An Introduction to OCaml
Stephen A. Edwards Columbia University Spring 2021
The Basics
Functions
Tuples, Lists, and Pattern Matching
User-Defined Types
Modules and Compilation
A Complete Interpreter in Three Slides
Exceptions; Directed Graphs
Standard Library Modules
An Endorsement?
A PLT student accurately summed up using OCaml:
Never have I spent so much time writing so little that does so much.
I think he was complaining, but I’m not sure. Other students have said things like
It’s hard to get it to compile, but once it compiles, it works.
Why OCaml?
It’s Great for Compilers
I’ve written compilers in C++, Python, Java, and OCaml, and it’s much easier in OCaml.
It’s Succinct
Would you prefer to write 10 000 lines of code or 5 000?
Its Type System Catches Many Bugs
It catches missing cases, data structure misuse, certain off-by-one errors, etc. Automatic garbage collection and lack of null pointers makes it safer than Java.
Lots of Libraries
All sorts of data structures, I/O, OS interfaces, graphics, support for compilers, etc.
Lots of Support
Many websites, free online books and tutorials, code samples, etc.
OCaml in One Slide
Apply a function to each list element; save results in a list
“Is recursive” Passing a function Case
Pattern Matching
splitting
Local name declaration
List support Recursion
Anonymous functions
# let rec map f = function [] -> []
|
| head :: tail ->
Polymorphic
let r = f map
head in f tail;;
Types inferred
r :: map
(’a -> ’b) -> ’a list -> ’b list
valmap: (’a->’b)->’alist->’blist
(function x -> x + 3)
# map (function x -> x + 3) [1;5;9];; -: intlist=[4;8;12]
The Basics
Hello World in OCaml: Interpret or Compile
Create a “hello.ml” file:
print_endline “Hello World!” Run it with the interpreter:
$ ocaml hello.ml
Hello World!
Compile a native executable and run:
$ ocamlopt -o hello hello.ml
$ ./hello
Hello World!
Use ocamlbuild (recommended):
$ ocamlbuild hello.native
$ ./hello.native
Hello World!
Hello World in OCaml: REPL
The interactive Read-Eval-Print Loop
$ ocaml
OCaml version 4.02.3
# print_endline “Hello World!”;;
Hello World!
– : unit = ()
# #use “hello.ml”;;
Hello World!
– : unit = ()
# #quit;;
$
Double semicolons ;; mean “I’m done with this expression” #quit terminates the REPL
Other directives enable tracing, modify printing, and display types and values
Use ledit ocaml or utop instead for better line editing (history, etc.)
Comments
OCaml
(* This is a multiline
comment in OCaml *)
(* Comments
(* like these *)
do nest
(* OCaml has no *)
(* single-line comments *)
C/C++/Java
/* This is a multiline
comment in C */
/* C comments
/* do not
nest */
// C++/Java also has
// single-line comments
*)
Basic Types and Expressions
# 42 + 17;;
– : int = 59
# 42.0 +. 18.3;;
– : float = 60.3
# 42 + 60.3;;
Error: This expression has type
float but an expression was
expected of type int
# 42 + int_of_float 60.3;;
– : int = 102
# true || (3 > 4) && not false;;
– : bool = true
# “Hello ” ^ “World!”;;
– : string = “Hello World!”
# String.contains “Hello” ’o’;;
– : bool = true
# ();;
– : unit = ()
# print_endline “Hello World!”;;
Hello World!
– : unit = ()
Integers
Floating-point numbers
Floating-point operators must be explicit (e.g., +.)
Only explicit conversions, promotions
(e.g., int_of_float)
Booleans Strings
The unit type is like “void” in C and Java
Standard Operators and Functions
+ – * / mod
+. -. *. /. **
ceil floor sqrt exp
log log10 cos sin
tan acos asin atan
not && ||
= <> == !=
< > <= >=
Integer arithmetic Floating-point arithmetic
Floating-point functions
Boolean operators
Structual comparison (polymorphic) Physical comparison (polymorphic)
Comparisons (polymorphic)
Structural vs. Physical Equality
==, != Physical equality compares pointers
#1==3;; -:bool= false
#1==1;; -:bool= true
=, <> Structural equality compares values
#1=3;; -:bool=false
#1=1;; -:bool=true
# 1.5 = 1.5;;
– : bool = true
# let f = 1.5 in f = f;;
– : bool = true
# “a” = “a”;;
– : bool = true
# 1.5 == 1.5;;
– : bool = false
(* Huh? *)
# let f = 1.5 in f == f;;
– : bool = true
# “a” == “a”;;
– : bool = false
(* Huh? *)
# let a = “hello” in a == a;;
– : bool = true
Use structural equality to avoid headaches
If-then-else
if expr1 then expr2 else expr3
If-then-else in OCaml is an expression. The else part is compulsory, expr1 must be Boolean, and the types of expr2 and expr3 must match.
