strtrs r sst - PowerPoint PPT Presentation

Pr❡❞✐❝t t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ ❛ s②st❡♠ ✶ ❛ss❡rt ❛ ♣r♦♣❡rt② ♦❢ ❛ s②st❡♠ ✷ ♣r♦✈❡ t❤❛t t❤❡ ❛ss❡rt✐♦♥ ❤♦❧❞s ❢♦r t❤❡ s②st❡♠ ◗ ✿ ❆r❡ t❤❡r❡ ❛❧❣♦r✐t❤♠s ❢♦r ✭✷✮ ❄ ❆ ✿ ■t ❞❡♣❡♥❞s ♦♥ ● t❤❡ ❝❧❛ss ♦❢ s②st❡♠s✱ ● t❤❡ ❝❧❛ss ♦❢ ♣r♦♣❡rt✐❡s✳ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸ ✴ ✾✼

Pr❡❞✐❝t t❤❡ ❜❡❤❛✈✐♦✉r ♦❢ ❛ s②st❡♠ ✶ ❛ss❡rt ❛ ♣r♦♣❡rt② ♦❢ ❛ s②st❡♠ ✷ ♣r♦✈❡ t❤❛t t❤❡ ❛ss❡rt✐♦♥ ❤♦❧❞s ❢♦r t❤❡ s②st❡♠ ◗ ✿ ❆r❡ t❤❡r❡ ❛❧❣♦r✐t❤♠s ❢♦r ✭✷✮ ❄ ❆ ✿ ■t ❞❡♣❡♥❞s ♦♥ ● t❤❡ ❝❧❛ss ♦❢ s②st❡♠s✱ ● t❤❡ ❝❧❛ss ♦❢ ♣r♦♣❡rt✐❡s✳ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸ ✴ ✾✼

✶ ❆①✐♦♠❛t✐❝ ♠❡t❤♦❞ s②st❡♠ ✐s ❞❡s❝r✐❜❡❞ ❜② ❛ s❡t ♦❢ s❡♥t❡♥❝❡s ✱ ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② ❛ s❡♥t❡♥❝❡ ✱ ❄ ✭ ❡✳❣✳ r❡s♦❧✉t✐♦♥ ✮ ✷ ▼♦❞❡❧✲t❤❡♦r❡t✐❝ ♠❡t❤♦❞ s②st❡♠ ✐s ❣✐✈❡♥ ❛s str✉❝t✉r❡ ✱ ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② ❛ s❡♥t❡♥❝❡ ✱ ❄ ❚✇♦ ✇❛②s ♦❢ s②st❡♠ ♠♦❞❡❧❧✐♥❣ ✭❛♠♦♥❣ ♦t❤❡rs✮ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹ ✴ ✾✼

✷ ▼♦❞❡❧✲t❤❡♦r❡t✐❝ ♠❡t❤♦❞ s②st❡♠ ✐s ❣✐✈❡♥ ❛s str✉❝t✉r❡ ✱ ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② ❛ s❡♥t❡♥❝❡ ✱ ❄ ❚✇♦ ✇❛②s ♦❢ s②st❡♠ ♠♦❞❡❧❧✐♥❣ ✭❛♠♦♥❣ ♦t❤❡rs✮ ✶ ❆①✐♦♠❛t✐❝ ♠❡t❤♦❞ ● s②st❡♠ ✐s ❞❡s❝r✐❜❡❞ ❜② ❛ s❡t ♦❢ s❡♥t❡♥❝❡s Φ ✱ ● ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② ❛ s❡♥t❡♥❝❡ ϕ ✱ ● Φ ⊢ ϕ ❄ ✭ ⇔ Φ ∪ {¬ ϕ } ⊢ ◻ ❡✳❣✳ r❡s♦❧✉t✐♦♥ ✮ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹ ✴ ✾✼

❚✇♦ ✇❛②s ♦❢ s②st❡♠ ♠♦❞❡❧❧✐♥❣ ✭❛♠♦♥❣ ♦t❤❡rs✮ ✶ ❆①✐♦♠❛t✐❝ ♠❡t❤♦❞ ● s②st❡♠ ✐s ❞❡s❝r✐❜❡❞ ❜② ❛ s❡t ♦❢ s❡♥t❡♥❝❡s Φ ✱ ● ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② ❛ s❡♥t❡♥❝❡ ϕ ✱ ● Φ ⊢ ϕ ❄ ✭ ⇔ Φ ∪ {¬ ϕ } ⊢ ◻ ❡✳❣✳ r❡s♦❧✉t✐♦♥ ✮ ✷ ▼♦❞❡❧✲t❤❡♦r❡t✐❝ ♠❡t❤♦❞ ● s②st❡♠ ✐s ❣✐✈❡♥ ❛s str✉❝t✉r❡ A ✱ ● ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② ❛ s❡♥t❡♥❝❡ ϕ ✱ ● A ⊧ ϕ ❄ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹ ✴ ✾✼

❚✇♦ ✇❛②s ♦❢ s②st❡♠ ♠♦❞❡❧❧✐♥❣ ✭❛♠♦♥❣ ♦t❤❡rs✮ ✶ ❆①✐♦♠❛t✐❝ ♠❡t❤♦❞ ● s②st❡♠ ✐s ❞❡s❝r✐❜❡❞ ❜② ❛ s❡t ♦❢ s❡♥t❡♥❝❡s Φ ✱ ● ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② ❛ s❡♥t❡♥❝❡ ϕ ✱ ● Φ ⊢ ϕ ❄ ✭ ⇔ Φ ∪ {¬ ϕ } ⊢ ◻ ❡✳❣✳ r❡s♦❧✉t✐♦♥ ✮ ✷ ▼♦❞❡❧✲t❤❡♦r❡t✐❝ ♠❡t❤♦❞ ● s②st❡♠ ✐s ❣✐✈❡♥ ❛s str✉❝t✉r❡ A ✱ ● ♣r♦♣❡rt② ✐s ❣✐✈❡♥ ❜② ❛ s❡♥t❡♥❝❡ ϕ ✱ ● A ⊧ ϕ ❄ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹ ✴ ✾✼

❊①❛♠♣❧❡ ✭❛ ♥❛✐✈❡ ♦♥❡✮ birth birth birth birth birth 1 2 3 4 5 6 ⋯ ❍❡r❜✐✈♦rs death death death death death eating eating eating eating eating birth birth birth birth birth 1 2 3 4 5 6 ⋯ Pr❡❞❛t♦rs death death death death death ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺ ✴ ✾✼

❊①❛♠♣❧❡ ✭❝♦♥t✐♥✉❡❞✮ ■♥t❡r❛❝t✐♦♥ Pr❡❞❛t♦rs ❍❡r❜✐✈♦rs ❡❛t✐♥❣ ❞❡❛t❤ ( e,d ) ❜✐rt❤ ( b, =) ❞❡❛t❤ ( d, =) ❜✐rt❤ (= ,b ) ❞❡❛t❤ (= ,d ) ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻ ✴ ✾✼

❊①❛♠♣❧❡ ✭s②♥❝❤r♦♥✐s❡❞ ♣r♦❞✉❝t✮ ⋮ ⋮ ⋮ ⋮ ( b, =) ( b, =) ( b, =) ⋯ ( 1 , 3 ) ( 2 , 3 ) ( 3 , 3 ) ( 4 , 3 ) ( d, =) ( d, =) ( d, =) (= , d ) (= , d ) (= , d ) (= , d ) (= , b ) (= , b ) (= , b ) (= , b ) ( e, d ) ( e, d ) ( e, d ) ( e, d ) ( b, =) ( b, =) ( b, =) ⋯ ( 1 , 2 ) ( 2 , 2 ) ( 3 , 2 ) ( 4 , 2 ) ( d, =) ( d, =) ( d, =) (= , d ) (= , d ) (= , d ) (= , d ) (= , b ) (= , b ) (= , b ) (= , b ) ( e, d ) ( e, d ) ( e, d ) ( e, d ) ( b, =) ( b, =) ( b, =) ⋯ ( 1 , 1 ) ( 2 , 1 ) ( 3 , 1 ) ( 4 , 1 ) ( d, =) ( d, =) ( d, =) ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼ ✴ ✾✼

❙tr✉❝t✉r❡s ❆❧♣❤❛❜❡t Σ ✇✐t❤ ❛r✐t✐❡s α ∶ Σ → IN A ∶ Σ → ⋃ n ∈ α ( Σ ) ℘ ( A n ) ❘❡❧❛t✐♦♥❛❧ str✉❝t✉r❡ ✭✶✮ ❝♦✉♥t❛❜❧❡ ♦r ✜♥✐t❡ ✉♥✐✈❡rs❡ A ❢♦r ❡❛❝❤ a ∈ Σ ✱ A ( a ) ✐s ❛ r❡❧❛t✐♦♥ ♦❢ ❛r✐t② α ( a ) A ( a ) ⊆ A α ( a ) ❙♣❡❝✐❛❧ ❝❛s❡s max ( α ( Σ )) ≤ 2 ↝ A ✐s ❛ ❞✐r❡❝t❡❞ ❣r❛♣❤ ✇❤❡r❡ ❜✐♥❛r② r❡❧❛t✐♦♥ s②♠❜♦❧s ❛r❡ ❡❞❣❡ ❧❛❜❡❧s ❛♥❞ ✉♥❛r② r❡❧❛t✐♦♥ s②♠❜♦❧s ❛r❡ ✈❡rt❡① ❧❛❜❡❧s α ( a ) > 2 ↝ A ( a ) ♠❛② ❜❡ s❡❡♥ ❛s ❛ s❡t ♦❢ a ✕❧❛❜❡❧❧❡❞ ❤②♣❡r❡❞❣❡s✱ ↝ A ✐s ❛ ✭❞✐r❡❝t❡❞✮ ❤②♣❡r❣r❛♣❤ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✽ ✴ ✾✼

