❲♦r❞ ❡q✉❛t✐♦♥s ✐♥ ♥♦♥❞❡t❡r♠✐♥✐st✐❝ ❧✐♥❡❛r s♣❛❝❡ ❆rt✉r ❏❡➺ ■♥st✐t✉t❡ ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦❧❧♦q✉✐✉♠ ♦♥ ❆✉t♦♠❛t❛✱ ▲❛♥❣✉❛❣❡s ❛♥❞ Pr♦❣r❛♠♠✐♥❣ ❲❛rs③❛✇❛✱ ✶✸✳✵✼✳✷✵✶✼
❲❡ ❡①t❡♥❞ t♦ ❛ ❀ ✐❞❡♥t✐t② ♦♥ ✳ ✐s ❛ s♦❧✉t✐♦♥ ✇♦r❞✳ ▲❡♥❣❤t✲♠✐♥✐♠❛❧ ✿ ♠✐♥✐♠✐s❡s ❚❤✐s ✐s ✐♠♣♦rt❛♥t ✉♥✐✜❝❛t✐♦♥ ✇♦r❞ ❝♦♠❜✐♥❛t♦r✐❝s ❤❡❧♣❢✉❧ ✐♥ ❡q✉❛t✐♦♥s ✐♥ ❢r❡❡ ❣r♦✉♣ ✭❛♥❞ ♦t❤❡r✮ ❲♦r❞ ❊q✉❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ●✐✈❡♥ ❡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✐♦♥❄
❲❡ ❡①t❡♥❞ t♦ ❛ ❀ ✐❞❡♥t✐t② ♦♥ ✳ ✐s ❛ s♦❧✉t✐♦♥ ✇♦r❞✳ ▲❡♥❣❤t✲♠✐♥✐♠❛❧ ✿ ♠✐♥✐♠✐s❡s ❚❤✐s ✐s ✐♠♣♦rt❛♥t ✉♥✐✜❝❛t✐♦♥ ✇♦r❞ ❝♦♠❜✐♥❛t♦r✐❝s ❤❡❧♣❢✉❧ ✐♥ ❡q✉❛t✐♦♥s ✐♥ ❢r❡❡ ❣r♦✉♣ ✭❛♥❞ ♦t❤❡r✮ ❲♦r❞ ❊q✉❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ●✐✈❡♥ ❡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✐♦♥❄ aX bX Y bbb = X abaabY bY S ( X ) = aa, S ( Y ) = bb
❲❡ ❡①t❡♥❞ t♦ ❛ ❀ ✐❞❡♥t✐t② ♦♥ ✳ ✐s ❛ s♦❧✉t✐♦♥ ✇♦r❞✳ ▲❡♥❣❤t✲♠✐♥✐♠❛❧ ✿ ♠✐♥✐♠✐s❡s ❚❤✐s ✐s ✐♠♣♦rt❛♥t ✉♥✐✜❝❛t✐♦♥ ✇♦r❞ ❝♦♠❜✐♥❛t♦r✐❝s ❤❡❧♣❢✉❧ ✐♥ ❡q✉❛t✐♦♥s ✐♥ ❢r❡❡ ❣r♦✉♣ ✭❛♥❞ ♦t❤❡r✮ ❲♦r❞ ❊q✉❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ●✐✈❡♥ ❡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✐♦♥❄ aX bX Y bbb = X abaabY bY S ( X ) = aa, S ( Y ) = bb aaabaabbbbb = aaabaabbbbbb
❚❤✐s ✐s ✐♠♣♦rt❛♥t ✉♥✐✜❝❛t✐♦♥ ✇♦r❞ ❝♦♠❜✐♥❛t♦r✐❝s ❤❡❧♣❢✉❧ ✐♥ ❡q✉❛t✐♦♥s ✐♥ ❢r❡❡ ❣r♦✉♣ ✭❛♥❞ ♦t❤❡r✮ ❲♦r❞ ❊q✉❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ●✐✈❡♥ ❡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✐♦♥❄ aX bX Y bbb = X abaabY bY S ( X ) = aa, S ( Y ) = bb aaabaabbbbb = aaabaabbbbbb ❲❡ ❡①t❡♥❞ S t♦ ❛ S : (Σ ∪ X ) ∗ → Σ ∗ ❀ ✐❞❡♥t✐t② ♦♥ Σ ✳ S ( U ) ✐s ❛ s♦❧✉t✐♦♥ ✇♦r❞✳ ▲❡♥❣❤t✲♠✐♥✐♠❛❧ S ✿ ♠✐♥✐♠✐s❡s | S ( U ) |