# if 3 = 4 then 42 else 17;;
– : int = 17
# if “a” = “a” then 42 else 17;;
– : int = 42
# if true then 42 else “17”;;
This expression has type string but is here used with type int
Naming Expressions with let
let name = expr1 in expr2 Bind name to expr1 in expr2 only
let name = expr Bind name to expr forever after # let x = 38 in x + 4;;
– : int = 42
# let x = (let y = 2 in y + y) * 10 in x;;
– : int = 40
# x + 4;;
Unbound value x
# let x = 38;;
val x : int = 38
# x + 4;;
– : int = 42
# let x = (let y = 2) * 10 in x;;
Error: Syntax error: operator expected.
# let x = 10 in let y = x;;
Error: Syntax error
Let is Not Assignment
Let can be used to bind a succession of values to a name.
This is not assignment: the value disappears in the end.
# let a = 4 in
let a = a + 2 in
let a = a * 2 in
a;;
– : int = 12
# a;;
Unbound value a
This looks like sequencing, but it is really data dependence.
Let is Really Not Assignment
OCaml picks up the values in effect where the function (or expression) is defined.
Global declarations are not like C’s global variables.
# let a = 5;;
val a : int = 5
# let adda x = x + a;;
val adda : int -> int =
# let a = 10;;
val a : int = 10
# adda 0;;
– : int = 5 (* adda sees a = 5 *)
# let adda x = x + a;;
val adda : int -> int =
# adda 0;;
– : int = 10 (* adda sees a = 10 *)
Functions
Functions
A function is just another type whose value can be defined with an expression.
# fun x -> x * x;;
– : int -> int =
# (fun x -> x * x) 5;; (* function application *)
– : int = 25
# fun x -> (fun y -> x * y);;
– : int -> int -> int =
# fun x y -> x * y;; (* shorthand *)
– : int -> int -> int =
# (fun x -> (fun y -> (x+1) * y)) 3 5;;
– : int = 20
# let square = fun x -> x * x;;
val square : int -> int =
# square 5;;
– : int = 25
# let square x = x * x;; (* shorthand *)
val square : int -> int =
# square 6;;
– : int = 36
Let is Like Function Application
let name = expr1 in expr2
(fun name -> expr2) expr1
Both mean “expr2, with name replaced by expr1”
# let a = 3 in a + 2;;
– : int = 5
# (fun a -> a + 2) 3;;
– : int = 5
Semantically equivalent; let is easier to read
Recursive Functions
OCaml
let rec gcd a b = if a = b then
a
else if a > b then
gcd (a – b) b else
gcd a (b – a)
C/C++/Java
int gcd(int a, int b) {
while (a != b) { if (a > b)
a -= b; else
let rec allows for recursion
Use recursion instead of loops
Tail recursion runs efficiently in OCaml
} }
b -= a; return a;
Recursive Functions
By default, a name is not visible in its defining expression.
# let fac n = if n < 2 then 1 else n * fac (n-1);;
Unbound value fac
The rec keyword makes the name visible.
# let rec fac n = if n < 2 then 1 else n * fac (n-1);;
val fac : int -> int =
# fac 5;;
– : int = 120
The and keyword allows for mutual recursion.
# let rec fac n = if n < 2 then 1 else n * fac1 n
and fac1 n = fac (n - 1);;
val fac : int -> int =
val fac1 : int -> int =
# fac 5;;
– : int = 120
First-Class and Higher Order Functions
First-class functions: name them, pass them as arguments
# let appadd = fun f -> (f 42) + 17;;
val appadd : (int -> int) -> int =
# let plus5 x = x + 5;;
val plus5 : int -> int =
# appadd plus5;;
– : int = 64
Higher-order functions: functions that return functions
# let makeInc i = fun x -> x + i;;
val makeInc : int -> int -> int =
# let i5 = makeInc 5;;
val i5 : int -> int =
# i5 10;;
– : int = 15
Tuples, Lists, and Pattern Matching
Tuples
Pairs or tuples of different types separated by commas.
Very useful lightweight data type, e.g., for function arguments.
