(* OCaml Lecture I *)
(* To start with we will use the top loop as a simple calculator. *)
3;; (* use ;; to end input. *)
3 + 4;; (* notice how types are *inferred* in output *)
let x = 3 + 4;; (* x forever more is 7 - no mutation by default in OCaml! *)
x + 5;;
let y = 5 in x + y;; (* this makes y a local variable: think C's { int y; return(x+y); } *)
(* y + 6 ;; *) (* errors because y was only defined *locally* in previous line *)
(* Boolean operations *)
true && false;;
true || false;;
1 = 2;; (* = not == for equality comparison *)
1 <> 2;; (* <> not != for not equal *)
(* int and float cannot be mixed without being explicit *)
1;;
1.;;
4 * 5;;
(* 4.0 * 1.5;; *) (* error -- '*' operator is only for integers *)
4.0 *. 1.5;; (* works -- '*.' is for floats *)
(float 4) *. 1.5;; (* use an explicit caast when you want to mix *)
(* Lists are easy to create and manipulate *)
[1; 2; 3];;
[1; 1+1; 1+1+1];;
["a"; "b"; "c"];;
(* [1; "a"];; *) (* error - All elements must have same type - HOMOGENEOUS *)
[];; (* empty list *)
(* Operations on lists. Lists are represented as BINARY TREES with left child a leaf. *)
0 :: [1; 2; 3];; (* :: is 'consing' an element to the front - fast *)
0 :: (1 :: (2 :: (3 :: [])));; (* equivalent to the above *)
[1; 2; 3] @ [4; 5];; (* appending lists - slower *)
let z = [2; 4; 6];;
let y = 0 :: z;;
z;; (* Observe z itself did not change -- lists are IMMUTABLE in OCaml *)
(* everything in OCaml returns values (i.e. is an 'expression') - no commands *)
if (x = 3) then (5 + 35) else 6;; (* (x==3)?5:6 in C *)
(if (x = 3) then 5 else 6) * 2;;
(if (x = 3) then 5.4 else 6) * 2;; (* type error: two branches of if must have same type *)
(* ====================================================================== *)
(* OCaml Lecture II *)
(* Tuples - fixed length lists, but types of each element CAN differ, unlike lists *)
(2, "hi");; (* type is int * string -- '*' is like "x" of set theory, a product *)
let tuple = (2, "hi");;
(1,1.1,'c',"cc");;
(* Defining and using functions *)
let squared x = x * x;; (* declare a function: "squared" is its name, "x" is its one parameter. return implicit. *)
squared 4;; (* call a function -- separate arguments with S P A C E S *)
(*
* - no return statement; value of the whole body-expression is what gets
* returned
* - type is printed as domain -> range
* - "officially", OCaml functions take only one argument - !
* multiple arguments can be encoded by some tricks (later)
*)
(* fibonacci series - 1 1 2 3 5 8 13 ... *)
let rec fib n = (* the "rec" keyword needs to be added to allow recursion (ugh) *)
if n <= 2 then
1
else
fib (n - 1) + fib (n - 2);;
fib 10;;
(* Anonymous functions: define a function as an expression *)
(* Similar to lambdas in Python, Java, C++, etc - all are based on the lambda calculus *)
let funny_add1 = (function x -> x + 1);; (* "x" is (sole) argument here -- one-argument functions only *)
funny_add1 3;;
((function x -> x + 1) 4) + 7;; (* a "->" function is an expression and can be used anywhere *)
((fun x -> x + 1) 4) + 7;; (* shorthand notation -- cut off the "ction" *)
(* multiple arguments - just leave spaces between multiple arguments in definition and use *)
let add x y = x + y;;
add 3 4;;
(add 3) 4;; (* same meaning as previous *)
let add3 = add 3;; (* don't need to give all arguments at once! Type of add is in fact int -> (int -> int) *)
add3 4;;
add3 20;;
(* Conclusion: add is a function taking an integer, and returning a FUNCTION which takes ints to ints.
So, add is a HIGHER-ORDER FUNCTION:
it either takes a function as an argument, or returns a function as result. *)
(* Observe 'int -> int -> int' is parenthesized as 'int -> (int -> int)' -- unusual RIGHT associativity *)
(* ******************************************************************** *)
(* Pattern matching: switch or case on steroids *)
(* A very cool and useful but not so common language feature; Haskell and Scala also have it *)
(* Basic pattern match with numbers, looks like switch more or less: *)
let mixemup n =
match n with
| 0 -> 4
| 5 -> 0
| y -> y + 1;; (* default case giving a name to the matched number, x *)
mixemup 3;; (* matches last case and x is bound to the value 3 *)
let five_oh y =
"Hawaii " ^ (match y with
0 -> "Zero"
| 5 -> "Five" (* notice the "|" separator between multiple patterns *)
| _ -> "Nothing") ^ "-O";; (* default case -- _ is a pattern matching anything and without a name for it *)
five_oh 5;;
(* List pattern matching - now things get interesting! *)
match ['h';'o'] with (* recall ['h';'o'] is really 'h' :: ('o' :: []) *)
| a :: (b :: c) -> "first"
| x :: y -> "second"
| _ -> "third";;
match ['h';'o';'p';' ';'h';'o';'p'] with
| x :: y -> y
| _ -> ['0'];;
match ["hi"] with (* ["hi"] is "hi" :: [] *)
| x :: (y :: z) -> "first"
| x :: y -> "second"
| _ -> "third";;
(* tuple pattern matching *)
let tuple = (2, "hi", 1.2);;
match tuple with
(f, s, th) -> s
;;
(* let pattern shorthand: a single pattern match to assign multiple values *)
let mypair = (2.2, 3.3);;
let (f, s) = mypair in f +. s;;
match mypair with (f,s) -> f +. s;; (* same behavior as above let *)
let getSecond t =
match t with
(f, s) -> s
;;
let s = getSecond (2, "hi");;
let s = getSecond mypair;;
let getSec t =
match t with
(f, s) -> (s :: [],4)
;;
(* warning - non-exhaustive pattern matching; avoid this *)
(* let getHead l =
match l with
head :: tail -> head;;
*)
(* getHead [];; *) (* OCaml warned before that this case is not covered: gives uncaught runtime exception *)
(* an error-free version *)
let getHead l =
match l with
| [] -> failwith "you dodo"
| head :: tail -> head
;;
(* Patterns in function definitions - this is the usual usage *)
let cadd p = match p with (x, y) -> x + y;;
(* Shorthand for the above (only works because there is only one pattern clause - special *)
let cadd (x, y) = x + y;; (* same result *)
(* This 'cadd' takes only a SINGLE argument - a tuple - more like how C etc functions are called *)
cadd (1, 2);;
(* compare above add to this earlier one above, recalling: *)
let add x y = x + y;;
add 1 2;; (* notice different calling convention compared to cadd *)
(* The difference between cadd and add can be observed in their types.
