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P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s✿ ❋❛st ❆❧❣♦r✐t❤♠s ❛♥❞ ◆❡✇ ❍❛r❞♥❡ss ❘❡s✉❧ts ❍❡♥♥✐♥❣ ❋❡r♥❛✉ 1 ❋❧♦r✐♥ ▼❛♥❡❛ 2 ❘♦❜❡rt ▼❡r❝❛➩ 2 , 3 ▼❛r❦✉s ▲✳ ❙❝❤♠✐❞ 1 1 ❚r✐❡r ❯♥✐✈❡rs✐t②✱ ●❡r♠❛♥② 2 ❑✐❡❧ ❯♥✐✈❡rs✐t②✱ ●❡r♠❛♥② 3 ❑✐♥❣✬s ❈♦❧❧❡❣❡✱ ▲♦♥❞♦♥✱ ❯❑ ❙❚❆❈❙ ✷✵✶✺

P❛tt❡r♥s ✇✐t❤ ❱❛r✐❛❜❧❡s ❋✐♥✐t❡ ❛❧♣❤❛❜❡t ♦❢ t❡r♠✐♥❛❧s Σ = { a , b , c , d } ❙❡t ♦❢ ✈❛r✐❛❜❧❡s X = { x 1 , x 2 , x 3 , . . . } α ∈ (Σ ∪ X ) + P❛tt❡r♥s w ∈ Σ + ❲♦r❞s h : X → Σ + ❙✉❜st✐t✉t✐♦♥ α = y 1 . . . y n ✱ h ( α ) = h ( y 1 ) . . . h ( y n ) ✱ ✇✐t❤ h ( a ) = a ✱ a ∈ Σ ✳

P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s ♣❛tt❡r♥ α ♠❛t❝❤❡s ✇♦r❞ w ⇐ ⇒ ∃ s✉❜st✐t✉t✐♦♥ h : h ( α ) = w ✳

P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s ♣❛tt❡r♥ α ♠❛t❝❤❡s ✇♦r❞ w ⇐ ⇒ ∃ s✉❜st✐t✉t✐♦♥ h : h ( α ) = w ✳ α = x 1 x 2 x 1 x 3 x 2 w = a b b b a a b b a a a b a b a

P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s ♣❛tt❡r♥ α ♠❛t❝❤❡s ✇♦r❞ w ⇐ ⇒ ∃ s✉❜st✐t✉t✐♦♥ h : h ( α ) = w ✳ α = a b b x 2 a b b x 3 x 2 w = a b b b a a b b a a a b a b a

P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s ♣❛tt❡r♥ α ♠❛t❝❤❡s ✇♦r❞ w ⇐ ⇒ ∃ s✉❜st✐t✉t✐♦♥ h : h ( α ) = w ✳ α = a b b b a a b b x 3 b a w = a b b b a a b b a a a b a b a

P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s ♣❛tt❡r♥ α ♠❛t❝❤❡s ✇♦r❞ w ⇐ ⇒ ∃ s✉❜st✐t✉t✐♦♥ h : h ( α ) = w ✳ α = a b b b a a b b a a a b a b a w = a b b b a a b b a a a b a b a

P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s ♣❛tt❡r♥ α ♠❛t❝❤❡s ✇♦r❞ w ⇐ ⇒ ∃ s✉❜st✐t✉t✐♦♥ h : h ( α ) = w ✳ α = x 1 a x 2 b x 2 x 1 x 2 w = b a c b a c b c b a c b c

P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s ♣❛tt❡r♥ α ♠❛t❝❤❡s ✇♦r❞ w ⇐ ⇒ ∃ s✉❜st✐t✉t✐♦♥ h : h ( α ) = w ✳ α = b a c b a x 2 b x 2 b a c b x 2 w = b a c b a c b c b a c b c

P❛tt❡r♥ ▼❛t❝❤✐♥❣ ✇✐t❤ ❱❛r✐❛❜❧❡s ♣❛tt❡r♥ α ♠❛t❝❤❡s ✇♦r❞ w ⇐ ⇒ ∃ s✉❜st✐t✉t✐♦♥ h : h ( α ) = w ✳ α = b a c b a c b c b a c b c w = b a c b a c b c b a c b c