# (42, “Arthur”);;
– : int * string = (42, “Arthur”)
# (42, “Arthur”, “Dent”);;
– : int * string * string = (42, “Arthur”, “Dent”)
# let p = (42, “Arthur”);;
val p : int * string = (42, “Arthur”)
# fst p;;
– : int = 42
# snd p;;
– : string = “Arthur”
# let trip = (“Douglas”, 42, “Adams”);;
val trip : string * int * string = (“Douglas”, 42, “Adams”)
# let (fname, _, lname) = trip in (lname, fname);;
– : string * string = (“Adams”, “Douglas”)
Lists
(* Literals *)
[];;
[1];;
[42; 16];;
(* The empty list *)
(* A singleton list *)
(* A list of two integers *)
(* cons: Put something at the beginning *)
7 :: [5; 3];; (* Gives [7; 5; 3] *)
[1; 2] :: [3; 4];; (* BAD: type error *)
(* concat: Append a list to the end of another *) [1;2]@[3;4];; (*Gives[1;2;3;4]*)
(* Extract first entry and remainder of a list *) List.hd [42; 17; 28];; (* = 42 *)
List.tl [42; 17; 28];; (* = [17; 28] *)
The elements of a list must all be the same type. :: is very fast; @ is slower—O(n)
Pattern: create a list with cons, then use List.rev.
Some Useful List Functions
Three great replacements for loops:
List.map f [a1; … ;an] = [f a1; … ;f an] Apply a function to each element of a list to produce another list.
List.fold_left f a [b1; …;bn] =
f (…(f (f a b1) b2)…) bn
Apply a function to a partial result and an element of the list to produce the next partial result.
List.iter f [a1; …;an] =
begin f a1; … ; f an; () end
Apply a function to each element of a list; produce a unit result.
List.rev [a1; …; an] = [an; … ;a1] Reverse the order of the elements of a list.
List Functions Illustrated
# List.map (fun a -> a + 10) [42; 17; 128];;
– : int list = [52; 27; 138]
# List.map string_of_int [42; 17; 128];;
– : string list = [“42”; “17”; “128”]
# List.fold_left (fun s e -> s + e) 0 [42; 17; 128];;
– : int = 187
# List.iter print_int [42; 17; 128];;
4217128- : unit = ()
# List.iter (fun n -> print_int n; print_newline ())
[42; 17; 128];;
42
17
128
– : unit = ()
# List.iter print_endline (List.map string_of_int [42; 17; 128]);;
42
17
128
– : unit = ()
Example: Enumerating List Elements
To transform a list and pass information between elements, use List.fold_left with a tuple:
# let (l, _) = List.fold_left
(fun (l, n) e -> ((e, n)::l, n+1)) ([], 0) [42; 17; 128]
in List.rev l;;
– : (int * int) list = [(42, 0); (17, 1); (128, 2)]
Result accumulated in the (l, n) tuple, List.rev reverses the result (built backwards) in the end. Can do the same with a recursive function, but List.fold_left separates list traversal from modification:
# let rec enum (l, n) = function
[] -> List.rev l
| e::tl -> enum ((e, n)::l, n+1) tl
in
enum ([], 0) [42; 17; 128];;
– : (int * int) list = [(42, 0); (17, 1); (128, 2)]
Pattern Matching
A powerful variety of multi-way branch that is adept at picking apart data structures. Unlike anything in C/C++/Java.
# let xor p = match p
with (false, false) -> false
| (false, true) -> true
| ( true, false) -> true
| ( true, true) -> false;;
val xor : bool * bool -> bool =
# xor (true, true);;
– : bool = false
A name in a pattern matches anything and is bound when the pattern matches. Each may appear only once per pattern.
# let xor p = match p
with (false, x) -> x
| ( true, x) -> not x;;
val xor : bool * bool -> bool =
# xor (true, true);;
– : bool = false
Case Coverage
The compiler warns you when you miss a case or when one is redundant (they are tested in order):
# let xor p = match p
with (false, x) -> x
| (x, true) -> not x;;
Warning P: this pattern-matching is not exhaustive.
Here is an example of a value that is not matched:
(true, false)
val xor : bool * bool -> bool =
# let xor p = match p
with (false, x) -> x
|(true, x)->notx
| (false, false) -> false;;
Warning U: this match case is unused.
val xor : bool * bool -> bool =
Wildcards
Underscore (_) is a wildcard that will match anything, useful as a default or when you just don’t care.
