General Computer Science 320201 GenCS I & II Lecture ... - Kwarc

- identity(5);
val it = 5 : int
Example 97 returning functions: (again, functions are normal values.)
- val constantly = fn k => (fn a => k);
- (constantly 4) 5;
val it = 4 : int
- fun constantly k a = k;
c○: Michael Kohlhase 68
One of the neat uses of higher-order function is that it is possible to re-interpret binary functions as
unary ones using a technique called “Currying” after the Logician Haskell Brooks Curry (∗(1900),
†(1982)). Of course we can extend this to higher arities as well. So in theory we can consider
n-ary functions as syntactic sugar for suitable higher-order functions.
Cartesian and Cascaded Procedures
We have not been able to treat binary, ternary,. . . procedures directly
Workaround 1: Make use of (Cartesian) products (unary functions on tuples)
Example 98 +: Z × Z → Z with +(〈3, 2〉) instead of +(3, 2)
fun cartesian_plus (x:int,y:int) = x + y;
cartesian_plus : int * int -> int
Workaround 2: Make use of functions as results
Example 99 +: Z → Z → Z with +(3)(2) instead of +(3, 2).
fun cascaded_plus (x:int) = (fn y:int => x + y);
cascaded_plus : int -> (int -> int)
Note: cascaded_plus can be applied to only one argument: cascaded_plus 1 is the function
(fn y:int => 1 + y), which increments its argument.
c○: Michael Kohlhase 69
SML allows both Cartesian- and cascaded functions, since we sometimes want functions to be
flexible in function arities to enable reuse, but sometimes we want rigid arities for functions as
this helps find programming errors.
Cartesian and Cascaded Procedures (Brackets)
Definition 100 Call a procedure Cartesian, iff the argument type is a product type, call it
cascaded, iff the result type is a function type.
Example 101 the following function is both Cartesian and cascading
- fun both_plus (x:int,y:int) = fn (z:int) => x + y + z;
val both_plus (int * int) -> (int -> int)
Convenient: Bracket elision conventions
e1 e2 e3 (e1 e2) e3 5 (procedure application associates to the left)
τ1 → τ2 → τ3 τ1 → (τ2 → τ3) (function types associate to the right)
SML uses these elision rules
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