Ling 5801: Lecture Notes 20 Cued Association Semantics

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1 Ling 58: Lecture Notes Cued Association Semantics Natural language sometimes uses complex generalizations over sets of objects.. Start with Generalized Quantifiers (Barwise & Cooper, 98) Here s a simple generalized quantifier analysis for the sentence Most cars have four wheels: It works like this python program: (most (λ u car u ) (λ v four (λ u5 wheel u 5 ) (λ v5 have v v 5 ))) U = [ c, c, c3, w, w, w3, w4, w5, w6, w7, w8 ] def most(f,g): def four(f,g): count,count =, count = for u in U: for u in U: if f(u): count+= if f(u) and g(u): count+= if f(u) and g(u): count+= return count > 4 return count >.5*count def car(u): return u in [ c, c, c3 ] def wheel(u): return u in [ w, w, w3, w4, w5, w6, w7, w8 ] def have(u,v): return (u,v) in [( c, w ),( c, w ),( c, w3 ),( c, w4 ), ( c, w5 ),( c, w6 ),( c, w7 ),( c, w8 )] most( lambda u: car(u), lambda v: four( lambda u5: wheel(u5), lambda v5: have(v,v5) ) ) Here the variables have clear meanings (indices just identify source word by token number): u is a variable over cars; v is a variable over things that have four wheels; u 5 is a variable over wheels; v 5 is a variable for each car v over things that car has.. Eventualities (Davidson, 967; Parsons, 99) Unfortunately this doesn t admit modification yet: Most cars have four wheels today. Fix this by adding variable u 3 over eventualities (where U is the universe of discourse): (most (λ u car u ) (λ v four (λ u5 wheel u 5 ) (λ v5 some (λ u3 have u 3 v v 5 today u 3 ) U)))

2 Add eventuality/proposition vars for other predicates (call them e, e 5 when u s are taken): (most (λ u some (λ e car e u ) U) (λ v four (λ u5 some (λ e5 wheel e 5 u 5 ) U) (λ v5 some (λ u3 have u 3 v v 5 some (λ u6 today u 6 u 3 ) U) U))) Now we can think of eventualities/propositions as objects (so let s give predicates gerundy names): (most (λ u some (λ e beingacar e u ) U) (λ v four (λ u5 some (λ e5 beingawheel e 5 u 5 ) U) (λ v5 some (λ u3 having u 3 v v 5 some (λ u6 beingtoday u 6 u 3 ) U) U))) The new variables still have (mostly) clear meanings: e is a variable for each car u over instances of it being a car (say it s wrecked and recycled); e 5 is a variable for each wheel u 5 over instances of it being a wheel (it used to be a doorstop); u 3 is a variable for each car v and wheel v 5 over instances of v having v 5 ; u 6 is... not so important (a variable over propositions, mainly just there for consistency). These extra variables let us quantify over things