❊①❛♠♣❧❡s ✶ ( e,d ) , ( b, − ) , ( d, − ) , ( − ,b )( − ,d ) ❛r❡ ❜✐♥❛r② r❡❧❛t✐♦♥s ↝ ❞✐r❡❝t❡❞ ❣r❛♣❤ ✷ t❡r♥❛r② r❡❧❛t✐♦♥ ✇✐t❤ t✉♣❧❡s ✭ ❢❛t❤❡r✱ ♠♦t❤❡r✱ ❝❤✐❧❞ ✮ ↝ ❡✈❡r② t✉♣❧❡ ✐s ❛ ❤②♣❡r❡❞❣❡ ↝ ❞✐r❡❝t❡❞ ❤②♣❡r❣r❛♣❤ Dev Esha F arukh Am y Ben Kani Cha y a Harish Ja y Gina Ila ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✾ ✴ ✾✼

❲♦r❦✐♥❣ ❡①❛♠♣❧❡ Σ = { likes , talks } α ( likes ) = 2 ✱ α ( talks ) = 3 A = { Amy , Ben , Chaya , Dora , Elil , Farukh } ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✵ ✴ ✾✼

A ( ❧✐❦❡s ) ❛s ❛ t❛❜❧❡ ❧✐❦❡s ❧✐❦✐♥❣ ❧✐❦❡❞ ❈❤❛②❛ ❆♠② ❊❧✐❧ ❆♠② ❆♠② ❆♠② ❈❤❛②❛ ❇❡♥ ❊❧✐❧ ❈❤❛②❛ ❆♠② ❇❡♥ ❈❤❛②❛ ❈❤❛②❛ ❊❧✐❧ ❊❧✐❧ ❆♠② ❈❤❛②❛ ❈❤❛②❛ ❉♦r❛ ❋❛r✉❦❤ ❆♠② ❆♠② ❉♦r❛ ❈❤❛②❛ ❊❧✐❧ ❋❛r✉❦❤ ❇❡♥ ❆♠② ❋❛r✉❦❤ ❈❤❛②❛ ❋❛r✉❦❤ ❋❛r✉❦❤ ❈❤❛②❛ ❇❡♥ ❆♠② ❉♦r❛ ❈❤❛②❛ ❋❛r✉❦❤ ❉♦r❛ ❇❡♥ ❈❤❛②❛ ❉♦r❛ ❉♦r❛ ❋❛r✉❦❤ ❊❧✐❧ ❇❡♥ ❉♦r❛ ❉♦r❛ ❋❛r✉❦❤ ❋❛r✉❦❤ ❋❛r✉❦❤ ❇❡♥ ❊❧✐❧ ❇❡♥ ❋❛r✉❦❤ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✶ ✴ ✾✼

❋✐rst✲♦r❞❡r q✉❡r✐❡s✿ ❧♦♦❦✐♥❣ ❢♦r s♦❧✉t✐♦♥s ♦❢ ❢♦r♠✉❧❛❡ likes ( x,x ) ¬∃ y likes ( x,y ) ✶ ✺ likes ( x,y ) ∧ x = y ✷ ✻ likes ( x,y ) ∧ likes ( y,x ) ∧ ¬ likes ( x,x ) ∀ y likes ( x,y ) ✸ ∃ x likes ( x,y ) ✹ ✼ ∀ y ( ∀ z likes ( y,z ) ⇒ likes ( x,y )) ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✷ ✴ ✾✼

[ ∣ likes ( x,x )∣ ] { ❆♠②✱ ❈❤❛②❛✱ ❉♦r❛✱ ❊❧✐❧✱ ❋❛r✉❦❤ } = ✶ ❆♠② = ❈❤❛②❛ ❉♦r❛ ❊❧✐❧ ❋❛r✉❦❤ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✸ ✴ ✾✼

[ ∣ likes ( x,y ) ∧ x = y ∣ ] { ✭❆♠②✱ ❆♠②✮✱ ✭❈❤❛②❛✱ ❈❤❛②❛✮✱ = ✷ ✭❉♦r❛✱ ❉♦r❛✮✱ ✭❊❧✐❧✱ ❊❧✐❧✮✱ ✭❋❛r✉❦❤✱ ❋❛r✉❦❤✮ ⑥ ❆♠② ❆♠② = ❈❤❛②❛ ❈❤❛②❛ ❉♦r❛ ❉♦r❛ ❊❧✐❧ ❊❧✐❧ ❋❛r✉❦❤ ❋❛r✉❦❤ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✹ ✴ ✾✼

[ ∣∀ y likes ( x,y )∣ ] { ❈❤❛②❛✱ ❋❛r✉❦❤ } = ✸ ❈❤❛②❛ = ❋❛r✉❦❤ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✺ ✴ ✾✼

[ ∣∃ x likes ( x,y )∣ ] ❆♠② = ✹ ❇❡♥ ❈❤❛②❛ ❉♦r❛ ❊❧✐❧ ❋❛r✉❦❤ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✻ ✴ ✾✼

✺ [ ∣¬∃ y likes ( x,y )∣ ] = ∅ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✼ ✴ ✾✼

[ ∣ likes ( x,y ) ∧ likes ( y,x ) ∧ ¬ likes ( x,x )∣ ] { ✭❇❡♥✱ ❆♠②✮✱ = ✻ ✭❇❡♥✱ ❈❤❛②❛✮✱ ✭❇❡♥✱ ❋❛r✉❦❤✮ ⑥ ❇❡♥ ❆♠② = ❇❡♥ ❈❤❛②❛ ❇❡♥ ❋❛r✉❦❤ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✽ ✴ ✾✼

[ ∣∀ y (∀ z likes ( y,z ) ⇒ likes ( x,y ))∣ ] { ❆♠②✱ ❇❡♥✱ = ✼ ❈❤❛②❛✱ ❉♦r❛✱ ❋❛r✉❦❤ ⑥ ❆♠② = ❇❡♥ ❈❤❛②❛ ❉♦r❛ ❋❛r✉❦❤ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✶✾ ✴ ✾✼

❊①♣❧✐❝✐t ✭❤②♣❡r✮❡❞❣❡s✿ ✐♥❝✐❞❡♥❝❡ str✉❝t✉r❡s ↝ ❘❡❧❛t✐♦♥❛❧ str✉❝t✉r❡ ✭✶✮ ✶ ✉♥✐✈❡rs❡ ✐♠♣❧✐❝✐t ✭❤②♣❡r✮❡❞❣❡s ❘❡❧❛t✐♦♥❛❧ str✉❝t✉r❡ ✭✷✮ ✷ ✉♥✐✈❡rs❡s ✿ A ✈❡rt ✱ A ❡❞❣ A = ⟨ A ✈❡rt , A ❧❛❜ ⟩ A ✈❡rt ∶ A ❡❞❣ → ⋃ n ∈ α ( Σ ) A n A ❧❛❜ ∶ A ❡❞❣ → Σ ✈❡rt s✳t✳ A ✈❡rt ( e ) ∈ A α ( A ❧❛❜ ( e )) ❢♦r ❡✈❡r② e ∈ A ❡❞❣ ✈❡rt ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✵ ✴ ✾✼

❘❡♠❛r❦ ❛❜♦✉t ✶st ♦r❞❡r ❛t♦♠✐❝ ❢♦r♠✉❧❛s ❋♦r ❡❛❝❤ a ∈ Σ ✇✐t❤ α ( a ) = n t②♣❡ ✶ str✉❝t✉r❡s a ( x 1 ,...,x n ) a ( x ) t②♣❡ ✷ str✉❝t✉r❡s ❡❞❣ a ( y,x 1 ,...,x n ) y r❛♥❣❡s ♦✈❡r A ❡❞❣ x 1 ,...,x n r❛♥❣❡ ♦✈❡r A ✈❡rt ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✶ ✴ ✾✼

❊①♣r❡ss✐♥❣ ♣r♦♣❡rt✐❡s ♦❢ ❛ str✉❝t✉r❡ ❆ ♣r♦♣❡rt② P ✐s ❡①♣r❡ss✐❜❧❡ ✐♥ ❛ ❧♦❣✐❝ L ✐❢ t❤❡r❡ ❡①✐sts ❛♥ L ✲s❡♥t❡♥❝❡ ϕ s✳t✳✱ ❢♦r ❡✈❡r② str✉❝t✉r❡ A ⇔ A ⊧ ϕ A ❤❛s ♣r♦♣❡rt② P ❆ r❡❧❛t✐♦♥ r ⊆ A n ✐s ❞❡✜♥❛❜❧❡ ✐♥ L ✐❢ t❤❡r❡ ❡①✐sts ❛♥ L ✲❢♦r♠✉❧❛ ϕ ( x ) ✇✐t❤ ❢r❡❡ ✈❛r✐❛❜❧❡s x = ( x 1 ,...,x n ) s✳t✳ r = { d ∈ A n ∣ A ⊧ ϕ ( d )} ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✷ ✴ ✾✼

❋❖ ✿ ✜rst ♦r❞❡r ❧♦❣✐❝ ❊①❛♠♣❧❡s ♦❢ ♣r♦♣❡rt✐❡s ❢♦r t②♣❡ ✶ str✉❝t✉r❡s ❋✐♥✐t❡ ❛✉t♦♠❛t♦♥ ♦✈❡r { a,b,c } ✐s ♥♦r♠❛❧✐③❡❞ ✿ Σ = { a,b,c,ι,f } ✇✐t❤ α ( a ) = α ( b ) = α ( c ) = 2 ✱ α ( ι ) = α ( f ) = 1 ∃ x ( ι ( x ) ∧ ∀ x ′ ( ι ( x ′ ) ⇒ x ′ = x ) ∧ ∀ z ¬( a ( z,x ) ∨ b ( z,x ) ∨ c ( z,x ))) ∧ ∃ y ( f ( y ) ∧ ∀ y ′ ( f ( y ′ ) ⇒ y ′ = y ) ∧ ∀ z ¬( a ( y,z ) ∨ b ( y,z ) ∨ c ( y,z ))) ❊✈❡r② r❡❧❛t✐♦♥ ✐s ✏❢✉♥❝t✐♦♥♥❛❧✑ ✇✳r✳t✳ t♦ ✐ts ❧❛st ❝♦♠♣♦♥❡♥t ✿ Σ = { f,g,h } ✇✐t❤ α ( f ) = 2 ✱ α ( g ) = α ( h ) = 3 ∀ x ∀ y 1 ∀ y 2 (( f ( x,y 1 ) ∧ f ( x,y 2 )) ⇒ y 1 = y 2 ) ∧ ∀ x ∀ y ∀ z 1 ∀ z 2 (( g ( x,y,z 1 ) ∧ g ( x,y,z 2 )) ⇒ z 1 = z 2 ) ∧ ∀ x ∀ y ∀ z 1 ∀ z 2 (( h ( x,y,z 1 ) ∧ h ( x,y,z 2 )) ⇒ z 1 = z 2 ) ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✸ ✴ ✾✼

