❖♥ ❘❡❝♦♠♣r❡ss✐♦♥ ❢♦r ❲♦r❞ ❊q✉❛t✐♦♥s ❆rt✉r ❏❡➺ ▼❡❡t✐♥❣ ♦♥ ❙tr✐♥❣ ❈♦♥str❛✐♥ts ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ✭▼❖❙❈❆✮ ✵✼✳✵✺✳✷✵✶✾
❲❡ ❡①t❡♥❞ t♦ ❛ ❀ ✐❞❡♥t✐t② ♦♥ ✳ ✐s ❛ s♦❧✉t✐♦♥ ✇♦r❞✳ ▲❡♥❣❤t✲♠✐♥✐♠❛❧ ✿ ♠✐♥✐♠✐s❡s ✳ ❯s✉❛❧❧②✿ ♥♦ ✱ ✐✳❡✳ ✳ ❲♦r❞ ❊q✉❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ✭❙❛t✐s✜❛❜✐❧✐t② ♦❢ ✇♦r❞ ❡q✉❛t✐♦♥s✮ ●✐✈❡♥ ❡q✉❛t✐♦♥ U = V ✱ ✇❤❡r❡ U, V ∈ (Σ ∪ X ) ∗ ✳ ■s t❤❡r❡ ❛ s✉❜st✐t✉t✐♦♥ S : X → Σ ∗ s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥❄ ✭❆❧s♦ ♠♦r❡ ❣❡♥❡r❛❧✿ ✜♥t✐❡❧② ♠❛♥② s♦❧✉t✐♦♥s✱ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛❧❧✱ ✳ ✳ ✳ ✮
❲❡ ❡①t❡♥❞ t♦ ❛ ❀ ✐❞❡♥t✐t② ♦♥ ✳ ✐s ❛ s♦❧✉t✐♦♥ ✇♦r❞✳ ▲❡♥❣❤t✲♠✐♥✐♠❛❧ ✿ ♠✐♥✐♠✐s❡s ✳ ❯s✉❛❧❧②✿ ♥♦ ✱ ✐✳❡✳ ✳ ❲♦r❞ ❊q✉❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ✭❙❛t✐s✜❛❜✐❧✐t② ♦❢ ✇♦r❞ ❡q✉❛t✐♦♥s✮ ●✐✈❡♥ ❡q✉❛t✐♦♥ U = V ✱ ✇❤❡r❡ U, V ∈ (Σ ∪ X ) ∗ ✳ ■s t❤❡r❡ ❛ s✉❜st✐t✉t✐♦♥ S : X → Σ ∗ s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥❄ ✭❆❧s♦ ♠♦r❡ ❣❡♥❡r❛❧✿ ✜♥t✐❡❧② ♠❛♥② s♦❧✉t✐♦♥s✱ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛❧❧✱ ✳ ✳ ✳ ✮ aX bX Y bbb = X abaaY bY S ( X ) = aa, S ( Y ) = bb
❲❡ ❡①t❡♥❞ t♦ ❛ ❀ ✐❞❡♥t✐t② ♦♥ ✳ ✐s ❛ s♦❧✉t✐♦♥ ✇♦r❞✳ ▲❡♥❣❤t✲♠✐♥✐♠❛❧ ✿ ♠✐♥✐♠✐s❡s ✳ ❯s✉❛❧❧②✿ ♥♦ ✱ ✐✳❡✳ ✳ ❲♦r❞ ❊q✉❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ✭❙❛t✐s✜❛❜✐❧✐t② ♦❢ ✇♦r❞ ❡q✉❛t✐♦♥s✮ ●✐✈❡♥ ❡q✉❛t✐♦♥ U = V ✱ ✇❤❡r❡ U, V ∈ (Σ ∪ X ) ∗ ✳ ■s t❤❡r❡ ❛ s✉❜st✐t✉t✐♦♥ S : X → Σ ∗ s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥❄ ✭❆❧s♦ ♠♦r❡ ❣❡♥❡r❛❧✿ ✜♥t✐❡❧② ♠❛♥② s♦❧✉t✐♦♥s✱ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛❧❧✱ ✳ ✳ ✳ ✮ aX bX Y bbb = X abaaY bY S ( X ) = aa, S ( Y ) = bb aaabaabbbbb = aaabaabbbbb
