(* ML-programming *)

(* Exercise 1:
Characters have type "char" and constants have the form #c, where c is a character. The functions "ord" and "chr" convert between characters and character codes. In most implementations if 0 <= k <= 255, then char(k) returns the character with code k. Conversely, ord(c) returns the (ASCII) integer code of character c. The following function convert the number 5 into the character '5':

- fun digit i = chr(i + ord #"0");
val digit = fn : int -> char

Try the following:

- ord (5);
- ord #"0";
- digit 5;
- digit 7;

The functions "str" and "String.sub" convert between characters and strings. If s is a string then String.sub(s, n) returns the nth character in s, counting from zero. We can define the digit function in a different way:

- fun digit i = String.sub("0123456789", i);
val digit = fn : int -> char

Try the following:

- str (digit 5);
- digit 5; 

For each version of "digit", try to guess what would be the answer when called with the following parameters (try next in the computer to check your guess):

- digit ~1;
- digit 10;

Play with other combinations of the functions in order to understand them better.
*)


(* Exercise 2:

      Write the function 
	fun abs (v) = ... ;
      that finds the absolute value of v  (i.e. -v if v is negative).
      Example: 
	- abs 17;
	val it = 17 : int
	- abs ~19;
	val it = 19 : int


*)

(* Exercise 3:
   Second degree polynomials  ax^2+bx+c=0 have solutions 
   (-b+Math.sqrt(b^2-4ac))/2a and (-b-Math.sqrt(b^2-4ac))/2a
   where  "Math.sqrt" is the square root function in ML.
   
   Write the function 
	fun annengrad (a, b, c) = ... ;
   that returns both solutions. 
   Avoid computing the square root more than once.

   Example: 

- annengrad(1.0, 2.0, ~3.0);
val it = (1.0,~3.0) : real * real
- annengrad(1.0, 0.0, ~49.0);
val it = (7.0,~7.0) : real * real


*)

(* Exercise 4:
   Write the function 
	fun power (a, b) = ... ;
   that computes the exponentiation a^b (i.a. a multiplied by itself b times).
      We may assume that a and b are integers and that b>=0.
      Example: 
	- power(2, 23);
	val it = 8388608 : int
	- power(~25, 2); 
	val it = 625 : int
*)

(* Exercise 5:
     (Note that "teller" and "nevner" is Norwegian for numerator and denominator.)

      A fraction can be represented by the record
	{teller=..., nevner=...} 
      Write the function 
	fun add ({teller=t1,nevner=n1}, {teller=t2,nevner=n2}) = ... ;
      that evaluates the sum of the two fractions.
      (Remember the rules for fraction additions: 
	a/b + c/d = (a*d + c*b)/(b*d)
      .)
      Example: 
	- add({teller=1,nevner=2}, {teller=3,nevner=4}); 
	val it = {nevner=8,teller=10} : {nevner:int, teller:int}
*)

(* Exercise 6:
   a) Write the function 
	fun max2 (a, b) = ... ;
      that returns the largest of the two integers a and b 
      Example:
	- max2(3, ~17);
	val it = 3 : int
	
   b) Write the function
	 fun max3 (a, b, c) = ... ;
      that returns the largest of the three integers a, b and c.
      Example:
	- max3(4, ~5, 19);
	val it = 19 : int

   c) Write the function
	fun max ... = ... ;
      that returns the largest integer in the list of integers given as a parameter.
      Example:
	- max [7];
	val it = 7 : int
	- max [5, ~4, ~17, 28, 13];
	val it = 28 : int
*)

(* Exercise 7:
   a) Write the function
	fun first ... = ... ;
      that returns the first element in the string list given as parameter.  
      (There exists a standard operator "hd" which does this but do not use it.)
      Example:
	- val tv = ["Det", "bodde", "en", "underlig"];
	val tv = ["Det","bodde","en","underlig"] : string list
	- first tv;
	val it = "Det" : string
      The answer should be "" (an empty string) if the list is empty. 			     

   b) Write the function
	fun last ... = ... ;
      that returns the last element in the string list given as parameter.
      Example:
	- last tv;
	val it = "underlig" : string
      The answer should be "" (an empty string) if the list is empty. 			     				  

   c) Write the function
	fun nth (n,x) = ... ;
      that retrieves element number n from the list x.  
      The answer should be "" (an empty string) if there is no element number n. 
      Example:
	- nth(1, tv);
	val it = "Det" : string
	- nth(3, tv);
	val it = "en" : string
	- nth(5, tv);
	val it = "" : string
 *)

(* Exercise 8:
   ("smaa" and "store" is Norwegian for small and large respectively) 
   a) Write the function
	fun smaa (v, x) = ...;
      that returns a list of the "small" integers in the list x,
      i.e. the ones less than or equal to v. 
      Example:
	- smaa(5, [4, 2, 8, ~2, 5, 12]);
	val it = [4,2,~2,5] : int list

   b) Write the function
	fun store (v, x) = ...;
      that works like a) but gives the "large" numbers, i.e. those that are larger than v.
      Example:
	- store(5, [4, 2, 8, ~2, 5, 12]); 
	val it = [8,12] : int list

   c) Write the function
	fun quicksort (x) = ... ;
      that sorts the integer list x using the quicksort algorithm. 
      (Quicksort amounts to selecting one element from the data and 
	then split the other values in two groups: "small" 
	(i.e. less than or equal to the chosen element) and "large".  
	The two groups are sorted separately and then merged
	afterwards.)
      Example:
	- quicksort [5, 4, 2, 8, ~2, 5, 12]; 
	val it = [~2,2,4,5,5,8,12] : int list
*)