Type Variables in ML

1
Type Variables in ML

• Until we know the type of a value (perhaps
  never!), we use a variable for its type
• Book uses, e.g., t1, tx, tf
• PL literature uses Greek letters
• l x.x a ? a
• (lambda (a b) b) ab ? b
• ML uses 'a, 'b, 'c, ...

gt fun foo(a,b) b val foo fn 'a 'b -gt 'b
2
ML Let's Write...

• sum int list -gt int
• member 'a 'a list -gt bool

Recall fun reverse(nil) nil
reverse(xt) reverse(t) _at_ x
3
ML Local environments using let

• Recall (?) mergesort Divide-and-conquer for
  sorting a list
• Split list into two sorted sublists, then merge
  them merge(1,3,5, 2,4,6) 1,2,3,4,5,6
• So first write merge

4
ML Local environments using let
gt fun merge(nil,M) M merge(L,nil)
L merge(xxs, yys) if (xint) lt
y then xmerge(xs,yys) else
ymerge(xxs,ys) val merge fn int list
int list -gt int list gt merge(1,3,5,
2,4,6) val it 1,2,3,4,5,6 int
list
Now write split
5
gt fun split(nil) (nil,nil) split(a)
(a, nil) split(abcs) let
val (M,N) split(cs) in (aM,
bN) end val split fn 'a list -gt 'a
list 'a list gt split(1,2,3,4,5) val it
(1,3,5,2,4) int list int list
Finally write mergeSort
6
gt fun mergeSort(nil) nil mergeSort(a)
a mergeSort(L) let val (M,N)
split(L) val M mergeSort(M) val
N mergeSort(N) in merge(M,N) end
val mergeSort fn int list -gt int list gt
mergeSort(5,4,3,2,1) val it 1,2,3,4,5
int list
7
ML Polymorphism

• Note that merge is the only function here
  explicitly declaring a type (int)

fun merge(nil,M) M merge(L,nil)
L merge(xxs, yys) if (xint) lt
y then xmerge(xs,yys) else
ymerge(xxs,ys)

• We can factor out the int comparison, and make
  merge and mergeSort polymorphic

8
ML Polymorphism
gt fun merge(nil,M,C) M ( C is comparator fun
) merge(L,nil,C) L merge(xxs,
yys,C) if C (x,y) then
xmerge(xs,yys,C) else
ymerge(xxs,ys,C) val merge fn 'a list
'a list ('a 'a -gt bool)
-gt 'a list
9
ML Polymorphism
gt fun mergeSort(nil,C) nil
mergeSort(a,C) a mergeSort(L,C)
let val (M,N) split(L) val M
mergeSort(M,C) val N mergeSort(N,C)
in merge(M,N,C) end val mergeSort fn
'a list ('a 'a -gt bool)
-gt 'a list
10
ML Polymorphism
gt fun intcmp(aint, b) a lt b val intcmp fn
int int -gt bool gt fun realcmp(areal, b) a
lt b val realcmp fn real real -gt bool gt
mergeSort(5,4,3,2,1, intcmp) val it
1,2,3,4,5 int list gt mergeSort(5.5, 4.4,
3.3, 2.2, 1.1, realcmp) val it
1.1,2.2,3.3,4.4,5.5 real list
11
ML Polymorphism

• Q. What's the point?
• A. ML supports functions as first -class
  objects (as in Scheme), while also providing
  strong type-checking (as in Java)
• Java supports this ability (abstract comparison)
  through the Comparable interface.