❊①❛♠♣❧❡ ♦❢ ❛ ♣r♦♣❡rt② ❢♦r t②♣❡ ✷ str✉❝t✉r❡s ❆ ❣r❛♣❤ ✐s s✐♠♣❧❡ ✿ ∀ e 1 ∀ e 2 ∀ x ∀ y ⋁ (( ❡❞❣ a ( e 1 ,x,y ) ∧ ❡❞❣ a ( e 2 ,x,y )) ⇒ e 1 = e 2 ) a ∈ Σ ❛ss✉♠✐♥❣ t❤❛t α ( Σ ) = { 2 } ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✹ ✴ ✾✼

▲✐♠✐t❛t✐♦♥s ♦❢ ❋❖ ■♠♣♦ss✐❜❧❡ t♦ ❡①♣r❡ss t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♣❛t❤ ❢r♦♠ ♦♥❡ ✈❡rt❡① t♦ ❛♥♦t❤❡r ■♥ ❢❛❝t✱ ✐♥ ❛ ❝❡rt❛✐♥ s❡♥s❡ ♦♥❧② ❧♦❝❛❧ ♣r♦♣❡rt✐❡s ♠❛② ❜❡ ❡①♣r❡ss❡❞✳ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✺ ✴ ✾✼

❍❛♥❢✬s✴●❛✐❢♠❛♥✬s✲❧✐❦❡ t❤❡♦r❡♠ r ✲❧♦❝❛❧ ❢♦r♠✉❧❛ ✭❛r♦✉♥❞ x ✮ ✿ q✉❛♥t✐✜❝❛t✐♦♥ ✐s ❜♦✉♥❞ t♦ ❛♥ r ✲♥❡✐❣❤❜♦✉r❤♦♦❞ ∃ y ( δ ( x,y ) ≤ r ∧ ϕ ) ∀ y ( δ ( x,y ) ≤ r ⇒ ϕ ) ♦r ✏ δ ( x,y ) ≤ r ✑ ✐s ❛ ✶st ♦r❞❡r ❢♦r♠✉❧❛✱ ✇❤❡r❡ ϕ ✐s ❡✐t❤❡r q✉❛♥t✐✜❡r✲❢r❡❡ ♦r r ✲❧♦❝❛❧ ❛r♦✉♥❞ x ❚❤❡♦r❡♠ ✭❙❝❤✇❡♥t✐❝❦ ❛♥❞ ❇❛rt❤❡❧♠❛♥♥✱ ✶✾✾✽✮ ❊✈❡r② ✜rst ♦r❞❡r ❢♦r♠✉❧❛ ✐s ❡q✉✐✈❛❧❡♥t t♦ ∃ x ∀ y ϕ ( x,y ) ✇❤❡r❡ ϕ ✐s r ✲❧♦❝❛❧ ❛r♦✉♥❞ x ❢♦r s♦♠❡ r ∈ IN ✳ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✻ ✴ ✾✼

❋❖✭❘✮ ✿ ✜rst ♦r❞❡r ❧♦❣✐❝ ✇✐t❤ r❡❛❝❤❛❜✐❧✐t② ❋❖ ✰ r❡❛❝❤❛❜✐❧✐t② ♣r❡❞✐❝❛t❡ r❡❛❝❤ ( x,y ) ♣❛t❤ ✇✐t❤ ❧❛❜❡❧ w ∈ Σ ∗ ✐♥ A ❢r♦♠ d ∈ A t♦ e ∈ A ✐s ❞❡✜♥❡❞ ✐♥❞✉❝t✐✈❡❧② ✭ 0 ✮ w = ε ✇❤❡♥ d = e ✭ n + 1 ✮ w = ua ✇❤❡♥ t❤❡r❡ ✐s ❛ ♣❛t❤ ✇✐t❤ ❧❛❜❡❧ u ❢r♦♠ d t♦ s♦♠❡ d ′ ∈ A s✳t✳ e = d ′ ✇❤❡♥ α ( a ) = 1 ❛♥❞ e ∈ A ( a ) ♣r♦✈✐❞❡❞ t❤❛t a ✐s ♥♦t t❤❡ ❧❛st ❧❡tt❡r ♦❢ u ✱ ✇❤❡♥ α ( a ) > 1 t❤❡r❡ ❡①✐sts ( d 1 ,...,d n ) ∈ A ( a ) s✳t✳ d ′ = d i ✱ e = d j ❢♦r s♦♠❡ i < j ✳ A ⊧ r❡❛❝❤ ( d,e ) t❤❡r❡ ✐s ❛ ♣❛t❤ ❢r♦♠ d t♦ e ✳ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✼ ✴ ✾✼

❊①❛♠♣❧❡ ✭r❡❛❝❤❛❜✐❧✐t②✮ α ( a ) = 3 α ( b ) = 4 α ( c ) = 1 α ( d ) = α ( e ) = 2 a 1 2 3 4 d A ⊧ r❡❛❝❤ ( 0 , 5 ) d b 0 A 5 e s♦♠❡ ♣❛t❤ ❧❛❜❡❧s ✿ dacb ✱ eedacb ✱ daddacb ✱ dacbacb ❊①❛♠♣❧❡ ✭♣r♦♣❡rt②✮ ◆♦ ❞❡❛❞❧♦❝❦ ♠❛② ♦❝❝✉r ✿ ∀ x ∀ y (( ι ( x ) ∧ r❡❛❝❤ ( x,y )) ⇒ ∃ z ( z ≠ y ∧ ⋁ a ( y,z ))) a ∈ Σ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✽ ✴ ✾✼

❋❖✭❘✱ s✉❜✮ ✿ ❛ s❧✐❣❤t ❡①t❡♥s✐♦♥ ♦❢ ❋❖✭❘✮ ❋❖ ✰ r❡❛❝❤❛❜✐❧✐t② ♣r❡❞✐❝❛t❡s ✿ r❡❛❝❤ Γ ( x,y ) ❢♦r ❡✈❡r② Γ ⊆ Σ A ⊧ r❡❛❝❤ Γ ( d,e ) t❤❡r❡ ✐s ❛ ♣❛t❤ ❢r♦♠ d t♦ e ❧❛❜❡❧❧❡❞ ❜② ❛ ✇♦r❞ ✐♥ Γ ∗ ✳ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✷✾ ✴ ✾✼

❊①❛♠♣❧❡ ✭♣r♦♣❡rt②✮ ❆♥ ♦✛✲❧✐♥❡ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ ♦✈❡r ❛♥ ✐♥♣✉t ❛❧♣❤❛❜❡t ∆ ❛ ✇♦r❦ t❛♣❡ ✇✐t❤ r❡❛❞✴✇r✐t❡ ❤❡❛❞✱ ❛♥ ✐♥♣✉t t❛♣❡ ✇✐t❤ r❡❛❞✲♦♥❧② ❤❡❛❞ t❤❛t ❝❛♥♥♦t ♠♦✈❡ ❜❛❝❦✇❛r❞s✳ ❚❤❡ ❜❡❤❛✈✐♦✉r ♦❢ ❛ ❚▼ ✐s r❡♣r❡s❡♥t❡❞ ❜② ✐ts tr❛♥s✐t✐♦♥ ❣r❛♣❤ ✈❡rt✐t❡s ❛r❡ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥s ♦❢ ❚▼ ❡✈❡r② ❡❞❣❡ ❝♦rr❡s♣♦♥❞s t♦ ❛ ♠♦✈❡ ❛♥ ❡❞❣❡ ✐s ❧❛❜❡❧❧❡❞ ❜② a ∈ ∆ ✐❢ a ✐s r❡❛❞ ♦♥ t❤❡ ✐♥♣✉t t❛♣❡ ❛♥ ❡❞❣❡ ✐s ❧❛❜❡❧❧❡❞ ❜② τ ✐❢ ❚▼ ♣❡r❢♦r♠s ❛ s✐❧❡♥t ♠♦✈❡ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ ❡①♣r❡ss❡s t❤❡ ♣r♦♣❡rt② t❤❛t ❛ ❚▼ ❞♦❡s ♥♦t ❧♦♦♣✳ ∀ x ∀ y (( ι ( x )∧ r❡❛❝❤ ( x,y )) ⇒ ¬∃ z ( z ≠ y ∧ r❡❛❝❤ { τ } ( y,z )∧ r❡❛❝❤ { τ } ( z,y ))) ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✵ ✴ ✾✼

▼♦r❡ ❣❡♥❡r❛❧ ❡①t❡♥s✐♦♥s ♦❢ ❋❖✭❘✮ ▲❡t L ⊆ ℘ ( Σ ∗ ) ❜❡ ❛ ❧❛♥❣✉❛❣❡ ❢❛♠✐❧② ❋❖✭ L ✮ ✿ ❋❖ ✰ r❡❛❝❤❛❜✐❧✐t② ♣r❡❞✐❝❛t❡s r❡❛❝❤ L ( x,y ) ❢♦r ❡✈❡r② ❧❛♥❣✉❛❣❡ L ∈ L A ⊧ r❡❛❝❤ L ( d,e ) ✇❤❡♥ t❤❡r❡ ✐s ❛ ♣❛t❤ ❢r♦♠ d t♦ e ❧❛❜❡❧❧❡❞ ❜② ❛ ✇♦r❞ ✐♥ L ✳ ▼❛✐♥❧② ❋❖✭❘❡❣✮ ❢♦r t❤❡ ❢❛♠✐❧② ❘❡❣ ♦❢ r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s ❋❖✭❈❋✮ ❢♦r t❤❡ ❢❛♠✐❧② ❈❋ ♦❢ ❝♦♥t❡①t✲❢r❡❡ ❧❛♥❣✉❛❣❡s ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✶ ✴ ✾✼

