HW4: Lambda Calculus Due: Friday 8/11/23 1. [Types] (4 points) What are the types of the following lambda expressions? (a) AP(e,t).AQ (e,t) . Vx(P(x) - Q(z)) (b) Are mouse(z) (c) Are.dye.chase(y, x) (d) AP(e,t) .LzP(z) 2. [Function Application] (3 points) Apply function application to the following lambda expres- sions. (a) [AP(e,t) .AQ (e,t). Vx(P(x) + Q(x))](Aye.cat(y)) (b) [Are.Aye.chase(y, x)](j) (c) [APle,t).LzP(z)](Are.mouse(z)) 3. [Derivation] (13 points) Derive the meanings of the following sentences/structures. Use your answers from Exercise 2 where needed; they're labeled in the tree as (2a), (2b), and (2c). (a) (1 point) (2b) Tom Are.Aye.chase(y, x) chased Jerry (b) (2 points) Tom Are.Aye.chase(y, 2) (2c) chased AP(e,t).LZP(z) Are.mouse(x) the mouse(c) (2 points) (2a) (2b) AP(e,t) .AQ (e,t) .Vi(P(x) - Q(z)) Aye.cat(y) Are.Aye.chase(y, I) chased Jerry Every cat (d) (3 points) (2a) AP(e.t) .AQ (e,t).Vx(P(x) - Q(x)) Aye.cat(y) Are.Aye.chase(y, I) (2c) Every cat chased AP(e,t) .LZP(z) Are mouse( the mouse (e) (5 points) Use Predicate Modification to derive the meaning of 'hungry cat'. Complete the derivation using function application everywhere else. (2b) Are.Aye.chase(y, I) APle,t) . AQ (e,t). Vx( P(x) - Q(z)) chased Jerry Every Aye hungry(y) Aye.cat(y) hungry cat