that happen Most cars work twice: and negate them Most cars do not work: Hobbs (985) calls this reification. (most (λ u some (λ e beingacar e u ) U) (λ v two (λ u3 working u 3 v ) U)) (most (λ u some (λ e beingacar e u ) U) (λ v none (λ u3 working u 3 v ) U)) We can reify quantifier predicates as well (but then those propositions will need quantifiers, etc... )..3 Cognitive States and Cued Associations (Anderson et al., 977) How is meaning implemented in the brain? We have durable, rapidly-mutable weight-based ( associative ) memory (Howard & Kahana, ). This associates cue and target referential cognitive states (generalizations over stimuli), w/o rehersal. We use it to store sentence meaning, not form (Sachs, 967; Jarvella, 97; Bransford & Franks, 97). We have cognitive states u t,v t and associative memory M, such that: M def = v t u t where: (v u) [i, j] def = v [i] u [ j] t

3 The associative memory can then be queried using u t to get (an approximation of) v t : where: (M u) [i] def = J j= M [i, j] u [ j] v t M u t Model dependency relations r t with label l t between cue and target u t and v t : M def = v t r t + r t l t + r t u t Cue with diagonal product: where: (diag(v) u) [i] def = v [i] u [i] or more simply (ignoring the approximation): t v t M diag(m l t ) M u t v t = (f lt u t ).4 Semantic Argument Dependencies and Elementary Prediations Define semantic argument dependencies in elementary predications (Copestake et al., 5): ( f u v v v... ) (f u)=v f (f u)=v (f u)=v (f 3 u)=v... For example, to define an eventuality u: (have u v v ) (f u)=v have (f u)=v (f u)=v Graphically: u have v v Also, quantifiers (where p is a proposition): (most p u v) (f p)=v most (f p)=u (f p)=v p most u v 3

4 .5 Scope Dependencies (Schuler & Wheeler, 4) Also use cued associations to define dependencies: (f v)=v Now we can define entire expressions, e.g. Cars usually have four wheels: p e u v u 3 p 4 e 5 u 5 v 5 most car have four wheel.6 Simple Translation to Lambda Calculus (Schuler & Wheeler, 4) Translate source predications Γ into lambda expression (w. Q as a set of quantifier functions):. Add a lambda term to Γ for each predication in Γ with no outd variables: Γ, ( f v v.. v.. v N ) ; Γ, (λ v f v v.. v.. v N ) ; f Q, u (f u)=v Γ (P). Conjoin lambda terms over the same variable in Γ (this combines modifier predications): Γ, (λ v φ), (λ v ψ) ; Γ, (λ v φ ψ) ; (C) 3. Move terms in Γ with no missing predications or outd variables to : Γ, (λ v ψ) ; Γ ; (λ v ψ), f Q ( f.. v..) Γ, u (f u)=v Γ (M) 4. Add translations τ f of quantifiers f in Γ over complete lambda terms in : Γ, ( f p u v) ; (λ u φ), (λ v ψ), Γ, (τ f (λ u φ) (λ v ψ)) ; (λ u φ), (λ v ψ), f Q (Q) Γ, (f v)=v, (τ f (λ u φ) (λ v ψ)) ; Γ, (λ v τ f (λ u φ) (λ v ψ)) ; (Q) For example, our graph: 4