❋❖✭❚❈✮ ✿ ✜rst ♦r❞❡r ❧♦❣✐❝ ✇✐t❤ tr❛♥s✐t✐✈❡ ❝❧♦s✉r❡ ❋❖ ✰ ❢♦r♠✉❧❛s [ TC x,y ϕ ( x,y,z )]( s,t ) ✇❤❡r❡ x,y,z ❛r❡ t✉♣❧❡s ♦❢ ♣❛✐r✇✐s❡ ❞✐st✐♥❝t ✈❛r✐❛❜❧❡s s,t ❛r❡ t✉♣❧❡s ♦❢ ✈❛r✐❛❜❧❡s ❛♥❞ ∣ x ∣ = ∣ y ∣ = ∣ s ∣ = ∣ t ∣ = n ❢r❡❡ ( ϕ ( x,y,z )) ❂ { x,y,z } ❢r❡❡ ([ TC x,y ϕ ( x,y,z )]( s,t )) ❂ { s,t,z } ( A ,d 1 ,d 2 ,d 3 ) ⊧ [ TC x,y ϕ ( x,y,z )]( s,t ) ✇❤❡♥ ( d 1 ,d 2 ) ❜❡❧♦♥❣s t♦ t❤❡ tr❛♥s✐t✐✈❡ ❝❧♦s✉r❡ ♦❢ t❤❡ r❡❧❛t✐♦♥ {( e 1 ,e 2 ) ∣ ( A ,e 1 ,e 2 ,d 3 ) ⊧ ϕ ( x,y,z )} ❋❖✭❚❈✮ ( n ) ✿ tr❛♥s✐t✐✈❡ ❝❧♦s✉r❡ ❛♣♣❧✐❡s t♦ r❡❧❛t✐♦♥s ♦♥ t✉♣❧❡s ♦❢ ❧❡♥❣t❤ ❛t ♠♦st n ✳ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✷ ✴ ✾✼

❊①❛♠♣❧❡ ♦❢ ❋❖✭❚❈✮ ( 1 ) ❢♦r♠✉❧❛ ❋♦r ❛ ❧❛♥❣✉❛❣❡ ❣✐✈❡♥ ❜② ❛ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥ R ✱ r❡❛❝❤ R ( x,y ) ✐s ❞❡✜♥❡❞ ✐♥❞✉❝t✐✈❡❧② ⎧ ⎪ ⎪ ⎪ x = y ✐❢ R = ε ⎪ ⎪ ⎪ ⎪ ⎪ a ( x,y ) ✐❢ R = a r❡❛❝❤ R ( x,y ) ∶⇔ r❡❛❝❤ R 1 ( x,y ) ∨ r❡❛❝❤ R 2 ( x,y ) ⎨ ✐❢ R = R 1 + R 2 ⎪ ⎪ ⎪ ∃ z ( r❡❛❝❤ R 1 ( x,z ) r❡❛❝❤ R 2 ( z,y ) , ✐❢ R = R 1 R 2 ⎪ ⎪ [ ❚❈ st r❡❛❝❤ P ( s,t )]( x,y ) ⎪ ⎪ ✐❢ R = P ∗ ⎪ ⎩ ❊①❛♠♣❧❡ ♦❢ ❋❖✭❚❈✮ ( 2 ) ❢♦r♠✉❧❛ ❋♦r tr❡❡s ✿ t✇♦ ♥♦❞❡s x ❛♥❞ y ❛r❡ ❛t t❤❡ s❛♠❡ ❞❡♣t❤ ✿ ∃ r ( ∀ z z ↛ r ∧ [ ❚❈ x 1 y 1 ,x 2 y 2 ( x 1 → x 2 ∧ y 1 → y 2 )]( rr,xy )) ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✸ ✴ ✾✼

❙❖ ✿ ✷♥❞ ♦r❞❡r ❧♦❣✐❝ ✶st ♦r❞❡r ✈❛r✐❛❜❧❡s ✿ r❛♥❣❡ ♦✈❡r ❡❧❡♠❡♥ts ♦❢ A x,y,z,... ✷♥❞ ♦r❞❡r ✈❛r✐❛❜❧❡s ✿ X,Y,Z,... r❛♥❣❡ ♦✈❡r r❡❧❛t✐♦♥s ♦♥ A ❡✈❡r② ✈❛r✐❛❜❧❡ X ❤❛s ✐ts ❛r✐t② α ( X ) ❙❖ ✐s t❤❡ ✉s✉❛❧ ❧♦❣✐❝ ❢♦r ♠❛t❤❡♠❛t✐❝s✳ ❊①❛♠♣❧❡ ♦❢ ❛ ❞❡✜♥❛❜❧❡ r❡❧❛t✐♦♥ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠✉❧❛ eqp ( X,Y ) s❛②s t❤❛t s❡ts X ❛♥❞ Y ❛r❡ ❡q✉✐♣♦t❡♥t✱ ∃ Z ( ∀ x ( X ( x ) ⇒ ∃ y ( Y ( y ) ∧ Z ( x,y )) ∧ ∀ y ( Y ( y ) ⇒ ∃ x ( X ( x ) ∧ Z ( x,y )) ∧ ∀ x ∀ y 1 ∀ y 2 (( Z ( x,y 1 ) ∧ Z ( x,y 2 )) ⇒ y 1 = y 2 ) ∧ ∀ x 1 ∀ x 2 ∀ y (( Z ( x 1 ,y ) ∧ Z ( x 2 ,y )) ⇒ x 1 = x 2 ) ) α ( X ) = α ( Y ) = 1 , α ( Z ) = 2 ✳ ✇❤❡r❡ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✹ ✴ ✾✼

▼❙❖ ✿ ♠♦♥❛❞✐❝ ✷♥❞ ♦r❞❡r ❧♦❣✐❝ ▼❙❖ 1 ✶st ♦r❞❡r ✈❛r✐❛❜❧❡s ✿ r❛♥❣❡ ♦✈❡r ❡❧❡♠❡♥ts ♦❢ A x,y,z,... ✷♥❞ ♦r❞❡r ✈❛r✐❛❜❧❡s ✿ X,Y,Z,... r❛♥❣❡ ♦✈❡r s✉❜s❡ts ♦❢ A ✈✐③✳✱ α ( X ) = 1 ❢♦r ❡✈❡r② X ▼❙❖ 2 ❡❛❝❤ x r❛♥❣❡s ❡✐t❤❡r ♦✈❡r ❡❧❡♠❡♥ts ♦❢ A ✈❡rt ♦r ♦✈❡r ❡❧❡♠❡♥ts ♦❢ A ❡❞❣ ❡❛❝❤ X r❛♥❣❡s ❡✐t❤❡r ♦✈❡r s✉❜s❡ts ♦❢ A ✈❡rt ♦r ♦✈❡r s✉❜s❡ts ♦❢ A ❡❞❣ ▼❙❖ 2 ✐s ❡q✉✐✈❛❧❡♥t t♦ s♦ ❝❛❧❧❡❞ ❣✉❛r❞❡❞ ✷♥❞ ♦r❞❡r ❧♦❣✐❝ ✭●❙❖✮✳ ∣ X ∣ ≥ ℵ 0 ▼❙❖ ω ❊①t❡♥s✐♦♥s ✿ ∣ X ∣ ✪ m = n ❢♦r m,n ∈ IN ▼❙❖ ❝ ▼❙❖ ✇✐t❤ ❝♦✉♥t✐♥❣ ❘❡str✐❝t✐♦♥s ✿ ❲▼❙❖ ❡✈❡r② X r❛♥❣❡s ♦✈❡r ✜♥✐t❡ s❡ts ✇❡❛❦ ▼❙❖ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✺ ✴ ✾✼

❊①❛♠♣❧❡ ❢♦r t②♣❡ ✶ str✉❝t✉r❡s ❆ ❞❡t❡r♠✐♥✐st✐❝ ❚▼ ❤❛❧ts ♦♥ ❡✈❡r② ✐♥♣✉t ✿ ∀ x ( ι ( x ) ⇒ (∃ y r❡❛❝❤ ( x,y ) ∧ ∀ z y ↛ z )) ✇❤❡r❡ r❡❛❝❤ ( x,y ) ✐s ∀ X ⎛ ⎝( x ∈ X ∧ ∀ z 1 ∀ z 2 (( z 1 ∈ X ∧ z 1 → z 2 ) ⇒ z 2 ∈ X )) ⇒ y ∈ X ⎞ ⎠ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✻ ✴ ✾✼

❊①❛♠♣❧❡ ❢♦r t②♣❡ ✷ str✉❝t✉r❡s ❆ ❣r❛♣❤ ✭✇✐t❤ ❛t ❧❡❛st ✷ ✈❡rt✐❝❡s✮ ❤❛s ❛ ❍❛♠✐❧t♦♥✐❛♥ ❝✐r❝✉✐t ✿ ∃ E ∀ x ⎛ ⎝∃ e 1 ∃ e 2 ∃ y ∃ z ( e 1 ≠ e 2 ∧ e 1 ∈ E ∧ e 2 ∈ E ∧ ❡❞❣ ( e 1 ,y,x ) ∧ ❡❞❣ ( e 2 ,x,z ) ∧ ∀ e 3 ∀ y ( e 3 ∈ E ⇒ (( ❡❞❣ ( e 3 ,y,x ) ⇒ e 3 = e 1 ) ∧ ( ❡❞❣ ( e 3 ,x,y ) ⇒ e 3 = e 2 )))) ∧ s❝ ( E )⎞ ⎠ ❊✈❡r② ✈❡rt❡① ❤❛s ❡①❛❝t❧② ♦♥❡ ✐♥❝♦♠✐♥❣ ❡❣❡ ✐♥ E ❛♥❞ ♦✉t❣♦✐♥❣ ❡❞❣❡ ✐♥ E ❛♥❞ t❤❡ ❣r❛♣❤ ✐s str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❜② E ✳ ❋♦r♠✉❧❛ s❝ ( E ) ✉s❡❞ ❛❜♦✈❡ ✐s ∀ x ∀ y ∀ X ⎛ ⎝( x ∈ X ∧ ∀ z 1 ∀ z 2 ∀ e (( z 1 ∈ X ∧ e ∈ E ∧ ❡❞❣ ( e,z 1 ,z 2 )) ⇒ z 2 ∈ X )) ⇒ y ∈ X ⎞ ⎠ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✼ ✴ ✾✼