# let xor p = match p
with (true, false) | (false, true) -> true
| _ -> false;;
val xor : bool * bool -> bool =
# xor (true, true);;
– : bool = false
# xor (false, false);;
– : bool = false
# xor (true, false);;
– : bool = true
# let logand p = match p
with (false, _) -> false
| (true, x) -> x;;
val logand : bool * bool -> bool =
# logand (true, false);;
– : bool = false
# logand (true, true);;
– : bool = true
Pattern Matching with Lists
# let length = function (* let length = fun p -> match p with *)
[] -> “empty”
| [_] -> “singleton”
| [_; _] -> “pair”
| [_; _; _] -> “triplet”
| hd :: tl -> “many”;;
val length : ’a list -> string =
# length [];;
– : string = “empty”
# length [1; 2];;
– : string = “pair”
# length [“foo”; “bar”; “baz”];;
– : string = “triplet”
# length [1; 2; 3; 4];;
– : string = “many”
Pattern Matching with when and as
The when keyword lets you add a guard expression:
# let tall = function
| (h, s) when h > 180 -> s ^ ” is tall”
| (_, s) -> s ^ ” is short”;;
val tall : int * string -> string =
# List.map tall [(183, “Stephen”); (150, “Nina”)];;
– : string list = [“Stephen is tall”; “Nina is short”]
The as keyword lets you name parts of a matched structure:
# match ((3,9), 4) with
(_ as xx, 4) -> xx
| _ -> (0,0);;
– : int * int = (3, 9)
Application: Length of a list
let rec length l =
if l = [] then 0 else 1 + length (List.tl l);;
Correct, but not very elegant. With pattern matching,
let rec length = function [] ->0
| _::tl -> 1 + length tl;;
Elegant, but inefficient because it is not tail-recursive (needs O(n) stack space). Common trick: use an argument as an accumulator.
let length l =
let rec helper len = function
[] -> len
| _::tl -> helper (len + 1) tl
in helper 0 l
This is the code for the List.length standard library function.
OCaml Can Compile This Efficiently
OCaml source code
let length list =
let rec helper len = function
[] -> len
| _::tl -> helper (len + 1) tl
in helper 0 list
Arguments in registers
Pattern matching reduced to a conditional branch
Tail recursion implemented with jumps
LSB of an integer always 1
ocamlopt generates this x86 assembly
camlLength__helper:
.L101:
addl $2, %eax
jmp .L101
.L100:
ret
camlLength__length:
movl %eax, %ebx
movl $camlLength__2, %eax
movl $1, %eax # len = 0
jmp camlLength__helper
cmpl $1, %ebx
je .L100
movl 4(%ebx), %ebx # get tail
# empty?
# len++
User-Defined Types
Type Declarations
A new type name is defined globally. Unlike let, type is recursive by default, so the name being defined may appear in the typedef.
type name = typedef Mutually-recursive types can be defined with and.
type name1 = typedef1 and name2 = typedef2
.
and namen = typedefn
Records
OCaml supports records much like C’s structs.
# type base = { x : int; y : int; name : string };;
type base = { x : int; y : int; name : string; }
# let b0 = { x = 0; y = 0; name = “home” };;
val b0 : base = {x = 0; y = 0; name = “home”}
# let b1 = { b0 with x = 90; name = “first” };;
val b1 : base = {x = 90; y = 0; name = “first”}
# let b2 = { b1 with y = 90; name = “second” };;
val b2 : base = {x = 90; y = 90; name = “second”}
# b0.name;;
– : string = “home”
# let dist b1 b2 =
let hyp x y = sqrt (float_of_int (x*x + y*y)) in
hyp (b1.x – b2.x) (b1.y – b2.y);;
val dist : base -> base -> float =
# dist b0 b1;;
– : float = 90.
# dist b0 b2;;
– : float = 127.279220613578559
Algebraic Types/Tagged Unions/Sum-Product Types
Vaguely like C’s unions, enums, or a class hierarchy: objects that can be one of a set of types. In compilers, great for trees and instructions.
