# CMSC 330, Fall 2013, Practice Problem 3 Solutions

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1 CMSC 330, Fall 2013, Practice Problem 3 Solutions 1. OCaml and Functional Programming a. Define functional programming Programs are expression evaluations b. Define imperative programming Programs change the value of variables c. Define higher-order functions Functions can be passed as arguments and returned as results d. Describe the relationship between type inference and static types Variable has a fixed type that can be inferred by looking at how variable is used in the code e. Describe the properties of OCaml lists Entity containing 0 or more elements of the same type. Type of list is determined by type of element. f. Describe the properties of OCaml tuples Entity containing 2 or more elements of possibly different types. Type of tuple is determined by type and number of elements. g. Define pattern variables in OCaml Variables making up patterns used by match h. Describe the usage of _ in OCaml Pattern variable that can match anything but does not add binding i. Describe polymorphism Function that can take different types for same formal parameter j. Write a polymorphic OCaml function let f x = x // a -> a, x can be of any type k. Describe variable binding A variable (symbol) is associated with a value in an expression (or environment) l. Describe scope Portion of program where variable binding is visible m. Describe lexical scoping Variable binding determined by nearest scope in text of program n. Describe dynamic scoping Variable binding determined by nearest runtime function invocation o. Describe environment Collection of variable bindings p. Describe closure Function code + environment pair, may be invoked as function q. Describe currying Functions consume one argument at a time, returning closures until all arguments are consumed

2 2. OCaml Types & Type Inference Give the type of the following OCaml expressions: a. [] // a list b. 1::[] // int list c. 1::2::[] // int list d. [1;2;3] // int list e. [[1];[1]] // int list list f. (1) // int g. (1, bar ) // int * string h. ([1,2], [ foo, bar ]) // (int * int) list * (string * string) list i. [(1,2, foo );(3,4, bar )] // (int * int * string) list j. let f x = 1 // a -> int k. let f (x) = x * // float -> float l. let f (x,y) = x // a * b -> a m. let f (x,y) = x+y // int * int -> int n. let f (x,y) = (x,y) // a * b -> a * b o. let f (x,y) = [x,y] // a * b -> ( a * b) list p. let f x y = 1 // a -> b -> int q. let f x y = x*y // int -> int -> int r. let f x y = x::y // a -> a list -> a list s. let f x = match x with [] -> 1 // a list -> int t. let f x = match x with (y,z) -> y+z // int * int -> int u. let f (x::_) -> x // a list -> a v. let f (_::y) = y // a list -> a list w. let f (x::y::_) = x+y // int list -> int x. let f = fun x -> x + 1 // int -> int y. let rec x = fun y -> x y // a -> b z. let rec f x = if (x = 0) then 1 else 1+f (x-1) // int -> int aa. let f x y z = x+y+z in f // int bb. let f x y z = x+y+z in f 1 2 // int -> int cc. let f x y z = x+y+z in f // int -> int -> int -> int dd. let rec f x = match x with // a list -> int [] -> 0 (_::t) -> 1 + f t ee. let rec f x = match x with // int list -> int [] -> 0 (h::t) -> h + f t ff. let rec f = function [] -> 0 (h::t) -> h + (2*(f t)) gg. let rec func (f, l1, l2) = match l1 with [] -> [] (h1::t1) -> match l2 with [] -> [f h1] (h2::t2) -> [f h1; f h2] // int list -> int // ( a -> b) * a list * a list -> b list

3 3. OCaml Types & Type Inference Write an OCaml expression with the following types: a. int list // [1] b. int * int // (1,1) c. int -> int // let f x = x+1 d. int * int -> int // let f (x,y) = x+y e. int -> int -> int // let f x y = x+y f. int -> int list -> int list // let f x y = (x+1)::y g. int list list -> int list // let f (x::_) = 1::x h. a -> a // let f x = x i. a * b -> a // let f (x,y) = x j. a -> b -> a // let f x y = x k. a -> b -> b // let f x y = y l. a list * b list -> ( a * b) list // let f (x::_,y::_) = [(x,y)] m. int -> (int -> int) // let f x y = x+y n. (int -> int) -> int // let f x = 1+(x 1) o. (int -> int) -> (int -> int) -> int // let f x y = 1+(x 1)+(y 1) p. ( a -> b) * ( c * c -> a) * c -> b // let f (x, y, z) = (x (y (z,z))) 4. OCaml Programs What is the value of the following OCaml expressions? If an error exists, describe the error. a. 2 ; 3 // 3 b. 2 ; // 7 c. (2 ; 3) + 4 // 7 d. if 1<2 then 3 else 4 // 3 e. let x = 1 in 2 // 2 f. let x = 1 in x+1 // 2 g. let x = 1 in x ; x+1 // 2 h. let x = (1, 2) in x ; x+1 // error: x has type int*int but used with int i. (let x = (1, 2) in x) ; x+1 // error: unbound value x j. let x = 1 in let y = x in y // 1 k. let x = 1 let y = 2 in x+y // syntax error: missing in l. let x = 1 in let x = x+1 in let x = x+1 in x // 3 m. let x = x in let x = x+1 in let x = x+1 in x // error: unbound value x n. let rec x y = x in 1 // error: x has type a -> b but used with b o. let rec x y = y in 1 // 1 p. let rec x y = y in x 1 // 1 q. let x y = fun z -> z+1 in x // fun y -> (fun z -> z+1) r. let x y = fun z -> z+1 in x 1 // fun z -> z+1 s. let x y = fun z -> z+1 in x 1 1 // 2 t. let x y = fun z -> x+1 in x 1 // error: unbound value x u. let rec x y = fun z -> x+1 in x 1 // error: x has type a -> b -> c but used with int