▲♦❣✐❝ ✿ s✉♠♠❛r② F O ⊊ ⊊ F O(R) F O(CF) ⊊ SO F O(R, sub) ⊊ ⊊ ⊊ MSO ω ⊊ 1 1 MSO ⊊ ⊊ ⊊ F O(Reg) ⊊ ⊊ MSO ω WMSO ⊊ MSO 1 ⊊ MSO 2 ⊊ 2 ⊊ 2 ⊆ MSO ⊊ O(TC) ( 1 ) ⊊ ⋯ ⊊ O(TC) ( n ) ⊊ ⋯ ⊊ ⋯ F F F O(TC) ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✽ ✴ ✾✼

❙tr✉❝t✉r❡ ❜✉✐❧❞✐♥❣ ♦♣❡r❛t✐♦♥s ❝❧❛ss ♦❢ str✉❝t✉r❡s S 1 S 2 ❧♦❣✐❝ L 1 L 2 f ∶ S 1 → S 2 ❙tr✉❝t✉r❡ ❜✉✐❧❞✐♥❣ ♦♣❡r❛t✐♦♥ ✇✐t❤ ❜❛❝❦✇❛r❞s tr❛♥s❧❛t✐♦♥ ♣r♦♣❡rt② ✿ ❋♦r ❡✈❡r② str✉❝t✉r❡ A ∈ S 1 ❛♥❞ ❢♦r ❡✈❡r② s❡♥t❡♥❝❡ ϕ ∈ L 2 A ⊧ f ♯ ( ϕ ) ⇔ f ( A ) ⊧ ϕ ✇❤❡r❡ f ♯ ∶ L 2 → L 1 ✐s ❡✛❡❝t✐✈❡✳ ■❢ ❚❤ L 1 ( A ) ✐s r❡❝✉rs✐✈❡✱ s♦ ✐s ❚❤ L 2 ( f ( A )) ✳ ❘❡♠❛r❦ f ✐s ( L 1 , L 2 ) ✲❝♦♠♣❛t✐❜❧❡ f ( A 1 ,..., A n ) ●❡♥❡r❛❧ ❝❛s❡ ✿ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✸✾ ✴ ✾✼

▼❙❖ ✐♥t❡r♣r❡t❛t✐♦♥s A ∈ Struct [ Σ ] ↦ B ∈ Struct [ Γ ] d ❞❡✜♥✐t✐♦♥ s❝❤❡♠❡ ✭♣❛r❛♠❡t❡r❧❡ss✮ ⟨ δ ( x ) , ( θ b ( x 1 ,...,x α ( b ) )) b ∈ Γ , ⟩ = d δ ( x ) ∈ Form ▼❙❖ [ Σ ] ✱ ✇❤❡r❡ ✭❞❡✜♥❡s ✉♥✐✈❡rs❡ B ♦❢ B ✮ θ b ( x 1 ,...,x α ( b ) ) ∈ Form ▼❙❖ [ Σ ] ✱ ✭❞❡✜♥❡s B ( b ) ✮ B = d ( A ) ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s ✿ ∶ = { d ∈ A ∣ A ⊧ δ ( d )} B B ( b ) ∶ = { d ∣ A ⊧ θ b ( d )} ❢♦r ❡✈❡r② b ∈ Γ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✵ ✴ ✾✼

❇❛❝❦✇❛r❞s tr❛♥s❧❛t✐♦♥ d ♯ ∶ Form ▼❙❖ [ Γ ] → Form ▼❙❖ [ Σ ] d ♯ ( x = y ) = x = y d ♯ ( x ∈ X ) = x ∈ X d ♯ ( b ( x 1 ,...,x α ( b ) )) θ b ( x 1 ,...,x α ( b ) ) = d ♯ (¬ ϕ ) ¬ d ♯ ( ϕ ) = d ♯ ( ϕ 1 ∨ ϕ 2 ) d ♯ ( ϕ 1 ) ∨ d ♯ ( ϕ 2 ) = d ♯ (∃ x ϕ ) ∃ x ( δ ( x ) ∧ d ♯ ( ϕ )) = d ♯ (∃ X ϕ ) ∃ X (∀ x ( x ∈ X ⇒ δ ( x )) ∧ d ♯ ( ϕ )) = ❆♥ ▼❙❖ ✐♥t❡r♣r❡t❛t✐♦♥ ✐s ▼❙❖✲❝♦♠♣❛t✐❜❧❡ ✭✈✐③✳ ✭▼❙❖✱ ▼❙❖✮✲❝♦♠♣❛t✐❜❧❡✮✳ ■❢ A ❤❛s ❛ ❞❡❝✐❞❛❜❧❡ ▼❙❖ t❤❡♦r②✱ t❤❡♥ d ( A ) ❤❛s ❛ ❞❡❝✐❞❛❜❧❡ ▼❙❖ t❤❡♦r②✳ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✶ ✴ ✾✼

❊①❛♠♣❧❡ { l,r } α ( l ) = α ( r ) = 2 r l ⋅ Σ = ⋅ ⋅ { a,b,c } α ( a ) = α ( b ) = α ( c ) = 2 r r l l = Γ ⟨ δ ( x ) ,θ a ( x,y ) ,θ b ( x,y ) ,θ c ( x,y )⟩ ⋅ ⋅ ⋅ ⋅ r l r l r l r l = d ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ δ ( x ) ∃ t (¬∃ z ( z � → t ∨ z → t ) ∧ l r = � ∀ X (( t ∈ X ∧ ∀ z 1 ∀ z 2 (( z 1 ∈ X ∧ z 1 → z 2 ) ⇒ z 2 ∈ X )) ⇒ l � ( x ∈ X ∨ ∃ z ( z ∈ X ∧ z → x )))) r � θ a ( x,y ) l = x � → y θ b ( x,y ) ∃ z 1 ∃ z 2 ( z 1 → z 2 ∧ z 1 → y ∧ z 2 → x ) z 1 l r l r r = � � � z 2 y θ c ( x,y ) r r = x � → y x ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✷ ✴ ✾✼

ε a c l r 0 1 b a c a c l l r r 00 01 10 11 b b a c a c a c a c l l l l r r r r 000 001 010 011 100 101 110 111 ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✸ ✴ ✾✼

ε a c 0 1 b a c 00 01 b a c 000 001 ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✹ ✴ ✾✼

1 01 001 0001 b b b c c c c ε 0 00 000 a a a ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✺ ✴ ✾✼

▼❙❖ tr❛♥s❞✉❝t✐♦♥s A ∈ Struct [ Σ ] ↦ B ∈ Struct [ Γ ] d ❞❡✜♥✐t✐♦♥ s❝❤❡♠❡ ✭♣❛r❛♠❡t❡r❧❡ss✮ ⟨ n, ( δ i ( x )) i ∈[ n ] , ( θ b,k ( x 1 ,...,x α ( b ) )) b ∈ Γ ,k ∈[ n ] α ( b ) ⟩ = d ✇❤❡r❡ n ∈ IN ✱ ✭♥✉♠❜❡r ♦❢ ❝♦♣✐❡s✮ δ i ( x ) ∈ Form ▼❙❖ [ Σ ] ✱ ✭❞❡✜♥❡ ✉♥✐✈❡rs❡ B ♦❢ B ✮ θ b,k ( x 1 ,...,x α ( b ) ) ∈ Form ▼❙❖ [ Σ ] ✱ ✭❞❡✜♥❡ B ( b ) ✮ B = d ( A ) ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s ✿ ∶ = {( d,i ) ∣ A ⊧ δ i ( d )} ⋃ B i ∈[ n ] B ( b ) ∶ = {(( d 1 ,i 1 ) ,..., ( d α ( b ) ,i α ( b ) )) ∣ A ⊧ θ b,i 1 ...i α ( b ) ( d 1 ,...,d α ( b ) )} ⋃ i 1 ...i α ( b ) ∈[ n ] α ( b ) ❢♦r ❡✈❡r② b ∈ Γ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✻ ✴ ✾✼

❊①❛♠♣❧❡ ♠❛♣s ❛♥② ✇♦r❞ u ∈ { a,b } ∗ t♦ u 3 d Σ = { s,a,b } α ( s ) = 1 α ( a ) = α ( b ) = 1 d = ⟨ 3 ,δ 1 ( x ) ✱ δ 2 ( x ) ,δ 3 ( x ) ,θ s, 11 ( x,y ) ,θ s, 12 ( x,y ) ,θ s, 13 ( x,y ) , θ s, 21 ( x,y ) ,θ s, 22 ( x,y ) ,θ s, 23 ( x,y ) ,θ s, 31 ( x,y ) , θ s, 32 ( x,y ) ,θ s, 33 ( x,y ) ,θ a, 1 ( x ) ,θ a, 2 ( x ) ,θ a, 3 ( x ) , θ b, 1 ( x ) ,θ b, 2 ( x ) ,θ b, 2 ( x )⟩ a b a t u v s s d a b a a b a a b a t 1 u 1 v 1 t 2 u 2 v 2 t 3 u 3 v 3 s s s s s s s s δ 1 ( x ) = δ 2 ( x ) = δ 3 ( x ) = tr✉❡ θ a, 1 ( x ) = θ a, 2 ( x ) = θ a, 3 ( x ) = a ( x ) θ s, 11 ( x,y ) = θ s, 22 ( x,y ) = θ s, 33 ( x,y ) = s ( x ) θ b, 1 ( x ) = θ b, 2 ( x ) = θ b, 3 ( x ) = b ( x ) θ s, 21 ( x,y ) = θ s, 32 ( x,y ) = θ s, 31 ( x,y ) = ❢❛❧s❡ θ s, 13 ( x,y ) = ❢❛❧s❡ θ s, 12 ( x,y ) = θ s, 23 ( x,y ) = ¬∃ z ( s ( x,z ) ∨ s ( z,y )) ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✼ ✴ ✾✼