# type seasons = Winter | Spring | Summer | Fall;;
type seasons = Winter | Spring | Summer | Fall
# let weather = function
Winter -> “Too Cold”
| Spring -> “Too Wet”
| Summer -> “Too Hot”
| Fall -> “Too Short”;;
val weather : seasons -> string =
# weather Spring;;
– : string = “Too Wet”
# let year = [Winter; Spring; Summer; Fall] in
List.map weather year;;
– : string list = [“Too Cold”; “Too Wet”; “Too Hot”; “Too Short”]
Simple Syntax Trees and an Interpreter
# type expr =
Lit of int
| Plus of expr * expr
| Minus of expr * expr
| Times of expr * expr;;
type expr =
Lit of int
| Plus of expr * expr
| Minus of expr * expr
| Times of expr * expr
# let rec eval = function
Lit(x) -> x
| Plus(e1, e2) -> (eval e1) + (eval e2)
| Minus(e1, e2) -> (eval e1) – (eval e2)
| Times(e1, e2) -> (eval e1) * (eval e2);;
val eval : expr -> int =
# eval (Lit(42));;
– : int = 42
# eval (Plus(Lit(17), Lit(25)));;
– : int = 42
# eval (Plus(Times(Lit(3), Lit(2)), Lit(1)));;
– : int = 7
Algebraic Type Rules
Each tag name must begin with a capital letter
# let bad1 = left | right;;
Syntax error
Tag names must be globally unique (required for type inference)
# type weekend = Sat | Sun;;
type weekend = Sat | Sun
# type days = Sun | Mon | Tue;;
type days = Sun | Mon | Tue
# function Sat -> “sat” | Sun -> “sun”;;
This pattern matches values of type days
but is here used to match values of type weekend
Algebraic Types and Pattern Matching
The compiler warns about missing cases:
# type expr =
Lit of int
| Plus of expr * expr
| Minus of expr * expr
| Times of expr * expr;;
type expr =
Lit of int
| Plus of expr * expr
| Minus of expr * expr
| Times of expr * expr
# let rec eval = function
Lit(x) -> x
| Plus(e1, e2) -> (eval e1) + (eval e2)
| Minus(e1, e2) -> (eval e1) – (eval e2);;
Warning P: this pattern-matching is not exhaustive.
Here is an example of a value that is not matched:
Times (_, _)
val eval : expr -> int =
The Option Type: A Safe Null Pointer Part of the always-loaded core library:
type ’a option = None | Some of ’a
This is a polymorphic algebraic type: ’a is any type. None is like a null pointer; Some is a non-null pointer. The compiler requires None to be handled explicitly.
# let rec sum = function
[] -> 0 (* base case *)
| None::tl -> sum tl (* handle the “null pointer” case *)
| Some(x)::tl -> x + sum tl;; (* normal case *)
val sum : int option list -> int =
# sum [None; Some(5); None; Some(37)];;
– : int = 42
Algebraic Types vs. Classes and Enums
Choice of Types Operations
Fields
Hidden fields Recursive Inheritance Case splitting
Algebraic Types
fixed extensible
ordered none yes none simple
Classes Enums
extensible fixed fixed extensible
named none supported none yes no supported none costly simple
An algebraic type is best when the set of types rarely change but you often want to add additional functions. Classes are good in exactly the opposite case.
Modules and Compilation
Modules
Each source file is a module and everything is public.
foo.ml
(* Module Foo *)
type t = { x : int ; y : int } let sum c = c.x + c.y
To compile and run these,
$ ocamlc -c foo.ml
(creates foo.cmi foo.cmo)
$ ocamlc -c bar.ml
(creates bar.cmi bar.cmo)
$ ocamlc -o ex foo.cmo bar.cmo
$ ./ex
333
bar.ml
(* The dot notation *)
letv={ Foo.x=1; Foo.y = 2 };;
print_int (Foo.sum v)
(* Create a short name *)
module F = Foo;; print_int (F.sum v)
(* Import every name from
a module with “open” *)
open Foo;; print_int (sum v)
Separating Interface and Implementation
stack.mli
type ’a t exception Empty
val create : unit -> ’a t
val push : ’a -> ’a t -> unit val pop : ’a t -> ’a
val top : ’a t -> ’a
val clear : ’a t -> unit
val copy : ’a t -> ’a t
val is_empty : ’a t -> bool val length : ’a t -> int
val iter : (’a -> unit) ->
’a t -> unit
stack.ml
type ’a t =
{ mutable c : ’a list }
exception Empty
let create () = { c = [] }
let clear s = s.c <- []
let copy s = { c = s.c }
let push x s = s.c <- x :: s.c
let pop s = match s.c with
hd::tl -> s.c <- tl; hd | [] -> raise Empty
let top s = match s.c with
hd::_ -> hd
| [] -> raise Empty
let is_empty s = (s.c = []) let length s = List.length s.c let iter f s = List.iter f s.c
A Complete Interpreter in Three Slides
The Scanner and AST