cadd : int * int -> int -- C/Java etc style functions
add : int -> int -> int -- the default OCaml style *)
let add = (+);; (* this syntax is how you can give any built-in infix operator a name as a function *)
(* using patterns in recursive functions: a function to reverse a list
note this does not mutate the list, it makes a new list that reverses original
We also want to study this function to show how its correctness is
justified by an induction argument
*)
let rec rev l =
match l with
[] -> []
| x :: xs -> rev xs @ [x]
;;
rev [1;2;3];; (* = 1 :: ( 2 :: ( 3 :: [])) *)
(* The Magic of Induction lets us prove it is correct:
*
* Theorem: rev reverses any list.
* Proof: by induction on the length of the list, say l.
* Case l = []: obviously rev [] = [] - CHECK!
* Case l is non-empty: then, l = x :: xs for some x and xs;
* we assume by INDUCTION that "rev xs" reverses the tail, xs;
* then, the result "rev xs @ [x]" will clearly be the reverse of the whole list. QED.
*)
(* Now let us unwrap the "magic"; its really just marching through natural numbers
* l = [] : rev [] = [], check!
* l = [3] : rev [3] = rev (3 :: []) = (rev []) @ [3] = (using previous line) [3]
* l = [2;3] : rev [2;3] = rev (2 :: [3]) = (rev [3]) @ [2] = (again by previous) [3;2]
* l = [1;2;3] : rev [1;2;3] = rev (1 :: [2;3]) = (rev [2;3]) @ [1] = (again by previous) [3;2;1]
*)
(* One more view, an unwrapping of the execution to equivalent code: *)
rev [1;2;3];;
(rev [2;3]) @ [1];;
((rev [3]) @ [2]) @ [1];;
(((rev []) @ [3]) @ [2]) @ [1];;
(([]@[3]) @ [2]) @ [1];;
(* ====================================================================== *)
(* OCaml Lecture III *)
(* We are going to cover variant types later but here are a few
simple examples *)
(* Variant types are like unions in C, and generalize enums of Java.
Unlike types up to now you need to declare them. *)
type comparison = LessThan | EqualTo | GreaterThan;;
let intcmp x y =
if x < y then LessThan else
if x > y then GreaterThan else EqualTo;;
(* Of course we will pattern match to take the data apart *)
match intcmp 4 5 with
| LessThan -> "less!"
| EqualTo -> "equal!"
| GreaterThan -> "greater!";;
(* Variants can also wrap arguments: they are more like C unions than Java enums *)
type 'a nullable = Null | NotNull of 'a;;
match NotNull(4) with
| Null -> "null!"
| NotNull(n) -> (string_of_int n)^" is not null!"
;;
(* Immutable declarations *)
(*
* All variable declarations in OCaml are IMMUTABLE -- value will never change
* helps in reasoning about programs, we know the variable's value is fixed
*)
(let y = 3 in
( let x = 5 in
( let f z = x + z in
( let x = y in (* this is a re-definition of x, NOT an assignment *)
(f (y - 1)) + x
)
)
)(* x is STILL 5 in the function body - thats what x was when f defined *)
)
;;
(* top loop is conceptually an open-ended series of lets which never close: compare following with previous *)
let y = 3;;
let x = 5;;
let f z = x + z;;
let x = y;; (* as in previous example, this is a nested definition, not assignment! *)
f (y-1) + x;;
(*
* Function definitions are similar, you can't mutate an existing definition.
*)
let f x = x + 1;;
let g x = f (f x);;
let shad = f;; (* make a new name for f *)
(* lets "change" f, say we made an error in its definition above *)
let f x = if x <= 0 then 0 else x + 1;;
g (-5);; (* g still refers to the initial f - !! *)
let g x = f (f x);; (* FIX: resubmit (identical) g code *)
g (-6);; (* now it works as expected *)
(* MORAL: When interactively editing a group of functions that call each other,
* re-submit ALL the functions to the top loop when you change any ONE
* of them. Otherwise you can have some functions using a now-shadowed version.
