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❈❛♥ ❲❡ ❆❞❞ ❙✉❜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❜❡ ◆PP✸ ≤ 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)✳

✾ ✴ ✹✷

❘❡❝♦r❞ s✉❜t②♣✐♥❣ ✐♥ ●❋✬s ❝♦♥❝r❡t❡ ❣r❛♠♠❛rs

❆❧t❤♦✉❣❤ ●❋✬s ❝♦♥❝r❡t❡ s②♥t❛① ❤❛s r❡❝♦r❞ s✉❜t②♣✐♥❣✱ t❤❡ ❤✐❞❞❡♥ lock✲✜❡❧❞s ♦❢ ✐♠♣❧❡♠❡♥t❛t✐♦♥ t②♣❡s ❜❧♦❝❦ ❛♣♣❛r❡♥t s✉❜t②♣✐♥❣s✿ ❊①❛♠♣❧❡ ❢r♦♠ ❈❛t❊♥❣✳❣❢ ≡ NP = {s : NPCase => Str ; a : Agr} ; Pron = {s : NPCase => Str ; sp : Case => Str ; a : Agr} ; ❆❝t✉❛❧ ✐♠♣❧❡♠❡♥t❛t✐♦♥ t②♣❡s ❛r❡ ❞✐✛❡r❡♥t≡ Lang> cc -unqual NP {s : NPCase => Str; a : Agr; lock NP : {}} Lang> cc -unqual Pron {s : NPCase => Str; a : Agr; sp : Case => Str; lock Pron : {}} ❍❡♥❝❡✱ ✇❡ ❞♦♥✬t ❤❛✈❡ Pron < NP✳ ■♥st❡❛❞✱ ●❋ ✉s❡s ❛ ❝♦❡r❝✐♦♥ UsePron : Pron -> NP t♦ ❞r♦♣ ✜❡❧❞s ❛♥❞ ❛❞❥✉st t❤❡ ❧♦❝❦✲✜❡❧❞✳ ■♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ UsePron : Pron -> NP≡ \p -> {s = p.s; a = p.a; lock NP : {} = <>}

✶✵ ✴ ✹✷

❊①❛♠♣❧❡✿ ❛❜str❛❝t ✇✐t❤ ❝♦❡r❝✐♦♥ ❢✉♥❝t✐♦♥≡ abstract Subtype = { cat A ; B ; C ; D ; fun b : B ; UseB : B -> A ; } ❊①❛♠♣❧❡✿ ❝♦♥❝r❡t❡ ✇✐t❤ ❇ ❁ ❆ ✈✐❛ ❝♦❡r❝✐♦♥≡ concrete SubtypeConc of Subtype = { lincat A = {s:Str; r:C} ; B = {s,t:Str; r:D} ; C = {c:Str} ; D = {c,d:Str} ; lin b = {s,t = "b"; r = lin D {c = "c" ; d = "d"}} ; UseB x = lin A {s = x.s; r = x.r} ; } ❈♦❡r❝✐♦♥ ❢✉♥❝t✐♦♥✱ ♦♠✐tt✐♥❣ ✜❡❧❞ t ❛♥❞ s✉❜✜❡❧❞ d ♦❢ ✜❡❧❞ r≡ Subtype> cc UseB b {s : Str = "b"; r : {c : Str; lock C : {}} = {c : Str = "c"; d : Str = "d"; lock D : {} = <>}; lock A : {} = <>}

✶✶ ✴ ✹✷

❍♦✇ t❤❡♥ ✐s r❡❝♦r❞ s✉❜t②♣✐♥❣ ✉s❡❞ ✐♥ ❝♦♥❝r❡t❡ ❣r❛♠♠❛rs ✕ ✇✐t❤♦✉t ❝♦❡r❝✐♦♥ ❢✉♥❝t✐♦♥s❄

❋✉♥❝t✐♦♥s ✇✐t❤ r❡❝♦r❞ ❛r❣✉♠❡♥t ❛♥❞ r❡s✉❧t t②♣❡s ❛r❡ ❞❡✜♥❡❞ ❛s ♦♣❡r❛t✐♦♥s✱ ❛s ✐♥ ❋r♦♠ ❘❡s❊♥❣✳❣❢ ≡

• per

Verb : Type = { s : VForm => Str ; isRefl : Bool } ; VP : Type = { s : VerbForms ; ... } ; predV : Verb -> VP = \verb -> { s = ... ; ... } ❚❤❡♥✱ ❞✐✛❡r❡♥t s✉❜t②♣❡s ♦❢ Verb ❛r❡ ✐♥tr♦❞✉❝❡❞ ❜② ❋r♦♠ ❈❛t❊♥❣✳❣❢ ≡ lincat V, VS, VQ, VA = Verb ; ❧❡❛❞✐♥❣ t♦ r❡❝♦r❞ t②♣❡s Verb ** { lock V : {} } ❡t❝✿ ■♠♣❧❡♠❡♥t❛t✐♦♥ t②♣❡s ♦❢ ❱✱ ❱❙✱ ❡t❝✳ < Verb≡ {s : VForm => Str; isRefl : Bool; lock V : {}} {s : VForm => Str; isRefl : Bool; lock VS : {}}

✶✷ ✴ ✹✷

❋✐♥❛❧❧②✱ ❛s V,VS,VA,VQ < Verb✱ predV ❝❛♥ ❜❡ ❛♣♣❧✐❡❞ t♦ ❛❧❧ ❦✐♥❞s ♦❢ ✈❡r❜s ✭✇✐t❤♦✉t ✉s✐♥❣ ❝♦❡r❝✐♦♥ ❢✉♥❝t✐♦♥s✮✿ ❋r♦♠ ❛❜str❛❝t✴❱❡r❜✳❣❢ ❛♥❞ ❱❡r❜❊♥❣✳❣❢ ≡ data UseV : V

• > VP ;
• - sleep

ComplVS : VS

• > S
• > VP ;
• - say that she runs

ComplVQ : VQ

• > QS -> VP ;
• - wonder who runs

ComplVA : VA

• > AP -> VP ;
• - they become red

lin UseV = predV ; ComplVS v s = insertExtra (conjThat ++ s.s) (predV v) ComplVQ v q = insertExtra (q.s ! QIndir) (predV v) ; ComplVA v ap = insertObj (ap.s) (predV v) ; ❘❡♠❛r❦✿ ■♥ t❤❡ ❛❜str❛❝t s②♥t❛①✱ ❛ ❝❛t❡❣♦r② Verb ❞♦❡s ♥♦t ❡①✐st✱ ❛♥❞ V,VS,VQ,VA ❛r❡ ❥✉st ❞✐✛❡r❡♥t ❝❛t❡❣♦r✐❡s✳

✶✸ ✴ ✹✷

❇❡s✐❞❡s t❤❡ r❡❝♦r❞ s✉❜t②♣✐♥❣✱ t❤❡ ❝♦♥❝r❡t❡ ❣r❛♠♠❛rs ✉s❡ ❝♦❡rs✐✈❡ s✉❜t②♣✐♥❣ t♦ ❡①t❡♥❞ ♣❛r❛♠❡t❡r t②♣❡s✿ ❢r♦♠ ❘❡s●❡r✳❣❢ ≡ param GenNum = GSg Gender | GPl ; NPForm = NPCase Case | NPPoss GenNum Case ; ✐♥ t❤✐s ❝❛s❡ ♠❛❦✐♥❣ ❞✐s❥♦✐♥t ✉♥✐♦♥s ❉●❡♥◆✉♠ ≃ ❉●❡♥❞❡r ∪ {P❧}, ❉◆P❋♦r♠ ≃ ❉❈❛s❡ ∪ (❉●❡♥◆✉♠ × ❉❈❛s❡) ■t ✇♦✉❧❞ s♦♠❡t✐♠❡s ❜❡ ♥✐❝❡ t♦ ❤❛✈❡ s✐♠♣❧❡ s✉❜s✉♠♣t✐♦♥s ❜❡t✇❡❡♥ ❞❛t❛t②♣❡s✱ s✉❝❤ ❛s ❢❛❦❡ ❝♦❞❡≡ subparam Case-Nom < Case ; fun ReflPron : { s: Case-Nom => Str ; a : Agr } ;