❋❖ ✐♥t❡r♣r❡t❛t✐♦♥s A ∈ Struct [ Σ ] ↦ B ∈ Struct [ Γ ] d ❞❡✜♥✐t✐♦♥ s❝❤❡♠❡ ✭♣❛r❛♠❡t❡r❧❡ss✮ ⟨ n,δ ( x ) , ( θ b ( x 1 ,...,x α ( b ) )) b ∈ Γ ⟩ d = ✇❤❡r❡ n ∈ IN ✭✐s t❤❡ ❞✐♠❡♥s✐♦♥ ♦❢ d ✮✱ δ ( x ) ∈ Form L [ Σ ] ✱ ✭❞❡✜♥❡s ✉♥✐✈❡rs❡ B ♦❢ B ✮ θ b ( x 1 ,...,x α ( b ) ) ∈ Form L [ Σ ] ✱ ✭❞❡✜♥❡s B ( b ) ✮ ❡✈❡r② B = d ( A ) ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s ✿ { d ∈ A n ∣ A ⊧ δ ( d )} ∶ = B B ( b ) ∶ = {( d 1 ,...,d α ( b ) ) ∣ A ⊧ θ b ( d 1 ,...,d α ( b ) )} ❢♦r ❡✈❡r② b ∈ Γ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✽ ✴ ✾✼

❆ss✉♠✐♥❣ t❤❛t ❢♦r♠✉❧❛s ♦❢ d ❛r❡ ✇r✐tt❡♥ ✐♥ ❛ ❧♦❣✐❝ L ✱ d ♯ ∶ Form ❋❖ [ Γ ] → Form L [ Σ ] ❋✐rst✱ t♦ ❡✈❡r② ✈❛r✐❛❜❧❡ x ✇❡ ❛ss♦❝✐❛t❡ ❛ t✉♣❧❡ ♦❢ ♣❛✐r✇✐s❡ ❞✐st✐♥❝t ♥❡✇ ✈❛r✐❛❜❧❡s x = x 1 ...x n ✳ ❚❤❡♥ ✿ d ♯ ( x = y ) x 1 = y 1 ∧ ⋯ ∧ x n = y n = d ♯ ( b ( x 1 ,...,x α ( b ) )) θ b ( x 1 ,...,x α ( b ) ) = d ♯ ( ¬ ϕ ) ¬ d ♯ ( ϕ ) = d ♯ ( ϕ 1 ∨ ϕ 2 ) d ♯ ( ϕ 1 ) ∨ d ♯ ( ϕ 2 ) = d ♯ ( ∃ xϕ ) ∃ x ( δ ( x ) ∧ d ♯ ( ϕ )) = ❆♥ ❋❖ ✐♥t❡r♣r❡t❛t✐♦♥ ✐s ( L , ❋❖ ) ✲❝♦♠♣❛t✐❜❧❡✳ ■❢ A ❤❛s ❛ ❞❡❝✐❞❛❜❧❡ L t❤❡♦r②✱ t❤❡♥ d ( A ) ❤❛s ❛ ❞❡❝✐❞❛❜❧❡ ❋❖ t❤❡♦r②✳ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✹✾ ✴ ✾✼

Pr♦❞✉❝ts ❲❡ ❛ss✉♠❡ t❤❛t max ( α ( Σ i )) ≤ 2 ✱ ❢♦r i ∈ [ n ] ✳ ✭■♥ t❤❡ ❝❛s❡ ♦❢ ❛r✐t② ❤✐❣❤❡r t❤❛♥ ✷ s❡✈❡r❛❧ ❞❡✜♥✐t✐♦♥s ♦❢ ♣r♦❞✉❝t ❛r❡ ♣♦ss✐❜❧❡✳✮ ▲❡t A i ∈ Struct [ Σ i ] ✳ ❚❤❡♥ C = ∏ A i i ∈[ n ] ✐s ❛ str✉❝t✉r❡ ♦✈❡r Γ ✱ ✇❤❡r❡ ✿ ∶ = ∏ i ∈[ n ] ( Σ ( 1 ) ∪ { tr✉❡ }) ∖ {( tr✉❡ ,..., tr✉❡ )} Γ ( 1 ) i ∶ = ∏ i ∈[ n ] ( Σ ( 2 ) ∪ { = }) ∖ { = ,..., = )} Γ ( 2 ) i ∶ = ∏ i ∈[ n ] A i C { d ∣ d i ∈ A i ( a i ) ❢♦r i ∈ [ n ]} , C ( a ) ∶ = a ∈ Γ ( 1 ) C ( b ) ∶ = {( d,d ′ ) ∣ ( d i ,d ′ i ) ∈ A i ( b i ) ❢♦r i ∈ [ n ]} , b ∈ Γ ( 2 ) ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✵ ✴ ✾✼

= { l } = { a } Σ ( 1 ) Σ ( 1 ) 1 2 = { f,g } = { b,c } Σ ( 2 ) Σ ( 2 ) Γ ( 1 ) = {( tr✉❡ ,a ) , ( l, tr✉❡ ) , ( l,a )} 1 2 Γ ( 2 ) = {( = ,b ) , ( = ,c ) , ( f, = ) , ( f,b ) , ( f,c ) , ( g, = ) , ( g,b ) , ( g,c )} c a b 0 1 c ( l,a ) ( = ,b ) ( l, true ) u0 u1 l ( f,b ) u ( f, = ) ( g, = ) ( g, = ) ( f, = ) g ( f,c ) ( g,c ) ( g,c ) ( f,c ) f ( g,b ) v v0 v1 ( = ,b ) ( true ,a ) c ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✶ ✴ ✾✼

❙②♥❝❤r♦♥✐③❡❞ ♣r♦❞✉❝t ❙②♥❝❤r♦♥✐③❛t✐♦♥ ❝♦♥str❛✐♥t ∆ ⊆ Γ ∆ = {( l,a ) , ( f,b ) , ( f,c ) , ( g,b ) , ( g,c )} ❊①❛♠♣❧❡ c a b 0 1 ( l,a ) l u0 u1 u ( f,b ) g f ( f,c ) ( g,c ) ( g,c ) ( f,c ) v ( g,b ) v0 v1 ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✷ ✴ ✾✼

Pr♦❞✉❝t ✐s ❋❖✲❝♦♠♣❛t✐❜❧❡ ❚❤❡♦r❡♠ ✭▼♦st♦✇s❦✐✱ ✶✾✺✷✮ ▲❡t C = A × B ✳ ❚❤❡♥ ❚❤ ❋❖ ( C ) ✐s ✉♥✐q✉❡❧② ❞❡t❡r♠✐♥❡❞ ❜②✱ ❛♥❞ ♠❛② ❜❡ ❡✛❡❝t✐✈❡❧② ❝♦♠♣✉t❡❞ ❢r♦♠ ❚❤ ❋❖ ( A ) ❛♥❞ ❚❤ ❋❖ ( B ) ✳ ■♥ ✶✾✺✾ ❋❡❢❡r♠❛♥ ❛♥❞ ❱❛✉❣❤t ❣❛✈❡ ❛ ♠♦r❡ ❣❡♥❡r❛❧ r❡s✉❧t ❢♦r ✐♥✜♥✐t❡ ♣r♦❞✉❝ts ∏ i ∈ I A i ✳ ❇✉t ❚❤ ❋❖✭❘✮ ( A ) ❛♥❞ ❚❤ ❋❖✭❘✮ ( B ) ❞❡❝✐❞❛❜❧❡ ❚❤ ❋❖✭❘✮ ( A × B ) ⇏ ❞❡❝✐❞❛❜❧❡ ❋♦r ✷✲❞✐♠❡♥s✐♦♥❛❧ ✐♥✜♥✐t❡ ❣r✐❞ G 2 = ⟨ IN , s✉❝ ⟩ × ⟨ IN , s✉❝ ⟩ ❚❤ ❋❖✭❘✮ ( G 2 ) ✭❤❡♥❝❡ ❚❤ ▼❙❖ ( G 2 ) ✮ ✐s ♥♦t r❡❝✉rs✐✈❡✳ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✸ ✴ ✾✼

s s s 0 1 2 3 s 1 s 1 s 2 (0,3) (1,3) (2,3) (3,3) 3 s 2 s 2 s 2 s 2 s ❙②♥❝❤r♦♥✐s❛t✐♦♥ ❝♦♥str❛✐♥t ∆ = { s 1 ,s 2 } s 1 s 1 s 1 ✇❤❡r❡ s 1 = ( s, = ) (0,2) (1,2) (2,2) (3,2) 2 s 2 = ( = ,s ) s 2 s 2 s 2 s 2 s s 1 s 1 s 1 (0,1) (1,1) (2,1) (3,1) 1 s 2 s 2 s 2 s 2 s s 1 s 1 s 1 (0,0) (1,0) (2,0) (3,0) 0 ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✹ ✴ ✾✼

❉✐s❥♦✐♥t s✉♠ ▲❡t A i ∈ Struct [ Σ i ] ✳ ❚❤❡♥ C = ∐ A i i ∈[ n ] ✐s ❛ str✉❝t✉r❡ ♦✈❡r Γ = ⋃ i ∈[ n ] Σ i ✱ ✇❤❡r❡ ✿ ∶ = ⋃ i ∈[ n ] A i × { i } C C ( a ) ∶ = {(( d 1 ,i ) ,..., ( d k ,i )) ∣ ( d 1 ,...,d k ) ∈ A i ( a )} ❢♦r a ∈ Σ i ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✺ ✴ ✾✼

❊①❛♠♣❧❡ c a b u v l l u 1 c u a b g g u 2 f f v 2 v v 1 ❚❤❡♦r❡♠ ✭❙❤❡❧❛❤✱ ✶✾✼✺✮ ❉✐s❥♦✐♥t s✉♠ ✐s ▼❙❖✲❝♦♠♣❛t✐❜❧❡ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✻ ✴ ✾✼

❯♥❢♦❧❞✐♥❣ ▲❡t A ∈ Struct [ Σ ] ❜❡ ❛ ❣r❛♣❤ ✭ max ( α ( Σ )) = 2 ✮ ❛♥❞ r ∈ A ✳ ❚❤❡ ✉♥❢♦❧❞✐♥❣ C = ❯♥ ( r, A ) ♦❢ A ❢r♦♠ r ❜❡❧♦♥❣s t♦ Struct [ Σ ] ❛♥❞ ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s ∶ = { d 0 a 1 ...a n d n ∣ d 0 = r, n ∈ IN , d i − 1 � → d i ❢♦r i ∈ [ n ]} a i C { {( u,uad ) ∣ u ∈ C, uad ∈ C, a ∈ Σ ( 2 ) , d ∈ A } α ( a ) = 2 C ( a ) ∶ = { ubd ∈ C ∣ u ∈ C, b ∈ Σ ( 2 ) , d ∈ A ( a )} α ( a ) = 1 ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✼ ✴ ✾✼

