1. CS321 Functional Programming 2 © JAS 2005 3-1 Type Checking Polymorphism in Haskell is implicit. ie the system can derive the types of all objects. This is different to the philosophy of languages like Pascal where all typing is explicit. The aim in this part of the course is to examine in more detail how the type of any object (function, or expression) can be derived.

2. CS321 Functional Programming 2 © JAS 2005 3-2 The basis for a type checking algorithm was worked out by Robin Milner (who used to be here at Swansea) in the mid 1970s. Much of the theoretical background to the type checking algorithm cames from work on the typed λ-calculus, in particular work done by Roger Hindley (who is still here at Swansea). People refer to the type checking algorithm as the Milner or Hindley-Milner type checking algorithm.

3. CS321 Functional Programming 2 © JAS 2005 3-3 Consider the following function in Haskell:- len lis = if lis == [] then 0 else 1 + len (tail lis) if:: Bool -> a -> a -> a []:: [b] 0, 1:: Int tail:: [c] -> [c] +:: Int -> Int -> Int lis:: d (==):: e -> e -> Bool len:: f

4. CS321 Functional Programming 2 © JAS 2005 3-4 len lis = if (==) lis [] e->e->Boold[b] Bool then 0 Int else (+) 1 len (tail lis) [c]->[c] d f [c] Int->Int->Int Int g Int Bool->Int->Int->Int d Int len::[c]->Int with e=d=[b];d=[c];f=[c]->g;g=Int

5. CS321 Functional Programming 2 © JAS 2005 3-5 Consider another example:- oops fn lis = if lis == [] then [] else (fn(head lis)):fn (tail lis) if:: bool->a->a->a (==):: b->b->bool [] 1 :: [c] [] 2 :: [d] (:) :: e->[e]->[e] head :: [f]->f tail :: [g]->[g] fn :: h lis :: i

6. CS321 Functional Programming 2 © JAS 2005 3-6 oops fn lis = if (==) lis [] b->b->Bool i [c] then [] [d] else (:) (fn(head lis)) (fn (tail lis)) [f]->f i h f [g]->[g] i h [g] b=i=[c];i=[f];i=[g];h=[g]->kh=f->j; f=[g]=i=[f]

7. CS321 Functional Programming 2 © JAS 2005 3-7 A Type Checking Algorithm We have derived type equivalences. This is essentially an example of term unification. This idea can be defined without reference to types. Let us assume that we are given a collection Ops of operators and a collection Vars of variables. The family Terms of terms is then defined:- Terms ::=Vars | (Ops, [Terms,...., Terms]) Thus an "unsugared'' version of the expression:- 1 + x * y is:- (+,[(1,[]),(*,[x,y])])

8. CS321 Functional Programming 2 © JAS 2005 3-8 Any function that assigns terms to variables is called a substitution. Every such function may be extended to act upon terms. If σ is a substitution then Σ is a function Terms -> Terms defined:- Σ(v) = σ(v) forall v є Vars Σ(o,[t 1,..t n ]) = (o,[Σ(t 1 ),..Σ(t n )]) forall є Ops Suppose σ assigns " x+y '' to " x '' and " 0 '' to " y '' then Σ(1+x*y) = 1+(x+y)*0

9. CS321 Functional Programming 2 © JAS 2005 3-9 Two substitutions σ and t may be combined:- τ;σ(v) = (Σ. τ)(v) = Σ(τ(v)) forall v є Vars A unifier for any two terms, t and t', is any substition σ such that Σ(t) = Σ(t'). The most general unifier of two terms is the unifier σ 0 of these terms such that for any other unifier σ 1 there exists a unification τ such that σ 1 = σ 0 ; τ. It can be proved that for any pair of unifiable terms a most general unifier (unique up to renaming of variables) exists. Term unification is the process of finding the most general unifier.