✶✹ ✴ ✹✷

❉❡♣❡♥❞❡♥t t②♣❡s ✐♥ t❤❡ ❛❜str❛❝t ❣r❛♠♠❛r

❚❤❡ ❛❜str❛❝t ❣r❛♠♠❛r ♦❢ ●❋ ❤❛s ❞❡♣❡♥❞❡♥t t②♣❡s✱ ❜✉t ♥♦ s✉❜t②♣❡s✳ ❚②♣❡ ❤②♣♦t❤❡s✐s≡ Hyp := (x : T) | ( : T) | T ❈♦♥t❡①t≡ G := | Hyp G ❇❛s✐❝ ❝❛t❡❣♦r② ❞❡❝❧❛r❛t✐♦♥≡ cat C G ; ❆ ❝❛t❡❣♦r② ❞❡❝❧❛r❛t✐♦♥ cat C (x1:T1) ... (xn:Tn) ✐♥tr♦❞✉❝❡s ❛ t②♣❡ ❝♦♥str✉❝t♦r ❈ ✿ ❚✶ ✲❃ ✳✳✳ ✲❃ ❚♥ ✲❃ ❚②♣❡ ❋♦r ❛✐ : ❚✐ : ❚②♣❡ ❛♥❞ ♥ > ✵✱ (C a1 ... an) ✐s ❛ ❞❡♣❡♥❞❡♥t t②♣❡✳

✶✺ ✴ ✹✷

• ❋✲❜♦♦❦✱ ❊①❡r❝✐s❡s ✻✲✹✯✱ ✻✲✺✯ ✱ ♣✳✶✸✷✳

❙✉❜❥❡❝t✲✈❡r❜ ❛❣r❡❡♠❡♥t ✐♥ ♥✉♠❜❡r ❝♦✉❧❞ ❜❡ ❜✉✐❧t ✐♥t♦ ♣r❡❞✐❝❛t✐♦♥✿ ◆✉♠❜❡r❆❣r✳❣❢ ≡ abstract NumberAgr = cat S ; Number ; NP Number ; VP Number fun Pred : (n:Number) -> NP n -> VP n -> S ; ❱❡r❜ t②♣❡s ❝♦✉❧❞ ❜❡ ♠❛❞❡ ❞❡♣❡♥❞❡♥t ♦♥ s✉❜❝❛t✲❢r❛♠❡s ✭❍P❙●✮✿ ❙✉❜❝❛t✳❣❢ ≡ abstract Subcat = { cat VSub ; VP ; Comps VSub ; fun Compl : (sub : VSub) -> V sub -> Comps sub -> VP ❇✉t✿ ❈✉rr❡♥t ❘●▲ ❞♦❡s ♥♦t ✉s❡ ❞❡♣❡♥❞❡♥t t②♣❡s ✭❛s ❢❛r ❛s ■ ❦♥♦✇✮✳

✶✻ ✴ ✹✷

❊①❛♠♣❧❡ ✭❙✉❜❝❛t ❢r❛♠❡s✮

❆❛r♥❡ ❘❛♥t❛✬s ❣r❛♠♠❛r ✐♥ ✏❚②♣❡s ❛♥❞ ❘❡❝♦r❞s ❢♦r Pr❡❞✐❝❛t✐♦♥✑ ♣❛r❛♠❡t❡r✐③❡s ♣❤r❛s❡ ❝❛t❡❣♦r✐❡s ❜② ❛ ❧✐st ♦❢ ❛r❣✉♠❡♥t t②♣❡s✳ ❲❤❡♥ ❝♦♠❜✐♥✐♥❣ ❡①♣r❡ss✐♦♥s✱ t❤❡ ❧✐st ♦❢ ❛r❣✉♠❡♥t t②♣❡s ✐s r❡❞✉❝❡❞ ❜② t❤❡ t②♣❡ ♦❢ t❤❡ ❛r❣✉♠❡♥t ❡①♣r❡ss✐♦♥✳ ❢r♦♠ ❆❘✬s ❣r❛♠♠❛r≡ cat Arg ; Args ; V Args ; VP Args ; ... fun ap, cl, cn, np, qcl, vp : Arg ; 0 : Args ; c : Arg -> Args -> Args ; -- lists UseV : (x:Args) -> V x -> VP x ; -- (simplified) ComplNP : (x:Args) -> VP (c np x) -> NP -> VP x ; ReflVP : (x:Args) -> VP (c np x) -> VP x ; ... ■s t❤✐s ❦✐♥❞ ♦❢ ♣❛r❛♠❡t❡r✐③❛t✐♦♥ ❛ s✉❜st✐t✉t❡ t♦ s✉❜❝❛t❡❣♦r✐❡s V x < V ❡t❝✳❄ ❚❤❡ ❝♦❞❡ ✐s s✐♠♣❧❡r t❤❛♥ ✇✐t❤ ❞✐✛❡r❡♥t ❝❛t❡❣♦r✐❡s V x ❡t❝✳

✶✼ ✴ ✹✷

❊①❛♠♣❧❡ ✭❆❞✈❡r❜✐❛❧ ❞✐♠❡♥s✐♦♥s✮

❉✐✛❡r❡♥t ✈❡r❜s ❛r❡ ♠♦❞✐✜❛❜❧❡ ✐♥ ❞✐✛❡r❡♥t ❛❞✈❡r❜✐❛❧ ❞✐♠❡♥s✐♦♥s✳ ▲❡t ❱s ❛♥❞ ❱Ps ❝❛rr② t❤❡ ❞✐♠❡♥s✐♦♥s ✐♥ ✇❤✐❝❤ t❤❡② ❝❛♥ ❜❡ ♠♦❞✐✜❡❞✱ ❛♥❞ ✇❤❡♥ ❝♦♠❜✐♥✐♥❣ ✇✐t❤ ❛ ♠♦❞✐✜❡r✱ r❡♠♦✈❡ t❤❡ ❞✐♠❡♥s✐♦♥ ❛t ❱P✳ Ps❡✉❞♦❝♦❞❡ ≡ cat Kind ; fun loc, dir, tmp, instr, mod : Kind cat Adv Kind ; V Kind ; V Kind Kind ; ... VP Kind ; VP Kind Kind ; ... fun here : Adv loc ; later : Adv tmp ; ... live : V loc mod ; -- live nicely in Paris travel : V dir instr ; -- travel to Malta by plane ModVP : (x, y : Kind) -> VP x y -> Adv y -> VP x ; ... ❚❤❡r❡ ✐s ❛ s✉❜❝❛t ❤✐❡r❛r❝❤② ❝♦♥✈❡rs❡ t♦ ⊆ ♦♥ s❡ts ♦❢ Kind✿ ✐❢ ❳ ⊆ ❨ ⊆ ❑✐♥❞✱ t❤❡♥ (subcat VP Y < VP X) ✭✉s✐♥❣ ❝✲❧✐sts ❳✮✳

✶✽ ✴ ✹✷

❊①❛♠♣❧❡ ✭◆✉♠❜❡r r❡str✐❝t✐♦♥s ✐♥ ◆Ps ❛♥❞ ❱Ps✮

◆♦t ♦♥❧② ✐♥ ●❋✱ ◆Ps ❤❛✈❡ ✐♥❤❡r❡♥t ♥✉♠❜❡r✱ ❣❡♥❞❡r✱ ♣❡rs♦♥✳ ❇✉t ❝♦♦r❞✐♥❛t❡❞ ◆Ps ❛❝t✉❛❧❧② ❞♦♥✬t✱ s♦ ●❋ ✉s❡s ❛rt✐✜❝✐❛❧ ✈❛❧✉❡s✿ ❛❣r❡❡♠❡♥t ✈❛❧✉❡s ♦❢ ✧❞✉ ♦❞❡r ✇✐r✧≡ Lang> cc -unqual (ConjNP or Conj (BaseNP (UsePron youSg Pron) (UsePron we Pron))).a Ag Fem Pl P1 ❆s ♥✉♠❜❡r ❛♥❞ ♣❡rs♦♥ ❛r❡ ♥❡❡❞❡❞ t♦ s❡❧❡❝t t❤❡ t❤❡ ✈❡r❜ ❢♦r♠✱ s✉❝❤ ◆Ps ❝❛♥♥♦t ❜❡ ✉s❡❞ ❛s s✉❜❥❡❝ts✿ ✭❞✉ ♦❞❡r ✇✐r✮ ✯✭♠✉ÿt ⑤ ♠üss❡♥✮ ❡s t✉♥ → ✭✭❞✉ ♦❞❡r ✇✐r✮✱ ❥❡♠❛♥❞✸P,❙❣✮ ♠✉ÿ ❡s t✉♥ ❆❧s♦✿ ✈❡r❜s ♠❛② ❞❡♠❛♥❞ t❤❡✐r s✉❜❥❡❝ts ✭♦r ♦❜❥❡❝ts✮ t♦ ❜❡ ♣❧✉r❛❧❀ ❞❡t❡r♠✐♥❡rs s♣❧✐t ✐♥t♦ ♠❛ss✲ ✈s✳ ✐♥❞✐✈✐❞✉❛❧✲❞❡t✬s ❛♥❞ ❝r❡❛t❡ ◆Ps ✐♥ sg r❡s♣✳ pl✳ ✭✯♠❛♥② ❣♦❧❞✱ ✯♠✉❝❤ ❞❛②s✮

