t t s - PowerPoint PPT Presentation

❈❛♥ ❲❡ ❆❞❞ ❙✉❜t②♣✐♥❣ t♦ ●❋❄ ❍❛♥s ▲❡✐ÿ ❧❡✐ss❅❝✐s✳✉♥✐✲♠✉❡♥❝❤❡♥✳❞❡ ❯♥✐✈❡rs✐tät ▼ü♥❝❤❡♥ ❈❡♥tr✉♠ ❢ür ■♥❢♦r♠❛t✐♦♥s✲ ✉♥❞ ❙♣r❛❝❤✈❡r❛r❜❡✐t✉♥❣ ✹t❤ ●r❛♠♠❛t✐❝❛❧ ❋r❛♠❡✇♦r❦ ❙✉♠♠❡r ❙❝❤♦♦❧ ●♦③♦✱ ❏✉❧② ✶✸✕✷✹✱ ✷✵✶✺ ✶ ✴ ✹✷

❲❡ ❢♦❝✉s ♦♥ s✉❜t②♣❡s ✐♥ ❛❜str❛❝t s②♥t❛① ✱ ✐✳❡✳ s✉❜t②♣❡s ✐♥ ❛❜str❛❝t r❡s♦✉r❝❡ ❣r❛♠♠❛rs r❛t❤❡r t❤❛♥ ❛♣♣❧✐❝❛t✐♦♥ ❣r❛♠♠❛rs✳ ●❋ ❛❞♠✐ts s✉❜t②♣❡s ✐♥ ❝♦♥❝r❡t❡ ❣r❛♠♠❛rs✱ ❜✉t ♥♦t ✐♥ ❛❜str❛❝t ♦♥❡s✳ • ❙✉❜t②♣❡s ✐♥ ❝♦♥❝r❡t❡ ❣r❛♠♠❛r✿ r❡❝♦r❞ s✉❜t②♣✐♥❣ • ❙✉❜t②♣❡s ✐♥ ❛❜str❛❝t ❣r❛♠♠❛r✿ ❞♦ t❤❡② ♠❛❦❡ ♠✉❝❤ s❡♥s❡❄ ❈♦♥t❡♥t✿ • ◆♦t✐♦♥s ♦❢ s✉❜t②♣✐♥❣✱ ❛♥❞ t❤❡✐r ✉s❡ ✐♥ ●❋ • ❊①❛♠♣❧❡s ✇✐t❤ ❞❡♣❡♥❞❡♥t t②♣❡s • ❊①❛♠♣❧❡s ♦❢ ♣♦ss✐❜❧❡ ✉s❡ ♦❢ s✉❜t②♣❡s ✐♥ ❛❜str❛❝t ❣r❛♠♠❛r • ■ss✉❡s ♦❢ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ s✉❜t②♣❡s ✐♥ ❛❜str❛❝t ❣r❛♠♠❛r • ❑♥♦✇♥ ❝♦♠♣❧❡①✐t② r❡s✉❧ts ✇✐t❤ r❡s♣❡❝t t♦ s✉❜t②♣✐♥❣ ✷ ✴ ✹✷

❇❛s✐❝ ✐❞❡❛ ♦❢ s✉❜t②♣❡s ❚❤❡ ❣❡♥❡r❛❧ ✐❞❡❛ ✭✐♥ ♦❜❥❡❝t✲♦r✐❡♥t❡❞ ♣r♦❣r❛♠♠✐♥❣✱ ❢♦r ❡①❛♠♣❧❡✮ ✐s✿ • ✐❢ ❜ : ❇ ❛♥❞ ❇ ≤ ❆ ✱ t❤❡♥ ❜ : ❆ ■❢ ❢✉♥❝t✐♦♥ t②♣❡s ❆ → ❈ ❝♦♥s✐st ♦❢ t♦t❛❧ ❢✉♥❝t✐♦♥s ♦♥❧②✱ t❤✐s ✐♠♣❧✐❡s • ✐❢ ❢ : ❆ → ❈ ❛♥❞ ❜ : ❇ ≤ ❆ ✱ t❤❡♥ ❢ ❛♣♣❧✐❡s t♦ ❜ ✱ ❛♥❞ ❢ ( ❜ ) : ❈ ✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❤❡♥ ❈ = ❜♦♦❧ ✿ ♦❜❥❡❝ts ♦❢ ❛ s✉❜t②♣❡ ❇ ≤ ❆ ❝❛♥ ❤❛✈❡ ❛❧❧ ♣r♦♣❡rt✐❡s t❤❛t ♦❜❥❡❝ts ♦❢ t②♣❡ ❝❛♥ ❆ ❤❛✈❡✱ ❛♥❞ ♣♦ss✐❜❧② s♦♠❡ ♠♦r❡✳ ■❢ t❤❡ ❢✉♥❝t✐♦♥s ❛r❡ s②♥t❛❝t✐❝ ❝♦♥str✉❝t✐♦♥s✱ t❤✐s ♠❡❛♥s✿ ❡①♣r❡ss✐♦♥s ♦❢ ❛ ❝❛t❡❣♦r② ❇ ≤ ❆ ❝❛♥ ❜❡ ✉s❡❞ ✐♥ ❡✈❡r② ❝♦♥str✉❝t✐♦♥ ✇❤❡r❡ t❤♦s❡ ♦❢ ❝❛t❡❣♦r② ❆ ❝❛♥ ❜❡ ✉s❡❞✳ ✸ ✴ ✹✷

