; booleans @@ -35,12 +36,11 @@ let isZero = \n. n (\x. false) true in let succ = \n s z. s (n s z) in let mult = \m n s. m (n s) in let length = Y (\length l. isNil l 0 (succ (length (tail l)))) in -let predecessor = \n. length (tail (n (\p. makeList meh p) nil)) in -let leq = ; (leq m n) will be true iff m is less than or equal to n - Y (\leq m n. isZero m true (isZero n false (leq (predecessor m)(predecessor n)))) in +let pred = \n. isZero n 0 (length (tail (n (\p. makeList meh p) nil))) in +let leq = \m n. isZero(n pred m) in let eq = \m n. and (leq m n)(leq n m) in -eq 3 3 +eq 2 2 yes no@@ -65,13 +65,13 @@ same length. That is, listLenEq mylist (makeList meh (makeList meh nil))) ~~> false -4. (Still easy) Now write the same function, but don't use the length function (hint: use `leq` as a model). +4. (Still easy) Now write the same function, but don't use the length function. 5. In assignment 2, we discovered that version 3-type lists (the ones that work like Church numerals) made it much easier to define operations -like map and filter. But now that we have recursion in our toolbox, +like `map` and `filter`. But now that we have recursion in our toolbox, reasonable map and filter functions for version 3 lists are within our -reach. Give definitions for such a map and a filter. +reach. Give definitions for `map` and a `filter` for verson 1 type lists. 6. Linguists analyze natural language expressions into trees. We'll need trees in future weeks, and tree structures provide good @@ -94,26 +94,29 @@ Then we have the following representations: Limitations of this scheme include the following: there is no easy way -to label a constituent (typically a syntactic category, S or NP or VP, +to label a constituent with a syntactic category (S or NP or VP, etc.), and there is no way to represent a tree in which a mother has a single daughter. When processing a tree, you can test for whether the tree contains only a numeral (in which case the tree is leaf node) by testing for whether the length of the list is less than or equal to 1. This will -be your base case for your recursive functions that operate on trees. +be your base case for your recursive functions that operate on these +trees. -Write a function that sums the number of leaves in a tree. +#Write a function that sums the number of leaves in a tree.# Expected behavior: -let t1 = (make-list 1 nil) -let t2 = (make-list 2 nil) -let t3 = (make-list 3 nil) -let t12 = (make-list t1 (make-list t2 nil)) -let t23 = (make-list t2 (make-list t3 nil)) -let ta = (make-list t1 t23) -let tb = (make-list t12 t3) -let tc = (make-list t1 (make-list t23 nil)) +

+ +let t1 = (make-list 1 nil) in +let t2 = (make-list 2 nil) in +let t3 = (make-list 3 nil) in +let t12 = (make-list t1 (make-list t2 nil)) in +let t23 = (make-list t2 (make-list t3 nil)) in +let ta = (make-list t1 t23) in +let tb = (make-list t12 t3) in +let tc = (make-list t1 (make-list t23 nil)) in count-leaves t1 ~~> 1 count-leaves t2 ~~> 2 @@ -123,27 +126,7 @@ count-leaves t23 ~~> 5 count-leaves ta ~~> 6 count-leaves tb ~~> 6 count-leaves tc ~~> 6 - -Write a function that counts the number of leaves. - - - - -[The following should be correct, but won't run in my browser: --let factorial = Y (\fac n. isZero n 1 (mult n (fac (predecessor n)))) in - -let reverse = - Y (\rev l. isNil l nil - (isNil (tail l) l - (makeList (head (rev (tail l))) - (rev (makeList (head l) - (rev (tail (rev (tail l))))))))) in - -reverse (makeList 1 (makeList 2 (makeList 3 nil))) --It may require more resources than my browser is willing to devote to -JavaScript.] +#Write a function that counts the number of leaves.#