5 p e u v u 3 p 4 e 5 u 5 v 5 most car have four wheel which consists of the following elementary predications and dependencies: (most p u v ), (f v 5 )=v, (four p 4 u 5 v 5 ), (car e u ), (wheel e 5 u 5 ), (have u 3 v v 5 ) translates like this (assume unbound eventuality variables have low existential ): (most p u v ), (f v 5 )=v, (four p 4 u 5 v 5 ), (car e u ), (wheel e 5 u 5 ), (have u 3 v v 5 ) ; P (most p u v ), (f v 5 )=v, (four p 4 u 5 v 5 ), (λ u car e u ), (λ u5 wheel e 5 u 5 ), (λ v5 have u 3 v v 5 ) ; M (most p u v ), (f v 5 )=v, (four p 4 u 5 v 5 ) ; (λ u car e u ), (λ u5 wheel e 5 u 5 ), (λ v5 have u 3 v v 5 ) Q (most p u v ), (f v 5 )=v, (four (λ u5 wheel e 5 u 5 ) (λ v5 have u 3 v v 5 )) ; (λ u car e u ), (λ u5...), (λ v5...) Q (most p u v ), (λ v four (λ u5 wheel e 5 u 5 ) (λ v5 have u 3 v v 5 )) ; (λ u car e u ), (λ u5...), (λ v5...) M (most p u v ) ; (λ v four (λ u5 wheel e 5 u 5 ) (λ v5 have u 3 v v 5 )), (λ u car e u ), (λ u5...), (λ v5...) Q (most (λ u car e u ) (λ v four (λ u5 wheel e 5 u 5 ) (λ v5 have u 3 v v 5 ))) ; (λ u...), (λ u5...), (λ v5...).7 Restriction Inheritance We need references to previous restrictor sets: Ten pumps work. Most of them are busy. p e u v u 3 p 4 v 5 u 6 ten pump work most busy But preferred reading is often reference to restrictor nuclear sets (pumps that work). We can handle this with inheritance dependencies ( restriction inheritance, rin): p e u v u 3 p 4 v 5 u 6 rin rin ten pump work most busy 5

6 Now v is a variable over the set of pumps that work, not just things that work. This is valid because most natural language quantifiers are conservative: Q(R, S ) = Q(R, R S )..8 Conjunction Inheritance Conjunctions also require inheritance ( conjunction inheritance, cin): p e u v u 3 p 4 e 5 u 5 v 5 u 6 v 6 p 7 e 8 u 8 v 8 rin rin rin cin cin cin cin most car have four wheel an engine.9 Full Translation to Lambda Calculus Translate source predications Γ into lambda expression (w. Q as a set of quantifier functions):. Add a lambda term to Γ for each predication in Γ with no outd variables or inheritances: Γ, ( f v v.. v.. v N ) ; Γ, (λ v f v v.. v.. v N ) ; f Q, u (f u)=v Γ, f {frin,f cin,f uin } ( f v)=u Γ (P). Conjoin lambda terms over the same variable in Γ (this combines modifier predications): Γ, (λ v φ), (λ v ψ) ; Γ, (λ v φ ψ) ; (C) 3. Move terms in Γ with no missing predications, outd variables or inheritances to : Γ, (λ v ψ) ; Γ ; (λ v ψ), f Q ( f.. v..) Γ, u (f u)=v Γ, f {frin,f cin,f uin } ( f v)=u Γ (M) 4. Add translations τ f of quantifiers f in Γ over complete lambda terms in : Γ, ( f p u v) ; (λ u φ), (λ v ψ), Γ, (τ f (λ u φ) (λ v ψ)) ; (λ u φ), (λ v ψ), f Q, f {f rin,f cin,f uin } ( f..)=v Γ, f Q ( f.. v..) Γ (Q) 6