*)
(* Mutually recursive functions *)
(* warm up to the next function - write a (useless) copy function on lists *)
let rec copy l =
match l with
| [] -> []
| hd :: tl -> hd::(copy tl);;
let result = copy [1;2;3;4;5;6;7;8;9;10]
(* copy is useless because immutable data can't be mutated so never a need to copy, ever! *)
(* Refine copy to flip back and forth between copying and not *)
let rec copyodd l = match l with
| [] -> []
| hd :: tl -> hd::(copyeven tl)
and (* keyword for mutually recursive functions *)
copyeven l = match l with
| [] -> []
| x :: xs -> copyodd xs;;
copyodd [1;2;3;4;5;6;7;8;9;10];;
copyeven [1;2;3;4;5;6;7;8;9;10];;
(* Using let .. in to hide functions *)
(* Here is a version that hides the skip function -- make both internal and export one *)
let copyodd ll =
( let rec
copyoddlocal l = match l with
| [] -> []
| hd :: tl -> hd::(copyevenlocal tl)
and
copyevenlocal l = match l with
| [] -> []
| x :: xs -> copyoddlocal xs
in
copyoddlocal ll
);;
copyodd [1;2;3;4;5;6;7;8;9;10];;
(* ******************************************************************** *)
(*
* Higher Order Functions -
* functions that either:
* take other functions as arguments
* or return functions as results
* or both
*
* Why?
* - "pluggable" programming by passing in and out chunks of code
* - greatly increases reusability of code since any varying
* code can be pulled out as a function to pass in
*)
(* Lets show the power by extracting out some pluggable code *)
(* Example: multiply each element of a list by ten *)
let rec timestenlist l =
match l with
[] -> []
| hd::tl -> (hd * 10) :: timestenlist tl;;
timestenlist [3;2;50];;
(* Example: append gobble to a list of words *)
let rec appendgobblelist l =
match l with
[] -> []
| hd::tl -> (hd ^"gobble") :: appendgobblelist tl;;
appendgobblelist ["have";"a";"good";"day"];;
("have" ^"gobble") :: ("a"^"gobble") :: appendgobblelist ["good";"day"];;
(* Notice there is a common pattern of "do an operation on each
list element". So lets pull out the "times ten" / "add gobble"
as a function parameter! This is in fact a classic example,
the map function *)
let rec map f l = (* Notice function f is an ARGUMENT here *)
match l with
[] -> []
| hd::tl -> (f hd) :: map f tl;;
map (fun x -> x * 10) [3;2;50];;
(* Note there is a built-in List.map since it is so common: *)
let middle = List.map (function s -> s^"gobble");;
middle ["have";"a";"good";"day"];;
map (fun (x,y) -> x + y) [(1,2);(3,4)];;
let flist = map (fun x -> (fun y -> x + y)) [1;2;4] ;; (* lists of functions! *)
(* What can you do with a list of functions? e.g. compose them *)
(* compose_list [f1;..;fn] v = f1 (... (fn v) ... ) *)
let rec compose_list lf v =
match lf with
| [] -> v
| hd :: tl -> hd(compose_list tl v);;
compose_list flist 0;;
(* another classic operator on lists *)
(* foldl f v [b1; ..;bn] is (... (f (f v b1) b2) ...) bn ) *)
let rec foldl f v l =
match l with
| [] -> v
| [hd] -> f v hd
| hd::tl -> f (foldl f v tl) hd;;
(* Note this is List.fold_left in OCaml library *)
(* summing elements of a list can now be succinctly coded: *)
foldl (fun x-> fun y -> x+y) 0 [1;2;3];;
(* Just to be clear..
named and anonymous functions are absolutely identical *)
let timesten x = x * 10;;
map timesten [1;2;3;34;56;90];;
(* Composition function g o f: *)
let compose g f = (fun x -> g (f x));;
let plus3 x = x+3;;
let times2 x = x*2;;
let times2plus3 = compose plus3 times2;;
times2plus3 10;;
(* equivalent but with anonymous functions: *)
compose (fun x -> x+3) (fun x -> x*2) 10;;
(* Equivalent notation for compose *)
let compose g f x = g (f x);;
let compose = (fun g -> (fun f -> (fun x -> g(f x))));;
(*
Parametric polymorphism -- a key feature of inferred types
*)
let id = fun x -> x;;
let id x = x;;
(* id applied to int returns an int *)
id 3;;
(* SAME id applied to bool returns a bool *)
id true;;
(* conclusion: the type of id ('a -> 'a) is PARAMETRIC, i.e.
the return type is parameterized by the type of the argument.
Same as Java's generics.
We saw several parametric functions above: *)
copyodd;; (* observe type is 'a list -> 'a list *)
map;; (* type is ('a -> 'b) -> 'a list -> 'b list *)
compose;; (* type is ('a -> 'b) -> ('c -> 'a) -> 'c -> 'b *)
(* ******************************************************************** *)
(*
* Self-study topic: in fact, only 'let'-defined functions are polymorphic
*)
(* Background: turning a let-defined function into an argument to a function *)
let add1 = fun x -> x + 1;;
add1 4;;
(* Equivalent way to do the above: define add1 as an anonymous function and pass in *)
(function addup -> addup 4)(fun x -> x + 1);;
(* Lets try to do the same refactoring for a polymorphic function *)
let id = function x -> x;;
match id (true) with
true -> id (3)
| false -> id(4);;
(* The below will error - variable mono_id is not defined by let so it can't be polymorphic
(function mono_id ->
match mono_id(true) with
true -> mono_id(3)
| false -> mono_id(4)) (fun x -> x);;
*)
(* If only used at one type its OK: *)
(function mono_id -> mono_id 4) id;; (* mono_id is solely of type int -> int, thats OK *)
(* ******************************************************************** *)
(* One topic left in higher-order functions ...