scanner.mll
{ open Parser }
rule token =
parse [’ ’ ’\t’ ’\r’ ’\n’] { token lexbuf }
| ’+’
| ’-’
| ’*’
| ’/’
| [’0’-’9’]+ as lit | eof
ast.mli
{PLUS}
{ MINUS }
{ TIMES }
{ DIVIDE }
{ LITERAL(int_of_string lit) } { EOF }
type operator = Add | Sub | Mul | Div
type expr =
Binop of expr * operator * expr
| Lit of int
The Parser
parser.mly
%{ open Ast %}
%token PLUS MINUS TIMES DIVIDE EOF
%token
%left TIMES DIVIDE %start expr
%type
%%
expr:
expr PLUS expr { Binop($1, Add, $3) }
| expr MINUS expr { Binop($1, Sub, $3) } | expr TIMES expr { Binop($1, Mul, $3) } | expr DIVIDE expr { Binop($1, Div, $3) } | LITERAL { Lit($1) }
The Interpeter
calc.ml
open Ast
let rec eval = function Lit(x) -> x
| Binop(e1, op, e2) ->
let v1 = eval e1 and v2 = eval e2 in match op with
Add -> v1 + v2 | Sub -> v1 – v2 | Mul -> v1 * v2 | Div -> v1 / v2
let _ =
let lexbuf = Lexing.from_channel stdin in
let expr = Parser.expr Scanner.token lexbuf in let result = eval expr in
print_endline (string_of_int result)
Compiling the Interpreter
$ ocamllex scanner.mll # create scanner.ml
8 states, 267 transitions, table size 1116 bytes
$ ocamlyacc parser.mly # create parser.ml and parser.mli $ ocamlc -c ast.mli # compile AST types
$ ocamlc -c parser.mli # compile parser types
$ ocamlc -c scanner.ml # compile the scanner
$ ocamlc -c parser.ml # compile the parser
$ ocamlc -c calc.ml # compile the interpreter
$ ocamlc -o calc parser.cmo scanner.cmo calc.cmo
$ ./calc
2*3+4*5
26
$
Compiling with ocamlbuild
$ ls
ast.mli calc.ml parser.mly scanner.mll
$ ocamlbuild calc.native # Build everything
Finished, 15 targets (0 cached) in 00:00:00.
$ ls
ast.mli _build calc.ml calc.native parser.mly scanner.mll $ ./calc.native
2*3+4*5
Ctrl-D
26
$ ocamlbuild -clean # Remove _build and all .native
Exceptions; Directed Graphs
Exceptions
# 5 / 0;;
Exception: Division_by_zero.
# try 5/0
with Division_by_zero -> 42;;
– : int = 42
# exception My_exception;;
exception My_exception
# try
if true then
raise My_exception
else 0
with My_exception -> 42;;
– : int = 42
Exceptions
# exception Foo of string;;
exception Foo of string
# exception Bar of int * string;;
exception Bar of int * string
# let ex b = try
if b then
raise (Foo(“hello”))
else
raise (Bar(42, ” answer”))
with Foo(s) -> “Foo: ” ^ s
| Bar(n, s) -> “Bar: ” ^ string_of_int n ^ s;;
val ex : bool -> unit =
# ex true;;
– : string = “Foo: hello”
# ex false;;
– : string = “Bar: 42 answer”
Application: Directed Graphs
let edges = [
(“a”, “b”); (“a”, “c”); (“a”, “d”); (“b”, “e”); (“c”, “f”); (“d”, “e”); (“e”, “f”); (“e”, “g”) ]
let rec successors n = function [] -> []
| (s, t) :: edges -> if s = n then
t :: successors n edges else
successors n edges
# successors “a” edges;;
– : string list = [“b”; “c”; “d”]
# successors “b” edges;;
– : string list = [“e”]
b acfeg
d
More Functional Successors
let rec successors n = function [] -> []
| (s, t) :: edges -> if s = n then
t :: successors n edges else
successors n edges
Our first example is imperative: performs “search a list,” which is more precisely expressed using the library function List.filter:
let successors n edges =
let matching (s,_) = s = n in
List.map snd (List.filter matching edges)
This uses the built-in snd function, which is defined as let snd (_,x) = x
Depth-First Search
b acfeg
d
let rec dfs edges visited = function [] -> List.rev visited
| n::nodes ->
if List.mem n visited then
dfs edges visited nodes
else
dfs edges (n::visited) ((successors n edges) @ nodes)
# dfs edges [] [“a”];;
– : string list = [“a”; “b”; “e”; “f”; “g”; “c”; “d”]
# dfs edges [] [“e”];;
– : string list = [“e”; “f”; “g”]
# dfs edges [] [“d”];;
– : string list = [“d”; “e”; “f”; “g”]
Topological Sort
b acfeg
d
Remember the visitor at the end.
let rec tsort edges visited = function [] -> visited
| n::nodes ->
let visited’ = if List.mem n visited then visited
else n :: tsort edges visited (successors n edges) in tsort edges visited’ nodes;;
# tsort edges [] [“a”];;
– : string list = [“a”; “d”; “c”; “b”; “e”; “g”; “f”]
# let cycle = [ (“a”, “b”); (“b”, “c”); (“c”, “a”) ];;
val cycle : (string * string) list = [(“a”, “b”); …]
# tsort cycle [] [“a”];;
Stack overflow during evaluation (looping recursion?).