4 v. let rec x y = fun z -> x+y in x 1 // error: x has type a -> b -> c but used with int w. let rec x y = fun z -> x y in x 1 // error: x has type a -> b but used with b x. let rec x y = fun z -> x z in x 1 // error: x has type a -> b but used with b y. let x y = y 1 in 1 // 1 z. let x y = y 1 in x // fun y -> (y 1) aa. let x y = y 1 in x 1 // error: 1 has type int but used with int -> a bb. let x y = y 1 in x fun z -> z + 1 // syntax error at x fun cc. let x y = y 1 in x (fun z -> z + 1) // 2 dd. let a = 1 in let f x y z = x+y+z+a in f // 7 ee. let a = 1 in let f x y z = x+y+z+a in f // error: (f 1 2) has type int -> int but used with int 5. OCaml Programming let rec map f l = match l with [] -> [] (h::t) -> (f h)::(map f t) let rec fold f a l = match l with [] -> a (h::t) -> fold f (f a h) t a. Write an OCaml function named fib that takes an int x, and returns the Fibonacci number for x. Recall that fib(0) = 0, fib(1) = 1, fib(2) = 1, fib(3) = 2. let rec fib x = if (x = 0) then 0 else if (x = 1) then 1 else (fib (x-1) + fib (x-2)) b. Write a function find_suffixes which applied to a list lst returns a list of all the suffixes of lst. For instance, suffixes [1;2;5] = [ [1;2;5] ; [2;5] ; [5] ] let rec suffix_helper (x, r) = match x with [] -> r (h::t) -> (suffix_helper (t, (h::t)::r)) let suffixes x = List.rev (suffix_helper (x, []))

5 c. Write an OCaml function named map_odd which takes a function f and a list lst, applies the function to every other element of the list, starting with the first element, and returns the result in a new list. let rec map_odd f l = match l with [] -> [] (x1::[]) -> [f x1] (x1::x2::t) -> (f x1)::(map_odd f t) d. Use map_odd and fib applied to the list [1;2;3;4;5;6;7] to calculate the Fibonacci numbers for 1, 3, 5, and 7. map_odd fib [1;2;3;4;5;6;7] e. Using map, write a function triple which applied to a list of ints lst returns a list with all elements of lst tripled in value. let triple x = map (fun x -> 3*x) x f. Using fold, write a function all_true which applied to a list of booleans lst returns true only if all elements of lst are true. let all_true lst = fold (fun a x -> (x = true) && (a = true)) true lst g. Using fold and anonymous helper functions, write a function product which applied to a list of ints lst returns the product of all the elements in lst. let product x = fold (fun a y -> a*y) 1 x h. Using fold and anonymous helper functions, write a function find_min which applied to a list of ints lst returns the smallest element in lst. let find_min x = fold (fun a y -> min a y) max_int x i. Using the fold function and anonymous helper functions, write a function count_vote which applied to a list of booleans lst returns a tuple (x,y) where x is the number of true elements and y is the number of false elements. let count_vote x = fold (fun (y,n) v -> if (v) then (y+1,n) else (y,n+1)) (0,0) x j. Using the function count_vote, write a function majority which applied to a list of booleans lst returns true if 1/2 or more elements of lst are true. let majority x = match (count_vote x) with (y,n) -> (y >= n)

6 6. OCaml Polymorphic Types Consider a OCaml module Bst that implements a binary search tree: module Bst = struct type bst = Empty Node of int * bst * bst let empty = Empty (* empty binary search tree *) let is_empty = function (* return true for empty bst *) Empty -> true Node (_, _, _) -> false let rec insert n = function (* insert n into binary search tree *) Empty -> Node (n, Empty, Empty) Node (m, left, right) -> if m = n then Node (m, left, right) else if n < m then Node(m, (insert n left), right) else Node(m, left, (insert n right)) (* Implement the following functions val min : bst -> int val remove : int -> bst -> bst val fold : ('a -> int -> 'a) -> 'a -> bst -> 'a val size : bst -> int *) let rec min = (* return smallest value in bst *) let rec remove n t = (* tree with n removed *) let rec fold f a t = (* apply f to nodes of t in inorder *) let size t = (* # of non-empty nodes in t *) end a. Is insert tail recursive? Explain why or why not. No, since the return value for recursive call to insert cannot be used as the return value of the original call to insert. The return value is used to create a Node data type first, and the Node value is returned. b. Implement min as a tail-recursive function. Raise an exception for an empty bst. Any reasonable exception is fine. let rec min = function Empty -> (raise (Failure "min")) Node (m, left, right) -> if (is_empty left) then m else min left

7 c. Implement remove. The result should still be a binary search tree. let rec remove n = function Empty -> Empty Node (m, left, right) -> if m = n then ( if (is_empty left) then right else if (is_empty right) then left else let x = min right in Node(x, left, remove x right) // OR // else let x = max left in // Node(x, remove x left, right) ) else if n < m then Node(m, (remove n left), right) else Node(m, left, (remove n right)) d. Implement fold as an inorder traversal of the tree so that the code List.rev (fold (fun a m -> m::a) [] t) will produce an (ordered) list of values in the binary search tree. let rec fold f a n = match n with Empty -> a Node (m, left, right) -> fold f (f (fold f a left) m) right e. Implement size using fold. let size t = fold (fun a m -> a+1) 0 t

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