Currying - idea due to logician Haskell Curry
*)
(* First lets recall how functions allow incremental arguments to be passed *)
let addC x y = x + y;;
addC 1 2;; (* recall this is the same as '(addC 1) 2' *)
let tmp = addC 1 in tmp 2;; (* the partial application of arguments - result is a function *)
(* an equivalent way to define addC, clarifying what the above means: *)
let addC = fun x -> (fun y -> x + y);;
(* yet another identical way .. *)
let addC x = fun y -> x + y;;
(addC 1) 2;; (* same result as above *)
(* Also recall the related non-Curry'ing version: use a pair of arguments instead *)
let addNC p =
match p with (x,y) -> x+y;;
(* recall this equivalent abbreviation *)
let addNC (x, y) = x + y;;
(* Notice how the type of the above differs from addC's type *)
addNC (3, 4);;
(* addNC 3;; *) (* error, need all or no arguments supplied *)
(*
* Fact: these two approaches to a 2-argument function are isomorphic:
*
* 'a * 'b -> 'c === 'a -> 'b -> 'c
*
* We now define two cool higher-order functions:
*
* curry - takes in non-curry'ing 2-arg function and returns a curry'ing version
*
* uncurry - takes in curry'ing 2-arg function and returns an non-curry'ing version
* Since we can then go back and forth between the two reps, they are
* ***ISOMORPHIC***
*)
let curry fNC = fun x -> fun y -> fNC (x, y);;
let uncurry fC = fun (x, y) -> fC x y;;
let newaddNC = uncurry addC;;
newaddNC (2,3);;
let newaddC = curry addNC;;
newaddC 2 3;;
(* Observe the types themselves pretty much specify the behavior: *)
(* curry : ('a * 'b -> 'c) -> 'a -> 'b -> 'c *)
(* uncurry : ('a -> 'b -> 'c) -> 'a * 'b -> 'c *)
let noop1 = curry (uncurry addC);; (* a no-op *)
let noop2 = uncurry (curry addNC);; (* another no-op; noop1 & noop2 together show isomorphism *)
(* End Currying topic *)
(* Misc OCaml *)
(* See http://caml.inria.fr/pub/docs/manual-ocaml/libref/Pervasives.html
for various functions available in the OCaml top-level.
See http://caml.inria.fr/pub/docs/manual-ocaml/stdlib.html for modules of extra
functions for lists, strings, integers, as well as sets, trees, etc structures *)
(* print_x for atomic types 'x', again no overloading in meaning here *)
print_string ("hi\n");;
(* raise a failure exception (more on exceptions later) *)
(failwith "BOOM!") + 3 ;;
(* you CAN also declare types, anywhere in fact *)
(* Put parens around any such declaration or it won't parse *)
let add (x: float) (y: float) = x +. y;;
let add (x: int) (y: int) = (((x:int) + y) : int);;
(* type abbreviations via "type" *)
type intpair = int * int;;
let f (p : intpair) = match p with
(l, r) -> l + r
;;
(2,3);; (* ocaml doesn't call this an intpair by default *)
f (2, 3);; (* still, can pass it to the function expecting an intpair *)
((2,3):intpair);; (* can also explicitly tag data with its type *)
(* ******************************************************************* *)
(* An old PL Homework 1 - lets work through some of it to get experience
with writing simple functional OCaml programs *)
(*
1. Write a function 'compose_funs' which takes a list of functions
[f1; ...; fn] and returns a function representing the composition
fn o .. o f1 of all of these.
*)
let rec compose_funs lf =
match lf with
[] -> (function x -> x)
| f :: fs -> (function x -> (compose_funs fs) (f x))
;;
(* equivalent alternate version - refactor to hoist out the "function x" *)
let rec compose_funs lf =
function x ->
(match lf with
[] -> x
| f :: fs -> (compose_funs fs) (f x)
)
;;
(* Yet another equivalent alternative - hoist x up one more level *)
let rec compose_funs lf x =
match lf with
[] -> x
| f :: fs -> (compose_funs fs) (f x)
;;
(* tests *)
let composeexample = compose_funs [(function x -> x+1); (function x -> x-1);
(function x -> x*3); (function x -> x-1)];;
composeexample 5;; (* should return 14 *)
(*
2. Write a function 'toUpperCase' which takes a list (l) of characters and
returns a list which has the same characters as l, but capitalized (if not
already).
Note: 1. Assume that the capital of characters other than alphabets
(A - Z or a - z), are the characters themselves e.g.
character corresponding capital character
a A
z Z
A A
1 1
% %
2. You can only use Char.code and Char.chr library functions.
You CANNOT use Char.uppercase.
*)
let toUpperChar c =
let c_code = Char.code c in
if c_code >= 97 && c_code <= 122 then
Char.chr (c_code - 32)
else c;;
let rec toUpperCase l =
match l with
[] -> []
| c :: cs -> toUpperChar c :: toUpperCase cs
;;
(* test *)
toUpperCase ['a'; 'q'; 'B'; 'Z'; ';'; '!'];;
(* should return ['A'; 'Q'; 'B'; 'Z'; ';'; '!'] *)
(* could have used map instead (note map is built in as List.map): *)
let toUpperCase l = List.map toUpperChar l ;;
(* could have also defined it even more simply - partly apply the Curried map : *)
let toUpperCase = List.map toUpperChar ;;
(*
3. Write a function 'nth' which takes a list (l) and index (n) and returns
the nth element of the list. If n is an invalid index i.e. n is negative or
l has less then (n + 1) elements then fail.
Note: indices start with 0 for the head of the list, 1 for the next element
and so on (similar to arrays).