✶✾ ✴ ✹✷

❙♦✱ t♦ ❜❡ ♣r❡❝✐s❡✱ ✇❡ s❡❡♠ t♦ ♥❡❡❞ s✉❜❝❛t❡❣♦r✐❡s✿

• ◆Ps ✇✐t❤ ✜①❡❞ ♥✉♠❜❡r✿ ◆Ps❣ ❛♥❞ ◆P♣❧✱ ✭✉s❛❜❧❡ ❛s s✉❜❥❡❝t ♦r

♦❜❥❡❝t✱ ✇❤❡♥ ♠❡❡t✐♥❣ ❛ ♣♦ss✐❜❧❡ ♥✉♠❜❡r ❝♦♥str❛✐♥t ♦❢ t❤❡ ✈❡r❜✮

• ◆Ps ✇✐t❤ ♥♦ ❞❡✜♥✐t❡ ♥✉♠❜❡r✿ ◆P♥♦♥❡ ✭✉s❛❜❧❡ ❛s ♦❜❥❡❝t✱✳ ✳ ✳ ✮
• ❱Ps ❝♦♥str❛✐♥✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✐ts ◆P✲❛r❣✉♠❡♥t✿ ❱P♣❧
• ❱Ps ♥♦t ❝♦♥str❛✐♥✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✐ts ◆P✲❛r❣✉♠❡♥t✿ ❱P❛♥②

❚❤✐s ❣✐✈❡s✿ ❱P❛♥② ≤ ❱P♣❧ ❛♥❞ ◆Ps❣, ◆P♣❧ ≤ ◆P♥♦♥❡✳ ❖r ❝❛♥ ✇❡ ❞♦ ✐t ✇✐t❤ ❞❡♣❡♥❞❡♥t t②♣❡s ❛♥❞ s❡✈❡r❛❧ ♣r❡❞✐❝❛t✐♦♥ r✉❧❡s❄ ❢r♦♠ ❉❡♣❘❡❝✐♣r♦❝❛❧s✳❣❢ ≡ cat Number ; fun sg, pl, any : Number ; cat NP Number ; V1 Number ; ... ; VP Number ; VPSlash Number Number ; S ; fun agree1 : V1 pl ;

• - subject must be pl

walk1 : V1 any ;

• - subject may be sg or pl

agree2 : V2 any any ;

• - to agree with sb.

mix2 : V2 any pl ;

• - object must be pl

✷✵ ✴ ✹✷

❢r♦♠ ❉❡♣❘❡❝✐♣r♦❝❛❧s✳❣❢ +≡ UseV1 : (n:Number) -> V1 n -> VP n ; PredVP : (n:Number) -> NP n -> VP n -> S ; -- n-agree PredVPany : (n:Number) -> NP n -> VP any -> S ; ... ComplV3 : (n,m,l:Number) -> V3 n m any -> NP l

• > VPSlash n m ;
• - reciprocal obj reduce VP’s arity and enforce pl

Reci2any : VPSlash any any -> VP pl ; Reci2pl : VPSlash any pl -> VP pl ; Reci3 : V3 any any any -> VPSlash any pl ;

• - (I|we) introduce them(pl) to each other

❲♦r❦s ♣❛rt❧②✱ ♥♦t ♣r❡❝✐s❡❧②✱ ❛s ■ ♠✐s✉s❡❞ ◆P❛♥② ❢♦r ◆P♥♦♥❡✳ ❉♦❛❜❧❡✱ ❜✉t ❝❧✉♠s②✱ ✐❢ ✇❡ ♥❡❡❞ ❞✐✛❡r❡♥t ❦✐♥❞s KindNP✱ KindVP ❛♥❞ ♠❛♥② ❢✉♥❝t✐♦♥s Pred k1 k2 : NP k1 -> VP k2 -> S✳

✷✶ ✴ ✹✷

❲❤❡r❡ ❝♦✉❧❞ ✇❡ ✉s❡ s✉❜t②♣❡s ✐♥ ❛❜str❛❝t s②♥t❛①❄

■❣♥♦r✐♥❣ t❤❡ ♣♦ss✐❜❧❡ ✉s❡❢✉❧♥❡ss ♦❢ s✉❜t②♣❡s ❢♦r ❛♣♣❧✐❝❛t✐♦♥ ❣r❛♠♠❛rs ✕ ✇❤② ✇♦✉❧❞ ✇❡ ✇❛♥t s✉❜t②♣❡s ✐♥ ❛❜str❛❝t s②♥t❛①❄

❊①❛♠♣❧❡ ✭❱❡r❜s✿ ❱✵ ❁ ❱❄✮

• ✵✲❛r② ✈❡r❜s ❛❞♠✐t ♦♥❧② ✐♠♣❡rs♦♥❛❧ ❝♦♥str✉❝t✐♦♥s✿

it rains (heavily) (today)

• ❣r❛♠♠❛r r✉❧❡s ✉s❡ s✉✐t❛❜❧❡ ❦✐♥❞s ♦❢ ✈❡r❜s ✭❈●✮

◆♦t✐❝❡✿ ●❋✬s ❘●❇ ❞♦❡s ♥♦t s❡♣❛r❛t❡ V0 ❢r♦♠ V✿ fun rain V0 : V ❱✵ ✇✐t❤ ✭❣♦♦❞✿✮ ✐♠♣❡rs♦♥❛❧ ❛♥❞ ✭❜❛❞✿✮ ♣❡rs♦♥❛❧ ❝♦♥str✉❝t✐♦♥≡ Lang> parse -cat=Cl "it rains" ImpersCl (UseV rain V0) PredVP (DetNP (DetQuant DefArt NumSg)) (UseV rain V0) PredVP (UsePron it Pron) (UseV rain V0)

✷✷ ✴ ✹✷

❊①❛♠♣❧❡ ✭❱✶ ✇✐t❤ ♣❛ss✐✈❡s ❁ ❱✶❄✮

• ●❡r♠❛♥ ✐♥tr❛♥s✐t✐✈❡ ❛❝t✐♦♥ ✈❡r❜s ❛❞♠✐t ❛ ♣❛ss✐✈❡✿

sie arbeiten ✕ es wird gearbeitet

• ♦t❤❡r ✐♥tr❛♥s✐t✈❡ ✈❡r❜s ❞♦♥✬t✿

die Sonne geht auf ✕ *es wird aufgegangen ▲✐❦❡✇✐s❡ ❢♦r ●❋✬s V2✿ s❤♦✉❧❞ t❤❡r❡ ❜❡ ❛ s✉❜t②♣❡ V2pass < V2❄ ❢r♦♠ ❛❜str❛❝t✴❱❡r❜✳❣❢ ≡

• - *Note*. the rule can be overgenerating, since
• - the $V2$ need not take a direct object.