❲♦r❞ ❊q✉❛t✐♦♥s ❉❡✜♥✐t✐♦♥ ●✐✈❡♥ ❡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✐♦♥❄ aX bX Y bbb = X abaabY bY S ( X ) = aa, S ( Y ) = bb aaabaabbbbb = aaabaabbbbbb ❲❡ ❡①t❡♥❞ S t♦ ❛ S : (Σ ∪ X ) ∗ → Σ ∗ ❀ ✐❞❡♥t✐t② ♦♥ Σ ✳ S ( U ) ✐s ❛ s♦❧✉t✐♦♥ ✇♦r❞✳ ▲❡♥❣❤t✲♠✐♥✐♠❛❧ S ✿ ♠✐♥✐♠✐s❡s | S ( U ) | ❚❤✐s ✐s ✐♠♣♦rt❛♥t ◮ ✉♥✐✜❝❛t✐♦♥ ◮ ✇♦r❞ ❝♦♠❜✐♥❛t♦r✐❝s ◮ ❤❡❧♣❢✉❧ ✐♥ ❡q✉❛t✐♦♥s ✐♥ ❢r❡❡ ❣r♦✉♣ ✭❛♥❞ ♦t❤❡r✮
❊①❛❝t s♣❛❝❡ ❝♦♠♣❧❡①✐t②❄ ❬✳ ✳ ✳ ❪ ●✉t✐❡rr❡③ ✬✾✽ ❊❳P❙P❆❈❊ P❧❛♥❞♦✇s❦✐ ✫ ❘②tt❡r ✬✾✽ ♥❡✇ ❛♣♣r♦❛❝❤ ✖ ✉s✐♥❣ ❝♦♠♣r❡ss✐♦♥ P❧❛♥❞♦✇s❦✐ ✬✾✾ P❙P❆❈❊ ❏✳ ✬✶✸ P❙P❆❈❊ ◆P✲❤❛r❞✱ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P ❚❤✐s t❛❧❦ ❲♦r❞ ❊q✉❛t✐♦♥s ❛r❡ ✐♥ ◆▲✐♥❙P❆❈❊ ❆❧❣♦r✐t❤♠s ❛♥❞ ❈♦♠♣❧❡①✐t② ▼❛❦❛♥✐♥ ✬✼✼ ✹◆❊❳P❚■▼❊
❊①❛❝t s♣❛❝❡ ❝♦♠♣❧❡①✐t②❄ ●✉t✐❡rr❡③ ✬✾✽ ❊❳P❙P❆❈❊ P❧❛♥❞♦✇s❦✐ ✫ ❘②tt❡r ✬✾✽ ♥❡✇ ❛♣♣r♦❛❝❤ ✖ ✉s✐♥❣ ❝♦♠♣r❡ss✐♦♥ P❧❛♥❞♦✇s❦✐ ✬✾✾ P❙P❆❈❊ ❏✳ ✬✶✸ P❙P❆❈❊ ◆P✲❤❛r❞✱ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P ❚❤✐s t❛❧❦ ❲♦r❞ ❊q✉❛t✐♦♥s ❛r❡ ✐♥ ◆▲✐♥❙P❆❈❊ ❆❧❣♦r✐t❤♠s ❛♥❞ ❈♦♠♣❧❡①✐t② ▼❛❦❛♥✐♥ ✬✼✼ ✹◆❊❳P❚■▼❊ ❬✳ ✳ ✳ ❪
❊①❛❝t s♣❛❝❡ ❝♦♠♣❧❡①✐t②❄ P❧❛♥❞♦✇s❦✐ ✫ ❘②tt❡r ✬✾✽ ♥❡✇ ❛♣♣r♦❛❝❤ ✖ ✉s✐♥❣ ❝♦♠♣r❡ss✐♦♥ P❧❛♥❞♦✇s❦✐ ✬✾✾ P❙P❆❈❊ ❏✳ ✬✶✸ P❙P❆❈❊ ◆P✲❤❛r❞✱ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P ❚❤✐s t❛❧❦ ❲♦r❞ ❊q✉❛t✐♦♥s ❛r❡ ✐♥ ◆▲✐♥❙P❆❈❊ ❆❧❣♦r✐t❤♠s ❛♥❞ ❈♦♠♣❧❡①✐t② ▼❛❦❛♥✐♥ ✬✼✼ ✹◆❊❳P❚■▼❊ ❬✳ ✳ ✳ ❪ ●✉t✐❡rr❡③ ✬✾✽ ❊❳P❙P❆❈❊