*)
let rec nth l n =
match l with
| [] -> failwith "list too short"
| x :: xs ->
if n = 0 then
x
else
nth xs (n - 1) (* eureka *)
;;
(* tests *)
nth [1;2;3] 0;; (* should return 1 *)
nth [1;2;3] 1;; (* should return 2 *)
nth [1;2;3] (-1);; (* should raise exception *)
nth [1;2;3] 3;; (* should raise exception *)
(*
4. Write a function 'partition' which takes a predicate (p) and a list
(l) as arguments and returns a tuple (l1, l2) such that l1 is the
list of all the elements of l that satisfy the predicate p and l2
is the list of all the elements of l that do NOT satisfy p. The
order of the elements in the input list (l) should be preserved.
Note: A predicate is any function which returns a boolean.
e.g. let isPositive n = (n > 0);;
*)
let rec partition p l =
match l with
|[] -> ([],[])
| hd :: tl ->
let (posl,negl) = partition p tl in
if (p hd)
then
(hd :: posl,negl)
else
(posl,hd::negl)
(* test *)
let isPositive n = n > 0;;
partition isPositive [1; -1; 2; -2; 3; -3];; (* should return *)
(* ([1; 2; 3], [-1; -2; -3]) *)
(*
5. Write a function 'diff' which takes in two lists l1 and l2 and returns
a list containing all elements in l1 not in l2.
Note: You will need to write another function 'contains x l' which checks
whether an element 'x' is contained in a list 'l' or not.
*)
let rec contains x l =
match l with
[] -> false
| y :: ys -> x = y || contains x ys
;;
let rec diff l1 l2 =
match l1 with
[] -> []
| x :: xs ->
if contains x l2 then
diff xs l2
else
x :: diff xs l2
;;
(* tests *)
contains 1 [1; 2; 3];; (* should return true *)
contains 5 [1; 2; 3];; (* should return false *)
diff [1;2;3] [3;4;5];; (* should return [1; 2] *)
diff [1;2] [1;2;3];; (* should return [] *)
(* ====================================================================== *)
(* OCaml Lecture IV *)
(* ******************************************************************* *)
(* Next topic: user-defined data-structures -- variants & records *)
(* Big change from above: we need to declare some types now. *)
(* Still more convenient than Java etc: declare a type once globally, infer thereafter *)
(*
* Variant Type Declaration
* - related to union types in C or enums in Java: "this OR that OR theother"
* - BUT like OCamls lists/tuples they are immutable data structures
* - each case of the union is identified by a name called 'Constructor'
* which serves for both
* - Constructing values of the variant type
* - inspecting them by pattern matching
* - Constructors must start with Capital Letter to distinguish from variables
* - type declarations needed but once they are in place type inference on them works
*)
(*
* variant type for doing mixed arithmetic (integers and floats)
*)
type mynumber = Fixed of int | Floating of float;; (* read "|" as "or", same as match *)
Fixed 5;; (* tag 5 as a Fixed *)
Floating 4.0;; (* tag 4.0 as a Floating *)
(* note constructors look like functions but they are not -- you always need to give arg *)
let pullout x =
match x with
| Fixed n -> n (* variants fit well into pattern matching syntax *)
| Floating z -> int_of_float z;;
pullout (Fixed 5);;
(* A non-trivial function using the above variant type *)
let add_num n1 n2 =
match (n1, n2) with (* note use of pair here to parallel-match on two variables *)
| (Fixed i1, Fixed i2) -> Fixed (i1 + i2)
| (Fixed i1, Floating f2) -> Floating(float i1 +. f2) (* need to coerce *)
| (Floating f1, Fixed i2) -> Floating(f1 +. float i2) (* ditto *)
| (Floating f1, Floating f2) -> Floating(f1 +. f2)
;;
add_num (Fixed 123) (Floating 3.14159);;
(*
* Enum types
* - special case of variant types,
* where all alternatives are constants
* - leave off the "of " business since there is no data
*)
type sign = Positive | Negative | Zero;;
let sign_int n = if n > 0 then Positive else if n = 0 then Zero else Negative;;
sign_int (-6);;
(* Multiple data items in a single clause? Use the pre-existing product type! *)
type complex = CZero | Nonzero of float * float;;
let com = Nonzero(3.2,11.2);;
(*
* Recursive data structures - a key use of variant types
*
* - Functional programming is fantastic for computing over tree-structured data
* - recursive types can refer to themselves in their own definition
* - similar in spirit to how C structs can be recursive (but, no pointer needed here)
*
* Example of binary trees with integers in nodes and empty leaves:
*)
type itree = ILeaf | INode of int * itree * itree;; (* type refers to itself! *)
INode(4,ILeaf,INode(2,INode(18,ILeaf,ILeaf),ILeaf));;
(* Better polymorphic version of above which allows any type at leaves: *)
type 'a btree = Leaf | Node of 'a * 'a btree * 'a btree;;
(* Observe how above type takes a (prefix) argument, 'a -- "btree" is a type function *)
(* example trees *)
let whack = Node("whack!",Leaf, Leaf);;
let bt = Node("fiddly ",
Node("backer ",
Leaf,
Node("crack ",
Leaf,
Leaf)),
whack);;
let bt2 = Node("fiddly ",
Node("backer ",
Leaf,
Node("crack ",
Leaf,
Leaf)),
whack);;
(* type error, must have uniform data: Node("fiddly",Node(0,Leaf,Leaf),Leaf);; *)
(* OCaml's lists could have been defined with a recursive type declaration *)
type 'a mylist = MtList | ColonColon of 'a * 'a mylist;;
let mylisteg = ColonColon(3,ColonColon(5,ColonColon(7,MtList)));; (* [3;5;7] *)
let rec double_list_elts ml =
match ml with
| MtList -> MtList (* vs [] -> [] *)
| ColonColon(hd,tl) -> ColonColon(hd * 2,double_list_elts tl);; (* vs mh :: mt -> .. *)
double_list_elts mylisteg;;
(*
* Functions on binary trees are similar to functions on lists: use recursion
*
*)
let rec add_gobble binstringtree =
match binstringtree with
Leaf -> Leaf
| Node(y, left, right) ->
Node(y^"gobble",add_gobble left,add_gobble right)
;;
let rec lookup x bintree =
match bintree with
| Leaf -> false
| Node(y, left, right) ->
if x = y then
true
else if x < y then
lookup x left
else
lookup x right
;;
lookup "whack!" bt;;
lookup "flack" bt;;
(* Recall we are still 100% IMMUTABLE: tree insert is functional insert, return the new tree *)
let rec insert x bintree =
match bintree with
Leaf -> Node(x, Leaf, Leaf)
| Node(y, left, right) ->
if x <= y then
Node(y, insert x left, right)
else
Node(y, left, insert x right)
;;
let goobt = insert "goober " bt;;
bt;; (* Yes, OCaml data structures are by default immutable! *)
let gooobt =
insert "slacker " goobt;;
(* END variants *)
(*
* Record Declarations
* - like tuples but with labels on fields.
* - similar to the structs of C/C++.
* - the types must be declared just like OCaml variants.
* - can be used in pattern matches as well.
* - again the fields are immutable by default
* - not used as often as structs of C, most data is a variant with tupled multiple arguments
*)
(* record to represent rational numbers *)
type ratio = {num: int; denom: int};;
let q = {num = 53; denom = 6};;
q.num;;
q.denom;;
(* pattern matching works of course *)
let rattoint r =
match r with
{num = n; denom = d} -> n / d;;
(* only one pattern matched so can again inline pattern in functions and lets *)
let rattoint {num = n; denom = d} =
n / d;;
(* equivalently can use dot projections, but happy path is usually patterns *)
let rattoint r =
r.num / r.denom;;
rattoint q;;
let add_ratio r1 r2 = {num = r1.num * r2.denom + r2.num * r1.denom; denom = r1.denom * r2.denom};;
add_ratio {num = 1; denom = 3} {num = 2; denom = 5};;
(* Somewhat annoying shadowing issue: there is one global namespace of record labels *)
type newratio = {num: int; coeff: float};; (* shadowing ratio's label num *)
fun x -> x.num;; (* requires x to be a newratio -- inference HAS to give one type. *)
(* one solution: declare type *)
fun (x : ratio) -> x.num;;
(* best solution: pattern match on full record *)
fun {num = _; denom = d} -> d;;
(* OCaml programmers often use tuples instead of records for conciseness *)
(* ********************************************************************** *)
(*
* State
* - variables in ML are NEVER directly mutable themselves; only (indirectly)
* mutable if they hold a
* - reference
* - mutable record
* - array
*
* - Indirect mutability - variable itself can't change,
* but what it points to can.
* - items are immutable unless their mutability is explicitly declared
*
* - DON'T use state unless its really needed (you can't use it in HW1 or most of HW2)
*)
(* References *)
let x = ref 4;; (* always have to declare initial value when creating a reference *)
(* Meaning of the above: x forevermore (i.e. forever unless shadowed) refers to a fixed cell.
The CONTENTS of that fixed call can change, but not x. *)
(* x + 1;; *) (* a type error ! *)
!x + 1;; (* need !x to get out the value; something like *x in C *)
x := 6;; (* assignment - x must be a ref cell. Returns () - goal is side effect *)
!x + 1;; (* Mutation ! *)
(* 'a ref is really implemented by a mutable record with one field, contents:
'a ref abbreviates the type { mutable contents: 'a }
-- the keyword mutable on a record field means it can mutate *)
let x = { contents = 4};; (* identical to x's definition above *)
x := 6;;
x.contents <- 7;; (* same effect as previous line: backarrow updates a field *)
!x + 1;;
x.contents + 1;; (* same effect as previous line *)
(* declaring your own mutable record: put "mutable" qualifier on field *)
type mutable_point = { mutable x: float; mutable y: float };;
let translate p dx dy =
p.x <- (p.x +. dx); (* observe use of ";" here to sequence effects *)
p.y <- (p.y +. dy) (* ";" is useless without side effects (think about it) *)
;;
let mypoint = { x = 0.0; y = 0.0 };;
translate mypoint 1.0 2.0;;
mypoint;;
(* observe: mypoint itself is immutable but it has two spots in it where we can mutate *)
(* tree with mutable nodes *)
type mtree = MLeaf | MNode of int * mtree ref * mtree ref
;;
(*
* - ONLY ref typed variables or mutable records may be assigned to
* - The notion of immutable variables is one of the great strengths of ML.
* - Note: 'let' doesn't turn into a mutation operator with 'ref's
*)
let x = ref 4;;
let f () = !x;; (* This is syntax for a 0-argument function in OCaml - it only takes () as argument *)
x := 234;;
f();;
let x = ref 6;; (* shadowing previous x definition, NOT an assignment to x !! *)
f ();;
(* Yes, we can even write a while loop ! *)
let x = ref 1 in
while !x < 10 do
print_int !x;
print_string "\n";
x := !x + 1
done;;
(* Fact: while loops are useless without mutation: either never loop or infinitely loop *)
(*
* Why is immutability good?
* - programmer can depend on the fact that something will never be mutated
* when writing code: permanent like mathematical definitions
*
* ML still lets you express mutation, but its extra so you only use it when
* its really needed
*
* Haskell has an even stronger separation of mutation, its all strictly "on top".
*)
(*
* Arrays
* - fairly self-explanatory
* - have to be initialized before using
* - in general there is no such thing as "uninitialized" in OCaml.