❙✉❜t②♣✐♥❣ ❢♦r ❜❛s✐❝ ♦❜❥❡❝ts✿ s✉❜s✉♠♣t✐♦♥ ■♥ ❛ ❈❋●✱ ❡①♣r❡ss✐♦♥ ❝❛t❡❣♦r✐❡s ❳ ❛r❡ ✐♥t❡r♣r❡t❡❞ ❛s str✐♥❣ s❡ts ❉ ❳ = ▲ ( ❳ ) ✱ s♦ ❇ ≤ ❆ ≤ str✐♥❣ ♠❡❛♥s ▲ ( ❇ ) ⊆ ▲ ( ❆ ) ⊆ ❉ str✐♥❣ ✳ ❍❡♥❝❡✿ ✐♥ ❛ ❈❋●✱ ❆ → ❇ ✶ · · · ❇ ♥ | ❈ ✶ · · · ❈ ❦ ❛♠♦✉♥ts t♦ s✉❜t②♣❡ ❛ss✉♠♣t✐♦♥s ❆ ≥ ❇ ✶ · · · ❇ ♥ ❛♥❞ ❆ ≥ ❈ ✶ · · · ❈ ❦ ❋♦r ❡①❛♠♣❧❡✱ ◆P → Pr♦♥ r❡s♣✳ Pr♦♥ ≤ ◆P ♠❡❛♥s✿ ❛ ♣r♦♥♦✉♥ ❝❛♥ ♦❝❝✉r ✇❤❡r❡❡✈❡r ❛ ♥♦✉♥ ♣❤r❛s❡ ❝❛♥ ♦❝❝✉r✳ ✹ ✴ ✹✷

❇✉t✿ Pr♦♥ ≤ ◆P ✐s♥✬t r❡❛❧❧② tr✉❡ ✐♥ ●❡r♠❛♥✿ • P♦ss❡ss✐✈❡ ◆P ❣❡♥ ✲❛ttr✐❜✉t❡s ✐♥ ❉❡t ◆ ◆P ❣❡♥ ♠✉st ♥♦t ❜❡ Pr♦♥ ✬s✿ ✭❛❧❧❡✮ ❞✐❡ ❲❡r❦❡ ✭❞❡s ❆✉t♦rs ⑤ ●♦❡t❤❡s ⑤ ✯s❡✐♥❡r✮ Pr♦♥✬s ❤❛✈❡ s♣❡❝✐❛❧ ♣♦ss❡ss✐✈❡ ❢♦r♠s ❛s ❞❡t❡r♠✐♥❡rs✿ ✭❛❧❧❡✮ s❡✐♥❡ ❲❡r❦❡ • ❆♥♦t❤❡r ♣♦ss❡ss✐✈❡ ❝♦♥str✉❝t✐♦♥ ❛♣♣❧✐❡s ✭❜❡tt❡r❄✮ t♦ ❛❧❧ ◆Ps✿ ❞✐❡ ❲❡r❦❡ ✭✭❄✮✈♦♥ ❞❡♠ ❆✉t♦r ⑤ ✈♦♥ ●♦❡t❤❡ ⑤ ✈♦♥ ✐❤♠✮ ❙♦✱ Pr♦♥ �≤ ◆P ✿ ✭▼❛②❜❡ ◆P P ✸ ≤ Pr♦♥ P ✸ ✳✮ ▲✐❦❡✇✐s❡ ◆ ❣❡♥ �≤ ◆P ❣❡♥ ♣❧ ✿ ♣❧ ❡✐♥ ❚❛❣ ✭❞❡s ●❧ü❝❦s ⑤ ✯●❧ü❝❦s✮ ❀ ❞✐❡ ❋r❡✉❞❡ ✭❞❡r ❋✐s❝❤❡ ⑤ ✯❋✐s❝❤❡✮ ✱ ❆ ♠♦r❡ ♣r❡❝✐s❡ ♣♦ss❡ss✐✈❡ r✉❧❡ ✇❡r❡✿ ◆P → ❉❡t ◆ ✭◆P✲Pr♦♥✲◆✮ ❣❡♥ ✺ ✴ ✹✷