PassV2 : V2 -> VP ;

• - be loved

❊①❛♠♣❧❡ ✭❉❡♣♦♥❡♥t ✈❡r❜s ❁ ❱❄✮

• ❞❡♣♦♥❡♥t ✈❡r❜s ✐♥ ▲❛t✐♥ ❛♥❞ ❆●r❡❡❦ ❧❛❝❦ ❛❝t✐✈❡ ❢♦r♠s ❛♥❞ ✉s❡

♣❛ss✐✈❡✴♠✐❞❞❧❡ ❢♦r♠s ✐♥st❡❛❞ ✭❤❡♥❝❡ ❤❛✈❡ ♥♦ ♣❛ss✐✈❡✮

• ❙❤♦✉❧❞ ✇❡ ❤❛✈❡ (subcat Vdep < V)❄

✷✸ ✴ ✹✷

❊①❛♠♣❧❡ ✭Pr♦♥s ❛♥❞ ♥♦✉♥ ♣❤r❛s❡s✿ ❘❡✢Pr♦♥ ❁ Pr♦♥ ❁ ◆P❄✮

• ❣r❛♠♠❛rs s✉❜s✉♠❡ ♣r♦♥♦✉♥s ✉♥❞❡r ♥♦✉♥ ♣❤r❛s❡s✱ ❜✉t✿
• r❡✢❡①✐✈❡ ❛♥❞ r❡❝✐♣r♦❝❛❧ ♣r♦♥♦✉♥s ❝❛♥♥♦t ❜❡ ✉s❡❞ ❛s s✉❜❥❡❝ts

(They | *Themselves) saw the movie (They | *each other) like (apples | each other) ❍❡♥❝❡✿ ❘❡❝✐Pr♦♥✱❘❡❢❧Pr♦♥ ≤ Pr♦♥✳ ❲❡ ❛❧r❡❛❞② s❛✇ Pr♦♥ ≤ ◆P✳ ❆✳ ❈♦♥❝❧✉s✐♦♥ s♦ ❢❛r✿ ❋♦r ❧❡①✐❝❛❧ ❝❛t❡❣♦r✐❡s✱ ✇❤❛t ✐♥t✉✐t✐✈❡❧② ♠❛② s❡❡♠ t♦ ❜❡ ❛ s✉❜t②♣❡ B ♦❢ ❝❛t❡❣♦r② A ❝♦rr❡s♣♦♥❞s ♦❢t❡♥ t♦ ❛ s✉❜s❡t ♦❢ ✇♦r❞s✱ ❧❛❝❦✐♥❣ s♦♠❡ ❜❡❤❛✈✐♦✉r ♦r ❤❛✈✐♥❣ s♦♠❡ s♣❡❝✐❛❧ ❜❡❤❛✈✐♦✉r✳ ❚❤❛t ✐s✱ ❆ = (❇ + . . .) ✐s ❛ ❞✐s❥♦✐♥t s✉♠ ✇✐t❤ s✉♠♠❛♥❞ ❇✱ ❉❆ = ❉❇ ˙ ∪ . . .

• ■❢ ❇s ❧❛❝❦ s♦♠❡ ❢♦r♠s ♦t❤❡r ❆s ❤❛✈❡✱ t❤❡♥ ❇ ≤ ❆✱ ❜✉t ♣❡r❤❛♣s

❆ ≤ ❇✳ ✭❘❡❧❢Pr♦♥ ≤ Pr♦♥ < ❘❡✢Pr♦♥✱ ❱❞❡♣ ≤ ❱ < ❱❞❡♣✮

• ■❢ ❇s ❡♥t❡r s♣❡❝✐❛❧ ❝♦♥str✉❝t✐♦♥s✱ t❤❡♥ ❆ ≤ ❇✳ ✭❱ ≤ ❱♣❛ss✮

✷✹ ✴ ✹✷

❇✳ ❋♦r ♣❤r❛s❛❧ ❝❛t❡❣♦r✐❡s✱ ✇❡ ❤❛✈❡ s❡❡♥ ❝❛♥❞✐❞❛t❡s ❢♦r ≤

• ❱Ps✱ ❱P❙❧❛s❤s ❤❛✈✐♥❣✴❧❛❝❦✐♥❣ ❛ ♣❧✉r❛❧✲❝♦♥str❛✐♥t ✭♣❡r ❛r❣✮
• ❱Ps✱❱P❙❧❛s❤s ♣✳♦r❞❡r❡❞ ❜② ♠♦❞✐✜❛❜✐❧✐t② ✐♥ ❛❞✈❡r❜ ❦✐♥❞s✶
• ◆Ps ✇✐t❤✴✇✐t❤♦✉t ❝❧❡❛r ♥✉♠❜❡r ❛♥❞ ♣❡rs♦♥ ❢❡❛t✉r❡✳

❚❤❡ ❧❛st ❡①❛♠♣❧❡ ❡①t❡♥❞s t♦ ♦t❤❡r ❝❛t❡❣♦r✐❡s✿

❊①❛♠♣❧❡ ✭❈♦♦r❞✐♥❛t❡❞ ❈s ❁ s✐♠♣❧❡ ❈s❄✮

■❢ ❈s ❤❛✈❡ ❛ ❣♦✈❡r♥✐♥❣ ❢❡❛t✉r❡✱ (C coord C) ❧❛❝❦s t❤❡ ❢❡❛t✉r❡✱ ✐❢ t❤❡ ❝♦♠♣♦♥❡♥t Cs ❞✐s❛❣r❡❡ ♦♥ ✐t✳ ❙♦ (C coord C) ✐s ♥♦t ✉s❛❜❧❡ ✐♥ ❡✈❡r② ❝♦♥t❡①t ✇❤❡r❡ C ✐s✳

• (ein oder der) (*kleiner|*kleine) Hund

❙✐♠✐❧❛r t♦ ◆P♥♦♥❡✱ ✇❡ ❤❛✈❡ ❉❡tstr♦♥❣, ❉❡t✇❡❛❦, ❉❡t♠✐①❡❞ < ❉❡t♥♦♥❡✳

✶❙✐♠✐❧❛r❧② ❢♦r ❱Ps ✇✐t❤ ❛❧t❡r♥❛t✐✈❡ ❝♦♠♣❧❡♠❡♥t ❢r❛♠❡s✳ ✷✺ ✴ ✹✷

❊①❛♠♣❧❡ ✭◆✉♠❜❡r r❡str✐❝t✐♦♥s✱ ✭❝♦♥t✳✮✮

❲❡ ❛rr✐✈❡❞ ❛t✿ ❱P❛♥② ≤ ❱P♣❧ ❛♥❞ ◆Ps❣, ◆P♣❧ ≤ ◆P♥♦♥❡✳

• ◆Ps ✇✐t❤ ✜①❡❞ ♥✉♠❜❡r✿ ◆Ps❣ ❛♥❞ ◆P♣❧✱ ✭✉s❛❜❧❡ ✇❤❡♥ ✳✳✳✮
• ◆Ps ✇✐t❤ ♥♦ ❞❡✜♥✐t❡ ♥✉♠❜❡r✿ ◆P♥♦♥❡ ✭✉s❛❜❧❡ ❛s ♦❜❥❡❝t✱✳ ✳ ✳ ✮
• ❱Ps ❝♦♥str❛✐♥✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✐ts ◆P✲❛r❣✉♠❡♥t✿ ❱P♣❧
• ❱Ps ♥♦t ❝♦♥str❛✐♥✐♥❣ t❤❡ ♥✉♠❜❡r ♦❢ ✐ts ◆P✲❛r❣✉♠❡♥t✿ ❱P❛♥②

❲❤✐❝❤ ♣r❡❞✐❝❛t✐♦♥ r✉❧❡s ❞♦ ✇❡ ♥❡❡❞ t♦ ❞✐st✐♥❣✉✐s❤❄ ◆P♥♦♥❡ ❱P♣❧ Pr❡❞❆♥②✿ ◆Ps❣✰◆P♣❧ ✲❃ ❱P❛♥② ✲❃ ❙ ✴ ❭ ⑤ Pr❡❞P❧✿ ◆P♣❧ ✲❃ ❱P♣❧ ✲❃ ❙ ◆Ps❣ ◆P♣❧ ❱P❛♥② ❈♦♠♣❧❆♥②✿ ❱P✴❛♥② ✲❃ ◆P♥♦♥❡ ✲❃ ❱P ❈♦♠♣❧P❧ ✿ ❱P✴♣❧ ✲❃ ◆P♣❧ ✲❃ ❱P ❍♦✇ ♠✉❝❤ ❞♦❡s t❤❡ ≤ s❛✈❡❄ ❈♦♥❝❧✉s✐♦♥✿ ❖♥❧② ✈❡r② ✢❛t ❤✐❡r❛r❝❤✐❡s❄