❊①❛♠♣❧❡ ● ● ● ● ● b b b b b b b b ● ● ● ● ● a a a b b ● ● ● b b b ● ● ● ● b b b b ● ● ● a a a ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✽ ✴ ✾✼

❚❤❡♦r❡♠ ✭❈♦✉r❝❡❧❧❡✱ ❲❛❧✉❦✐❡✇✐❝③✱ ✶✾✾✺✮ ❚❤❡ ✉♥❢♦❧❞✐♥❣ ✐s ▼❙❖✲❝♦♠♣❛t✐❜❧❡ ❋♦r ❡✈❡r② ❣r❛♣❤ A ❛♥❞ r ∈ A ✱ ❢♦r ❡✈❡r② ▼❙❖ s❡♥t❡♥❝❡ ψ ♦♥ ♠❛② ❝♦♥str✉❝t ❛♥ ▼❙❖ s❡♥t❡♥❝❡ ϕ s✉❝❤ t❤❛t A ⊧ ϕ ⇔ ❯♥ ( r, A ) ⊧ ψ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✺✾ ✴ ✾✼

◗✉❡st✐♦♥ ✿ ❲❤✐❝❤ ♦❢ t❤❡ ✷ str✉❝t✉r❡s ✐s s✐♠♣❧❡r ❄ ● ● ● ● ● b b b b b b b b ● ● ● ● ● a a a b b ● ● ● b b b ● ● ● ● b b b b ● ● ● a a a ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✵ ✴ ✾✼

❆♥s✇❡r ✿ ▼❙❖ ✐♥t❡r♣r❡t❛t✐♦♥ ⟨ δ ( x ) ,θ a ( x,y ) ,θ b ( x,y )⟩ ● δ ( x ) = tr✉❡ θ a ( x,y ) � b a = → y x θ b ( x,y ) → y ∨ (¬∃ z ( x → z ∨ x → z ) b a b = x � � � ● ● ∧ r❡❛❝❤ ab ∗ ( y,x )) b b ● ● ● b b b b a n b 2 ( n + 1 ) b ● ● ● b b b b b ● ● ● a a a ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✶ ✴ ✾✼

⋅ r l � � � � � � r ✉♥❢♦❧❞✐♥❣ l ⋅ → ⋅ ⋅ r r l l ⋅ ⋅ ⋅ ⋅ r r r r l l l l ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✷ ✴ ✾✼

■t❡r❛t✐♦♥ ▲❡t A ∈ Struct [ Σ ] ❚❤❡ ✐t❡r❛t✐♦♥ ♦❢ A ✐s ❛ str✉❝t✉r❡ A ∗ ♦✈❡r Σ ∪ {♯ , & } α (♯) = 2 α ( & ) = 1 A ∗ ✐s ❞❡✜♥❡❞ ❛s ❢♦❧❧♦✇s ✿ t❤❡ ✉♥✐✈❡rs❡ ♦❢ A ∗ ✐s t❤❡ s❡t A ∗ ♦❢ ❛❧❧ ✇♦r❞s ♦✈❡r A A ∗ ( a ) ∶ = {( wd 1 ,...,wd n ) ∣ ( d 1 ,...,d n ) ∈ A ( a )} ❢♦r a ∈ Σ ( n ) A ∗ (♯) ∶ = {( w,wd ) ∣ w ∈ A ∗ ,d ∈ A } A ∗ ( & ) ∶ = { wdd ∣ w ∈ A ∗ ,d ∈ A } ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✸ ✴ ✾✼

❊①❛♠♣❧❡ ⋅ 0 2 b Shelah-Stupp a a 1 ⋅ ⋅ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✹ ✴ ✾✼

❊①❛♠♣❧❡ ⋅ 0 2 b Shelah-Stupp a a ♯ 1 ♯ ♯ 00 02 b a a 01 ⋅ ⋅ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✹ ✴ ✾✼

❊①❛♠♣❧❡ ⋅ 0 2 b Shelah-Stupp a a ♯ 1 ♯ ♯ ♯ ♯ ♯ 00 02 10 12 b b a a a a 01 11 ⋅ ⋅ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✹ ✴ ✾✼

❊①❛♠♣❧❡ ⋅ 0 2 b Shelah-Stupp a a ♯ ♯ 1 ♯ ♯ ♯ ♯ ♯ ♯ ♯ 00 02 10 12 20 22 b b b a a a a a a 01 11 21 ⋅ ⋅ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✹ ✴ ✾✼

❊①❛♠♣❧❡ ⋅ 0 2 b Shelah-Stupp a a ♯ ♯ 1 ♯ ♯ ♯ ♯ ♯ ♯ ♯ 00 02 10 12 20 22 b b b a a a a a a ♯ 01 11 21 ♯ ♯ 000 002 b a a 001 ⋅ ⋅ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✹ ✴ ✾✼

❊①❛♠♣❧❡ ⋅ 0 2 b Shelah-Stupp a a ♯ ♯ 1 ♯ ♯ ♯ ♯ ♯ ♯ ♯ 00 02 10 12 20 22 b b b a a a a a a ♯ 01 11 21 ♯ ♯ 000 002 b ♯ ♯ a a ♯ 001 010 012 b a a 011 ⋅ ⋅ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✹ ✴ ✾✼

❊①❛♠♣❧❡ ⋅ 0 2 b Shelah-Stupp a a ♯ ♯ 1 ♯ ♯ ♯ ♯ ♯ ♯ ♯ 00 02 10 12 20 22 b b b a a a a a a ♯ 01 11 21 ♯ ♯ ♯ ♯ ♯ 000 002 b ♯ ♯ 020 022 a a b ♯ 001 a a 021 010 012 b a a 011 ⋅ ⋅ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✹ ✴ ✾✼

❊①❛♠♣❧❡ ⋅ 0 2 b Shelah-Stupp a a ♯ hnik & ♯ 1 Mu ♯ ♯ ♯ ♯ ♯ ♯ ♯ 00 02 10 12 20 22 b b b a a a a a a ♯ 01 11 21 ♯ ♯ ♯ ♯ ♯ 000 002 b ♯ ♯ 020 022 a a b ♯ 001 a a 021 010 012 b a a 011 ⋅ ⋅ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✹ ✴ ✾✼

❚❤❡♦r❡♠ ✭▼✉❝❤♥✐❦✱ ✶✾✽✹✮ ❚❤❡ ✉♥❢♦❧❞✐♥❣ ✐s ▼❙❖✲❝♦♠♣❛t✐❜❧❡ ❋♦r ❡✈❡r② ❣r❛♣❤ A ❛♥❞ ❡✈❡r② ▼❙❖ s❡♥t❡♥❝❡ ψ ♦♥ ♠❛② ❝♦♥str✉❝t ❛♥ ▼❙❖ s❡♥t❡♥❝❡ ϕ s✉❝❤ t❤❛t A ∗ ⊧ ψ A ⊧ ϕ ⇔ ❋✉❧❧ ♣r♦♦❢ ❜② ❲❛❧✉❦✐❡✇✐❝③ ✐♥ ✶✾✾✻ Pr♦♣♦s✐t✐♦♥ ❚❤❡ ✉♥❢♦❧❞✐♥❣ ❯♥ ( r, A ) ❢r♦♠ ❛♥ ▼❙❖ ❞❡✜♥❛❜❧❡ r ∈ A ✐s ▼❙❖✲✐♥t❡r♣r❡t❛❜❧❡ ✐♥ A ∗ ✳ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✺ ✴ ✾✼

❙tr✉❝t✉r❡ ❜✉✐❧❞✐♥❣ ♦♣❡r❛t✐♦♥s ✿ s✉♠♠❛r② ♦♣❡r❛t✐♦♥ ❝♦♠♣❛t✐❜✐❧✐t② ▼❙❖ ✐♥t❡r♣r❡t❛t✐♦♥ ▼❙❖ ▼❙❖ tr❛♥s❞✉❝t✐♦♥ ▼❙❖ ❋❖ ✐♥t❡r♣r❡t❛t✐♦♥ ✭ L , ❋❖✮ ♣r♦❞✉❝t ❋❖ ❞✐s❥♦✐♥t s✉♠ ▼❙❖ ✉♥❢♦❧❞✐♥❣ ▼❙❖ ✐t❡r❛t✐♦♥ ▼❙❖ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✻ ✴ ✾✼

❉❡s❝r✐❜✐♥❣ ❛♥ ✐♥✜♥✐t❡ str✉❝t✉r❡ ❲❡ ✇❛♥t ❛ ✜♥✐t❡ ❞❡s❝r✐♣t✐♦♥ t❤❛t ❝❛♥ ❜❡ ❝♦❞❡❞ ♦♥ ❛ ❝♦♠♣✉t❡r✳ ● ❧✐♥❡❛r str✉❝t✉r❡s ● ✜①♣♦✐♥t ❡q✉❛t✐♦♥s ❡✳❣✳ x = f ( x,g ( x )) tr❡❡s ● ❣r❛♣❤s r❡❧❛t✐♦♥s ❣r❛♣❤s✱ tr❛♥s✐t✐♦♥ ❣r❛♣❤s ● ❛r❜✐tr❛r② str✉❝t✉r❡s r❡❧❛t✐♦♥ ❣r❛♣❤s ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✼ ✴ ✾✼

← � ⋯ a � → q i state register instru tion 1 instru tion 2 �nite program ⋮ instru tion n / ⋯ ⋯ 1 10 q i 12 ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✽ ✴ ✾✼