*)
let arrhi = Array.make 100 "";; (* size and initial value are the params here *)
let arr = [| 4; 3; 2 |];; (* another way to make an array *)
arr.(0);; (* access (unfortunately already used [] for lists in the syntax) *)
arr.(0) <- 55;; (* update *)
arr;;
(* Exceptions: pretty standard and mostly Java-like *)
exception Foo;; (* This is a new form of top-level declaration, along with let, type *)
let f () = raise Foo;; (* note no need to "raises Foo" in functions type as in Java *)
f ();;
exception Bar;;
let g _ = (* note that unlike Java there is no "raises" in the type of g *)
(try
f ()
with
Foo -> 5 | Bar -> 3) + 4;; (* Use power of pattern matching in handlers *)
g ();;
(* exceptions that pass up an argument - syntax is sort of like a variant *)
exception Goo of string;;
let f _ = raise (Goo "keyboard on fire");;
f ();;
let g () =
try
f ()
with
Foo -> ()
| Goo s ->
(print_string("exception raised: ");
print_string(s);print_string("\n"))
;;
g ();;
(* There are a few built-in exceptions we are already using *)
failwith "Oops";; (* Generic code failure - exception is named Failure *)
invalid_arg "This function works on non-empty lists only";; (* Invalid_argument exception *)
(* END Exceptions *)
(* ====================================================================== *)
(* OCaml Lecture V *)
(* OCaml Modules - structures and functors *)
(* Modules in programming languages
- a module is a larger level of program abstraction: functional units or library.
- e.g. Java package, C directory w/ header files
Some principles of modules:
- Modules have names they can be referenced by.
- A module contains code declarations: functions, classes, types, etc.
- The module has an interface in which it
* imports some things (e.g. other modules) from the outside and
* exports some things it has declared for outsiders to use;
* **hides** other things for internal use only
-- hiding is a key feature, users don't get overwhelmed
The C "module" system
- Informal use of files and filesystem directories as modules
- .h file declaring what is externally visible for a module
Problems with C modules
- There is a global space of function names, so there can be name clashes
- There is no strict relation enforced between the .c and .h files
* bad programmers can write really ugly code
The Java module system: packages
- A cleaner version of the C spirit of module
- Directory is explicitly a package; allows for nested packages
- Implicit export interface via public decls on classes/methods
- private/protected for information hiding
- Separate namespace for each package avoids name clashes
- Have to have at least the .class files of the imports around to compile
*)
(* OCaml Modules *)
(* We already saw some modules in action above, e.g. with List.map.
This is an invocation of the map function in the system List module. *)
List.map (fun x -> x ^"gobble")["Have";"a";"good";"day"];;
(* Now, lets look into how we can build our own modules *)
(*
OCaml module definitions are called **structures**
- collections of related definitions (functions, types, other structures,
exceptions, values, ...) given a name
*)
(* lets make a very simple functional set data structure *)
module FSet =
struct
exception NotFound (* any top-level definable can be included in a module *)
type 'a set = 'a list (* sets are just lists but make a new type to keep them distinct *)
let emptyset : 'a set = []
let rec add x (s: 'a set) = ((x :: s) : ('a set)) (* observe this is a FUNCTIONAL set - RETURN new *)
let rec remove x (s: 'a set) =
match s with
| [] -> raise NotFound
| hd :: tl ->
if hd = x then (tl: 'a set)
else hd :: remove x tl
let rec contains x (s: 'a set) =
match s with
| [] -> false
| hd :: tl ->
if x = hd then true else contains x tl
end
;;
(* observe what is printed in the top loop when the above is entered: a module *signature* is inferred
- The types of structs are called signatures
- they are the interfaces for structures, something like Java interfaces
- signatures can be used to hide some things from outside users
*)
(* Using modules: its something like Java packages, qualify the name *)
let mySet = FSet.add 5 [];; (* notice the return type here: int FSet.set -- includes module name *)
let myNextSet = FSet.add 22 mySet;;
FSet.contains 22 mySet;;
FSet.remove 5 myNextSet;;
open FSet;; (* puts an implicit "FSet." in front of all things in FSet; may shadow existing names *)
add "a" ["b"];;
contains "a" ["a"; "b"];;
(* ********************************************************************** *)
(* OCaml's module signatures and using them for information hiding *)
module type GROWINGSET = (* define a module type (signature) with no remove; not very useful *)
sig
exception NotFound
type 'a set = 'a list
val emptyset : 'a set
val add : 'a -> 'a set -> 'a set
val contains : 'a -> 'a set -> bool
end
;;
module GrowingSet = (FSet: GROWINGSET);; (* constrain a structure to have that signature *)
GrowingSet.add "a" ["b"];;
(* GrowingSet.remove;; *) (* Error: remove in struct but not in signature! *)
FSet.remove;; (* This is still fine, remember we are not mutating FSet when making GrowingSet *)
(* Now lets do some useful hiding. Hiding types is possible and allows "black box" data structures
- can be good software engineering practice to enforce hiding of internals *)
module type HIDDENSET =
sig
type 'a set (* hide the type 'a list here by not giving 'a set definition in signature *)
val emptyset : 'a set
val add: 'a -> 'a set -> 'a set
val remove : 'a -> 'a set -> 'a set
val contains: 'a -> 'a set -> bool
end
;;
module HiddenSet = (FSet: HIDDENSET);;
(* HiddenSet.add 3 [];; *) (* Error: [] not a set since we HID the fact that sets are really lists *)
let hs = HiddenSet.add 5 (HiddenSet.add 3 HiddenSet.emptyset);; (* now it works - result means type is abstract *)
HiddenSet.contains 5 hs;;
(* Also can declare signature along with module *)
module HFSet :
sig
type 'a set
val emptyset : 'a set
val add: 'a -> 'a set -> 'a set
val remove : 'a -> 'a set -> 'a set
val contains: 'a -> 'a set -> bool
end =
struct
exception NotFound (* any top-level definable can be included in a module *)
type 'a set = 'a list (* sets are just lists but make a new type to keep them distinct *)
let emptyset : 'a set = []
let rec add x (s: 'a set) = ((x :: s) : ('a set)) (* observe this is a FUNCTIONAL set - RETURN new *)
let rec remove x (s: 'a set) =
match s with
| [] -> raise NotFound
| hd :: tl ->
if hd = x then (tl: 'a set)
else hd :: remove x tl
let rec contains x (s: 'a set) =
match s with
| [] -> false
| hd :: tl ->
if x = hd then true else contains x tl
end
;;
let hs = HFSet.add 5 (HFSet.add 3 HFSet.emptyset);; (* now it works - result means type is abstract *)
(* ********************************************************************** *)
(* Separate Compilation with OCaml
- We can program OCaml in a cc / javac like way - use ocamlc instead of ocaml.