❈♦❡r❝✐✈❡ s✉❜t②♣✐♥❣ ❙✉❜t②♣✐♥❣ ♦♥ ❜❛s❡ ❞♦♠❛✐♥s✱ ❧✐❦❡ ❜♦♦❧ ≤ ✐♥t ✐♥ ♣r♦❣r❛♠♠✐♥❣✱ ✐s ✉♥❝♦♠♠♦♥ ✐♥ ●❋ ✭✐❢ ✐t ❡①✐sts ❛t ❛❧❧✮✳ ❲❤❛t ♦❝❝✉rs ✈❡r② ♦❢t❡♥ ✐s s✉❜t②♣✐♥❣ ✈✐❛ ❛ ❝♦❡r❝✐♦♥ ❢✉♥❝t✐♦♥✱ ❇ ≤ ❆ ⇐ ⇒ ❉ ❇ ⊆ ❝ ❉ ❆ , ♠❡❛♥✐♥❣ ❝ : ❉ ❇ → ❉ ❆ . ❖❢t❡♥✱ ❝ ✐s ✐♥❥❡❝t✐✈❡ ♦r ❡✈❡♥ ❛ ❝♦♥str✉❝t♦r✱ s✉❝❤ ❛s ❯s❡◆ ✿ ◆ ✲❃ ❈◆ ❀ ❯s❡Pr♦♥ ✿ Pr♦♥ ✲❃ ◆P❀ ■♠♣❱P ✿ ❱P ✲❃ ■♠♣ ❀ ❆♥ ❛❞✈❛♥t❛❣❡✭❄✮ ✐s t❤❛t ❞✐✛❡r❡♥t ❝♦❡rs✐♦♥s ❝❛♥ ❝❛✉s❡ ❇ ≤ ❆ ✱ ❛s ✐♥ � ❢r♦♠ ❘❛♥t❛ ✷✵✶✹ �≡ fun Decl : Cl -> S ; Quest : Cl -> S ✻ ✴ ✹✷

❙✉❜t②♣✐♥❣ ❢♦r r❡❝♦r❞s✿ ✏❝♦♠♣♦♥❡♥t ♦♠✐tt✐♥❣✑ ❝♦❡r❝✐♦♥ ❆ r❡❝♦r❞ t②♣❡ ρ = { ✐ : τ ✐ | ✐ ∈ ■ } ✱ ✇❤❡r❡ ■ ⊆ ▲❛❜ ✐s ❛ ✜♥✐t❡ s❡t ♦❢ ❧❛❜❡❧s✱ ✐s ✐♥t❡r♣r❡t❡❞ ❛s t❤❡ s❡t ♦❢ ❛❧❧ ❞❡♣❡♥❞❡♥t ❢✉♥❝t✐♦♥s � ❉ ρ = { ❢ : ■ → ❉ τ ✐ | ❢ ( ✐ ) ∈ ❉ τ ✐ , ✶ ≤ ✐ ≤ ♥ } . ✐ ∈ ■ ❲r✐t❡ ❢ ∈ ❉ ρ ❛s { ✐ = ❛ ✐ | ✐ ∈ ■ , ❛ ✐ ∈ ❉ τ ✐ } ✇❤❡r❡ ❛ ✐ = ❢ ( ✐ ) ✳ ❋♦r r❡❝♦r❞ t②♣❡s✱ s✉❜t②♣❡s ♠❛② ❤❛✈❡ ❛❞❞✐t✐♦♥❛❧ ✜❡❧❞s ❛♥❞ ✜❡❧❞s ✇✐t❤ s♠❛❧❧❡r t②♣❡s ❏ ⊇ ■ , τ ✐ ≤ σ ✐ ❢♦r ❛❧❧ ✐ ∈ ■ ❇ := { ❥ : τ ❥ | ❥ ∈ ❏ } ≤ { ✐ : σ ✐ | ✐ ∈ ■ } =: ❆ ( r❡❝ ≤ ) ❍❡r❡ t❤❡ ✐♥t❡r♣r❡t❛t✐♦♥ ❉ ❇ ✐s s✉❜s✉♠❡❞ ❜② ❉ ❆ ✈✐❛ ❛ ❝♦❡r❝✐♦♥ ❝ ❇ ≤ ❆ ⇐ ⇒ { ❝ ( ❢ ) | ❢ ∈ ❉ ❇ } ⊆ ❉ ❆ ⇐ ⇒ : ❉ ❇ ⊆ ❝ ❉ ❆ ✇❤❡r❡ ❝ ❝♦❡r❝❡s ❢ ❜② ❝ ❇ , ❆ ( ❢ )( ✐ ) = ❝ τ ✐ ,σ ✐ ( ❢ ↾ ■ ( ✐ )) ❢♦r ✐ ∈ ■ ✳ ✼ ✴ ✹✷

