For each ocaml value or function definition below

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For each OCaml value or function definition below, fill in the blank where the type annotation could go or write “ill typed” if there is a type error. If an expression can have multiple types, give the most generic one. We have done the first one for you. Some of these definitions refer to functions from the Map1 module, which has the following abstract interface: module type MAP = sig type (’a, ’b) map val empty : (’a, ’b) map val is_empty : (’a, ’b) map -> bool val mem : ’a -> (’a, ’b) map -> bool val find : ’a -> (’a, ’b) map -> ’b val add : ’a -> ’b -> (’a, ’b) map -> (’a, ’b) map val remove : ’a -> (’a, ’b) map -> (’a, ’b) map val from_list : (’a * ’b) list -> (’a, ’b) map val bindings : (’a, ’b) map -> (’a * ’b) list end module Map1 : MAP = struct ... end ;; open Map1 let x : ______ (int,string) map _____________ = add 120 "is fun" empty let a : _______int list list ________________ = (2::[])::[] let b : ________ill typed____________________ = 2 + "three" let c : ________(int, bool) map______________ = add 3 true empty let d : ___(int,bool) map -> (int, bool) map_ = add 3 true let e : ____ill typed________________________ = mem 3 [1;2;3] let f : ____(int -> int) -> int______________ = fun (g:int -> int) -> g 3 let g : _____int -> int -> int_______________ = fun (x:int) (y:int) -> x + y let h : _____(int,(int,int) map) map_________ = add 3 (from_list [(1,2)]) empty Grading scheme: 2 points per answer: 0 if wrong, 2 if right. 6
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4. Binary Search Trees (17 points) Recall the definition of generic binary trees and the binary search tree insert function: type ’a tree = | Empty | Node of ’a tree * ’a * ’a tree let rec insert (t:’a tree) (n:’a) : ’a tree = begin match t with | Empty -> Node(Empty, n, Empty) | Node(lt, x, rt) -> if x = n then t else if n < x then Node (insert lt n, x, rt) else Node(lt, x, insert rt n) end a. (5 points) Circle the trees that satisfy the binary search tree invariant . (Note that we have omitted the Empty nodes from these pictures.) (a) (b) (c) (d) (e) 4 2 2 2 2 / \ / \ / \ \ \ 2 5 5 6 5 6 5 5 \ / \ / \ / \ 6 4 4 4 6 4 4 Answer: (a), (d) b. (12 points) For each definition below, circle the letter of the tree above that it constructs or “none of the above”. let t1 : int tree = insert (Node(Node(Empty, 5 Empty), 2, Node(Empty, 6, Empty))) 4 (a) (b) (c) (d) (e) none of the above Answer: (b) let t2 : int tree = insert (insert (insert (insert Empty 4) 2) 5) 6 (a) (b) (c) (d) (e) none of the above Answer: (a) let t3 : int tree = insert (insert (insert (insert Empty 2) 5) 4) 6 (a) (b) (c) (d) (e) none of the above Answer: (d) 7
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let t4 : int tree = insert (insert (insert (insert Empty 5) 2) 4) 6 (a) (b) (c) (d) (e) none of the above Answer: none of the above 8
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5. Lists and Binary Trees (20 points)
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