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lecture01/notes.ml

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+(*
+  COMPILE = ocamlc -o outfile infile.ml
+
+  UTOP = #use "filename.ml";;
+*)
+
+(*
+  Ocaml is thankfully garbage collected
+
+  Expressions must end with double semi colon
+  e.g. let name = "Bread";;
+
+  expressions can be evaluated to a value
+
+  ocaml is a statically typed and strongly typed language
+  meaning that at compile time the type of a variable will
+  be known. The compiler uses type inference to determine the
+  type of a variable that you dont need to specify;
+
+  e.g.
+*)
+
+(* string type *)
+"Hello World";;
+
+(* character type *)
+'A';;
+
+(* integer type *)
+50;;
+
+(* floating type *)
+20.3;;
+
+(*
+  OPERATIONS
+
+  you have basic operations such as
+  +, -, *, /, that works with integer types
+
+  to do these operations with floating point
+  numbers you need to use the following
+
+  +., -., *., /., notice the "." after the operator
+
+  -----------------------------------------------
+
+  MODULE does not use the % sign
+
+  use "mod" e.g.
+
+  3 mod 2;; - : int = 1 (* evaluates to 1 *)
+
+  ----------------------------------------------
+
+  NOT uses the word "not" instead of "!" e.g.
+
+  not (1 < 2);; - : bool = false
+*)
+
+(*
+  COMPARATORS
+
+  standard and, or comparisons e.g.
+
+  a && b, a || b
+
+  you can use >, <, but the comparator uses a
+  single equal sign. e.g.
+
+  1. = 2.;; - : bool = false (* evaluates to false *)
+
+  -----------------------------------------------
+
+  NOT EQUALS
+
+  to check for inequality use "<>", e.g.
+
+  1. <> 2.;; - bool = true (* evaluates to true since 1. and 2. are not equal *)
+*)
+
+(*
+  STRING CONCATONATION
+
+  use the "^" (carrot operator) to concat strings e.g.
+*)
+"Braeden" ^ " " ^ "Sowinski"
+
+(*
+  FUNCTIONS
+
+  define a function square that takes in a
+  parameter x and returns x * x
+*)
+let square x = x * x;;
+square 5;;
+
+(* evaluates to 6 *)
+let x = 1 in x + 5;;
+
+let add x y = x + y;;
+
+(*
+  arrows are right associative
+
+    name : input -> input -> return type
+  val add : int -> int -> int = <fun>
+
+  so this can be translated to int -> (int -> int)
+
+  this means that every function essentially takes in 1 argument
+  that then returns another functoin that takes in another argument
+*)
+
+(* valid *)
+add 1 2;;
+
+(* valid *)
+(add 1) 2;;
+
+(* invalid *)
+(*add (1 2);;*)
+
+(*
+  Ocaml is functional and there are no loops
+  such as for or while,
+
+  all variables are immutable
+*)
+
+let x = 1;;
+let x = 2;;
+(*
+x = 3;; (* cannot reassign, this evaluates as a comparison
+          therefore variables are immutable *)
+*)
+(*
+  RECURSION
+
+  recursion is related to mathematical induction,
+  you typically have a proposition, e.g. P, where we
+  know P(1) is true, and we assume P(k) is true
+  where we can proove that P(k + 1) is true
+
+  to declare a recursive function you need "rec"
+*)
+let rec factorial n = if n = 0 then 1 else n * factorial (n - 1);;
+let r = (factorial 5);;
+print_int r;;
+print_endline "";;
+
+(*
+  tail-recursive
+
+  a recursive function is tail-recursive if the recursive
+  call is the last thing we do, for example, factorial is not
+  tail-recursive, because we need to multiply n to the recursive call.
+
+  non tail-recursive functions are bad since there is potential to bloew
+  through the stack.
+
+  lets consider factorial 3 = 3 * fact 2
+                            = 3 * (2 * fact 1)
+                            = 3 * (2 * (1 * fact 0))
+                            = 3 * (2 * 1)
+                            = 3 * 2
+                            = 6
+
+  we can see that this is not tail recursive as we need to store a value for
+  each part of the iteration, we can solve this by rewriting the function a little
+  using an accumulator
+*)
+let rec fact n acc = if n = 0 then acc else fact (n - 1) (n * acc);;
+
+(* we can have primes, e.g. function f, and function f prime or f' *)
+
+let factorial' n = fact n 1;;
+let r = (factorial 5);;
+print_int r;;
+print_endline "";;
+
+let factorial' = fact 1;;
+
+(*
+  consider the tail-recursive version of factorial, fact'
+
+  the signature is fact' n acc
+
+  fact' 1 3 = fact' 3 2
+            = fact' 6 1
+            = fact' 6 0
+
+  as you can see, the stack would not grow
+
+  lets combine the functoins into a single one
+  with nesting
+*)
+(* final tail-recursive version *)
+let fact n =
+  let rec fact' acc i =
+    if i = 0 then acc
+    else fact' (i * acc) (i - 1)
+  in
+  fact' 1 n;; (* automatically sets the accumulator to 1 *)
+
+let r = fact 5;;
+print_int r;;
+print_endline "";;
+
+(* example of a documentation comment *)
+
+(** [gcd a b] returns the greatest common divisor of [a] and [b]
+  * Requires:[a > 0] and [b > 0]
+  *)
+let rec gcd a b =
+  if a mod b = 0 then b
+  else gcd b (a mod b);;
+(*
+let r = gcd 12 18;;
+print_int r;;
+print_endline "";; *)
+
+let rec fib n =
+  if n = 0 then 0
+  else if n = 1 then 1
+  else fib (n - 1) + fib (n - 2);;
+
+let r = fib 10;;
+print_int r;;
+print_endline "";;
+
+(* tail-recursive version *)
+let rec fib n =
+  let rec fib' i a b =
+    if i = n then a
+    else fib' (i + 1) b (a + b)
+  in
+  fib' 0 0 1;;
+
+let r = fib 10;;
+print_int r;;
+print_endline "";;
+
+(*
+  MUTALLY-RECURSIVE FUNCTION
+*)
+let rec even n =
+  if n = 0 then true
+  else odd (n - 1)
+and odd n =
+  if n = 0 then false
+  else even (n - 1);;
+
+(*
+  useful stuff
+
+  float_of_int 4
+  int_of_string "123"
+*)
+
+let square_root x =
+  let good_enough y = abs_float (x -. y *. y) < 0.00000000001 in
+  let rec aux y =
+    if good_enough y then y
+    else aux (0.5 *. (y +. x /. y))
+  in
+  aux 1.;;
+
+print_float (square_root 2.);;
+print_endline "";;
+
+(*
+  Consider this
+
+  create an exponential function
+  where it calculates e^x
+
+  using recursion
+*)