❖❜❥❡❝ts ♦❢ ❛ s✉❜t②♣❡ ❝♦♥t❛✐♥ ♠♦r❡ ❛♥❞ ♠♦r❡ ❞❡t❛✐❧❡❞ ✏✐♥❢♦r♠❛t✐♦♥✑✳ � ❊①❛♠♣❧❡s ❢r♦♠ ❈❛t❊♥❣✳❣❢ �≡ NP = {s : NPCase => Str ; a : Agr} ; Pron = {s : NPCase => Str ; sp : Case => Str ; a : Agr} ; Ord = { s : Case => Str } ; Num = {s : Case => Str ; n : Number ; hasCard : Bool} ; Card = {s : Case => Str ; n : Number} ; Subj = {s : Str} ; Prep = {s : Str; isPre : Bool} ; ❆s r❡❝♦r❞ t②♣❡s✱ t❤✐s ❣✐✈❡s Pron < NP ✱ Num < Card < Ord ✱ Prep < Subj ✳ ❇✉t✱ ♦❢ ❝♦✉rs❡✱ ●❋ ❞♦❡s ♥♦t ✉s❡ ♣r❡♣♦s✐t✐♦♥s ❛s s✉❜❥✉♥❝t✐♦♥s ❡t❝✳ ✽ ✴ ✹✷

❙✉❜t②♣✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s✿ ❢ : ( ❆ → ❇ ) ≤ ( ❆ ′ → ❇ ′ ) ♠❛♣s ❛❧❧ ❡❧❡♠❡♥ts ♦❢ ❆ ≥ ❆ ′ t♦ ✈❛❧✉❡s ♦❢ ❇ ≤ ❇ ′ ■❢ ( ❆ → ❇ ) = { ❢ | ∀ ❛ : ❆ , ❢ ( ❛ ) : ❇ } ✐s t❤❡ s❡t ♦❢ t♦t❛❧ ❢✉♥❝t✐♦♥s✱ t❤❡♥ → ✐s ❝♦♥tr❛✈❛r✐❛♥t ✐♥ ✐ts ❛r❣✉♠❡♥t ❛♥❞ ❝♦✈❛r✐❛♥t ✐♥ t❤❡ t❛r❣❡t✿ ❆ ′ ≤ ❆ ❇ ≤ ❇ ′ ( ❆ → ❇ ) ≤ ( ❆ ′ → ❇ ′ ) ( → ≤ ) ❊①♣❧✿ ■♥ t❤❡ ❘●▲ ♦❢ ●❋✱ V2 = V ** {c2:Case} < V ✳ ❍❡♥❝❡ • ❛♥② ❢ : ❈ → ❱ ✷ ✐s ❛ ❢ : ❈ → ❱ • ❛♥② ❣ : ❱ → ❈ ✐s ❛ ❣ : ❱ ✷ → ❈ ✳ PassV2 : V2 -> VP ✐s ♥♦t ❛♣♣❧✐❝❛❜❧❡ t♦ ❛r❜✐tr❛r② v:V ✳ � ❊①❛♠♣❧❡ ❢r♦♠ ❱❡r❜✳❣❢✱ ✉s✐♥❣ ❝♦❡rs✐✈❡ s✉❜②♣✐♥❣ �≡ fun SlashV2a : V2 -> VPSlash ; -- aka V2 < VPSlash ReflVP : VPSlash -> VP ; ❍❡♥❝❡ ReflVP ✐s ❛♣♣❧✐❝❛❜❧❡ t♦ v2:V2 ✱ ✈✐❛ (SlashV2a v2) ✳ ✾ ✴ ✹✷