- Key invariant: each OCaml module is a separate .ml file
- Syntax of module **body** is identical
- No header "module XX = struct .. end" is included in .ml module file
- Name of module is capped name of file: fSet.ml defines module FSet
- File fSet.mli holds the signature of module FSet
if there is no file set.mli thats OK; you have nothing hidden
- Use ocamlc to compile and link to an executable: similar to C/C++
- main program that starts running is any non-values defined in the module(s)
- Also need to compile the .mli files! (unlike .h files)
Here is how the ocamlc compiler makes object files
.ml --ocamlc -c --> .cmo
.mli --ocamlc -c --> .cmi
Then to make the binary
.cmo's --ocamlc -o mybinary --> mybinary
You need any dependent .cmi's for modules you refer to before you can ocamlc -c them.
*)
module FSett: sig (* contents of file fSett.mli *) end
= struct (* contents of file fSett.ml *) end;;
module Main: sig (* contents of main.mli *) end
= struct (* contents of main.ml *) end;;
(* See http://pl.cs.jhu.edu/pl/ocaml/code/sep.zip for the example we cover in lecture.
We will follow http://pl.cs.jhu.edu/pl/ocaml/code/sep_compile/readme.txt in particular.
See the ocaml manual Chapter 8 for the full documentation *)
(* ********************************************************************** *)
(* Loading object file modules into the top loop *)
(* It is possible to mix ocamlc and ocaml for debugging: load .cmo files into top loop. *)
(* for the following to work need to first #cd to sep_compile/. My computer's version:
#cd "/Users/scott/pl/ocaml/code/sep_compile";;
*)
#load "fSet.cmo"
;;
FSet.emptyset;;
(* ********************************************************************** *)
(*
Functors
A "function" from structures to structures
Allows a module to be parameterized and so instantiated in multiple ways
- think of it as the ability to "plug in" a code module
In object-oriented languages, object polymorphism gives you much of this ability
- the "Animal" variable can have a Dog, Cat, Fish, etc plugged in to it
But, OCaml has no object polymorphism and something is needed to support this
General functors are found only in a few languages
*)
(* have this type above: type comparison = LessThan | EqualTo | GreaterThan *)
(* here is a kind of struct that we can take as a parameter; in Java we would just use an interface Comparable *)
module type ORDERED_TYPE =
sig
type t
val compare: t -> t -> comparison
end;;
(* Here is a functor version of a set, you feed in a struct with the set element ordering defined on it *)
module FSetFunctor =
functor (Elt: ORDERED_TYPE) ->
struct
type element = Elt.t (* import the type of elements from the structure *)
type set = element list
let empty = []
let rec add x s =
match s with
[] -> [x]
| hd::tl ->
match Elt.compare x hd with
EqualTo -> s
| LessThan -> x :: s
| GreaterThan -> hd :: add x tl
let rec contains x s =
match s with
[] -> false
| hd::tl ->
match Elt.compare x hd with
EqualTo -> true
| LessThan -> false
| GreaterThan -> contains x tl
end;;
(* Here is a concrete ordering we can feed in, one over ints *)
module OrderedInt =
struct
type t = int
let compare x y =
if x = y then
EqualTo
else
if x < y then
LessThan
else
GreaterThan
end;;
(* Here is how we feed it in, instantiating the functor to give a structure *)
module OrderedIntSet = FSetFunctor(OrderedInt);; (* note how this looks like a function application *)
let myOrderedIntSet = OrderedIntSet.add 5 OrderedIntSet.empty;;
OrderedIntSet.contains 3 myOrderedIntSet;;
(* We can do the same thing for a string comparison *)
module OrderedString =
struct
type t = string
let compare x y =
if x = y then EqualTo
else if x < y then LessThan
else GreaterThan
end;;
module OrderedStringSet = FSetFunctor(OrderedString);; (* a DIFFERENT instantiation of same *)
let myOrderedStringSet = OrderedStringSet.add "abc" OrderedStringSet.empty;;
(* Functors also have signatures; there can also be type abstraction in a functor signature *)
module type SETFUNCTOR = (* below is the syntax for a signature of a functor *)
functor (Elt: ORDERED_TYPE) ->
sig
type element = Elt.t (* concrete *)
type set (* abstract *)
val empty : set
val add : element -> set -> set
val contains : element -> set -> bool
end;;
module AbstractSet = (FSetFunctor : SETFUNCTOR);; (* slap that sig on a functor *)
module AbstractIntSet = AbstractSet(OrderedInt);;
AbstractIntSet.add 5 AbstractIntSet.empty;;
(* end functors topic *)