▼♦t✐✈❛t✐♦♥ ▲❡❛r♥✐♥❣ t❤❡♦r② ✭✐♥❞✉❝t✐✈❡ ✐♥❢❡r❡♥❝❡✱ P❆❈ ❧❡❛r♥✐♥❣✮✱ ❧❛♥❣✉❛❣❡ t❤❡♦r② ✭♣❛tt❡r♥ ❧❛♥❣✉❛❣❡s✮✱ ❝♦♠❜✐♥❛t♦r✐❝s ♦♥ ✇♦r❞s ✭✇♦r❞ ❡q✉❛t✐♦♥s✱ ✉♥❛✈♦✐❞❛❜❧❡ ♣❛tt❡r♥s✱ ❛♠❜✐❣✉✐t② ♦❢ ♠♦r♣❤✐s♠s✱ ❡q✉❛❧✐t② s❡ts✮✱ ♣❛tt❡r♥ ♠❛t❝❤✐♥❣ ✭♣❛r❛♠❡t❡r✐s❡❞ ♠❛t❝❤✐♥❣✱ ✭❣❡♥❡r❛❧✐s❡❞✮ ❢✉♥❝t✐♦♥ ♠❛t❝❤✐♥❣✮✱ ♠❛t❝❤t❡st ❢♦r r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ✇✐t❤ ❜❛❝❦r❡❢❡r❡♥❝❡s ✭t❡①t ❡❞✐t♦rs ✭❣r❡♣✱ ❡♠❛❝s✮✱ ♣r♦❣r❛♠♠✐♥❣ ❧❛♥❣✉❛❣❡ ✭P❡r❧✱ ❏❛✈❛✱ P②t❤♦♥✮✮✱ ❞❛t❛❜❛s❡ t❤❡♦r②✳

❇❛❞ ♥❡✇s✿ ▼❛t❝❤ r❡♠❛✐♥s ❤❛r❞ ✐❢ ♥✉♠❡r✐❝❛❧ ♣❛r❛♠❡t❡rs ❛r❡ r❡str✐❝t❡❞ ✭❢❡✇ ❡①❝❡♣t✐♦♥s✮✿ ✐❢ ♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s ♦r ✇♦r❞ ❧❡♥❣t❤ ❜♦✉♥❞❡❞ ✭tr✐✈✐❛❧✮✳ ▼❛t❝❤ ▼❛t❝❤ st✐❧❧ ❤❛r❞ ✐❢ ❛❧♣❤❛❜❡t s✐③❡ ✱ ❡❛❝❤ ✈❛r✐❛❜❧❡ ❤❛s ❛t ♠♦st ♦❝❝✉rr❡♥❝❡s✱ ❢♦r ❡✈❡r② ✳ ●♦♦❞ ♥❡✇s✿ ❚r❛❝t❛❜❧❡ ✐❢ str✉❝t✉r❡ ♦❢ ♣❛tt❡r♥s ✐s r❡str✐❝t❡❞✳ ❈♦♠♣❧❡①✐t② ▼❛t❝❤✐♥❣ Pr♦❜❧❡♠ ✭ ▼❛t❝❤ ✮ ●✐✈❡♥ ❛ ♣❛tt❡r♥ α ✱ ❛ ✇♦r❞ w ✳ ❉♦❡s α ♠❛t❝❤ w ✭✐✳ ❡✳✱ ∃ h : h ( α ) = w ✮❄ ▼❛t❝❤ ✐s ✭✐♥ ❣❡♥❡r❛❧✮ ◆P✲❝♦♠♣❧❡t❡✳