7 Γ, (f v)=v, (τ f (λ u φ) (λ v ψ)) ; Γ, (λ v τ f (λ u φ) (λ v ψ)) ; u (f u)=u Γ Γ, (f u)=u, (f v)=v, (τ f (λ u φ) (λ v ψ)) ; Γ, (f u)=u, (λ v τ f (λ u (λ u φ) v ) (λ v ψ)) ; (Q) (Q3) 5. Add a lambda term to Γ for each inheritance that is empty or from complete term in : Γ, ( f v)=u ; Γ, ( f v)=u, (λ u True) ; f {f rin, f cin }, f Q ( f.. u..) Γ (I) Γ, ( f v)=u ; (λ u φ), Γ, (λ v (λ u φ) v) ; (λ u φ), f {f rin, f cin }, u (f u)=u Γ (I) Γ, (f u)=u, ( f v)=u ; (λ u φ), Γ, (f u)=u, (f v)=u, (λ v (λ u φ) v) ; (λ u φ), f {f rin, f cin }, v (f v)=v Γ (I3) Γ, (f u)=u, (f v)=v, ( f v)=u ; (λ u φ), Γ, (f u)=u, (f v)=v, (λ v (λ u (λ u φ) v ) v) ; (λ u φ), f {f rin, f cin } (I4) 7

8 For example, our graph: p e u v u 3 p 4 e 5 u 5 v 5 u 6 v 6 p 7 e 8 u 8 v 8 rin rin rin cin cin cin cin most car have four wheel an engine which consists of the following elementary predications and dependencies: (m p u v ), (f r v )=u, (f s v 5 )=v, (f s v 8 )=v, (f p 4 u 5 v 5 ), (a p 7 u 8 v 8 ), (f r v 5 )=u 5, (f r v 8 )=u 8, (f c v 5 )=v 6, (f c v 8 )=v 6, (w e 5 u 5 ), (e e 8 u 8 ), (f c u 5 )=u 6, (f c u 8 )=u 6, (c e u ), (h e 3 v v 6 ) translates like this (assume unbound eventuality variables have low existential ): (m p u v ), (f r v )=u, (f s v 5 )=v, (f s v 8 )=v, (f p 4 u 5 v 5 ), (a p 7 u 8 v 8 ), (f r v 5 )=u 5, (f r v 8 )=u 8, (f c v 5 )=v 6, (f c v 8 )=v 6, (w e 5 u 5 ), (e e 8 u 8 ), (f c u 5 )=u 6, (f c u 8 )=u 6, (c e u ), (h e 3 v v 6 ) ; P (m p u v ), (f r v )=u, (f s v 5 )=v, (f s v 8 )=v, (f p 4 u 5 v 5 ), (a p 7 u 8 v 8 ), (f r v 5 )=u 5, (f r v 8 )=u 8, (f c v 5 )=v 6, (f c v 8 )=v 6, (w e 5 u 5 ), (e e 8 u 8 ), (f c u 5 )=u 6, (f c u 8 )=u 6, (λ u c e u ), (λ v6 h e 3 v v 6 ) ; I (m p u v ), (f r v )=u, (f s v 5 )=v, (f s v 8 )=v, (f p 4 u 5 v 5 ), (a p 7 u 8 v 8 ), (f r v 5 )=u 5, (f r v 8 )=u 8, (f c v 5 )=v 6, (f c v 8 )=v 6, (w e 5 u 5 ), (e e 8 u 8 ), (f c u 5 )=u 6, (f c u 8 )=u 6, (λ u c e u ), (λ v6 h e 3 v v 6 ), (λ u6 ) ; M (m p u v ), (f r v )=u, (f s v 5 )=v, (f s v 8 )=v, (f p 4 u 5 v 5 ), (a p 7 u 8 v 8 ), (f r v 5 )=u 5, (f r v 8 )=u 8, (f c v 5 )=v 6, (f c v 8 )=v 6, (w e 5 u 5 ), (e e 8 u 8 ), (f c u 5 )=u 6, (f c u 8 )=u 6 ; (λ u c e u ), (λ v6 h e 3 v v 6 ), (λ u6 ) I (m