Functional Programming Basics Correctness Clarity Efficiency cs 776

Functional Programming Basics Correctness > Clarity > Efficiency cs 776 (Prasad) L 1 FP 1

• Function Definition • Equations ; Recursion • Higher-order functions • Function Application • Computation by expression evaluation • Choices : parameter passing 8 Reliability 8 Types 8 Strong typing, Polymorphism, ADTs. 8 Garbage Collection cs 776 (Prasad) L 1 FP 2

Imperative Style vs Functional Style • Imperative programs – Description of WHAT is to be computed is inter-twined with HOW it is to be computed. – The latter involves organization of data and the sequencing of instructions. • Functional Programs – Separates WHAT from HOW. – The former is programmer’s responsibility; the latter is interpreter’s/compiler’s responsibility. cs 776 (Prasad) L 1 FP 3

• Functional Style – Value to be computed: a+b+c • Imperative Style – Recipe for computing the value • Intermediate Code – T : = a + b; T : = T + c; – T : = b + c; T : = a + T; • Accumulator Machine – Load a; Add b; Add c; • Stack Machine – Push a; Push b; cs 776 (Prasad) Add; L 1 FP Push c; Add; 4

GCD : functional vs imperative fun gcd(m, n) = if m=0 then n else gcd(n mod m, m); function gcd(m, n: int) : int; var pm: int; begin while m<>0 do begin pm : = m; m : = n mod m; end; return n end; cs 776 (Prasad) L 1 FP n : = pm 5

Pitfalls : Sequencing (define (factorial n) (define (iter prod counter) (if (> counter n) prod (iter (* counter prod) (+ counter 1) ) )) (iter 1 1) ) cs 776 (Prasad) L 1 FP 6

(define (factorial n) (let ((prod 1)(counter 1)) (define (iter) (if (> counter n) prod (begin (set! prod (* counter prod)) (set! counter (+ 1 counter)) (iter)) )) )) cs 776 (Prasad) L 1 FP 7

Function • A function f from domain A to co-domain B, denoted f : A -> B, B is a map that associates with every element a in A, a unique element b in B, denoted f(a). – Cf. Relation, multi-valued function, partial function, … – In mathematics, the term “function” usually refers to a total function; in computer science, the term “function” usually refers to a partial function. cs 776 (Prasad) L 1 FP 8

Representation of functions • Intensional : Rule of calculation fun double n = 2 * n; fun double n = n + n; • Extensional : Behavioral (Table) • Equality: f = g iff for all x: f(x) = g(x) cs 776 (Prasad) L 1 FP 9

Expression Evaluation : Reduction fun double x = x + x; double ( 3 * 2) double(6) 6+6 (3*2) + o 6 + (3 * 2) 6 + o Applicative-Order Normal-Order (call by value) (call by name) cs 776 (Prasad) L 1 FP Lazy (call by need) 10

Role of variable • In functional style, a variable stands for an arbitrary value, and is used to abbreviate an infinite collection of equations. • In imperative style, a variable is a location that can hold a value, and which can be changed through an assignment. x : = x + 1; 0+0=0 0+1=1 … for all x : 0 + x = x cs 776 (Prasad) � Functional variable can be viewed as assign-only- once imperative variable. L 1 FP 11

Referential Transparency • The only thing that matters about an expression is its value, and any subexpression can be replaced by any other expression equal in value. • The value of an expression is independent of its position only provided we remain within the scopes of the definitions which apply to the names occurring in the expression. cs 776 (Prasad) L 1 FP 12

Examples let x = 5 in x + let x = 4 in x + x; val y = 2; val y = 6; var x : int; begin x : = x + 2; end; address of x cs 776 (Prasad) x : = x + 1; value stored in location for x L 1 FP 13

(x=2) / (x+y>2) (2+y>2) vs fun f (x : int) : int ; begin y : = y + 1; return ( x + y) end; (y=0) / (z=0) / (f(y)=f(z)) = false (y=0) / (z=0) / (f(z)=f(z)) =/= (y=0) / (z=0) / (f(z)=1) cs 776 (Prasad) L 1 FP 14

• Common sub-expression elimination is an “incorrect optimization” without referential transparency. – In functional style: E + E = let x = E in x + x – In imperative style: return (x++ + x++) =/= y : = x++; return (y + y) • Parallel evaluation of sub-expressions possible with referential transparency. cs 776 (Prasad) L 1 FP 15

Strict vs Non-strict • A function is strict if it returns welldefined results only when the inputs are well-defined. – E. g. , In C, “+” and “*” are strict, while “&&” and “||” are not. – E. g. , In Ada, “and” and “or” are strict, while “and then” and “or else” are not. – E. g. , constant functions are non-strict if called by name, but are strict if called by value. cs 776 (Prasad) L 1 FP 16

Benefits of Programming in a Functional Language • Convenient to code symbolic computations and list processing applications. • Automatic storage management • Improves program reliability. • Enhances programmer productivity. • Abstraction through higher-order functions and polymorphism. • Facilitates code reuse. • Ease of prototyping using interactive development environments. cs 776 (Prasad) L 1 FP 17

Summary Programming Languages Imperative Functional C, Pascal Logic Prolog Dynamically Typed (Meta-programming) Statically Typed (Type Inference/Reliable) LISP, Scheme Lazy Eval / Pure Haskell cs 776 (Prasad) L 1 FP Eager Eval / Impure SML 18