✷✻ ✴ ✹✷

❊①❛♠♣❧❡ ✭❙❈ ❛s s✉❜❥❡❝ts ✕ ❢♦r ✈❡r❜s ♦❢ s✉✐t❛❜❧❡ ❦✐♥❞✮

❙❡♥t❡♥❝❡s✱ q✉❡st✐♦♥s✱ ✐♥✜♥✐t✐✈❛❧ ♣❤r❛s❡s ❝❛♥ ❜❡ ✉s❡❞ ❛s s✉❜❥❡❝ts✿ ❢r♦♠ ❛❜str❛❝t✴❙❡♥t❡♥❝❡✳❣❢✱ ✇✐t❤ ❝♦❡r❝✐✈❡ s✉❜t②♣✐♥❣≡ data EmbedS : S

• > SC ;
• - that she goes

EmbedQS : QS -> SC ;

• - who goes

EmbedVP : VP -> SC ;

• - to go

PredSCVP : SC -> VP -> Cl ; -- that she goes is good ❇✉t ❡❛❝❤ ♦❢ t❤❡ t❤r❡❡ ❦✐♥❞s ♦❢ SC ❝❛♥ ❜❡ s✉❜❥❡❝ts ♦❢ ♣❛rt✐❝✉❧❛r ✈❡r❜s ♦♥❧②✱ s♦ t❤❡ r✉❧❡ PredSCVP ✐s ♦✈❡r❣❡♥❡r❛t✐♥❣✳ ■♥ ❢❛❝t✱ t❤❡ s❡♠❛♥t✐❝ ❞♦♠❛✐♥s ❛r❡ ❢❛✐r❧② ❞✐✛❡r❡♥t✱ s❛t✐s❢②✐♥❣ ❛ ❞♦♠❛✐♥ ❡q✉❛t✐♦♥ ❧✐❦❡ ❉❙❈ ≃ ❉❙ + ❉◗ + ❉❱P ≃ ❉t + ❉❡∗ + ❉(❡→t)

✷✼ ✴ ✹✷

❚♦ r❡str✐❝t ♦✈❡r❣❡♥❡r❛t✐♦♥✱ ♦♥❡ ♦✉❣❤t t♦ ❤❛✈❡ ❞❡♣❡♥❞❡♥t ❝❛t❡❣♦r✐❡s✿ ❝❛t❡❣♦r✐❡s ❙❈ ❛♥❞ ❱P s❡♣❛r❛t❡❞ ✐♥t♦ t❤r❡❡ ❦✐♥❞s≡ cat Kind ; fun sK, qsK, vpK : Kind ; cat SC Kind ; VP Kind ; fun EmbedS : S -> SC sK ; EmbedQS : QS -> SC qsK ; EmbedVP : VP -> SC vpK ; PredSCVP : (k : Kind) -> SC k -> VP k -> Cl ; ❖❢ ❝♦✉rs❡✱ (VP k) ❤❛❞ t♦ ❜❡ ❜✉✐❧❞ ❢r♦♠ (V k)✱ (V2 k) ❡t❝✳ ❚❤❡ (VP k) ❛r❡ s♣❡❝✐❛❧ VPs ✇✐t❤ SC s✉❜❥❡❝t✱ s♦ (❱P ❦) < ❱P✱ r❛t❤❡r ❱P ≃ (❱P s❑) + (❱P qs❑) + (❱P ✈♣❑) + (❱P ♥♦♠) + . . .

✷✽ ✴ ✹✷

❙✉❜t②♣✐♥❣ ✐♥ ❍P❙●

❍P❙● ❝♦♠❡s ✇✐t❤ ❛ ❤✐❡r❛r❝❤② ♦❢ s✐❣♥ s♦rts ❛♥❞ ❝♦♥str❛✐♥ts ♦♥ s✐❣♥s✳ ❚❤❡ s♦rt ❤✐❡r❛r❝❤② ✐s ❛ ✜♥✐t❡ tr❡❡ (❙, ≤)❀ ✐ts r♦♦t ✐s ≤✲♠❛①✐♠❛❧✳ ❊❛❝❤ ♥♦❞❡ σ ∈ ❙ ✐♥ t❤❡ tr❡❡ ✐s ♣❛rt✐t✐♦♥❡❞ ❜② ✐t✬s ✐♠♠❡❞✐❛t❡ ♣r❡❞❡❝❡ss♦rs σ✶, . . . , σ♥ < σ✱ ✐✳❡✳ ❉σ ≃ ❉σ✶ + . . . + ❉σ♥ ✐s t❤❡ ❞✐s❥♦✐♥t s✉♠ ♦❢ ✐ts ✐♠♠❡❞✐❛t❡ s✉❜s♦rts✳ ❍P❙●✿ ❡①♣❧❡t✐✈❡ ❛♥❞ r❡❢❡r❡♥t✐❛❧ ◆Ps ❛s s✉❜t②♣❡s ♦❢ ◆P✿ ◆P ≃ ◆P[❡①♣❧] + ◆P[r❡❢ ] ❚❤❡♥✱ ✇✐t❤ s✉❜t②♣✐♥❣ ❢♦r ❢✉♥❝t✐♦♥s✱ ◆P[❡①♣❧] < ◆P ❱ ◆P := (◆P → ❙) < (◆P[❡①♣❧] → ❙) =: ❱ ❡①♣❧ (→≤) ❇✉t✿ ❞♦❡s ❱ ◆P < ❱ r❡❢ , ❱ ❡①♣❧ ♠❡❛♥ ♠♦r❡ t❤❛♥ ❱ ◆P = ∅❄

✷✾ ✴ ✹✷

Pr♦❜❧❡♠s ✇✐t❤ s✉❜t②♣✐♥❣ ✐♥ t❤❡ ❛❜str❛❝t ❣r❛♠♠❛r

P✶ ❙✉❜t②♣❡ ❞❡❝❧❛r❛t✐♦♥✿ (cat Pron < NP) ✈❡rs✉s ❝♦❡rs✐♦♥ ❝♦♥str✉❝t✐♦♥✿ (UsePron : Pron -> NP)✳ ❲✐t❤ s✉❜t②♣❡ ❞❡❝❧❛r❛t✐♦♥s ✇❡ ♠✐❣❤t ♦♠✐t t❤❡ ❝♦❡rs✐♦♥ ❝♦♥str✉❝t♦rs ✐♥ ❛❜str❛❝t s②♥t❛① tr❡❡s ✕ ✐❢ t❤❡② ❛r❡ ✉♥✐q✉❡✳ ❇✉t✿ ✐❢ ❇ ≤ ❆ ≤ ❈ ❛♥❞ ❇ ≤ ❆′ ≤ ❈✱ ❤♦✇ t♦ ❝♦❡r❝❡ ❇s t♦ ❈s❄ P✷ ❲❤✐❝❤ ♣r♦♣❡rt✐❡s ♦❢ < ❤❛✈❡ t♦ ❜❡ ❝❤❡❝❦❡❞ ✇❤❡♥ ❝♦♠♣✐❧✐♥❣ ❛♥ ❛❜str❛❝t ❣r❛♠♠❛r ✇✐t❤ s✉❜t②♣❡ ❞❡❝❧❛r❛t✐♦♥s❄ ❏✉st ❛♥t✐s②♠♠❡tr② ♦❢ t❤❡ r❡✢❡①✐✈❡ tr❛♥s✐t✐✈❡ ❝❧♦s✉r❡ ≤∗❄ ❍♦✇ ❡①♣❡♥s✐✈❡ ✐s t❤❛t❄ P✸ ❈❛♥ ❛♥ ❛❜str❛❝t (cat A < B) ❛❧✇❛②s ❜❡ ✐♠♣❧❡♠❡♥t❡❞ ❜② r❡❝♦r❞ s✉❜t②♣✐♥❣❄ ✭❆♥❞ ❞♦ ✇❡ ❤❛✈❡ t♦ ❜❧♦❝❦ t❤❡ ❝♦♥✈❡rs❡❄ ■✳❡✳ ❞✐st✐♥❣✉✐s❤ (lincat A < B) ✐♠♣❧❡♠❡♥t✐♥❣ (cat A < B) ❢r♦♠ str✉❝t✉r❛❧ r❡❝♦r❞ s✉❜t②♣✐♥❣ ✭✐♥❞❡♣❡♥❞❡♥t ♦❢ ❛❜str❛❝t <✮❄