p u v ), (f r v )=u, (f s v 5 )=v, (f s v 8 )=v, (f p 4 u 5 v 5 ), (a p 7 u 8 v 8 ), (f r v 5 )=u 5, (f r v 8 )=u 8, (f c v 5 )=v 6, (f c v 8 )=v 6, (w e 5 u 5 ), (e e 8 u 8 ), (λ u5 ), (λ u8 ) ; (λ u c e u ), (λ v6 h e 3 v v 6 ), (λ u6 ) P (m p u v ), (f r v )=u, (f s v 5 )=v, (f s v 8 )=v, (f p 4 u 5 v 5 ), (a p 7 u 8 v 8 ), (f r v 5 )=u 5, (f r v 8 )=u 8, (f c v 5 )=v 6, (f c v 8 )=v 6, (λ u5 w e 5 u 5 ), (λ u8 e e 8 u 8 ), (λ u5 ), (λ u8 ) ; (λ u c e u ), (λ v6 h e 3 v v 6 ), (λ u6 ) C (m p u v ), (f r v )=u, (f s v 5 )=v, (f s v 8 )=v, (f p 4 u 5 v 5 ), (a p 7 u 8 v 8 ), (f r v 5 )=u 5, (f r v 8 )=u 8, (f c v 5 )=v 6, (f c v 8 )=v 6, (λ u5 w e 5 u 5 ), (λ u8 e e 8 u 8 ) ; (λ u c e u ), (λ v6 h e 3 v v 6 ), (λ u6 ) M (m p u v ), (f r v )=u, (f s v 5 )=v, (f s v 8 )=v, (f p 4 u 5 v 5 ), (a p 7 u 8 v 8 ), (f r v 5 )=u 5, (f r v 8 )=u 8, (f c v 5 )=v 6, (f c v 8 )=v 6 ; (λ u5 w e 5 u 5 ), (λ u8 e e 8 u 8 ), (λ u c e u ), (λ v6 h e 3 v v 6 ), (λ u6 ) I (m p u v ), (f r v )=u, (f s v 5 )=v, (f s v 8 )=v, (f p 4 u 5 v 5 ), (a p 7 u 8 v 8 ), (λ v5 w e 5 u 5 ), (λ v8 e e 8 u 8 ), (λ v5 h e 3 v v 5 ), (λ v8 h e 3 v v 8 ) ; (λ u5 w e 5 u 5 ), (λ u8 e e 8 u 8 ), (λ u c e u ), (λ v6 h e 3 v v 6 ), (λ u6 ) C (m p u v ), (f r v )=u, (f s v 5 )=v, (f s v 8 )=v, (f p 4 u 5 v 5 ), (a p 7 u 8 v 8 ), (λ v5 w e 5 v 5 h e 3 v v 5 ), (λ v8 e e 8 v 8 h e 3 v v 8 ) ; (λ u5 w e 5 u 5 ), (λ u8 e e 8 u 8 ), (λ u c e u ), (λ v6 h e 3 v v 6 ), (λ u6 ) M (m p u v ), (f r v )=u, (f s v 5 )=v, (f s v 8 )=v, (f p 4 u 5 v 5 ), (a p 7 u 8 v 8 ) ; (λ v5 w e 5 v 5 h e 3 v v 5 ), (λ v8 e e 8 v 8 h e 3 v v 8 ), (λ u5 w e 5 u 5 ), (λ u8 e e 8 u 8 ), (λ u c e u ), (λ v6 h e 3 v v 6 ), (λ u6 ) Q (m p u v ), (f r v )=u, (f (λ u5 w e 5 u 5 ) (λ v5 w e 5 v 5 h e 3 v v 5 )), (a (λ u8 e e 8 u 8 ) (λ v8 e e 8 v 8 h e 3 v v 8 )) ; (λ v5...), (λ v8...), (λ u5...), (λ u8...), (λ u c e u ), (λ v6 h e 3 v v 6 ), (λ u6 ) Q (m p u v ), (f r v )=u, (λ v f (λ u5 w e 5 u 5 ) (λ v5 w e 5 v 5 h e 3 v v 5 )), (λ v a (λ u8 e e 8 u 8 ) (λ v8 e e 8 v 8 h e 3 v v 8 )) ; (λ v5...), (λ v8...), (λ u5...), (λ u8...), (λ u c e u ), (λ v6 h e 3 v v 6 ), (λ u6 ) I (m p u v ), (λ v c e v ), (λ v f (λ u5 w e 5 u 5 ) (λ v5 w e 5 v 5 h e 3 v v 5 )), (λ v a (λ u8 e e 8 u 8 ) (λ v8 e e 8 v 8 h e 3 v v 8 )) ; (λ v5...), (λ v8...), (λ u5...), (λ u8...), (λ u c e u ), (λ v6 h e 3 v v 6 ), (λ u6 ) C (m p u v ), (λ v c e v f (λ u5 w e 5 u 5 ) (λ v5 w e 5 v 5 h e 3 v v 5 ) a (λ u8 e e 8 u 8 ) (λ v8 e e 8 v 8 h e 3 v v 8 )) ; (λ v5...), (λ v8...), (λ u5...), (λ u8...), (λ u c e u ), (λ v6 h e 3 v v 6 ), (λ u6 ) M (m