Better Topological Sort
exception Cyclic of string
let tsort edges seed =
let rec sort path visited = function
[] -> visited | n::nodes ->
if List.mem n path then raise (Cyclic n) else let v’ = if List.mem n visited then visited else
n :: sort (n::path) visited (successors n edges) in sort path v’ nodes
in
sort [] [] [seed] # tsort edges “a”;;
– : string list = [“a”; “d”; “c”; “b”; “e”; “g”; “f”]
# tsort edges “d”;;
– : string list = [“d”; “e”; “g”; “f”]
# tsort cycle “a”;;
Exception: Cyclic “a”.
Standard Library Modules
Maps
Balanced trees for implementing dictionaries. Ask for a map with a specific kind of key; values are polymorphic.
# module StringMap = Map.Make(String);;
module StringMap :
sig
type key = String.t
type ’a t = ’a Map.Make(String).t
val empty : ’a t
val is_empty : ’a t -> bool
val add : key -> ’a -> ’a t -> ’a t
val find : key -> ’a t -> ’a
val remove : key -> ’a t -> ’a t
val mem : key -> ’a t -> bool
val iter : (key -> ’a -> unit) -> ’a t -> unit
val map : (’a -> ’b) -> ’a t -> ’b t
val mapi : (key -> ’a -> ’b) -> ’a t -> ’b t
val fold : (key -> ’a -> ’b -> ’b) -> ’a t -> ’b -> ’b
val compare : (’a -> ’a -> int) -> ’a t -> ’a t -> int
val equal : (’a -> ’a -> bool) -> ’a t -> ’a t -> bool
end
Maps
# let mymap = StringMap.empty;; (* Create empty map *)
val mymap : ’a StringMap.t =
# let mymap = StringMap.add “Douglas” 42 mymap;; (* Add pair *)
val mymap : int StringMap.t =
# StringMap.mem “foo” mymap;;
– : bool = false
# StringMap.mem “Douglas” mymap;;
– : bool = true
# StringMap.find “Douglas” mymap;;
– : int = 42
# let mymap = StringMap.add “Adams” 17 mymap;;
val mymap : int StringMap.t =
# StringMap.find “Adams” mymap;;
– : int = 17
# StringMap.find “Douglas” mymap;;
– : int = 42
# StringMap.find “Slarti” mymap;;
Exception: Not_found.
(* Is “foo” there? *)
(* Is “Douglas” there? *)
(* Get value *)
Maps
Fully functional: Map.add takes a key, a value, and a map and returns a new map that also includes the given key/value pair.
Needs a totally ordered key type. Pervasives.compare usually does the job (returns −1, 0, or 1); you may supply your own.
module StringMap = Map.Make(struct
type t = string
let compare x y = Pervasives.compare x y
end)
Uses balanced trees, so searching and insertion is O(logn).
Depth-First Search Revisited
Previous version
let rec dfs edges visited = function [] -> List.rev visited
| n::nodes ->
if List.mem n visited then
dfs edges visited nodes
else
dfs edges (n::visited) ((successors n edges) @ nodes)
was not very efficient, but good enough for small graphs. Would like faster visited test and successors query.
Depth-First Search Revisited
Second version:
use a Map to hold a list of successors for each node
use a Set (valueless Map) to remember of visited nodes
module StringMap = Map.Make(String) module StringSet = Set.Make(String)
Depth-First Search Revisited
let top_sort_map edges =
(* Create an empty successor list for each node *) let succs = List.fold_left
(fun map (s,d) ->
StringMap.add d [] (StringMap.add s [] map)
) StringMap.empty edges in
(* Build the successor list for each source node *)
let succs = List.fold_left (fun succs (s, d) ->
let ss = StringMap.find s succs
in StringMap.add s (d::ss) succs) succs edges
in
(* Visit recursively, storing each node after visiting successors *)
let rec visit (order, visited) n = if StringSet.mem n visited then
(order, visited) else
let (order, visited) = List.fold_left visit (order, StringSet.add n visited) (StringMap.find n succs)
in (n::order, visited) in
(* Visit the source of each edge *)
fst (List.fold_left visit ([], StringSet.empty) (List.map fst edges)
)
Imperative Features
# 0 ; 42;; (* “;” means sequencing *)
Warning S: this expression should have type unit.
– : int = 42
# ignore 0 ; 42;; (* ignore is a function: ’a -> unit *)
– : int = 42
# () ; 42;; (* () is the literal for the unit type *)
– : int = 42
# print_endline “Hello World!”;; (* Print; result is unit *)
Hello World!
