SML Practice Problems B

credit: Pavel Lepin and Charilaos Skiadas

Coursera Week 3 Extra Practice Problems

Code To Implement

file:	src/main/sml/practice2/practice2.sml
functions:	pass_or_fail has_passed number_passed number_misgraded ...

Previous Practice Problems With Pattern Matching

Consider any of the extra Practice Problems from Section 1 and redo them using pattern matching.

Student Grades

Use these type definitions:

type student_id = int
type grade = int (* must be in 0 to 100 range *)
type final_grade = { id : student_id, grade : grade option }
datatype pass_fail = pass | fail

Note that the grade might be absent (presumably because the student unregistered from the course).

pass_or_fail

Write a function pass_or_fail of type {grade : int option, id : 'a} -> pass_fail that takes a final_grade (or, as the type indicates, a more general type) and returns pass if the grade field contains SOME i for an i≥75 (else fail).

has_passed

Using pass_or_fail as a helper function, write a function has_passed of type {grade : int option, id : 'a} -> bool that returns true if and only if the the grade field contains SOME i for an i≥75.

number_passed

Using has_passed as a helper function, write a function number_passed that takes a list of type final_grade (or a more general type) and returns how many list elements have passing (again, ≥75) grades.

number_misgraded

Write a function number_misgraded of type (pass_fail * final_grade) list -> int that indicates how many list elements are "mislabeled" where mislabeling means a pair (pass,x) where has_passed x is false or (fail,x) where has_passed x is true.

Tree

Use these type definitions:

datatype 'a tree = leaf 
                 | node of { value : 'a, left : 'a tree, right : 'a tree }
datatype flag = leave_me_alone | prune_me

tree_height

Write a function tree_height that accepts an 'a tree and evaluates to a height of this tree. The height of a tree is the length of the longest path to a leaf. Thus the height of a leaf is 0.

sum_tree

Write a function sum_tree that takes an int tree and evaluates to the sum of all values in the nodes.

gardener

Write a function gardener of type flag tree -> flag tree such that its structure is identical to the original tree except all nodes of the input containing prune_me are (along with all their descendants) replaced with a leaf.

List and Option

Re-implement various functions provided in the SML standard libraries for lists and options. See http://sml-family.org/Basis/list.html and http://sml-family.org/Basis/option.html. Good examples include last, take, drop, concat, getOpt, and join.

Natural Numbers

Use this type definition for natural numbers:

datatype nat = ZERO | SUCC of nat

A "natural" number is either zero, or the "successor" of a another integer. So for example the number 1 is just SUCC ZERO, the number 2 is SUCC (SUCC ZERO), and so on.

is_positive

Write is_positive : nat -> bool, which given a "natural number" returns whether that number is positive (i.e. not zero).

pred

Write pred : nat -> nat, which given a "natural number" returns its predecessor. Since 0 does not have a predecessor in the natural numbers, throw an exception Negative (will need to define it first).

nat_to_int

Write nat_to_int : nat -> int, which given a "natural number" returns the corresponding int. For example, nat_to_int (SUCC (SUCC ZERO)) = 2.

int_to_nat

Write int_to_nat : int -> nat which given an integer returns a "natural number" representation for it, or throws a Negative exception if the integer was negative.

---

---

(Do not use nat_to_int or int_to_natfor the following functions -- it makes them too easy.)

add

Write add : nat * nat -> nat to perform addition.

sub

Write sub : nat * nat -> nat to perform subtraction. (Hint: Use pred.)

mult

Write mult : nat * nat -> nat to perform multiplication. (Hint: Use add.)

less_than

Write less_than : nat * nat -> bool to return true when the first argument is less than the second.

intSet

The remaining problems use this datatype, which represents sets of integers:

datatype intSet = 
  Elems of int list (*list of integers, possibly with duplicates to be ignored*)
| Range of { from : int, to : int }  (* integers from one number to another *)
| Union of intSet * intSet (* union of the two sets *)
| Intersection of intSet * intSet (* intersection of the two sets *)

isEmpty

Write isEmpty : intSet -> bool that determines if the set is empty or not.

contains

Write contains: intSet * int -> bool that returns whether the set contains a certain element or not.

toList

Write toList : intSet -> int list that returns a list with the set's elements, without duplicates.

Test

file:	unit_test_practice2.sml
source folder:	src/test/sml/practice2