p u v ) ; (λ v c e v f (λ u5 w e 5 u 5 ) (λ v5 w e 5 v 5 h e 3 v v 5 ) a (λ u8 e e 8 u 8 ) (λ v8 e e 8 v 8 h e 3 v v 8 )), (λ v5...), (λ v8...), (λ u5...), (λ u8...), (λ u c e u ), (λ v6 h e 3 v v 6 ), (λ u6 ) Q (m (λ u c e u ) (λ v c e v f (λ u5 w e 5 u 5 ) (λ v5 w e 5 v 5 h e 3 v v 5 ) a (λ u8 e e 8 u 8 ) (λ v8 e e 8 v 8 h e 3 v v 8 ))) ; (λ v...), (λ v5...), (λ v8...), (λ u5...), (λ u8...), (λ u c e u ), (λ v6 h e 3 v v 6 ), (λ u6 ) The result, with existential quantifiers over eventualities filled in (note inheritances now explicit): (most (λ u some (λ e car e u )) (λ v some (λ e car e v ) four (λ u5 some (λ e5 wheel e 5 u 5 )) (λ v5 some (λ e5 wheel e 5 v 5 ) some (λ e3 have e 3 v v 5 )) an (λ u8 some (λ e8 engine e 8 u 8 )) (λ v8 some (λ e8 engine e 8 v 8 ) some (λ e3 have e 3 v v 8 )))) 8

9 . Donkey sentences (Kamp, 98) E.g. Every farmer who owns a donkey beats it: rin p e u v u 3 p 4 e 5 u 5 v 5 u 6 p 7 v 7 rin rin every farmer own a donkey beat sg which consists of the following elementary predications and dependencies: (e p u v ), (f r v )=u, (f e u ), (f s v 5 )=u, (a p 4 u 5 v 5 ), (f s v 7 )=v, (s p 7 v 5 v 7 ), (b u 6 v v 7 ), (f r v 7 )=v 5, (o u 3 u v 5 ), (f r v 5 )=u 5, (d e 5 u 5 ) translates like this (assume unbound eventuality variables have low existential ): (e p u v ), (f r v )=u, (f e u ), (f s v 5 )=u, (a p 4 u 5 v 5 ), (f s v 7 )=v, (s p 7 v 5 v 7 ), (b u 6 v v 7 ), (f r v 7 )=v 5, (o u 3 u v 5 ), (f r v 5 )=u 5, (d e 5 u 5 ) ; P (e p u v ), (f r v )=u, (f e u ), (f s v 5 )=u, (a p 4 u 5 v 5 ), (f s v 7 )=v, (s p 7 v 5 v 7 ), (b u 6 v v 7 ), (f r v 7 )=v 5, (o u 3 u v 5 ), (f r v 5 )=u 5, (λ u5 d e 5 u 5 ) ; M (e p u v ), (f r v )=u, (f e u ), (f s v 5 )=u, (a p 4 u 5 v 5 ), (f s v 7 )=v, (s p 7 v 5 v 7 ), (b u 6 v v 7 ), (f r v 7 )=v 5, (o u 3 u v 5 ), (f r v 5 )=u 5 ; (λ u5 d e 5 u 5 ) I (e p u v ), (f r v )=u, (f e u ), (f s v 5 )=u, (a p 4 u 5 v 5 ), (f s v 7 )=v, (s p 7 v 5 v 7 ), (b u 6 v v 7 ), (f r v 7 )=v 5, (o u 3 u v 5 ), (λ v5 d e 5 v 5 ) ; (λ u5 d e 5 u 5 ) P (e p u v ), (f r v )=u, (f e u ), (f s v 5 )=u, (a p 4 u 5 v 5 ), (f s v 7 )=v, (s p 7 v 5 v 7 ), (b u 6 v v 7 ), (f r v 7 )=v 5, (λ v5 o u 3 u v 5 ), (λ v5 d e 5 v 5 ) ; (λ u5 d e 5 u 5 ) C (e p u v ), (f r v )=u, (f e u ), (f s v 5 )=u, (a p 4 u 5 v 5 ), (f s v 7 )=v, (s p 7 v 5 v 7 ), (b u 6 v v 7 ), (f r v 7 )=v 5, (λ v5 o u 3 u v 5 d e 5 v 5 ) ; (λ u5 d e 5 u 5 ) M (e p u v ), (f r v )=u, (f e u ), (f s v 5 )=u, (a p 4 u 5 v 5 ), (f s v 7 )=v, (s p 7 v 5 v 7 ), (b u 6 v v 7 ), (f r v 7 )=v 5 ; (λ v5 o u 3 u v 5 d e 5 v 5 ), (λ u5 d e 5 