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❆✈♦✐❞✐♥❣ ❦✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

▼✐❝❤❛ë❧ ❘❛♦

❈◆❘❙ ✲ ❊◆❙ ▲②♦♥ ▲■P ✲ éq✉✐♣❡ ▼❈✷ ♠✐❝❤❛❡❧✳r❛♦❅❡♥s✲❧②♦♥✳❢r ❤tt♣✿✴✴♣❡rs♦✳❡♥s✲❧②♦♥✳❢r✴♠✐❝❤❛❡❧✳r❛♦✴

▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

❦✲❛❜❡❧✐❛♥ ❡q✉✐✈❛❧❡♥❝❡

❚✇♦ ✇♦r❞s ✉ ❛♥❞ ✈ ❛r❡ ❦✲❛❜❡❧✐❛♥ ❡q✉✐✈❛❧❡♥t ✐❢✿ ❢♦r ❡✈❡r② ✇ ✇✐t❤ ∣✇∣ ≤ ❦✱ ∣✉∣✇ = ∣✈∣✇✳ ❊q✉✐✈❛❧❡♥t❧②✿ ❢♦r ❡✈❡r② ✇ ✇✐t❤ ∣✇∣ = ❦✱ ∣✉∣✇ = ∣✈∣✇ ❛♥❞ ✉[✶ ∶ ❦ − ✶] = ✈[✶ ∶ ❦ − ✶] ❛♥❞ ✉[∣✉∣ − ❦ + ✷ ∶ ∣✉∣] = ✈[∣✈∣ − ❦ + ✷ ∶ ∣✈∣]✳ ♣qr ✐s ❛ ❦✲❛❜❡❧✐❛♥ ❝✉❜❡ ✐❢ ♣✱ q✱ r ❛r❡ ♣❛✐r✇✐s❡ ❦✲❛❜❡❧✐❛♥ ❡q✉✐✈❛❧❡♥t✳

▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

Pr♦❜❧❡♠s ★ ✶✻

Pr♦❜❧❡♠ ❆r❡ ✷✲❛❜❡❧✐❛♥ ❝✉❜❡ ❛✈♦✐❞❛❜❧❡ ♦♥ ❛ ❜✐♥❛r② ❛❧♣❤❛❜❡t ❄ ❨❡s Pr♦❜❧❡♠ ❋♦r ✇❤✐❝❤ ❦ ❛r❡ ❦✲❛❜❡❧✐❛♥ sq✉❛r❡s ❛✈♦✐❞❛❜❧❡ ♦♥ ❛ t❡r♥❛r② ❛❧♣❤❛❜❡t ❄ ❦ ✸

▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

Pr♦❜❧❡♠s ★ ✶✻

Pr♦❜❧❡♠ ❆r❡ ✷✲❛❜❡❧✐❛♥ ❝✉❜❡ ❛✈♦✐❞❛❜❧❡ ♦♥ ❛ ❜✐♥❛r② ❛❧♣❤❛❜❡t ❄ ⇒ ❨❡s Pr♦❜❧❡♠ ❋♦r ✇❤✐❝❤ ❦ ❛r❡ ❦✲❛❜❡❧✐❛♥ sq✉❛r❡s ❛✈♦✐❞❛❜❧❡ ♦♥ ❛ t❡r♥❛r② ❛❧♣❤❛❜❡t ❄ ⇒ ❦ = ✸

▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

✷✲❛❜❡❧✐❛♥✲❝✉❜❡✲❢r❡❡ ❜✐♥❛r② ✇♦r❞

▲❡t ❤ ∶ Σ∗

✸ → Σ∗ ✷ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✹✼✲✉♥✐❢♦r♠ ♠♦r♣❤✐s♠✳

❤ ∶ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ✵ → ✵✵✶✵✵✶✵✶✵✵✶✵✶✶✵✵✶✵✵✶✵✶✵✵✶✵✵✶✶✵✵✶✵✵✶✶✵✵✶✵✶✶✵✶✵✶✶ ✶ → ✵✵✶✵✵✶✶✵✵✶✵✵✶✶✵✵✶✶✵✶✶✵✵✶✶✵✶✶✵✵✶✵✵✶✶✵✵✶✶✵✶✶✵✶✵✶✶ ✷ → ✵✵✶✶✵✶✶✵✶✵✶✶✵✶✵✵✶✵✶✶✵✶✵✶✶✵✶✵✵✶✵✶✵✵✶✵✵✶✶✵✶✶✵✶✵✶✶ ❚❤❡♦r❡♠ ❋♦r ❡✈❡r② ❛❜❡❧✐❛♥✲❝✉❜❡✲❢r❡❡ ✇♦r❞ ✇ ∈ Σ∗