10. CS321 Functional Programming 2 © JAS 2005 3-10 How do we find the most general unifier, mgu, of two terms, t and t‘? Need to consider both variables and operators. If t is a variable that does not occur in t' then mgu(t,t') is the substitution [t'/t] of term t' for t. The reverse is true if t' is a variable that does not exist in t. If t is a variable that does occur in t' there is no unifier unless t=t'. For operators then it is clear that for two terms no unifier can exist unless the operators are equal. If the operators are equal then a unifier can exist iff the argument lists are equal in length and the arguments are pairwise unifiable by the same substitution.

11. CS321 Functional Programming 2 © JAS 2005 3-11 Consider the two terms:- (o,[t1,....,tn]) (o,[t'1,....,t'n]) and let the following substitutions exist σ 1 = mgu (t 1,t' 1 ) σ 2 = mgu (Σ 1 t 2,Σ 1 t' 2 ) σ 3 = mgu (Σ 2 (Σ 1 t 3 ),Σ 2 (Σ 1 t' 3 )).......... σ n = mgu (Σ n-1 (Σ 2 (Σ 1 t n )),Σ n-1 (Σ 2 (Σ 1 t' n ))) then mgu( t, t‘ ) = σ 1 ; σ 2 ;......;σ n

12. CS321 Functional Programming 2 © JAS 2005 3-12 To write a program to perform the type checking we would need a type to represent a type expression: type Tvname = String data Texp = Tvar Tvname | Tcons String [Texp] arrow :: Texp -> Texp -> Texp arrow t1 t2 = Tcons "arrow" [t1,t2] int :: Texp int = Tcons "int" [] cross :: Texp -> Texp -> Texp cross t1 t2 = Tcons "cross" [t1,t2] list :: Texp -> Texp list t = Tcons "list" [t]

13. CS321 Functional Programming 2 © JAS 2005 3-13 A subsitution function maps type variable names to type expressions type Subst = Tvname -> Texp We can extend this to apply to type expressions sub_type :: Subst -> Texp -> Texp sub_type phi (Tvar tvn) = phi tvn sub_type phi (Tcons tcn ts) = Tcons tcn (map (sub_type phi) ts) Composition of substitutions is possible scomp :: Subst -> Subst -> Subst scomp sub2 sub1 tvn = sub_type sub2 (sub1 tvn)

14. CS321 Functional Programming 2 © JAS 2005 3-14 We define an identity substitution id_subst :: Subst id_subst tvn = Tvar tvn such that sub_type id_subst t = t A delta substitution is one that affects only one variable and all other substitutions can be built from the identity substitution by composition on the left with delta substitutions. delta :: Tvname -> Texp -> Subst delta tvn t tvn' = if tvn == tvn' then t else Tvar tvn'

15. CS321 Functional Programming 2 © JAS 2005 3-15 Assume a data type to allows to return success (a Subst ) or failure data Reply a = OK a | FAILURE A unification function will be applied to pairs of Type expressions unify :: Subst->(Texp,Texp)->Reply Subst unify phi ((Tvar tvn),t)= unify phi ((Tcons tcn ts),(Tvar tvn))= unify phi ((Tcons tcn ts),(Tcons tcn' ts')) =

16. CS321 Functional Programming 2 © JAS 2005 3-16 Applying type checking to Haskell Need to extend from simple function abstractions and applications to include (for example) Pattern Matching for each case the type of the pattern(s) must be consistent Guards each guard must be of type Bool Type Classes

17. CS321 Functional Programming 2 © JAS 2005 3-17 Type Checking Classes – An Example member :: Eq a => [a] -> a -> Bool member []y = False member (x:xs)y = (x == y) || member xs y e :: Ord b => [[b]] member e [a]->a->Bool [[b]] [b] -> Bool [a]=[[b]]; (Eq[b],Ord b) (Eq b, Ord b) Ord b Recall instance Eq => Eq [a]… class Eq a => Ord a… Ord b => [b] -> Bool

18. CS321 Functional Programming 2 © JAS 2005 3-18 member [id] Eq a => [a]->a->Bool [a->a] [a->a] not an instance of Eq