– : unit = ()
# print_string “Hello ” ; print_endline “World!”;;
Hello World!
– : unit = ()
# print_int 42 ; print_newline ();;
42
– : unit = ()
# print_endline (“Hello ” ^ string_of_int 42 ^ ” world!”);;
Hello 42 world!
– : unit = ()
Hash Tables
let equal x y = x = y
let hash = Hashtbl.hash
end);;
module StringHash :
sig
type key = string
type ’a t
val create : int -> ’a t
val clear : ’a t -> unit
val copy : ’a t -> ’a t
val add : ’a t -> key -> ’a -> unit
val remove : ’a t -> key -> unit
val find : ’a t -> key -> ’a
val find_all : ’a t -> key -> ’a list
val replace : ’a t -> key -> ’a -> unit
val mem : ’a t -> key -> bool
val iter : (key -> ’a -> unit) -> ’a t -> unit
val fold : (key -> ’a -> ’b -> ’b) -> ’a t -> ’b -> ’b
val length : ’a t -> int
end
# module StringHash = Hashtbl.Make(struct
type t = string
(* type of keys *)
(* use structural comparison *)
(* generic hash function *)
Hash Tables
# let hash = StringHash.create 17;; (* initial size estimate *)
val hash : ’_a StringHash.t =
# StringHash.add hash “Douglas” 42;; (* modify the hash table *)
– : unit = ()
# StringHash.mem hash “foo”;;
– : bool = false
# StringHash.mem hash “Douglas”;;
– : bool = true
# StringHash.find hash “Douglas”;;
– : int = 42
# StringHash.add hash “Adams” 17;;
– : unit = ()
# StringHash.find hash “Adams”;;
– : int = 17
# StringHash.find hash “Douglas”;;
– : int = 42
# StringHash.find hash “Slarti”;;
Exception: Not_found.
(* is “foo” there? *)
(* is “Douglas” there? *)
(* Get value *)
(* Add another key/value *)
Arrays
# let a = [| 42; 17; 19 |];;
val a : int array = [|42; 17; 19|]
# let aa = Array.make 5 0;;
val aa : int array = [|0; 0; 0; 0; 0|]
(* Array literal *)
(* Fill a new array *)
# a.(0);;
– : int = 42
# a.(2);;
– : int = 19
# a.(3);;
Exception: Invalid_argument “index out of bounds”.
# a.(2) <- 20;;
- : unit = ()
# a;;
- : int array = [|42; 17; 20|]
# let l = [24; 32; 17];;
val l : int list = [24; 32; 17]
# let b = Array.of_list l;;
val b : int array = [|24; 32; 17|]
(* Arrays are mutable! *)
(* Array from a list *)
# let c = Array.append a b;;
val c : int array = [|42; 17; 20; 24; 32; 17|]
(* Random access *)
(* Concatenation *)
Arrays vs. Lists
Random access Appending Mutable
Arrays Lists
O(1) O(n) O(n) O(1) Yes No
Useful pattern: first collect data of unknown length in a list then convert it to an array with Array.of_list for random queries.
DFS with Arrays
Second version used a lot of mem, find, and add calls on the string map, each O(logn). Can we do better?
Solution: use arrays to hold adjacency lists and track visiting information.
Basic idea: number the nodes, build adjacency lists with numbers, use an array for tracking visits, then transform back to list of node names.
DFS with Arrays 1/2
let top_sort_array edges =
(* Assign a number to each node *) let map, nodecount =
List.fold_left
(fun nodemap (s, d) ->
let addnode node (map, n) =
if StringMap.mem node map then (map, n) else (StringMap.add node n map, n+1)
in
addnode d (addnode s nodemap) ) (StringMap.empty, 0) edges
in
let successors = Array.make nodecount [] in
let name = Array.make nodecount “” in
(* Build adjacency lists and remember the name of each node *)
List.iter
(fun (s, d) ->
let ss = StringMap.find s map in
let dd = StringMap.find d map in successors.(ss) <- dd :: successors.(ss); name.(ss) <- s;
name.(dd) <- d;
) edges;
DFS with Arrays 2/2
(* Visited flags for each node *)
let visited = Array.make nodecount false in
(* Visit each of our successors if we haven’t done so yet *) (* then record the node *)
let rec visit order n =
if visited.(n) then order else (
visited.(n) <- true;
n :: (List.fold_left visit order successors.(n)) )
in
(* Compute the topological order *)
let order = visit [] 0 in
(* Map node numbers back to node names *)
List.map (fun n -> name.(n)) order