✸✱ ❤(✇) ✐s

✷✲❛❜❡❧✐❛♥✲❝✉❜❡✲❢r❡❡✳

▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

❙❦❡t❝❤ ♦❢ ♣r♦♦❢ ✭✶✮

ψ(✇) = (∣✇∣✵,∣✇∣✶,∣✇∣✷)❚✳ ψ′(✇) = (∣✇∣✵✵,∣✇∣✵✶,∣✇∣✶✶)❚✳ ▲❡t ▼ s✳t✳ ▼ψ(①) = ψ′(❤(①)✵) ❢♦r ❡✈❡r② ① ∈ Σ✸✳ ▼ = ⎛ ⎜ ⎝ ✶✵ ✾ ✺ ✶✻ ✶✹ ✶✼ ✺ ✶✵ ✽ ⎞ ⎟ ⎠ . ❲❡ ❤❛✈❡ ψ′(❤(✇)✵) = ▼ψ(✇) ❢♦r ❡✈❡r② ✇♦r❞ ✇ ∈ Σ∗

✸✳

▼−✶ = ⎛ ⎜ ⎝

✺✽ ✺✶✼ ✷ ✹✼ −✽✸ ✺✶✼ ✹✸ ✺✶✼ −✺ ✹✼ ✾✵ ✺✶✼ −✾✵ ✺✶✼ ✺ ✹✼ ✹ ✺✶✼

⎞ ⎟ ⎠ .

▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

❙❦❡t❝❤ ♦❢ ♣r♦♦❢ ✭✷✮

✇ = ❛♣′❜q′❝r′❞✱ ♣ = ♣✶♣✷♣✸✱ q = q✶q✷q✸✱ r = r✶r✷r✸✱ ❤(❛) = ✉♣✶✱ ❤(♣′) = ♣✷✱ ❤(❜) = ♣✸q✶✱ ❤(q′) = q✷✱ ❤(❝) = q✸r✶✱ ❤(r′) = r✷✱ ❤(❞) = r✸✈✳

• ❡♥❡r❛❧ ❝❛s❡✿ ✉ ♣✶✱ ♣✸✱ q✶✱ q✸✱ r✶✱ r✸ ❛♥❞ ✈ ❛r❡ ♥♦t ❡♠♣t②✳

✭❖t❤❡r ❝❛s❡s ❛r❡ tr❡❛t❡❞ s✐♠✐❧❛r❧②✳✮ ❙✐♥❝❡ ♣qr ✐s ❛ ✷✲❛❜❡❧✐❛♥✲❝✉❜❡✱ t❤❡ ✜rst ❧❡tt❡r ♦❢ ♣✶✱ q✶✱ r✶ ❤❛s t❤❡ s❛♠❡ ✜rst ❧❡tt❡r✱ ❛♥❞ ♣✸✱ q✸✱ r✸ ❤❛s t❤❡ s❛♠❡ ❧❛st ❧❡tt❡r✳ ❲❡ ❤❛✈❡ ψ′(♣) = ψ′(♣✶♣✸) + ▼ψ(♣′)✱ ψ′(q) = ψ′(q✶q✸) + ▼ψ(q′) ❛♥❞ ψ′(r) = ψ′(r✶r✸) + ▼ψ(r′) ❚❤❡ ❝♦♥❞✐t✐♦♥ ψ′(♣) = ψ′(q) = ψ′(r) ❜❡❝♦♠❡✿ ▼−✶ψ′(♣✶♣✸) + ψ(♣′) = ▼−✶ψ′(q✶q✸) + ψ(q′) = ▼−✶ψ′(r✶r✸) + ψ(r′)✳ ◆♦t❡✿ t❤✐s ❝♦♥❞✐t✐♦♥ ❝❛♥♥♦t ❜❡ ❢✉❧✜❧❧❡❞ ✐❢ ▼−✶(ψ′(♣✶♣✸) − ψ′(q✶q✸))✱ ♦r ▼−✶(ψ′(♣✶♣✸) − ψ′(r✶r✸)) ❛r❡ ♥♦t ✐♥t❡❣❡r ✈❡❝t♦rs✳

▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

❙❦❡t❝❤ ♦❢ ♣r♦♦❢ ✭✸✮

❇② ❝♦♠♣✉t❡r✱ ✇❡ ❝❤❡❝❦ t❤❛t ❢♦r ❡✈❡r② ❛,❜,❝,❞ ∈ Σ✱ ❛♥❞ ❢♦r ❡✈❡r② ♥♦♥✲❡♠♣t② ✉,♣✶,♣✸,q✶,q✸,r✶,r✸,✈ ∈ Σ∗ s✉❝❤ t❤❛t✿ ✉♣✶ = ❤(❛)✱ ♣✸q✶ = ❤(❜)✱ q✸r✶ = ❤(❝)✱ r✸✈ = ❤(❞)✱ ♣✶✱ q✶✱ r✶ ❤❛✈❡ t❤❡ s❛♠❡ ✜rst ❧❡tt❡r✱ ♣✸✱ q✸✱ r✸ ❤❛✈❡ t❤❡ s❛♠❡ ❧❛st ❧❡tt❡r✱ ▼−✶(ψ′(♣✶♣✸) − ψ′(q✶q✸)) ❛♥❞ ▼−✶(ψ′(♣✶♣✸) − ψ′(r✶r✸)) ❛r❡ ✐♥t❡❣❡r ✈❡❝t♦rs✱ t❤❡♥ ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥ ❤♦❧❞s✿ ψ(❛♣′) = ψ(❜q′) = ψ(❝r′)✱ t❤❛t ✐s✿ ▼−✶ψ′(♣✶♣✸)−ψ(❛) = ▼−✶ψ′(q✶q✸)−ψ(❜) = ▼−✶ψ′(r✶r✸)−ψ(❝), ψ(♣′❜) = ψ(q′❝) = ψ(r′❞)✱ ψ(❛♣′) = ψ(❜q′) = ψ(❝r′❞)✱ ψ(❛♣′) = ψ(❜q′❝) = ψ(r′❞)✳ ■♥ ❛❧❧ ❝❛s❡s✱ ✇ ❤❛s ❛♥ ❛❜❡❧✐❛♥ ❝✉❜❡✳ ❈♦♥tr❛❞✐❝t✐♦♥✳

▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

✷✲❛❜❡❧✐❛♥✲❝✉❜❡✲❢r❡❡ ❜✐♥❛r② ✇♦r❞ ✭♥♦♥✲✉♥✐❢♦r♠✮

▲❡t ❤ ∶ Σ∗

✸ → Σ∗ ✷ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦r♣❤✐s♠✳

❤ ∶ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ✵ → ✵✵✶✵✵✶✵✶✵✵✶✶✵✵✶✵✵✶✵✶✵✵✶✵✵✶✶✵✵✶✵✵✶✶✵✵✶✶✵✶✶ ✶ → ✵✶✵✶✶✵✶✶✵✵✶✶✵✶✶✵✵✶✶✵✵✶✵✵✶✶✵✵✶✶✵✶✶ ✷ → ✵✶✵✶✶✵✶✵✵✶✵✶✵✵✶✵✶✶✵✵✶✶✵✶✶ ❚❤❡♦r❡♠ ❋♦r ❡✈❡r② ❛❜❡❧✐❛♥✲❝✉❜❡✲❢r❡❡ ✇♦r❞ ✇ ∈ Σ∗

✸✱ ❤(✇) ✐s

✷✲❛❜❡❧✐❛♥✲❝✉❜❡✲❢r❡❡✳

▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡

✸✲❛❜❡❧✐❛♥✲sq✉❛r❡✲❢r❡❡ t❡r♥❛r② ✇♦r❞

▲❡t ❤ ∶ Σ∗

✹ → Σ∗ ✸ ❜❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✷✺✲✉♥✐❢♦r♠ ♠♦r♣❤✐s♠✳

❤ ∶ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ✵ → ✵✶✵✷✵✶✷✵✷✶✵✶✷✵✶✵✷✵✶✷✶✵✷✶✷ ✶ → ✵✶✵✷✶✵✶✷✵✶✵✷✶✷✵✶✷✶✵✶✷✵✷✶✷ ✷ → ✵✶✵✷✶✵✶✷✶✵✷✶✷✵✷✶✵✷✵✶✷✵✷✶✷ ✸ → ✵✶✷✶✵✷✵✶✷✵✷✶✵✷✵✶✷✶✵✶✷✵✷✶✷ ❚❤❡♦r❡♠ ❋♦r ❡✈❡r② ❛❜❡❧✐❛♥✲sq✉❛r❡✲❢r❡❡ ✇♦r❞ ✇ ∈ Σ∗

✹✱ ❤(✇) ✐s

✸✲❛❜❡❧✐❛♥✲sq✉❛r❡✲❢r❡❡✳

▼✐❝❤❛ë❧ ❘❛♦ ❆✈♦✐❞✐♥❣ ❦✲❛❜❡❧✐❛♥ ❝✉❜❡✴sq✉❛r❡