Functional Programming

Functional Programming

Organization of Programming Languages-Cheng (Fall 2004)

Big Picture What we’ve learned so far:

Imperative Programming LanguagesVariables, binding, scoping, reference environment,

etc What’s next:

Functional Programming LanguagesSemantics of Programming LanguagesControl FlowData TypesLogic ProgrammingSubroutines, Procedures: Control AbstractionData Abstraction and Object-Orientation


Design of Programming Languages

Design of imperative languages is based directly on the von Neumann architectureEfficiency is the primary concern, rather than

the suitability of the language for software development

Other designs:Functional programming languages

Based on mathematical functions

Logic programming languagesBased on formal logic (specifically, predicate

calculus)


Design View of a Program

If we ignore the details of computation (“how” of computation) and focus on the result being computed (“what” of computation)A program becomes a “black box” for obtaining

output from input

ProgramInput Output


Functional Programming Language

Functional programming is a style of programming in which the primary method of computation is the application of functions to arguments

f(x)x y

Argument or variable

Function Output


Historical Background

1940s:

Alonzo Church and Haskell Curry developed the lambda calculus, a simple but powerful mathematical theory of functions.

6



1960s:

John McCarthy developed Lisp, the first functional language. Some influences from the lambda calculus, but still retained variable assignments.

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1978:

John Backus publishes award winning article on FP, a functional language that emphasizes higher-order functions and calculating with programs.

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Mid 1970s:

Robin Milner develops ML, the first of the modern functional languages, which introduced type inference and polymorphic types.

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Late 1970s - 1980s:

David Turner develops a number of lazy functional languages leading up to Miranda, a commercial product.

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1988:

A committee of prominent researchers publishes the first definition of Haskell, a standard lazy functional language.

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1999:

The committee publishes the definition of Haskell 98, providing a long-awaited stable version of the language.

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Basic Idea of Functional Programming

Think of programs, procedures and functions as being represented by the mathematical concept of a function

A mathematical function: {elts in domain} → {elts in range} Example: sin(x)

domain set: set of all real numbers range set: set of real numbers from –1 to +1

In mathematics, variables always represent actual values

There is no such thing as a variable that models a memory location

Since there is no concept of memory location or l-value of variables, a statement such as (x = x + 1) makes no sense

A mathematical function defines a value, rather than specifying a sequence of operations on values in memory


Imperative Programming

Summing the integers 1 to 10 in Java:

total = 0;

for (i = 1; i 10; ++i)

total = total + i;

The computation method is variable assignment

Each variable is associated with a memory location 14



Summing the integers 1 to 10 in Haskell:

sum [1..10]

The computation method is function application.

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Function definition vs function applicationFunction definition

Declaration describing how a function is to be computed using formal parameters

Function applicationA call to a declared function using actual

parameters or arguments

In mathematics, a distinction is often not clearly made between a variable and a parameterThe term independent variable is often used

for both actual and formal parameters


Functional Programming Language

The objective of the design of a FPL is to mimic mathematical functions to the greatest extent possibleImperative language: operations are completed

and results are stored in variables for later useManagement of variables is source of complexity for

imperative programming

FPL: variables are not necessary Without variables, iterative constructs are

impossibleRepetition must be provided by recursion

rather than iterative constructs


Example: GCD

Find the greatest common divisor (GCD) between 12 and 30Divisors for 12: 1, 2, 3, 4, 6, 12Divisors for 30: 1, 2, 3, 5, 6, 10, 15, 30

GCD(12, 30) is equal to 6

Algorithm:u=30, v=12 30 mod 12 = 6u=12, v=6 12 mod 6 = 0u=6, v=0 since v=0, GCD=u=6


Computing GCDint gcd(int u, int v){

int temp;while (v != 0) {

temp = v;v = u % v;u = temp;

}return u;

}

* Computing GCD using iterative construct

int gcd(int u, int v){

if (v==0)return u;

elsereturn gcd(v, u %

v);}

* Computing GCD using recursion


Fundamentals of FPLs Referential transparency:

In FPL, the evaluation of a function always produces the same result given the same parameters

E.g., a function rand, which returns a (pseudo) random value, cannot be referentially transparent since it depends on the state of the machine (and previous calls to itself)

Most functional languages are implemented with interpreters


Using Imperative PLs to support FPLs

LimitationsImperative languages often have restrictions

on the types of values returned by the functionExample: in Fortran, only scalar values can be

returnedMost of them cannot return functionsThis limits the kinds of functional forms that can be

provided

Functional side-effectsOccurs when the function changes one of its

parameters or a global variable


Characteristics of Pure FPL Pure FP languages tend to

Have no side-effectsHave no assignment statementsOften have no variables!Be built on a small, concise frameworkHave a simple, uniform syntaxBe implemented via interpreters rather than

compilers


Functional Programming Languages

FPL provides a set of primitive functions, a set of functional forms to construct complex

functions, a function application operation, some structure to represent data

In functional programming, functions are viewed as values themselves, which can be computed by other functions and can be parameters to other functionsFunctions are first-class values