✸✵ ✴ ✹✷

P✹ ❲❤✐❝❤ s✉❜t②♣✐♥❣s (cat A < B) ❤♦❧❞ ❢♦r ♠❛♥② ❧❛♥❣✉❛❣❡s❄ ❊①❛♠♣❧❡✿ ❘❡✢❡①✐✈❡ ✈❡r❜s✿ ❧❛♥❣✉❛❣❡ ❞❡♣❡♥❞❡♥t✱ ✐♥❤❡r❡♥t ✈s✳ r❡✢❡①✐✈❡ ✉s❡✱ ❞✐✛❡r❡♥t ❢♦r♠s✿ ❊♥❣❧✐s❤≡ to enjoy oneself : V[refl]

• - obj reflexive

to blow one’s nose : V2[refl’] -- poss reflexive

• ❡r♠❛♥≡

sich freuen : V[refl]

• - inh. reflexive

sich[acc] schneuzen : V[refl]

• - inh. reflexive

sich[dat] die Nase putzen : V3[nom,dat,acc] ❊①❛♠♣❧❡✿ ❆❈■✲✈❡r❜s ❛r❡ ❞✐✛❡r❡♥t ✐♥ ❞✐✛❡r❡♥t ❧❛♥❣✉❛❣❡s ✭❏♣♥✿ ♦♥❧② ✏❧❛ss❡♥✑✱ ♠❛♥② ▲❛♥❣s✿ ♣❡r❝❡♣t✐♦♥ ✈❡r❜s✮

✸✶ ✴ ✹✷

■♠♣❧❡♠❡♥t✐♥❣ s✉❜t②♣❡s ♦❢ ❛❜str❛❝t s②♥t❛①

• ❋ ❤❛s✱ ✐❢ ❛❧❧ ❧✐♥❡❛r✐③❛t✐♦♥ ❝❛t❡❣♦r✐❡s ❛r❡ r❡❝♦r❞ t②♣❡s✿

❛❜str❛❝t ❝♦♥❝r❡t❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❝❛t ❆ ❧✐♥❝❛t ❆ ❂ ④✳✳✳⑥ ❆ ❂ ④✳✳✳❀ ❧♦❝❦❴❆✿④⑥⑥ ■t s❡❡♠s ♥❛t✉r❛❧ t♦ ✐♠♣❧❡♠❡♥t s✉❜❝❛t❡❣♦r② ❞❡❝❧❛r❛t✐♦♥s ❛s ❢♦❧❧♦✇s✿

❛❜str❛❝t ❝♦♥❝r❡t❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ s✉❜❝❛t ❇ ❁ ❆ ❧✐♥❝❛t ❇ ❁ ❆ ❇ ❂ ④✳✳✳❀ ❧♦❝❦❴❇✱❧♦❝❦❴❆✿④⑥⑥

• ❚❤❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ t②♣❡ ♦❢ B ✇♦✉❧❞ ❤❛✈❡ t♦ ❝♦❧❧❡❝t t❤❡

❧♦❝❦❴❆✲❧❛❜❡❧s ♦❢ ❛❧❧ ✭✐♠♠❡❞✐❛t❡✮ s✉♣❡r❝❛ts A > B ♦❢ B✳

• ❚❤❡ ❝♦♠♣✐❧❡r ♦✉❣❤t t♦ ❝❤❡❝❦ t❤❛t t❤❡ (subcat B < A)✲

❞❡❝❧❛r❛t✐♦♥s s♣❛♥ ❛ ♣❛rt✐❛❧ ♦r❞❡r < ♦♥ t❤❡ ❝❛t❡❣♦r✐❡s✳

• ❚❤❡ r❡✢❡①✐✈❡ tr❛♥s✐t✐✈❡ ❝❧♦s✉r❡ <∗ ♦❢ < ✇❡r❡ ♥❡❡❞❡❞ t♦ ❝❤❡❝❦

❛♣♣❧✐❝❛t✐♦♥s ♦❢ lin✲❢✉♥❝t✐♦♥s✳

✸✷ ✴ ✹✷

❙✉❜t②♣❡s ❢♦r ❞❡♣❡♥❞❡♥t t②♣❡s❄

❈✉rr❡♥t❧②✱ ●❋ ✐❣♥♦r❡s t❤❡ ❦✐♥❞ ❛r❣✉♠❡♥t k ♦❢ ❛ ❞❡♣❡♥❞❡♥t ❝❛t❡❣♦r② (A k) ❜♦t❤ ✐♥ t❤❡ lincat ❛♥❞ ✐♥ t❤❡ lock✲✜❡❧❞ ❢♦r (A k)✿

❛❜str❛❝t ❝♦♥❝r❡t❡ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ❝❛t ❑✐♥❞ ❀ ❧✐♥❝❛t ❑✐♥❞ ❂ ④✳✳✳⑥ ❑✐♥❞ ❂ ④✳✳✳❀❧♦❝❦❴❑✐♥❞✿④⑥⑥ ❝❛t ❆ ❑✐♥❞ ❧✐♥❝❛t ❆ ❴ ❂ ④✳✳✳⑥ ❆ ❂ ④✳✳✳❀ ❧♦❝❦❴❆✿④⑥⑥

■t ❢♦❧❧♦✇s t❤❛t ❧✐♥❡❛r✐③❛t✐♦♥ ❢✉♥❝t✐♦♥s lin f : (A k) -> C ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦❢ k✳ ❲✐t❤ subcat✱ ♦♥❡ ✇♦✉❧❞ ♣r♦❜❛❜❧② ♥❡❡❞ ❧✐♥❝❛t ❆ ❦ ❂ ④✳✳⑥ ❛♥❞ ❧♦❝❦✲✜❡❧❞s lock (A k) ❞❡♣❡♥❞✐♥❣ ♦♥ k✳

✸✸ ✴ ✹✷

❙❤♦✉❧❞ ❞❡♣❡♥❞❡♥t t②♣❡ ❝♦♥str✉❝t♦rs ❜❡ ♠♦♥♦t♦♥❡❄ ■♥ ✇❤✐❝❤ s❡♥s❡❄ s✉❜❝❛t ❑✐♥❞ ❁ ❑✐♥❞✬ ❀ ❝❛t ❆ ❑✐♥❞✬ ❀ ❝❛t ❆ ❑✐♥❞ ❀ (❝❛t <) s✉❜❝❛t ❑✐♥❞ ❁ ❑✐♥❞✬ ❀ ❝❛t ❆ ❑✐♥❞✬ s✉❜❝❛t ✭❆ ❑✐♥❞✮ ❁ ✭❆ ❑✐♥❞✬✮ (s✉❜❝❛t <) ❋♦r (❝❛t <)✱ ♥♦t❡ t❤❛t ❑✐♥❞ ⊆❝ ❑✐♥❞′ ✐s ♥♦t ✐♥❥❡❝t✐✈❡✳ ■❢ ❝(❦✶) = ❝(❦✷)✱ s❤♦✉❧❞ A k1 ❛♥❞ A k2 ❜❡ ❞✐✛❡r❡♥t ♦r ❡q✉❛❧❄ ❉♦ ✇❡ ❜② (s✉❜❝❛t <) r❡❛❧❧② ✇❛♥t s✉❜❝❛t ❑✐♥❞ ❁ ❑✐♥❞✬ ❀ ❝❛t ❆ ❑✐♥❞✬ ❀ ❢✉♥ ❦✿❑✐♥❞✱ ❦✬✿❑✐♥❞✬ ❀ s✉❜❝❛t ❆ ❦ ❁ ❆ ❦✬ ?