●♦♦❞ ♥❡✇s✿ ❚r❛❝t❛❜❧❡ ✐❢ str✉❝t✉r❡ ♦❢ ♣❛tt❡r♥s ✐s r❡str✐❝t❡❞✳ ❈♦♠♣❧❡①✐t② ▼❛t❝❤✐♥❣ Pr♦❜❧❡♠ ✭ ▼❛t❝❤ ✮ ●✐✈❡♥ ❛ ♣❛tt❡r♥ α ✱ ❛ ✇♦r❞ w ✳ ❉♦❡s α ♠❛t❝❤ w ✭✐✳ ❡✳✱ ∃ h : h ( α ) = w ✮❄ ▼❛t❝❤ ✐s ✭✐♥ ❣❡♥❡r❛❧✮ ◆P✲❝♦♠♣❧❡t❡✳ ❇❛❞ ♥❡✇s✿ ▼❛t❝❤ r❡♠❛✐♥s ❤❛r❞ ✐❢ ♥✉♠❡r✐❝❛❧ ♣❛r❛♠❡t❡rs ❛r❡ r❡str✐❝t❡❞ ✭❢❡✇ ❡①❝❡♣t✐♦♥s✮✿ ◮ ▼❛t❝❤ ∈ P ✐❢ ♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s ♦r ✇♦r❞ ❧❡♥❣t❤ ❜♦✉♥❞❡❞ ✭tr✐✈✐❛❧✮✳ ◮ ▼❛t❝❤ st✐❧❧ ❤❛r❞ ✐❢ ⋆ ❛❧♣❤❛❜❡t s✐③❡ 2 ✱ ⋆ ❡❛❝❤ ✈❛r✐❛❜❧❡ ❤❛s ❛t ♠♦st 2 ♦❝❝✉rr❡♥❝❡s✱ ⋆ | h ( x ) | ≤ 3 ❢♦r ❡✈❡r② x ✳

❈♦♠♣❧❡①✐t② ▼❛t❝❤✐♥❣ Pr♦❜❧❡♠ ✭ ▼❛t❝❤ ✮ ●✐✈❡♥ ❛ ♣❛tt❡r♥ α ✱ ❛ ✇♦r❞ w ✳ ❉♦❡s α ♠❛t❝❤ w ✭✐✳ ❡✳✱ ∃ h : h ( α ) = w ✮❄ ▼❛t❝❤ ✐s ✭✐♥ ❣❡♥❡r❛❧✮ ◆P✲❝♦♠♣❧❡t❡✳ ❇❛❞ ♥❡✇s✿ ▼❛t❝❤ r❡♠❛✐♥s ❤❛r❞ ✐❢ ♥✉♠❡r✐❝❛❧ ♣❛r❛♠❡t❡rs ❛r❡ r❡str✐❝t❡❞ ✭❢❡✇ ❡①❝❡♣t✐♦♥s✮✿ ◮ ▼❛t❝❤ ∈ P ✐❢ ♥✉♠❜❡r ♦❢ ✈❛r✐❛❜❧❡s ♦r ✇♦r❞ ❧❡♥❣t❤ ❜♦✉♥❞❡❞ ✭tr✐✈✐❛❧✮✳ ◮ ▼❛t❝❤ st✐❧❧ ❤❛r❞ ✐❢ ⋆ ❛❧♣❤❛❜❡t s✐③❡ 2 ✱ ⋆ ❡❛❝❤ ✈❛r✐❛❜❧❡ ❤❛s ❛t ♠♦st 2 ♦❝❝✉rr❡♥❝❡s✱ ⋆ | h ( x ) | ≤ 3 ❢♦r ❡✈❡r② x ✳ ●♦♦❞ ♥❡✇s✿ ❚r❛❝t❛❜❧❡ ✐❢ str✉❝t✉r❡ ♦❢ ♣❛tt❡r♥s ✐s r❡str✐❝t❡❞✳

◆♦t❛t✐♦♥ var( α ) ❙❡t ♦❢ ✈❛r✐❛❜❧❡s ♦❝❝✉rr✐♥❣ ✐♥ ♣❛tt❡r♥ α ✳ | α | x ◆✉♠❜❡r ♦❢ ♦❝❝✉rr❡♥❝❡s ♦❢ ✈❛r✐❛❜❧❡ x ✐♥ ♣❛tt❡r♥ α ✳