Lambda Expressions Function definitions are often written as a function

name, followed by list of parameters and mapping expression:Example: cube (x) x * x * x

A lambda expression provides a method for defining nameless functions

(x) x * x * xNameless functions are useful, e.g., functions

that are produced for intermediate application to a parameter list do not need a name, for it is applied only at the point of its construction


Lambda Expressions

Function applicationLambda expressions are applied to

parameter(s) by placing the parameter(s) after the expression

e.g. ((x) x * x * x)(3) evaluates to 27

Lambda expressions can have more than one parameter

e.g. (x, y) x / y


Functional Forms A functional form, is a function that either

takes functions as parameters, yields a function as its result, or both

Also known as higher-order function

Function Composition: A functional form that takes two functions as parameters and

yields a function whose result is a function whose value is the first actual parameter function applied to the result of the application of the second

Example: h f ° g

which means h (x) f ( g ( x))

If f(x) x + 2, g(x) 3 * x, then h(x) (3 * x) + 2


Functional Forms Construction: A functional form that takes a list of

functions as parameters and yields a list of the results of applying each of its parameter functions to a given parameter

For f (x) x * x * x and g (x) x + 3,

[f, g] (4) yields (64, 7)

Apply-to-all: A functional form that takes a single function as a parameter and yields a list of values obtained by applying the given function to each element of a list of parameters

For h (x) x * x * x,

( h, (3, 2, 4)) yields (27, 8, 64)


LISP Stands for LISt Processing

Originally, LISP was a typeless language and used for symbolic processing in AI

Two data types: atoms (consists of symbols and numbers) lists (that consist of atoms and nested lists)

LISP lists are stored internally as single-linked lists Lambda notation is used to specify nameless functions

(function_name (LAMBDA (arg_1 … arg_n) expression))


LISP Function definitions, function applications, and data all

have the same form Example:

If the list (A B C) is interpreted as data, it is a simple list of three atoms, A, B, and C

If it is interpreted as a function application, it means that the function named A is applied to the two parameters, B and C

This uniformity of data and programs gives functional languages their flexibility and expressive power programs can be manipulated dynamically


LISP The first LISP interpreter appeared only as a demonstration of the

universality of the computational capabilities of the notation

Example (early Lisp):(defun fact (n) (cond ((lessp n 2) 1)(T (times n (fact (sub1 n))))))

The original intent was to have a notation for LISP programs that would be as close to Fortran’s as possible, with additions when necessary This notation was called M-notation, for meta-notation

McCarthy believed that list processing could be used to study computability, which at the time was usually studied using Turing machines McCarthy thought that symbolic lists was a more natural model of

computation than Turing machines Common Lisp is the ANSI standard Lisp specification All LISP structures, including code and data, have uniform structure

Called S-expressions


LISP

EVAL: a universal function that could evaluate any other functions

LISP 1.5:Awkward—though elegantly uniform—syntax.Dynamic scope rule Inconsistent treatment of functions as

arguments—because of dynamic scoping


Scheme A mid-1970s dialect of LISP, designed to be

cleaner, more modern, and simpler version than the contemporary dialects of LISP

Uses only static scoping

Functions are first-class entitiesThey can be the values of expressions and

elements of listsThey can be assigned to variables and passed

as parameters


Scheme Arithmetic: +, -, *, /, ABS, SQRT

(+ 5 2) yields 7

Most interpreters use prefix notations:(+ 3 5 7) means add 3, 5, and 7(+ (* 3 5) (- 10 6))Advantages:

Operator/function can take arbitrary number of arguments

No ambiguity, because operator is always the leftmost element and the entire combination is delimited by parentheses


Scheme QUOTE

takes one parameter; returns the parameter without evaluation

QUOTE is required because the Scheme interpreter, named EVAL, always evaluates parameters to function applications before applying the function.