✸✹ ✴ ✹✷

■♥ t❤❡ ●❋✲❧✐st ❞✐s❝✉ss✐♦♥ ✐♥ ▼❛r❝❤✭❄✮ ❆❛r♥❡ ✇❛♥t❡❞✿ ❝❛r < ✈❡❤✐❝❧❡ ◆P ❝❛r < ◆P ✈❡❤✐❝❧❡ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ❢♦r r❡❝♦r❞ t②♣❡s ρ✱ ❛ ❞❡♣❡♥❞❡♥t t②♣❡ ❝♦♥str✉❝t♦r ❈ : ρ → ❚②♣❡ ♦✉❣❤t t♦ ❜❡ ♠♦♥♦t♦♥❡✿ r : ρ, t : τ, ρ ≤ τ : ❚②♣❡ ❈r ≤ ❈t (❞❡♣ ≤) ❲✐t❤ ❞❡♣❡♥❞❡♥t t②♣❡ ❝♦♥str✉❝t♦r ❇ : ❆ → ❚②♣❡✱ ✇❡ ❝❛♥ ♠❛❦❡ ✭♦r❞❡r❡❞✮ ❝♦♥t❡①ts (❛ : ❆) (❜ : ❇(❛)) ❜✉t ♥♦t r❡❝♦r❞ t②♣❡s {❛ : ❆, ❜ : ❇(❛)}✿ t❤❡ s❡t ♦❢ ❧❛❜❡❧s ✐♥ ❛ r❡❝♦r❞ ✐s ✉♥♦r❞❡r❡❞✳

✸✺ ✴ ✹✷

❚②♣❛❜✐❧✐t② ✇✐t❤ s✉❜t②♣✐♥❣

▲❡t ❚② ❜❡ t❤❡ s❡t ♦❢ s✐♠♣❧❡ t②♣❡s σ, τ := ι | (σ → τ)✳ ▲❡t ≤ ❜❡ ❛ ♣❛rt✐❛❧ ♦r❞❡r ♦♥ t❤❡ s❡t ♦❢ ❛t♦♠✐❝ t②♣❡s ι✱ ❡①t❡♥❞❡❞ t♦ →✲t②♣❡s ❜② σ′ ≤ σ τ ≤ τ ′ (σ → τ) ≤ (σ′ → τ ′) (→ ≤) ❚❤❡♦r❡♠ ✭❏✳▼✐t❝❤❡❧❧✮ ❚②♣❛❜✐❧✐t② ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ t②♣✐♥❣ r✉❧❡s ① : σ ∈ Γ Γ ⊢ ① : σ (❱❛r) Γ ⊢ t : τ, τ ≤ ρ Γ ⊢ t : ρ (❙✉❜) Γ, ① : σ ⊢ t : τ Γ ⊢ λ①t : (σ → τ) (❆❜s) Γ ⊢ t : (σ → τ), Γ ⊢ s : σ Γ ⊢ (t · s) : τ (❆♣♣) r❡❞✉❝❡s t♦ t❤❡ s✉❜t②♣❡ s❛t✐s✜❛❜✐❧✐t② ♣r♦❜❧❡♠ ✭s❡❡ ❜❡❧♦✇✮✳ Pr♦♦❢ ✐❞❡❛✿ P✉s❤ ✭❙✉❜✮ t♦ t❤❡ ❧❡❛✈❡s ♦❢ t❤❡ ❞❡r✐✈❛t✐♦♥ tr❡❡✳

✸✻ ✴ ✹✷

❙✉❜t②♣❡ s❛t✐s✜❛❜✐❧✐t② ♣r♦❜❧❡♠ ❙❙(≤) Pr♦❜❧❡♠

• ✐✈❡♥ ❛ ✜♥✐t❡ s❡t ❊ ⊆ ❚②(❱❛r) × ❚②(❱❛r)✱ ✐s t❤❡r❡ ❛

s♦❧✉t✐♦♥ ❙ : ❱❛r → ❚② s✉❝❤ t❤❛t ❙σ ≤ ❙τ, ❢♦r ❡❛❝❤ (σ, τ) ∈ ❊? ❋♦r ❛t♦♠✐❝ ✰ ❢✉♥❝✐♦♥❛❧ t②♣❡s✿

• ▼✐t❝❤❡❧❧ ✭✶✾✾✷✮✿ ❙❙(≤) ✐s ❞❡❝✐❞❛❜❧❡ ✐♥ ◆❊❳P❚■▼❊✳
• ❚✐✉r②♥✱ ❲❛♥❞ ✭✶✾✾✸✮✿ ❙❙(≤) ✐s ❞❡❝✐❞❛❜❧❡ ✐♥ ❉❊❳P❚■▼❊✳
• ❚✐✉r②♥ ✭✶✾✾✷✮ ❙❙(≤) ✐s P❙P❆❈❊ ❤❛r❞ ❢♦r ≤ ❂

✷ ✸

✟✟ ❍ ❍

✵ ✶

• ❚✐✉r②♥ ✭✶✾✾✷✮ ■❢ ≤ ✐s ❛ ❞✐s❥♦✐♥t ✉♥✐♦♥ ♦❢ ❧❛tt✐❝❡s✱ t❤❡♥ ❙❙(≤)

✐s ✐♥ P❚■▼❊✳

• ❇❡♥❦❡ ✭✶✾✾✸✮ ■❢ ≤ ❤❛s t❤❡ ❍❡❧❧❡② ♣r♦♣❡rt② ✭❣❡♥❡r❛❧✐③❡s ❧❛tt✐❝❡s

❛♥❞ tr❡❡s✮✱ t❤❡♥ ❙❙(≤) ✐s ✐♥ P❚■▼❊✳

✸✼ ✴ ✹✷

• ❑♦③❡♥✴P❛❧s❜❡r❣✴❙❝❤✇❛rt③❜❛❝❤ ✭✶✾✾✹✮ ❲✐t❤♦✉t ≤ ♦♥ ❛t♦♠✐❝

t②♣❡s✱ ❜✉t ❛ ❧❛r❣❡st t②♣❡ ⊤✱ ❙❙ ❢♦r →✲t②♣❡s ✐s ✐♥ P❚■▼❊ ❘❡♠❛r❦✿ ❲✐t❤ t②♣❡s ⊤, ⊥✱ ✇❡ ❣❡t ✏str❛♥❣❡✑ s♦❧✉t✐♦♥s✱ ❧✐❦❡ ⊥ → ⊤✳ ❋♦r ♦❜❥❡❝t s✉❜t②♣✐♥❣✱ ❤❛✈✐♥❣ ❛ t②♣❡ ⊤ = {} = { ✐ : σ✐ | ✐ ∈ ∅ }✱ ❛♥❞ ■ ⊆ ❏ ✜♥✐t❡ { ✐ : σ✐ | ✐ ∈ ❏ } ≤ { ✐ : σ✐ | ✐ ∈ ■ } (♦❜❥ ≤)

• P❛❧s❜❡r❣ ✭✶✾✾✺✮ ❙❙(≤) ❢♦r ♦❜❥❡❝t t②♣❡s ✐s P❚■▼❊ ❝♦♠♣❧❡t❡✳

❙❙(≤) ✐s P❚■▼❊ ❡q✉✐✈❛❧❡♥t t♦ t②♣❡ r❡❝♦♥str✉❝t✐♦♥ ❢♦r ❖❖▲s

✸✽ ✴ ✹✷

❋♦r r❡❝♦r❞ s✉❜t②♣✐♥❣✱ ■ ⊆ ❏ ✜♥✐t❡, σ✐ ≤ τ✐ ❢♦r ❛❧❧ ✐ ∈ ■ { ✐ : σ✐ | ✐ ∈ ❏ } ≤ { ✐ : τ✐ | ✐ ∈ ■ } (r❡❝ ≤) ❛♥❞ s②st❡♠s ✉s✐♥❣ ❛t ❧❡❛st r❡❝♦r❞✲t②♣❡s✿

• ❱♦r♦❜②♦✈ ✭✶✾✾✽✮✿ ❙❙ ✐s ◆P✲❤❛r❞ ❡✈❡♥ ✇✐t❤♦✉t ❛t♦♠✐❝ t②♣❡s✱ ✐❢

✇❡ ❤❛✈❡ t②♣❡ ❝♦♥str✉❝t♦rs → ❛♥❞ {✐ : τ✐, . . .} (= {} = ⊤✮✳

• ❱♦r♦❜②♦✈ ✭✶✾✾✽✮✿ ❙❙ ✐s ◆P✲❤❛r❞ ✇✐t❤ ❛ s✐♥❣❧❡ ❛t♦♠✐❝ t②♣❡ ❛♥❞

❥✉st t❤❡ {✐ : τ✐, . . .} (= ⊤✮ t②♣❡ ❝♦♥str✉❝t♦r✳

• ❱♦r♦❜②♦✈ ✭✶✾✾✽✮✿ ❙❙ ✐s ◆P✲❤❛r❞ ✇✐t❤ t❤❡ {✐ : τ✐, . . .} (= ⊤✮

t②♣❡ ❝♦♥str✉❝t♦r ❛♥❞ s♦♠❡ ♦t❤❡r t②♣❡ ❝♦♥str✉❝t♦r ✇✐t❤ ✏str✉❝t✉r❛❧✑ s✉❜t②♣✐♥❣ ✭✐✳❡✳ ❝♦♠♣❛r❛❜❧❡ t②♣❡s ❤❛✈❡ t❤❡ s❛♠❡ t♦♣✲❧❡✈❡❧ ❝♦♥str✉❝t♦r✮

