Higher-Order Functions

Higher-Order Functions L2HOF

Higher-Order Functions • A function that takes a function as argument and/or returns a function as result is called a higher-order function ora functional. • ML/Scheme treat functions as first-class (primitive) values. • Can be supplied as input to functions. • Not allowed in Ada. • Can be created through expression evaluation. • Not allowed in C++/Java/LISP. • Can be stored in data structures. L2HOF

Nested Functional + Static Scoping fun f x = let val g = fn y => 8 * x + y in g end; val h = f 5; h 2; • Breaks stack-based storage allocation; Requires heap-based storage allocation and garbage collection (Closures) L2HOF

ML function definitions fun elem_to_list [] = [] | elem_to_list (h::t) = [h] :: (elem_to_list t) elem_to_list [“a”] = [[“a”]] fun inc_each_elem [] = [] | inc_each_elem (h::t)= (h+1) :: (inc_each_elem t) inc_each_elem [1,2,3] = [2,3,4] L2HOF

Abstracting Patterns of Recursion fun map f [] = [] | map f (h::t) = (f h)::(map f t) fun elem_to_list x = map (fn x => [x]) x val inc_each_elem = map (fn x => x+1) L2HOF

Functionals : Reusable modules Libraries usually contain functionals that can be customized. sort order list Can be used for: descending_order_sort ascending_order_sort sort_on_length sort_lexicographically sort_on_name sort_on_salary ... L2HOF

Orthogonality • Instead of defining related but separate functions such as remove-if remove-if-not process_till_p process_till_not_p define one generalized functionalcomplement. • Refer to: CommonLISP vs Scheme L2HOF

Currying fun plus (x, y) = x + y fun add x y = x + y plus (5,3) = 8 add 5 3 = 8 add 5 = (fn x => 5+x) • Curried functions are higher-order functions that consume their arguments lazily. • All ML functions are curried ! L2HOF

Significance of Curried Functions • Supports partial evaluation. • Useful for customizing (instantiating) general purpose higher-order functions (generics). • Succinct definitions. • Denotational (Semantics) Specifications. • One-argument functions sufficient. • All ML functions are unary. L2HOF

fun curry f = fn x => (fn y => f(x,y)) fun uncurry f = fn (x,y) => (f x y) curry (uncurry f) = f uncurry (curry f) = f (Higher-order functions) L2HOF

Classification of functions • Primitive + , * , - , etc. • Higher-order • Hierarchical (sort order list),(filter pred list) • Self-applicable map , id (E.g., (map (map f)) ) fun double f = f o f; composition L2HOF

From Spec. to Impl. • Spec: (sqrt x) >= 0 /\ (sqrt x)^2 = x • Computable Spec: (sqrt x) >= 0 /\ abs((sqrt x)^2 - x) < eps • Improving Approximations (Newton’s method): y(n+1) = (y(n) + [x / y(n)]) / 2 L2HOF

fun improve x y = ( y + x / y) / 2.0 ; val eps = 1.0e~5; fun satis x y = abs( y*y - x) < eps ; fun until p f y = if p y then y else until p f (f y) ; fun sqrt x = until (satis x) (improve x) 1.0; L2HOF

ML : Meta Language • Initial Domains of Application • Formal Specification of Languages • Theorem Proving • Theorems are values (terms) of an ADT. • Theorems can only be constructed using the functions supported by the ADT (“no spoofing”). • ML is a mathematically clean modern programming language for the construction of readable and reliable programs. • Executable Specification. L2HOF

Symbolic Values • Recursive Definitions • Higher-Order Functions • Strongly Typed • Static Type Inference • Polymorphic Type System • Automatic Storage Management • Pattern Matching • ADTs; Modules and Functors • Functional + Imperative features L2HOF

fun len [] = 0 | len (x::xs) = 1 + len xs (define (len xs) (if (null? xs) 0 (+ 1 (len (cdr xs))) ) ) L2HOF

Higher-Order Functions