How does ML deal with +?

Transcription

1 How does ML deal with +? Moscow ML version 2.00 (June 2000) - op +; > val it = fn : int * int -> int ; > val it = 2 : int ; > val it = 2.0 : real - false + false;! Overloaded + cannot be applied to! argument(s) of type bool c Copyright 2009 Norman Ramsey. All Rights Reserved. 1

2 What about + in a function? - fun twice x = x + x; > val twice = fn : int -> int - twice 1; > val it = 2 : int - twice 1.0;! Type clash: expression of type! real! cannot have type! int c Copyright 2009 Norman Ramsey. All Rights Reserved. 2

3 How does ML deal with =? Moscow ML version 2.00 (June 2000) - op =; > val a it = fn : a * a -> bool - 1 = 1; > val it = true : bool = 1.0; > val it = true : bool - false = false; > val it = true : bool - (fn x => x) = (fn x => x);! Type clash: match rule of type a -> b! cannot have equality type c c Copyright 2009 Norman Ramsey. All Rights Reserved. 3

4 What about = in a function? - fun selfeq x = (x = x); > val a selfeq = fn : a -> bool - selfeq 1; > val it = true : bool - selfeq 1.0; > val it = true : bool - selfeq false; > val it = true : bool - selfeq (fn x => x);! Type clash: match rule of type! a -> b! cannot have equality type c c Copyright 2009 Norman Ramsey. All Rights Reserved. 4

5 What is going on here? Ad-hoc polymorphism + can be applied to more than one type But not to every type May use different code on different types Often called overloading, found in C, C++, Ada,... In ML, overloading is second class: user-defined functions not overloaded c Copyright 2009 Norman Ramsey. All Rights Reserved. 5

6 Equality is especially sticky In ML, a specially constrained type variable: eqtype may be instantiated only by type that admits equality Example of generic polymorphism (same code on every type) [owing to run-time cleverness] First-class construct (user code can participate) Totally wrong for abstract types (e.g., sets as lists) c Copyright 2009 Norman Ramsey. All Rights Reserved. 6

7 Three famous problems Overloading most pressing for Arithmetic Equality Printing c Copyright 2009 Norman Ramsey. All Rights Reserved. 7

8 One famous solution: type classes Haskell type class: group of overloaded operators Simplified, (ML-ified) example class Eq by a in Eq requires val op == : a * a -> bool Now == : a.( a in Eq) => a * a -> bool Form of a Haskell type scheme is α 1,...,α n.(c 1,...,c n ) => τ where c i has the form αin Class c Copyright 2009 Norman Ramsey. All Rights Reserved. 8

9 More examples Numbers (simplified, ML-ified) [indentation significant] class Num by a in Num requires a in Eq, val op + : a * a -> a, val op * : a * a -> a, val negate : a -> a fun twice x = x + x Now have + : a.( a in Num) => a * a -> a * : a.( a in Num) => a * a -> a negate : a.( a in Num) => a -> a twice : a.( a in Num) => a -> a c Copyright 2009 Norman Ramsey. All Rights Reserved. 9

10 Still more examples class Show by a in Show requires val show : a -> string Yields show : a.( a in Show) => a -> string Used by Haskell read-eval-print loop c Copyright 2009 Norman Ramsey. All Rights Reserved. 10

11 Putting a type into a class instance bool in Eq by val op == = primeqbool instance int in Eq by val op == = primeqint instance int in Num by val op + = primaddint, val op * = primmulint, val negate = primnegint c Copyright 2009 Norman Ramsey. All Rights Reserved. 11

12 An unbounded number of instances instance a * b in Eq when a in Eq, b in Eq by fun (x, y) == (x, y ) = x == x andalso y == y instance a list in Eq when a in Eq by fun nil == nil = true (x::xs) == (y::ys) = x == y andalso xs == ys _ == _ = false So automatically (int * int) list * bool is in Eq. instance a list in Show when a in Show by fun show [] = "[]" show (x::xs) = "[" ˆ show x ˆ concat (map (fn x => "," ˆ show x) xs) ˆ "]" c Copyright 2009 Norman Ramsey. All Rights Reserved. 12