✸✾ ✴ ✹✷

❙✉❜t②♣❡ s❛t✐s✜❛❜✐❧✐t② ❢♦r ●❋❄

■s t❤❡r❡ ❛ ❙❙ ♣r♦❜❧❡♠ ❢♦r ❛❜str❛❝t s②♥t❛① ♦❢ ●❋ ✰ subcat❄

• ❋ ❝❛♥ ✐♥tr♦❞✉❝❡
• ❛t♦♠✐❝ t②♣❡s C:Type✱ ✈✐❛ ❞❡❝❧❛r❛t✐♦♥s cat C✱
• ❢✉♥❝t✐♦♥ t②♣❡s Arrow A C : Type✱ ✈✐❛ ❞❡❝❧❛r❛t✐♦♥s cat

Arrow ( :A) ( :C)✳ ❲❤❛t ✐s ♦r s❤♦✉❧❞ ❜❡ ✐♥t❡♥❞❡❞ ❜② ❛ ✏str✉❝t✉r❛❧✑ s✉❜t②♣❡ ❞❡❝❧❛r❛t✐♦♥ ❝❛t ❈ ❆✶ ✳✳✳ ❆♥ ❁ ❈ ❇✶ ✳✳✳ ❇♠ ❢♦r ❞❡♣❡♥❞❡♥t t②♣❡s❄ ❲♦✉❧❞ ✐t ✐♠♣❧② ♠ = ♥ ❛♥❞ t❤❡ ♣r❡♠✐s❡ ♦❢ ❆✶ ≤ ❇✶ : ❚②♣❡, . . . , ❆♥ ≤ ❇♥ : ❚②♣❡ ❈ ❆✶ . . . ❆♥ ≤ ❈ ❇✶ . . . ❇♥ : ❚②♣❡ (❞❡♣ ≤) ❖r s❤♦✉❧❞ ❛♥② ❞❡♣❡♥❞❡♥t t②♣❡ ❝♦♥str✉❝t♦r C ❜❡ ❛ss✉♠❡❞ ♠♦♥♦t♦♥❡ ✐♥ t❤✐s s❡♥s❡✱ ✇✐t❤♦✉t t❤❡ ♥❡❡❞ ♦❢ ❛ ❞❡❝❧❛r❛t✐♦♥❄

✹✵ ✴ ✹✷

❈♦♥❝❧✉s✐♦♥ ❄

❲❡ ❧♦♦❦❡❞ ❛t ♣♦ss✐❜❧❡ ✉s❡s ♦❢ s✉❜t②♣❡s ✐♥ t❤❡ ❛❜str❛❝t s②♥t❛① ❛♥❞ ❝♦♠♣❛r❡❞ t❤❡♠ ✇✐t❤ ❞❡♣❡♥❞❡♥t t②♣❡s ❢♦r ✉s❡ ✐♥ r❡s♦✉r❝❡ ❣r❛♠♠❛rs✿

• ❯s✐♥❣ ❞❡♣❡♥❞❡♥t t②♣❡s t♦ s♣❧✐t ❛ ❝❛t❡❣♦r② ❈ ✐♥t♦ ❞✐s❥♦✐♥t ♣❛rts

❈ ≃ (❈ ❦✶) + . . . + (❈ ❦♥) ❛♥❞ ✭✐❢ t❤❛t ✐s ♣♦ss✐❜❧❡✮ s♣❧✐t ❝♦♥str✉❝t✐♦♥s ✉♥✐❢♦r♠❧② ❢ : (❦ : ❑✐♥❞) → (❈ ❦) → . . . → ❉ r❡❞✉❝❡s ♦✈❡r❣❡♥❡r❛t✐♦♥ ❛♥❞ r❡s✉❧ts ✐♥ ♣❛r❛♠❡tr✐❝ ❝♦❞❡✳

• ▲❛♥❣✉❛❣❡✲✐♥❞❡♣❡♥❞❡t s✉❜t②♣❡ r❡❧❛t✐♦♥s s❡❡♠ r❛r❡✱ ❛♥❞ ♠❛✐♥❧②

r❡❧❛t❡❞ t♦ ✭❛✮ ❝♦♥str❛✐♥ts ♦♥ ❛r❣✉♠❡♥ts ♦❢ ❝♦♥str✉❝t✐♦♥s ✭❱P❛♥② < ❱P♣❧✮ ✭❜✮ ❧✐♠✐t❡❞ ✉s❛❜✐❧✐t② ♦❢ ❝♦♦r❞✐♥❛t✐♦♥s ❞✉❡ t♦ ❢❡❛t✉r❡ ❝♦♥✢✐❝s ✭◆Ps❣ < ◆P♥♦♥❡✮✳ ❙✉❜❝❛t ❤✐❡r❛r❝❤✐❡s ✐♥ s②♥t❛① s❡❡♠ ♥♦t ✈❡r② ❞❡❡♣✱ s♦ ❤♦✇ ❜✐❣ ✐s t❤❡ ❣❛✐♥ ✐❢ ✇❡ ❤❛✈❡ t❤❡♠ ✐♥ t❤❡ ❛❜str❛❝t s②♥t❛①❄

✹✶ ✴ ✹✷

❘❡❢❡r❡♥❝❡s✿ ✶✳ ▲✳ ❈❛r❞❡❧❧✐✿ ❆ ❙❡♠❛♥t✐❝s ♦❢ ▼✉❧t✐♣❧❡ ■♥❤❡r✐t❛♥❝❡✳ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✼✻✱ ♣✳✶✸✽✲✶✻✹✱ ✶✾✽✽✳ ✷✳ ❏✳ ❚✐✉r②♥✿ ❙✉❜t②♣❡ ■♥❡q✉❛❧✐t✐❡s✳ Pr♦❝✳ ▲■❈❙✬✾✷✱ ♣✳✸✵✽✲✸✶✺✱ ✶✾✾✷✳ ✸✳ ❉✳ ❑♦③❡♥✱ ❏✳ P❛❧s❜❡r❣✱ ▼✳ ❙❝❤✇❛r③❜❛❝❤✿ ❊✣❝✐❡♥t ■♥❢❡r❡♥❝❡ ♦❢ P❛rt✐❛❧ ❚②♣❡s✳ ❏✳ ❈♦♠♣t✳ ❙②st✳ ❙❝✐✳ ✹✾✱ ♣✳✸✵✻✲✸✵✷✹✱ ✶✾✾✹✳ ✹✳ ❏✳ P❛❧s❜❡r❣✿ ❊✣❝✐❡♥t ■♥❢❡r❡♥❝❡ ♦❢ ❖❜❥❡❝t ❚②♣❡s✳ ■♥❢♦r♠❛t✐♦♥ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✶✷✸✱ ♣✳ ✶✾✽✲✷✵✾✱ ✶✾✾✺✳ ✺✳ ❙✳ ❱♦r♦❜②♦✈✿ ❙✉❜t②♣✐♥❣ ❋✉♥❝t✐♦♥❛❧ ✰ ◆♦♥✲❊♠♣t② ❘❡❝♦r❞ ❚②♣❡s✳ Pr♦❝✳ ❈❙▲ ✶✾✾✽✳ ✻✳ ❆✳ ❘❛♥t❛✿ ❚②♣❡s ❛♥❞ ❘❡❝♦r❞s ❢♦r Pr❡❞✐❝❛t✐♦♥✳ Pr♦❝✳ ❊❆❈▲ ❲♦r❦s❤♦♣ ♦♥ ❚②♣❡ ❚❤❡♦r② ❛♥❞ ◆❛t✉r❛❧ ▲❛♥❣✉❛❣❡ ❙❡♠❛♥t✐❝s✱ ♣✳✶✲✾✱ ✷✵✶✹✳

✹✷ ✴ ✹✷