❲♦r❞ ❊q✉❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ✭❙❛t✐s✜❛❜✐❧✐t② ♦❢ ✇♦r❞ ❡q✉❛t✐♦♥s✮ ●✐✈❡♥ ❡q✉❛t✐♦♥ U = V ✱ ✇❤❡r❡ U, V ∈ (Σ ∪ X ) ∗ ✳ ■s t❤❡r❡ ❛ s✉❜st✐t✉t✐♦♥ S : X → Σ ∗ s❛t✐s❢②✐♥❣ t❤❡ ❡q✉❛t✐♦♥❄ ✭❆❧s♦ ♠♦r❡ ❣❡♥❡r❛❧✿ ✜♥t✐❡❧② ♠❛♥② s♦❧✉t✐♦♥s✱ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ❛❧❧✱ ✳ ✳ ✳ ✮ aX bX Y bbb = X abaaY bY S ( X ) = aa, S ( Y ) = bb aaabaabbbbb = aaabaabbbbb ❲❡ ❡①t❡♥❞ S t♦ ❛ S : (Σ ∪ X ) ∗ → Σ ∗ ❀ ✐❞❡♥t✐t② ♦♥ Σ ✳ S ( U ) ✐s ❛ s♦❧✉t✐♦♥ ✇♦r❞✳ ▲❡♥❣❤t✲♠✐♥✐♠❛❧ S ✿ ♠✐♥✐♠✐s❡s | S ( U ) | ✳ ❯s✉❛❧❧②✿ ♥♦ S ( X ) = ǫ ✱ ✐✳❡✳ S : X → Σ + ✳
▲❡♥❣t❤ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✭❧❡♥❣t❤ ✮✿ ❝♦♠♣r❡ss✐❜❧❡ t♦ ✳ ✷◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ❛♥❞ ❘②tt❡r✱ ✶✾✾✽❪ ❚❤❡ s✐③❡ ♦❢ t❤❡ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✐s ❛t ♠♦st ❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧✳ ◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ P❙P❆❈❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ ❚❤❡ s❛♠❡✱ ❜✉t s✐♠♣❧❡r✳ ❬❏✳ ✷✵✶✸❪ ❈♦♠♣r❡ss✐♦♥ ❛♥❞ ✇♦r❞ ❡q✉❛t✐♦♥s ❖♥❧② ◆P✲❤❛r❞✳ ❆♥❞ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P✳ ❙♦❧✉t✐♦♥s ❛t ♠♦st ❡①♣♦♥❡♥t✐❛❧❄ ❆❧❣♦r✐t❤♠s ❛♥❞ ❈♦♠♣❧❡①✐t② ▼❛❦❛♥✐♥✬s ❛❧❣♦r✐t❤♠ ✶✾✼✼ ❍✐❣❤ ❝♦♠♣❧❡①✐t② ❬❊❳P❙P❆❈❊ ✬✾✽❪✱ ❞✐✣❝✉❧t ♣r♦♦❢✳
▲❡♥❣t❤ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✭❧❡♥❣t❤ ✮✿ ❝♦♠♣r❡ss✐❜❧❡ t♦ ✳ ✷◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ❛♥❞ ❘②tt❡r✱ ✶✾✾✽❪ ❚❤❡ s✐③❡ ♦❢ t❤❡ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✐s ❛t ♠♦st ❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧✳ ◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ P❙P❆❈❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ ❚❤❡ s❛♠❡✱ ❜✉t s✐♠♣❧❡r✳ ❬❏✳ ✷✵✶✸❪ ❖♥❧② ◆P✲❤❛r❞✳ ❆♥❞ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P✳ ❙♦❧✉t✐♦♥s ❛t ♠♦st ❡①♣♦♥❡♥t✐❛❧❄ ❆❧❣♦r✐t❤♠s ❛♥❞ ❈♦♠♣❧❡①✐t② ▼❛❦❛♥✐♥✬s ❛❧❣♦r✐t❤♠ ✶✾✼✼ ❍✐❣❤ ❝♦♠♣❧❡①✐t② ❬❊❳P❙P❆❈❊ ✬✾✽❪✱ ❞✐✣❝✉❧t ♣r♦♦❢✳ ❈♦♠♣r❡ss✐♦♥ ❛♥❞ ✇♦r❞ ❡q✉❛t✐♦♥s
