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

Problems 1-4 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 \verb|pass_or_fail|pass_or_fail of type \verb|{grade : int option, id : 'a} -> pass_fail|{grade : int option, id : ’a} -> pass_fail that takes a \verb|final_grade|final_grade (or, as the type indicates, a more general type) and returns \verb|pass|pass if the grade field contains \verb|SOME|\;iSOMEi for an i\geq 75i≥75 (else \verb|fail|fail).

has_passed

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

number_passed

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

number_misgraded

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

tree

Problems 5-7 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 \verb|tree_height|tree_height that accepts an \verb|'a tree|’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 \verb|0|0.

sum_tree

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

gardener

Write a function \verb|gardener|gardener of type \verb|flag tree -> flag tree|flag tree -> flag tree such that its structure is identical to the original tree except all nodes of the input containing \verb|prune_me|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 \verb|last|last, \verb|take|take, \verb|drop|drop, \verb|concat|concat, \verb|getOpt|getOpt, and \verb|join|join.

natural numbers

Problems 9-16 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 \verb|SUCC ZERO|SUCC ZERO, the number 2 is \verb|SUCC (SUCC ZERO)|SUCC (SUCC ZERO), and so on.

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

10. Write \verb|pred : nat -> nat|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 \verb|Negative|Negative (will need to define it first).

11. Write \verb|nat_to_int : nat -> int|nat_to_int : nat -> int, which given a "natural number" returns the corresponding \verb|int|int. For example, \verb|nat_to_int (SUCC (SUCC ZERO)) = 2|nat_to_int (SUCC (SUCC ZERO)) = 2. (Do not use this function for problems 13-16 -- it makes them too easy.)

12. Write \verb|int_to_nat : int -> nat|int_to_nat : int -> nat which given an integer returns a "natural number" representation for it, or throws a \verb|Negative|Negative exception if the integer was negative. (Again, do not use this function in the next few problems.)

13. Write \verb|add : nat * nat -> nat|add : nat * nat -> nat to perform addition.

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

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

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

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 *)

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

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

19. Write \verb|toList : intSet -> int list|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