣❡❝t✐✈❡ ❤❡❧♣s ✇✐t❤ t❤❡ ❛❝❝✉r❛t❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❤②❜r✐❞ s❡t✲✈❡❝t♦r ❛✉t♦♠❛t❛ ♠♦❞❡❧✳ ◗✉✐t❡ ❛ ❢❡✇ q✉❡st✐♦♥s r❡♠❛✐♥ t♦ ❜❡ ❛♥s✇❡r❡❞✳✳✳ ❈❛♥ ✇❡ ❝❤❛r❛❝t❡r✐s❡ t❤❡ ♣r❡s❤❡❛✈❡s t❤❛t ❛r❡ ♠♦♥♦✲❝♦❧✐♠✐ts ♦❢ r❡♣r❡s❡♥t❛❜❧❡s❄ ✭s♦♠❡ ♣❛rt✐❛❧ r❡s✉❧ts✱ ❡✳❣✳ ✇❡ ♣r♦✈❡❞ t❤❛t t❤❡② ♣r❡s❡r✈❡ ❡q✉❛❧✐s❡rs✱ ❜✉t t❤❛t ✐s ♥♦t s✉✣❝✐❡♥t✮ ❍♦✇ ❞♦ ✇❡ ❡✛❡❝t✐✈❡❧② ♠✐♥✐♠✐s❡ ❤②❜r✐❞✲s❡t✲✈❡❝t♦r ❛✉t♦♠❛t❛❄

✷✾ ✴ ✸✶

❈♦♥❝❧✉s✐♦♥s

❆❞❥✉♥❝t✐♦♥s ❜❡t✇❡❡♥ ♦✉t♣✉t ❝❛t❡❣♦r✐❡s ❧✐❢t t♦ ❛❞❥✉♥❝t✐♦♥s ❢♦r ✏❛❞❥♦✐♥t tr❛♥s♣♦s❡✑ ❧❛♥❣✉❛❣❡s✳ ❯♥✐❢②✐♥❣ ❡①♣❧❛♥❛t✐♦♥ ❢♦r ❼ ❞❡t❡r♠✐♥✐③❛t✐♦♥ ♦❢ ◆❋❆s ❼ ❣❡♥❡r❛❧✐s❡❞ ♣♦✇❡rs❡t ❝♦♥str✉❝t✐♦♥ ❼ r❡✈❡rs✐♥❣ ❛✉t♦♠❛t❛ ❲❤❛t ♦t❤❡r ✉s❡s ❝❛♥ ✇❡ ✜♥❞ ❢♦r t❤❡ ✏♠✐♥✐♠✐③❛t✐♦♥ ✇❤❡❡❧✑❄ ❼ s②♥t❛❝t✐❝ ♠♦♥♦✐❞s✱ ❛❧❣❡❜r❛s ❼ ♠✐♥✐♠✐③❛t✐♦♥ ❜② ❞✉❛❧✐t② ❼ s②♥t❛❝t✐❝ s♣❛❝❡s ✇✐t❤ ✐♥t❡r♥❛❧ ♠♦♥♦✐❞s ❬●❡❤r❦❡✱ P✳✱ ❘❡❣❣✐♦✱ ■❈❆▲P✬✶✻✱ ▲■❈❙✬✶✼❪ ❼ ♠✐♥✐♠✐③❛t✐♦♥ ♦❢ s✉❜s❡q✉❡♥t✐❛❧ tr❛♥s❞✉❝❡rs ✭à ❧❛ ❈❤♦✛r✉t✮

✸✵ ✴ ✸✶

❘❡❢❡r❡♥❝❡s

❬❈♦❧❝♦♠❜❡t✱ P✳✱ ❆❈▼ ❙■●▲❖● ❛♣r✐❧ ✷✵✶✼❪ ❆✉t♦♠❛t❛ ❛♥❞ ♠✐♥✐♠✐③❛t✐♦♥✳ ❬❈♦❧❝♦♠❜❡t✱ P✳✱ ▼❋❈❙ ✷✵✶✼❪ ❆✉t♦♠❛t❛ ✐♥ t❤❡ ❈❛t❡❣♦r② ♦❢ ●❧✉❡❞ ❱❡❝t♦r ❙♣❛❝❡s ❬❈♦❧❝♦♠❜❡t✱ P✳✱ ❈❆▲❈❖ ✷✵✶✼❪ ❆✉t♦♠❛t❛ ▼✐♥✐♠✐③❛t✐♦♥✿ ❛ ❋✉♥❝t♦r✐❛❧ ❆♣♣r♦❛❝❤

✸✶ ✴ ✸✶