1. Crash course on SML Wojciech Moczydłowski SML – functional programming language Complete formal semantics Extremely convenient for writing compilers, interpreters, provers and 611 homeworks OCaml, based on SML, is practical and used in industry

2. SML - ideology 3 worlds – expressions, values and types. Values have types, i.e. 0 : int, “asd” : string Expressions have types as well, i.e. 3+4 : int, fact : int -> int Expressions evaluate to values

3. SML types Some types: –int, string, char –int*int, int*string –int -> int, int*int -> int –int -> int -> int –int list –‘a list –‘a -> ‘b -> ‘a

4. Interpreter - 2+2; val it = 4 : int -23+17; val it = 40 : int -“611 rules”; val it = “611 rules” : string - ~17 + 17; val it = 0 : int -(17, “a”); val it = (17, “a”) : int * string

5. Definitions -val i = 10; val i = 10 : int -val j = i + i; val j = 20 : int -val s = Int.toString(j); val s = “20” : string -val t = (i, i + 1, i + 2); val t = (10, 11, 12) : int * int *int -val q = #2 t; val q = 11 : int

6. Datatypes -datatype Bool = True | False -datatype Color = Red | Black -datatype Nat = Zero | S of Nat -True; val it = True : Bool -S(S(Zero)); val it = S(S(Zero)) : Nat

7. Functions fun add(n, m : Nat) : Nat = case n of Zero => m | S(x) => S(add(x, m)); fun mult(n, m) = case n of Zero => Zero |S(x) => add(n, mult(x, m)) Evaluation – call by value. All functions have one argument – add : Nat * Nat -> Nat mult : Nat * Nat -> Nat

8. Anonymous functions As in lambda calculus, a function can be specified “on fly” -(fn x => x + 17) 26; val it = 43 : int; Large parts of lambda calculus can be expressed.

9. Polymorphism -val K = fn x => fn y => x; val K = fn : ‘a -> ‘b -> ‘a ‘a ‘b are type variables. -K 6 4; val it = 6 : int -K “611” 170; val it = “611” : string -K K K;

10. Polymorphism -val I = fn x => x; -val S = fn x => fn y => fn z => (x z) (y z); -val Ap = fn f => fn x => f x; However, the following won’t type-check: -val Y = fn f => (fn x => f (x x)) (fn x => f (x x)); -(fn x => x x) (fn x => x x);

11. Datatypes reloaded datatype ‘a option = NONE | SOME of ‘a; -NONE; val it = NONE : ‘a option; -SOME; val it = fn : ‘a -> ‘a option -SOME 5; val it = SOME 5 : int option; -SOME “Cornell”; val it = SOME “Cornell” : string option;

12. Datatypes -fun div(m, n) = if n = 0 then NONE else SOME (m div n); -datatype (‘a, ‘b) Either = Left of ‘a | Right of ‘b;

13. Recursive polymorphic datatypes -datatype ‘a Tree = Null | Node of (‘a Tree) * ‘a * (‘a Tree) -Node (Node(Null, 5, Null), 3, Null); -fun sum(t) = case t of Null => 0 | Node (a, b, c) => sum(a) + b + sum(c)

14. RPD’s fun size(t) = case t of Null => 0 |Node(t1, _, t2) => size t1 + (size t2) + 1 fun add(t, n) = case t of Null => Null |Node(t1, m, t2) => Node(add(t1, n), m + n, add(t2, n))

15. RPD’s fun mul(t, n) = case t of Null => Null |Node(t1, m, t2) => Node(mul(t1, n), m * n, mul(t2, n)) In general?

16. RPD’s fun mapTree(t, f) = case t of Null => Null |Node(t1, m, t2) => Node(mapTree(t1, f), f m, mapTree(t2, f)) -fun add(t, n) = mapTree(t, fn m => m + n); -val mul = fn (t, n) => mapTree (t, fn m => n * m);

17. RPD’s datatype ‘a list = nil | cons of (‘a * ‘a list); Notation: [] denotes nil (x::xs) denotes cons (x, xs)

18. Lists Some lists: [] (1::(2::(3::nil))) (“a”::”b”::”c”::nil) Another notation: [a, b, c] = a::b::c::nil [1,2,3], [“a”, “b”, “c”]

19. Lists -fun length(l) = case l of [] => 0 |(x::xs) => 1 + length xs; -fun hd [] = raise Fail “Empty” |hd(x::xs) = x; -fun tl [] = raise Fail “Empty” |tl (x::xs) = xs;

20. Lists -fun map(f, l) = case l of [] => [] |(x::xs) => f x ::(map f xs);

21. Modules signature SET = sig type ‘‘a set val empty : ‘‘a set val insert : ‘‘a -> ‘‘a set -> ‘‘a set val elem : ‘‘a -> ‘‘a set -> bool end

22. Modules structure Set :> SET= struct type ’’a set = ‘‘a list val empty : ‘‘a set = [] val insert : ‘‘a -> ‘‘a set -> ‘‘a set = fn x => fn y => x::y val elem : ‘‘a -> ‘‘a set -> bool =... end

23. Accessing structure elements -Set.insert, Set.add.... -open Set; -insert, add,...

24. Standard library Many useful structures List :> LIST hd, tl, last, map, find, filter... Option :> OPTION option, isSome, valOf, filter... String :> STRING size, isSubstring, ^, <=...