SML Practice Problems C

credit: Pavel Lepin and Charilaos Skiadas

Coursera Week 3 Extra Practice Problems

Code To Implement

file:	src/main/sml/practice3/practice3.sml
functions:	compose_opt do_until factorial fixed_point map2 app_all partition unfold map filter foldl map_tree fold_tree filter_tree

compose_opt

Write a function compose_opt : (’b -> ’c option) -> (’a -> ’b option) -> ’a -> ’c option that composes two functions with "optional" values. If either function returns NONE, then the result is NONE.

do_until

Write a function do_until : (’a -> ’a) -> (’a -> bool) -> ’a -> ’a. do_until f p x will apply f to x and f again to that result and so on until p x is false. Example: do_until (fn x => x div 2) (fn x => x mod 2 <> 1) will evaluate to a function of type int->int that divides its argument by 2 until it reaches an odd number. In effect, it will remove all factors of 2 its argument.

factorial

Use do_until to implement factorial.

fixed_point

Use \verb|do_until|do_until to write a function \verb|fixed_point: (a -> a) -> a -> a|fixed_point: (’’a -> ’’a) -> ’’a -> ’’a that given a function \verb|f|f and an initial value \verb|x|x applies \verb|f|f to \verb|x|x until \verb|f x = x|f x = x. (Notice the use of to indicate equality types.)

map2

Write a function \verb|map2 : ('a -> 'b) -> 'a * 'a -> 'b * 'b|map2 : (’a -> ’b) -> ’a * ’a -> ’b * ’b that given a function that takes \verb|'a|’a values to \verb|'b|’b values and a pair of \verb|'a|’a values returns the corresponding pair of \verb|'b|’b values.

app_all

Write a function \verb|app_all : ('b -> 'c list) -> ('a -> 'b list) -> 'a -> 'c list|app_all : (’b -> ’c list) -> (’a -> ’b list) -> ’a -> ’c list, so that: \verb|app_all f g x|app_all f g x will apply \verb|f|f to every element of the list \verb|g x|g x and concatenate the results into a single list. For example, for \verb|fun f n = [n, 2 * n, 3 * n]|fun f n = [n, 2 * n, 3 * n], we have \verb|app_all f f 1 = [1, 2, 3, 2, 4, 6, 3, 6, 9]|app_all f f 1 = [1, 2, 3, 2, 4, 6, 3, 6, 9].

foldr

Implement \verb|List.foldr|List.foldr (see http://sml-family.org/Basis/list.html#SIG:LIST.foldr:VAL).

partition

Write a function \verb|partition : ('a -> bool) -> 'a list -> 'a list * 'a list|partition : (’a -> bool) -> ’a list -> ’a list * ’a list where the first part of the result contains the second argument elements for which the first element evaluates to true and the second part of the result contains the other second argument elements. Traverse the second argument only once.

unfold

Write a function \verb|unfold : ('a -> ('b * 'a) option) -> 'a -> 'b list|unfold : (’a -> (’b * ’a) option) -> ’a -> ’b list that produces a list of \verb|'b|’b values given a "seed" of type \verb|'a|’a and a function that given a seed produces \verb|SOME|SOME of a pair of a \verb|'b|’b value and a new seed, or \verb|NONE|NONE if it is done seeding. For example, here is an elaborate way to count down from 5: \verb|unfold (fn n => if n = 0 then NONE else SOME(n, n-1)) 5 = [5, 4, 3, 2, 1]|unfold (fn n => if n = 0 then NONE else SOME(n, n-1)) 5 = [5, 4, 3, 2, 1].

factorial

Use \verb|unfold|unfold and $\verb|foldl|$$ to implement factorial.

map

Implement \verb|map|map using \verb|List.foldr|List.foldr.

filter

Implement \verb|filter|filter using \verb|List.foldr|List.foldr.

foldl

Implement \verb|foldl|foldl using \verb|foldr|foldr on functions. (This is challenging.)

map_tree, fold_tree, filter_tree

Define a (polymorphic) type for binary trees where data is at internal nodes but not at leaves. Define \verb|map|map and \verb|fold|fold functions over such trees. You can define \verb|filter|filter as well where we interpret a "false" as meaning the entire subtree rooted at the node with data that produced false should be replaced with a leaf.

Test

file:	unit_test_practice3.sml
source folder:	src/test/sml/practice3