❊①❛♠♣❧❡s ♦❢ ✭✐♥t❡r♥❛❧✮ ❝♦♥✜❣✉r❛t✐♦♥s ❚❤❡r❡ ❛r❡ ♦♥❧② ❜❧❛♥❦s t♦ ♦♥ t❤❡ ❧❡❢t ❛♥❞ ♦♥ t❤❡ r✐❣❤t ♦❢ ♦❢ 1012 ✳ 10 q 12 ❚❤❡ ❤❡❛❞ ✐s ♦♥ t❤❡ ✷♥❞ 1 ✳ ❚❤❡ ❝✉rr❡♥t st❛t❡ ✐s q ✳ 1012 p ❚❤❡ ❤❡❛❞ ✐s ♦♥ t❤❡ ✶st ❜❧❛♥❦ ❛❢t❡r 1012 ✳ ❚❤❡ ❝✉rr❡♥t st❛t❡ ✐s p ✳ 1012 r ❚❤❡ ❤❡❛❞ ✐s ♦♥ t❤❡ ✷♥❞ ❜❧❛♥❦ ❛❢t❡r 1012 ✳ ❚❤❡ ❝✉rr❡♥t st❛t❡ ✐s r ✳ ❚❤❡ ❤❡❛❞ ✐s ♦♥ t❤❡ ✶st 1 ✳ ❚❤❡ ❝✉rr❡♥t st❛t❡ ✐s q ✳ q 1012 q 1012 ❚❤❡ ❤❡❛❞ ✐s ♦♥ t❤❡ ❜❧❛♥❦ ✐♠♠❡❞✐❛t❡❧② ❜❡❢♦r❡ 1012 ✳ ❚❤❡ ❝✉rr❡♥t st❛t❡ ✐s q ✳ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✻✾ ✴ ✾✼

❊①❛♠♣❧❡s ♦❢ tr❛♥s✐t✐♦♥s � τ → 1 p 022 ❚❤❡ ❤❡❛❞ ♣r✐♥ts 2 ❛♥❞ ♠♦✈❡s ❧❡❢t✳ ◆♦ ✐♥♣✉t ✐s r❡❛❞ 10 q 12 ✭s✐❧❡♥t tr❛♥s✐t✐♦♥✮✳ ❚❤❡ ♥❡✇ st❛t❡ ✐s p ✳ b � → 101 q 20 1012 p ❚❤❡ ❤❡❛❞ ♣r✐♥ts 0 ❛♥❞ ♠♦✈❡s ❧❡❢t ✇❤✐❧❡ b ✐s r❡❛❞ ♦♥ t❤❡ ✐♥♣✉t t❛♣❡✳ ❚❤❡ ♥❡✇ st❛t❡ ✐s q ✳ a � → 10 q 1 101 p 2 ❚❤❡ ❤❡❛❞ ♣r✐♥ts ❜❧❛♥❦ ❛♥❞ ♠♦✈❡s ❧❡❢t ✇❤✐❧❡ a ✐s r❡❛❞ ♦♥ t❤❡ ✐♥♣✉t t❛♣❡✳ ❚❤❡ ♥❡✇ st❛t❡ ✐s q ✳ τ � → 1012 2 p 1012 r ❚❤❡ ❤❡❛❞ ♣r✐♥ts 2 ❛♥❞ ♠♦✈❡s r✐❣❤t✳ ◆♦ ✐♥♣✉t ✐s r❡❛❞✳ ❚❤❡ ♥❡✇ st❛t❡ ✐s p ✳ a � → q q 1012 012 ❚❤❡ ❤❡❛❞ ♣r✐♥ts ❜❧❛♥❦ ❛♥❞ ♠♦✈❡s ❧❡❢t ✇❤✐❧❡ a ✐s r❡❛❞ ♦♥ t❤❡ ✐♥♣✉t t❛♣❡✳ ❚❤❡ st❛t❡ r❡♠❛✐♥s ✉♥❝❤❛♥❣❡❞✳ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✵ ✴ ✾✼

� � � ❚r❛♥s✐t✐♦♥ ❣r❛♣❤s ✈❡rt✐❝❡s ✿ ✐♥t❡r♥❛❧ ❝♦♥✜❣✉r❛t✐♦♥s ❡❞❣❡s ✿ tr❛♥s✐t✐♦♥s ❜❡t✇❡❡♥ ❝♦♥✜❣✉r❛t✐♦♥s ✭st❛♥❞❛r❞ tr❛♥s✐t✐♦♥ ❣r❛♣❤✮ τ ∗ ✲❝❧♦s✉r❡s ♦❢ s✉❝❤ tr❛♥s✐t✐♦♥s ✭♦❜s❡r✈❛❜❧❡ tr❛♥s✐t✐♦♥ ❣r❛♣❤✮ M ♠❛❝❤✐♥❡ ✭❛✉t♦♠❛t♦♥✮ tg Σ ( M ) ✐ts tr❛♥s✐t✐♦♥ ❣r❛♣❤ tG Σ ( M ) ✐ts ♦❜s❡r✈❛❜❧❡ tr❛♥s✐t✐♦♥ ❣r❛♣❤ a a a � u 2 τ � u 3 τ � u 4 a u 1 u 1 u 2 u 3 u 4 tg Σ ( M ) tG Σ ( M ) ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✶ ✴ ✾✼

❘❡❧❛t✐♦♥ ❣r❛♣❤ ♦❢ ❛ ♠❛❝❤✐♥❡ ✐♥♣✉t✴♦✉t♣✉t ♠❛❝❤✐♥❡s ✭❛✉t♦♠❛t❛✮ ❜❡❣✐♥ ❡♥❞ q 0 f ✲ ✛ ✲ ❄ ❄ ... ... ❝ ❛ ❛ ❛ ❜ ❛ ❜ ❛ ❜ ❜ ✐♥♣✉t ✇♦r❞ u ♦✉t♣✉t ✇♦r❞ v ♣❛✐r ( u, v ) ✐s ❛❝❝❡♣t❡❞ ❜② t❤❡ ♠❛❝❤✐♥❡ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✷ ✴ ✾✼

❘❡❧❛t✐♦♥ ❣r❛♣❤ ♦❢ ❛ tr❛♥s❞✉❝❡r ✜♥✐t❡ tr❛♥s❞✉❝❡r ✛ ... ❜ ❛ ❛ ❝ ❛ u ✻ ✻ r❡❛❞ ❤❡❛❞ r❡❛❞ ❤❡❛❞ q 0 f ✲ ✇r✐t❡ ❤❡❛❞ ✇r✐t❡ ❤❡❛❞ ❄ ❄ ✛ ... ❝ ❛ v ♣❛✐r ( u, v ) ✐s ❛❝❝❡♣t❡❞ ❜② t❤❡ tr❛♥s❞✉❝❡r ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✸ ✴ ✾✼

❘❡❧❛t✐♦♥ ❣r❛♣❤s ( u, v ) ↦ � a → v u ✶ ❢❛♠✐❧② ♦❢ ♠❛❝❤✐♥❡s ( M a ∣ a ∈ Σ ) ✭r❡s♣✳ tr❛♥s❞✉❝❡rs ( T a ∣ a ∈ Σ ) ✮ ✷ ♠❛❝❤✐♥❡ ✭r❡s♣✳ tr❛♥s❞✉❝❡r✮ ✇✐t❤ ✜♥❛❧ st❛t❡s ❧❛❜❡❧❧❡❞ ✐♥ Σ tg Σ ( M ) tg Σ ( T ) ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✹ ✴ ✾✼

❘❡❧❛t✐♦♥ ❛♥❞ tr❛♥s✐t✐♦♥ ❣r❛♣❤s ✿ s✉♠♠❛r② ♦❜s❡r✈❛❜❧❡ tr❛♥s✐t✐♦♥ ❣r❛♣❤s ♠❡♠♦r② r❡str✐❝t✐♦♥ ♠❛❝❤✐♥❡ ❣r❛♣❤ ✐♥✜♥✐t❡✱ ♥♦ r❡str✐❝t✐♦♥ ❚✉r✐♥❣ r❡❝✉rs✐✈❡❧② ❡♥✉♠❡r❛❜❧❡ ❧✐♥❡❛r✐❧② ❜♦✉♥❞❡❞ ❧✐♥❡❛r✐❧② ❜♦✉♥❞❡❞ ❧✐♥❡❛r✐❧② ❜♦✉♥❞❡❞ ♣✉s❤❞♦✇♥ st♦r❡ ♣✉s❤❞♦✇♥ ♣✉s❤❞♦✇♥ ✜♥✐t❡ ✜♥✐t❡ ✜♥✐t❡ r❡❧❛t✐♦♥ ❣r❛♣❤s ♠❛❝❤✐♥❡ ❣r❛♣❤ ❚✉r✐♥❣ r❡❝✉rs✐✈❡❧② ❡♥✉♠❡r❛❜❧❡ ❧✐♥❛✐r❧② ❜♦✉♥❞❡❞ ❧✐♥❡❛r✐❧② ❜♦✉♥❞❡❞ tr❛♥s❞✉❝❡r r❛t✐♦♥❛❧ s②♥❝❤r♦♥✐s❡❞ tr❛♥s❞✉❝❡r ❛✉t♦♠❛t✐❝ ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✺ ✴ ✾✼

●r❛♣❤s ♦❢ ❧❛❜❡❧❧❡❞ str✐♥❣ r❡✇r✐t✐♥❣ s②st❡♠s { l i � → r i ∣ l i ,r i ∈ Γ ∗ , i ∈ I } a L = ∶ = {( xly, xry ) ∣ x,y ∈ Γ ∗ , l → r ∈ L } a a � → � L g Σ ( L ) ∶ = ⋃ a ∈ Σ L � → a ❚♦ ✐♥❝r❡❛s❡ t❤❡ ❡①♣r❡ss✐✈❡ ♣♦✇❡r✱ ♦♥❡ ✉s❡s r❡❣✉❧❛r r❡str✐❝t✐♦♥ ♦♥ ✈❡rt✐❝❡s t♦ K ∈ R eg ( Γ ∗ ) ✿ g Σ ( L , K ) ∶ = g Σ ( L )∣ K = g Σ ( L ) ∩ ( K × Σ × K ) ✳ L ✜♥✐t❡ s✉❜❝❧❛s ♦❢ ❛✉t♦♠❛t✐❝ ❣r❛♣❤s L r❡❝♦❣♥✐s❛❜❧❡ s✉❜❝❧❛s ♦❢ r❛t✐♦♥❛❧ ❣r❛♣❤s ❚❑ ✭❯◆❈ ✫ ■▼❙❝✮ ▲♦❣✐❝ ✫ str✉❝t✉r❡s ✼✻ ✴ ✾✼