❚❤❡ s✐③❡ ♦❢ t❤❡ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✐s ❛t ♠♦st ❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧✳ ◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ P❙P❆❈❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ ❚❤❡ s❛♠❡✱ ❜✉t s✐♠♣❧❡r✳ ❬❏✳ ✷✵✶✸❪ ❖♥❧② ◆P✲❤❛r❞✳ ❆♥❞ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P✳ ❙♦❧✉t✐♦♥s ❛t ♠♦st ❡①♣♦♥❡♥t✐❛❧❄ ❆❧❣♦r✐t❤♠s ❛♥❞ ❈♦♠♣❧❡①✐t② ▼❛❦❛♥✐♥✬s ❛❧❣♦r✐t❤♠ ✶✾✼✼ ❍✐❣❤ ❝♦♠♣❧❡①✐t② ❬❊❳P❙P❆❈❊ ✬✾✽❪✱ ❞✐✣❝✉❧t ♣r♦♦❢✳ ❈♦♠♣r❡ss✐♦♥ ❛♥❞ ✇♦r❞ ❡q✉❛t✐♦♥s ◮ ▲❡♥❣t❤ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✭❧❡♥❣t❤ N ✮✿ ❝♦♠♣r❡ss✐❜❧❡ t♦ poly (log N ) ✳ ✷◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ❛♥❞ ❘②tt❡r✱ ✶✾✾✽❪
P❙P❆❈❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ ❚❤❡ s❛♠❡✱ ❜✉t s✐♠♣❧❡r✳ ❬❏✳ ✷✵✶✸❪ ❖♥❧② ◆P✲❤❛r❞✳ ❆♥❞ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P✳ ❙♦❧✉t✐♦♥s ❛t ♠♦st ❡①♣♦♥❡♥t✐❛❧❄ ❆❧❣♦r✐t❤♠s ❛♥❞ ❈♦♠♣❧❡①✐t② ▼❛❦❛♥✐♥✬s ❛❧❣♦r✐t❤♠ ✶✾✼✼ ❍✐❣❤ ❝♦♠♣❧❡①✐t② ❬❊❳P❙P❆❈❊ ✬✾✽❪✱ ❞✐✣❝✉❧t ♣r♦♦❢✳ ❈♦♠♣r❡ss✐♦♥ ❛♥❞ ✇♦r❞ ❡q✉❛t✐♦♥s ◮ ▲❡♥❣t❤ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✭❧❡♥❣t❤ N ✮✿ ❝♦♠♣r❡ss✐❜❧❡ t♦ poly (log N ) ✳ ✷◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ❛♥❞ ❘②tt❡r✱ ✶✾✾✽❪ ◮ ❚❤❡ s✐③❡ N ♦❢ t❤❡ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✐s ❛t ♠♦st ❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧✳ ◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪
❚❤❡ s❛♠❡✱ ❜✉t s✐♠♣❧❡r✳ ❬❏✳ ✷✵✶✸❪ ❖♥❧② ◆P✲❤❛r❞✳ ❆♥❞ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P✳ ❙♦❧✉t✐♦♥s ❛t ♠♦st ❡①♣♦♥❡♥t✐❛❧❄ ❆❧❣♦r✐t❤♠s ❛♥❞ ❈♦♠♣❧❡①✐t② ▼❛❦❛♥✐♥✬s ❛❧❣♦r✐t❤♠ ✶✾✼✼ ❍✐❣❤ ❝♦♠♣❧❡①✐t② ❬❊❳P❙P❆❈❊ ✬✾✽❪✱ ❞✐✣❝✉❧t ♣r♦♦❢✳ ❈♦♠♣r❡ss✐♦♥ ❛♥❞ ✇♦r❞ ❡q✉❛t✐♦♥s ◮ ▲❡♥❣t❤ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✭❧❡♥❣t❤ N ✮✿ ❝♦♠♣r❡ss✐❜❧❡ t♦ poly (log N ) ✳ ✷◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ❛♥❞ ❘②tt❡r✱ ✶✾✾✽❪ ◮ ❚❤❡ s✐③❡ N ♦❢ t❤❡ ♠✐♥✐♠❛❧ s♦❧✉t✐♦♥ ✐s ❛t ♠♦st ❞♦✉❜❧② ❡①♣♦♥❡♥t✐❛❧✳ ◆❊❳P❚■▼❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪ ◮ P❙P❆❈❊ ❬P❧❛♥❞♦✇s❦✐ ✶✾✾✾❪
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