13 Example: complex numbers datatype a complex = C of a * a instance a complex in Eq when a in Eq by fun (C (r,c)) == (C (r,c )) = r == r andalso c == c instance a complex in Num when a in Num by fun (C (r,c)) + (C (r,c )) = C (r+r, c+c ) fun (C (r,c)) * (C (r,c )) = C (r*r -c*c, r*c +r *c) fun negate (C (r,c)) = C (negate r, negate c) instance a complex in Show when a in Show by show (C (r,c)) = show r ˆ "+" ˆ show c ˆ "i" Qdemo> C (1, 2.0) + C (3, 4) i Qdemo> C (1, 2.0) * C (3, 4) i Qdemo> C (1, 2) * C (3, 4) -5+10i c Copyright 2009 Norman Ramsey. All Rights Reserved. 13

14 Part II: Stupid Cool type-class tricks QuickCheck: random test generator Define functions, properties believed true fun rotate (C (r, c)) = C (-c, r) val proprotate : int complex -> bool = fn c => (rotate o rotate o rotate o rotate) c == c Now run tests automatically: Qdemo> quickcheck proprotate OK, passed 100 tests. c Copyright 2009 Norman Ramsey. All Rights Reserved. 14

15 More QuickCheck val proprev : int list -> bool = fn xs => rev (rev xs) == xs val proprevapp : int list -> int list -> bool = fn xs => fn ys => rev ys) == rev rev ys (*wrong*) Qdemo> quickcheck proprev OK, passed 100 tests. Qdemo> quickcheck proprevapp Falsifiable, after 0 tests: [2] [-2,1] c Copyright 2009 Norman Ramsey. All Rights Reserved. 15

16 What QuickCheck does QuickCheck Figures out # of arguments to each function Generates random test inputs Runs tests Reports bugs All as an ordinary Haskell library How? Type classes! quickcheck : ( a in Testable) => a -> unit c Copyright 2009 Norman Ramsey. All Rights Reserved. 16

17 What is testable? Testable property takes random to Boolean: class Test by a in Test requires val prop : a -> rand -> bool We ll need random (arbitrary) inputs: class Arby by a in Arby requires val arby : rand -> a Test a function (repeat to Curry!): instance ( a -> b) in Test when a in Arby, b in Test by prop f r = prop (f (arby r)) r Testing final result is trivial instance bool in Test by prop b r = b c Copyright 2009 Norman Ramsey. All Rights Reserved. 17

18 QuickCheck has lots of testable types instance int in Arby by arby r = Random.range (-20, 20) r instance a list in Arby when a in Arby by arby r = let val n = Random.range (0, 20) r in List.tabulate (n, fn i => arby r) end So what about proprevapp : int list -> int list -> bool? c Copyright 2009 Norman Ramsey. All Rights Reserved. 18

19 Many uncurried functions are OK too Create arbitrary tuples instance a * b in Arby when a in Arby, b in Arby by arby r = (arby r, arby r) instance a * b * c in Arby when a in Arby, b in Arby, c in Arby by arby r = (arby r, arby r, arby r) (*... and so on... *) c Copyright 2009 Norman Ramsey. All Rights Reserved. 19

20 How is it implemented: Dictionary passing Class declaration for Num becomes type a numdict = { + : a * a -> a, - : a * a -> a, negate : a -> a } Constraint a in Num is satisfied by a witness of type a numdict. c Copyright 2009 Norman Ramsey. All Rights Reserved. 20

21 Finding a witness Instance declaration creates witness, e.g., val intnumdict = { + = primaddint, * = primmulint, negate = primnegint } c Copyright 2009 Norman Ramsey. All Rights Reserved. 21

22 Using the witness Fun negate : ( a in Num) => a -> a becomes fun negate numdict x = # negate numdict x Fun twice : ( a in Num) => a -> a becomes fun twice numdict x = # + numdict (x, x) of type a numdict -> a -> a. c Copyright 2009 Norman Ramsey. All Rights Reserved. 22

23 Use a witness to get a witness fun complexnumdict anumdict = { negate = fn (C (r, c)) => C (negate anumdict r, negate anumdict c), * = fn (C (r, c)) => fn (C (r, c )) => C (op - anumdict (op * anumdict...)),... } c Copyright 2009 Norman Ramsey. All Rights Reserved. 23

24 How it is implemented Extended version of Milner s type inference: Infer types with constraints Simplify constraints using class hierarchy, instance rules Every time a constrained value is instantiated, find a witness using instance rules Finding of witness is a logic program (will do with Prolog) c Copyright 2009 Norman Ramsey. All Rights Reserved. 24

25 Summary of type classes Invented to solve arithmetic, equality, printing Far exceeded inventors expectations Logic programming in the type system Get the compiler to write lots of code for you c Copyright 2009 Norman Ramsey. All Rights Reserved. 25