QUOTE is used to avoid parameter valuation when it is not appropriate

QUOTE can be abbreviated with the apostrophe prefix operator

'(A B) is equivalent to (QUOTE (A B)) (list a b c) versus (list 'a 'b 'c)


Scheme LIST takes any number of parameters; returns a list with the

parameters as elements

CAR takes a list parameter; returns the first element of that list (CAR '(A B C)) yields A (CAR '((A B) C D)) yields (A B)

CDR takes a list parameter; returns the list after removing its first element (CDR '(A B C)) yields (B C) (CDR '((A B) C D)) yields (C D)

CONS returns a new list that includes the first parameter as its first element and the second parameter as remainder of its result (CONS ’(A D) '(B C)) returns ((A D) B C)

-


Scheme: Predicate Functions #T and () are true and false respectively EQ? returns #T if both parameters are atoms and are the same

(EQ? 'A 'A) yields #T (EQ? 'A '(A B)) yields () Note that if EQ? is called with list parameters, the result is not

reliable Also, EQ? does not work for numeric atoms

LIST? returns #T if the parameter is a list; otherwise ()

NULL? returns #T if the parameter is an empty list; otherwise () Note that NULL? returns #T if the parameter is ()

Numeric Predicate Functions=, <>, >, <, >=, <=, EVEN?, ODD?, ZERO?


Lambda Expressions

Form is based on notation(LAMBDA (L) (CAR (CDR L)))L is called a bound variableLambda expressions can be applied

((LAMBDA(L) (CAR (CDR L))) '((A B) C D))


Constructing Functions

DEFINE - Two forms:To bind a symbol to an expression

(DEFINE pi 3.141593)(DEFINE two_pi (* 2 pi))

To bind names to lambda expressions(DEFINE (cube x) (* x x x)) then you can use: (cube x)


Evaluation Process

Evaluation process (for normal functions): Parameters are evaluated, in no particular

orderThe values of the parameters are substituted

into the function bodyThe function body is evaluatedThe value of the last expression in the body is

the value of the function


Control Flow

Selection- the special form, IF(IF predicate then_exp else_exp)

(IF (<> count 0) (/ sum count) 0 )

(DEFINE (factorial n)

( IF ( = n 0) 1 (* n (factorial(- n 1) ) )

)

)


Control Flow

Multiple Selection - the special form, COND

(COND (predicate_1 expr {expr}) (predicate_1 expr {expr}) ... (predicate_1 expr {expr}) (ELSE expr {expr}) )

Returns the value of the last expr in the first pair whose predicate evaluates to true


Example Scheme Function

member takes an atom and a list; returns #T if atom is in list; () otherwise

(DEFINE (member atm lis)

(COND

((NULL? lis) '())

((EQ? atm (CAR lis)) #T)

((ELSE (member atm (CDR lis)))

))



equalsimp returns #T if the two simple lists are equal; ()

otherwise

(DEFINE (equalsimp lis1 lis2)

(COND

((NULL? lis1) (NULL? lis2))

((NULL? lis2) '())

((EQ? (CAR lis1) (CAR lis2))

(equalsimp (CDR lis1) (CDR lis2)))

(ELSE '())

))



equalreturns #T if the two general lists are equal; ()

otherwise

(DEFINE (equal lis1 lis2) (COND

((NOT (LIST? lis1)) (EQ? lis1 lis2))

((NOT (LIST? lis2)) '())

((NULL? lis1) (NULL? lis2))

((NULL? lis2) '())

((equal (CAR lis1) (CAR lis2))

(equal (CDR lis1) (CDR lis2)))

(ELSE '())

))



Append takes two lists as parameters; returns the first

parameter list with the elements of the second parameter list appended at the end

(DEFINE (append lis1 lis2)

(COND

((NULL? lis1) lis2)

(ELSE (CONS (CAR lis1)

(append (CDR lis1) lis2)))

))


Scheme Functional Forms Composition

The previous examples have used it Apply to All

one form in Scheme is mapcar Applies the given function to all elements of the

given list; result is a list of the results

(DEFINE mapcar fun lis)

(COND

((NULL? lis) '())

(ELSE (CONS (fun (CAR lis))

(mapcar fun (CDR lis))))

))


EVAL Scheme interpreter itself is a function named EVAL

The EVAL function can also be called directly by Scheme programs It is possible for Scheme to create expressions and

calls EVAL to evaluate them

Example: adding a list of numbers Cannot apply “+” directly on a list because “+” takes

any number of numeric atoms as arguments (+ 1 2 3 4) is OK (+ (1 2 3 4)) is not


EVAL One approach:

((DEFINE (adder lis) (COND ((NULL? lis) 0) (ELSE (+ (CAR lis) (adder (CDR lis))))

Another approach: ((DEFINE (adder lis) (COND ((NULL? lis) 0) (ELSE (EVAL (CONS '+ lis))) ))

(adder ‘(3 4 6)) causes adder to build the list (+ 3 4 6) The list is then submitted to EVAL, which invokes +

and returns the result 13


Scheme: Output Functions

Output Utility Functions:(DISPLAY expression)

Example: (DISPLAY (car ‘(A B C)))

(NEWLINE)

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Functional Programming