◆♦♥✲❈r♦ss P❛tt❡r♥s ✿ ✐s ♥♦t ♣♦ss✐❜❧❡✳ ❊✳ ❣✳✱ ❙tr✉❝t✉r❛❧ ❘❡str✐❝t✐♦♥s ♦❢ P❛tt❡r♥s ❘❡❣✉❧❛r P❛tt❡r♥s ✿ | α | x = 1 ✱ x ∈ var( α ) ✳ ❊✳ ❣✳✱ α = ab x 1 x 2 b x 3 aaa x 4 b ✳

❙tr✉❝t✉r❛❧ ❘❡str✐❝t✐♦♥s ♦❢ P❛tt❡r♥s ❘❡❣✉❧❛r P❛tt❡r♥s ✿ | α | x = 1 ✱ x ∈ var( α ) ✳ ❊✳ ❣✳✱ α = ab x 1 x 2 b x 3 aaa x 4 b ✳ ◆♦♥✲❈r♦ss P❛tt❡r♥s ✿ α = . . . x . . . y . . . x . . . ✐s ♥♦t ♣♦ss✐❜❧❡✳ ❊✳ ❣✳✱ α = x 1 aba x 1 a x 1 x 2 x 2 ba x 2 x 3 x 3 bb x 3 a x 3

P❛tt❡r♥ ✇✐t❤ ❇♦✉♥❞❡❞ ❙❝♦♣❡ ❈♦✐♥❝✐❞❡♥❝❡ ❉❡❣r❡❡ ✿ ❙❝♦♣❡ ✭♦❢ ✮✿ s❤♦rt❡st ❢❛❝t♦r ❝♦♥t❛✐♥✐♥❣ ❛❧❧ ♦❝❝✳ ♦❢ ✱ ❙❝♦♣❡ ❝♦✐♥❝✐❞❡♥❝❡ ❞❡❣r❡❡✿ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ❝♦✐♥❝✐❞✐♥❣ s❝♦♣❡s✳ ❙tr✉❝t✉r❛❧ ❘❡str✐❝t✐♦♥s ♦❢ P❛tt❡r♥s k ✲❘❡♣❡❛t❡❞✲❱❛r✐❛❜❧❡ P❛tt❡r♥s ✿ |{ x ∈ var( α ) | | α | x ≥ 2 }| ≤ k ✳ ❊✳ ❣✳✱ α = x 1 ab x 2 a x 2 a x 3 ba x 2 bb x 4 x 2 x 5 ✐s ❛ 1 ✲r❡♣❡❛t❡❞✲✈❛r✐❛❜❧❡ ♣❛tt❡r♥✳

❙tr✉❝t✉r❛❧ ❘❡str✐❝t✐♦♥s ♦❢ P❛tt❡r♥s k ✲❘❡♣❡❛t❡❞✲❱❛r✐❛❜❧❡ P❛tt❡r♥s ✿ |{ x ∈ var( α ) | | α | x ≥ 2 }| ≤ k ✳ ❊✳ ❣✳✱ α = x 1 ab x 2 a x 2 a x 3 ba x 2 bb x 4 x 2 x 5 ✐s ❛ 1 ✲r❡♣❡❛t❡❞✲✈❛r✐❛❜❧❡ ♣❛tt❡r♥✳ P❛tt❡r♥ ✇✐t❤ ❇♦✉♥❞❡❞ ❙❝♦♣❡ ❈♦✐♥❝✐❞❡♥❝❡ ❉❡❣r❡❡ ✿ ❙❝♦♣❡ ✭♦❢ x ✮✿ s❤♦rt❡st ❢❛❝t♦r ❝♦♥t❛✐♥✐♥❣ ❛❧❧ ♦❝❝✳ ♦❢ x ✱ ❙❝♦♣❡ ❝♦✐♥❝✐❞❡♥❝❡ ❞❡❣r❡❡✿ ♠❛①✐♠✉♠ ♥✉♠❜❡r ♦❢ ❝♦✐♥❝✐❞✐♥❣ s❝♦♣❡s✳ α 1 = x 1 x 2 x 1 x 3 x 2 x 3 x 1 x 2 x 3 scd( α 1 ) = 3 α 2 = x 1 x 2 x 1 x 1 x 2 x 3 x 2 x 3 x 3 scd( α 2 ) = 2