u 5 ) I4 (e p u v ), (f r v )=u, (f e u ), (f s v 5 )=u, (a p 4 u 5 v 5 ), (f s v 7 )=v, (s p 7 v 5 v 7 ), (b u 6 v v 7 ), (λ v7 o u 3 v v 7 d e 5 v 7 ) ; (λ v5 o u 3 u v 5 d e 5 v 5 ), (λ u5 d e 5 u 5 ) P (e p u v ), (f r v )=u, (f e u ), (f s v 5 )=u, (a p 4 u 5 v 5 ), (f s v 7 )=v, (s p 7 v 5 v 7 ), (λ v7 b u 6 v v 7 ), (λ v7 o u 3 v v 7 d e 5 v 7 ) ; (λ v5 o u 3 u v 5 d e 5 v 5 ), (λ u5 d e 5 u 5 ) C (e p u v ), (f r v )=u, (f e u ), (f s v 5 )=u, (a p 4 u 5 v 5 ), (f s v 7 )=v, (s p 7 v 5 v 7 ), (λ v7 b u 6 v v 7 o u 3 v v 7 d e 5 v 7 ) ; (λ v5 o u 3 u v 5 d e 5 v 5 ), (λ u5 d e 5 u 5 ) M (e p u v ), (f r v )=u, (f e u ), (f s v 5 )=u, (a p 4 u 5 v 5 ), (f s v 7 )=v, (s p 7 v 5 v 7 ) ; (λ v7 b u 6 v v 7 o u 3 v v 7 d e 5 v 7 ), (λ v5 o u 3 u v 5 d e 5 v 5 ), (λ u5 d e 5 u 5 ) Q (e p u v ), (f r v )=u, (f e u ), (f s v 5 )=u, (a p 4 u 5 v 5 ), (f s v 7 )=v, (s (λ v5 o u 3 u v 5 d e 5 u 5 ) (λ v7 b u 6 v v 7 o u 3 v v 7 d e 5 v 7 )) ; (λ v7...), (λ v5...), (λ u5...) Q3 (e p u v ), (f r v )=u, (f e u ), (f s v 5 )=u, (a p 4 u 5 v 5 ), (λ v s (λ v5 o u 3 v v 5 d e 5 u 5 ) (λ v7 b u 6 v v 7 o u 3 v v 7 d e 5 v 7 )) ; (λ v7...), (λ v5...), (λ u5...) Q (e p u v ), (f r v )=u, (f e u ), (f s v 5 )=u, (a (λ u5 d e 5 u 5 ) (λ v5 o u 3 u v 5 d e 5 v 5 )), (λ v s (λ v5 o u 3 v v 5 d e 5 u 5 ) (λ v7 b u 6 v v 7 o u 3 v v 7 d e 5 v 7 )) ; (λ v7...), (λ v5...), (λ u5...) Q (e p u v ), (f r v )=u, (f e u ), (λ u a (λ u5 d e 5 u 5 ) (λ v5 o u 3 u v 5 d e 5 v 5 )), (λ v s (λ v5 o u 3 v v 5 d e 5 u 5 ) (λ v7 b u 6 v v 7 o u 3 v v 7 d e 5 v 7 )) ; (λ v7...), (λ v5...), (λ u5...) P (e p u v ), (f r v )=u, (λ u f e u ), (λ u a (λ u5 d e 5 u 5 ) (λ v5 o u 3 u v 5 d e 5 v 5 )), (λ v s (λ v5 o u 3 v v 5 d e 5 u 5 ) (λ v7 b u 6 v v 7 o u 3 v v 7 d e 5 v 7 )) ; (λ v7...), (λ v5...), (λ u5...) C (e p u v ), (f r v )=u, (λ u f e u a (λ u5 d e 5 u 5 ) (λ v5 o u 3 u v 5 d e 5 v 5 )), (λ v s (λ v5 o u 3 v v 5 d e 5 u 5 ) (λ v7 b u 6 v v 7 o u 3 v v 7 d e 5 v 7 )) ; (λ v7...), (λ v5...), (λ u5...) M (e p u v ), (f r v )=u, (λ v s (λ v5 o u 3 v v 5 d e 5 u 5 ) (λ v7 b u 6 v v 7 o u 3 v v 7 d e 5 v 7 )) ; (λ u f e u a (λ u5 d e 5 u 5 ) (λ v5 o u 3 u v 5 d e 5 v 5 )), (λ v7...), (λ v5...), (λ u5...) I (e p u v ), (λ v f e u... s (λ v5 o u 3 v v 5 d e 5 u 5 ) (λ v7 b u 6 v v 7 o u 3 v v 7 d e 5 v 7 )) ; (λ u f e u a (λ u5 d e 5 u 5 ) (λ v5 o u 3 u v 5 d e 5 v 5 )), (λ v7...), (λ v5...), (λ u5...) M (e p u v ) ; (λ v f e u... s (λ v5 o u 3 v v 5 d e 5 u 5 ) (λ v7 b u 6 v v 7 o u 3 v v 7 d e 5 v 7 )), (λ u f e u a (λ u5 d e 5 u 5 ) (λ v5 o u 3 u v 5 d e 5 v 5 )), (λ v7...), (λ v5...), (λ u5...) Q (e (λ u f e u a (λ u5 d e 5 u 5 ) (λ v5 o u 3 u v 5 d e 5 v 5 )) (λ v f e u... s (λ v5 o u 3 v v 5 d e 5 u 5 ) (λ v7 b u 6 v v 7 o u 3 v v 7 d e 5 v 7 ))) ; (λ v...), (λ u...), (λ v7...), (λ v5...), (λ u5...) Not satisfied when farmers own multiple donkeys because singular pronoun unsatisfied. Generates strong reading with plural ( Every farmer that owns some donkeys beats them ). 9

10 . Summation anaphora (Kamp, 98) 6. Add union of conjunct constraints to complete lambda terms in : Γ, (f uin v)=u ; Γ, (f cin v)=u, (f union v)=u ; (U) Γ, (f union v)=u ; (λ v ψ), Γ, (λ u (λ v ψ) u) ; (λ v ψ), (λ u...) Γ (U) Γ, (λ u φ), (f union v)=u ; (λ v ψ), Γ, (λ u φ (λ v ψ) u) ; (λ v ψ), (U3) Γ, (λ u φ) ; (λ u ψ), Γ ; (λ u φ ψ), v (f union v)=u Γ (U4) References Anderson, J. A., Silverstein, J. W., Ritz, S. A., & Jones, R. S. (977). Distinctive features, categorical perception and probability learning: Some applications of a neural model. Psychological Review, 84, Barwise, J., & Cooper, R. (98). Generalized quantifiers and natural language. Linguistics and Philosophy, 4. Bransford, J. D., & Franks, J. J. (97). The abstraction of linguistic ideas. Cognitive Psychology,, Copestake, A., Flickinger, D., Pollard, C., & Sag, I. (5). Minimal recursion semantics: An introduction. Research on Language and Computation, Davidson, D. (967). The logical form of action sentences. In N. Rescher (Ed.), The logic of decision and action (pp. 8 94). Pittsburgh: University of Pittsburgh Press. Hobbs, J. R. (985). Ontological promiscuity. In Proc. ACL (pp. 6 69). Howard, M. W., & Kahana, M. J. (). A distributed representation of temporal context. Journal of Mathematical Psychology, 45, Jarvella, R. J. (97). Syntactic processing of connected speech. Journal of Verbal Learning and Verbal Behavior,, Kamp, H. (98). A theory of truth and semantic representation. In J. A. G. Groenendijk, T. M. V. Janssen, & M. B. J. Stokhof (Eds.), Formal methods in the study of language: Mathematical centre tracts 35 (pp. 77 3). Amsterdam: Mathematical Center. Marr, D. (98). Vision: A computational investigation into the human representation and processing of visual information. W.H. Freeman and Company.

11 Parsons, T. (99). Events in the semantics of English. MIT Press. Sachs, J. (967). Recognition memory for syntactic and semantic aspects of connected discourse. Perception and Psychophysics,, Schuler, W., & Wheeler, A. (4). Cognitive compositional semantics using continuation dependencies. In Third joint conference on lexical and computational semantics (*SEM 4).