❊①❛❝t s♣❛❝❡ ❝♦♠♣❧❡①✐t②❄ P❧❛♥❞♦✇s❦✐ ✬✾✾ P❙P❆❈❊ ❏✳ ✬✶✸ P❙P❆❈❊ ◆P✲❤❛r❞✱ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P ❚❤✐s t❛❧❦ ❲♦r❞ ❊q✉❛t✐♦♥s ❛r❡ ✐♥ ◆▲✐♥❙P❆❈❊ ❆❧❣♦r✐t❤♠s ❛♥❞ ❈♦♠♣❧❡①✐t② ▼❛❦❛♥✐♥ ✬✼✼ ✹◆❊❳P❚■▼❊ ❬✳ ✳ ✳ ❪ ●✉t✐❡rr❡③ ✬✾✽ ❊❳P❙P❆❈❊ P❧❛♥❞♦✇s❦✐ ✫ ❘②tt❡r ✬✾✽ ♥❡✇ ❛♣♣r♦❛❝❤ ✖ ✉s✐♥❣ ❝♦♠♣r❡ss✐♦♥
❊①❛❝t s♣❛❝❡ ❝♦♠♣❧❡①✐t②❄ ❏✳ ✬✶✸ P❙P❆❈❊ ◆P✲❤❛r❞✱ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P ❚❤✐s t❛❧❦ ❲♦r❞ ❊q✉❛t✐♦♥s ❛r❡ ✐♥ ◆▲✐♥❙P❆❈❊ ❆❧❣♦r✐t❤♠s ❛♥❞ ❈♦♠♣❧❡①✐t② ▼❛❦❛♥✐♥ ✬✼✼ ✹◆❊❳P❚■▼❊ ❬✳ ✳ ✳ ❪ ●✉t✐❡rr❡③ ✬✾✽ ❊❳P❙P❆❈❊ P❧❛♥❞♦✇s❦✐ ✫ ❘②tt❡r ✬✾✽ ♥❡✇ ❛♣♣r♦❛❝❤ ✖ ✉s✐♥❣ ❝♦♠♣r❡ss✐♦♥ P❧❛♥❞♦✇s❦✐ ✬✾✾ P❙P❆❈❊
❊①❛❝t s♣❛❝❡ ❝♦♠♣❧❡①✐t②❄ ◆P✲❤❛r❞✱ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P ❚❤✐s t❛❧❦ ❲♦r❞ ❊q✉❛t✐♦♥s ❛r❡ ✐♥ ◆▲✐♥❙P❆❈❊ ❆❧❣♦r✐t❤♠s ❛♥❞ ❈♦♠♣❧❡①✐t② ▼❛❦❛♥✐♥ ✬✼✼ ✹◆❊❳P❚■▼❊ ❬✳ ✳ ✳ ❪ ●✉t✐❡rr❡③ ✬✾✽ ❊❳P❙P❆❈❊ P❧❛♥❞♦✇s❦✐ ✫ ❘②tt❡r ✬✾✽ ♥❡✇ ❛♣♣r♦❛❝❤ ✖ ✉s✐♥❣ ❝♦♠♣r❡ss✐♦♥ P❧❛♥❞♦✇s❦✐ ✬✾✾ P❙P❆❈❊ ❏✳ ✬✶✸ P❙P❆❈❊
❊①❛❝t s♣❛❝❡ ❝♦♠♣❧❡①✐t②❄ ❚❤✐s t❛❧❦ ❲♦r❞ ❊q✉❛t✐♦♥s ❛r❡ ✐♥ ◆▲✐♥❙P❆❈❊ ❆❧❣♦r✐t❤♠s ❛♥❞ ❈♦♠♣❧❡①✐t② ▼❛❦❛♥✐♥ ✬✼✼ ✹◆❊❳P❚■▼❊ ❬✳ ✳ ✳ ❪ ●✉t✐❡rr❡③ ✬✾✽ ❊❳P❙P❆❈❊ P❧❛♥❞♦✇s❦✐ ✫ ❘②tt❡r ✬✾✽ ♥❡✇ ❛♣♣r♦❛❝❤ ✖ ✉s✐♥❣ ❝♦♠♣r❡ss✐♦♥ P❧❛♥❞♦✇s❦✐ ✬✾✾ P❙P❆❈❊ ❏✳ ✬✶✸ P❙P❆❈❊ ◮ ◆P✲❤❛r❞✱ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P
❚❤✐s t❛❧❦ ❲♦r❞ ❊q✉❛t✐♦♥s ❛r❡ ✐♥ ◆▲✐♥❙P❆❈❊ ❆❧❣♦r✐t❤♠s ❛♥❞ ❈♦♠♣❧❡①✐t② ▼❛❦❛♥✐♥ ✬✼✼ ✹◆❊❳P❚■▼❊ ❬✳ ✳ ✳ ❪ ●✉t✐❡rr❡③ ✬✾✽ ❊❳P❙P❆❈❊ P❧❛♥❞♦✇s❦✐ ✫ ❘②tt❡r ✬✾✽ ♥❡✇ ❛♣♣r♦❛❝❤ ✖ ✉s✐♥❣ ❝♦♠♣r❡ss✐♦♥ P❧❛♥❞♦✇s❦✐ ✬✾✾ P❙P❆❈❊ ❏✳ ✬✶✸ P❙P❆❈❊ ◮ ◆P✲❤❛r❞✱ ❜❡❧✐❡✈❡❞ t♦ ❜❡ ✐♥ ◆P ◮ ❊①❛❝t s♣❛❝❡ ❝♦♠♣❧❡①✐t②❄
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