Functional Programming
Chapter 14
History of Functional Languages
• Lisp is the second oldest language
• Motivated by a need to do symbolic, rather
than numerical, manipulations.
• Common LISP and Scheme are the two
most popular dialects of the traditional
LISP.
• Haskell is a modern functional language
that adds strong typing to the more usual
features.
History/Characteristics
• LISt Processing
• The linked list is the major data structure
• Both programs and data can be
represented this way
– Programs are able to manipulate code just
like any other data
– Learning programs, …
Statement of Purpose
• “The Lisp language is primarily for
symbolic data processing. It has been
used for symbolic calculations in
differential and integral calculus, electrical
circuit design, mathematic logic, game
playing, and other fields of artificial
intelligence.”, 1965, from the LISP 1.5
Programmer’s Manual
Background: Functions
• A function f from a set X (the domain) to a
set Y (the range) maps each element x in
X to a unique element y in Y.
• For example, f(x) = x2 maps the set of real
numbers into the set of positive real
numbers (ignoring imaginary numbers).
– i.e., the domain X is the set of all real
numbers.
Background: Functional
Composition
• If f is a function from X to Y and g is a function from
Y to Z, then (g ◦ f) (x) is a function from X to Z
defined as
(g ◦ f) (x) = g(f(x)), for all x in X
Simply put, it means to replace the x in g(x) with f(x).
• Example:
– f(x) = x2 + x
– g(x) = 2x + 1
– g ◦ f = 2(x2 + x) + 1
Overview
• Functional programming mirrors
mathematical functions:
– domain = input, range = output
– A computation maps inputs to outputs
• Variables are mathematical symbols; not
associated with memory locations.
– Compare to imperative programming
Overview
• Pure functional programming is state-free:
no assignment statements.
• Programs in pure functional languages
consist of composite functions; output of
each function is input to another.
• Today, most functional languages have
some imperative statements.
14.1 Functions and the Lambda
Calculus
• Typical mathematical function: Square(n) = n2
– funcName(args) = func-definition: an
expression
• Square : R  R maps from reals to reals
– domain and range are the set of real numbers
• A function is total if it is defined for all values
of its domain. Otherwise, it is partial. e.g.,
Square is total.
Referential Transparency
• Referential transparency: a function’s result
depends only upon the values of its arguments
and not on any previous computation or the
order of evaluation for its arguments.
• This is in contrast to the view of functions in
imperative languages, where function values are
based on arguments, order of evaluation, and
can also have side effects (change state).
– Pure functional languages have no state in the sense
of state in imperative programs & so no side effects.
Referential Transparency
• In imperative programming a statement such as
x = x + y;
means “add the value of in memory cell x to the
value in memory cell y and replace the value of x
by the value of the function”
– The name x is used in two different ways here;
lhs (a memory address) versus rhs (a value)
• Functional programs accept and return data, but
have no side effects and do not have the
concept of a variable as a memory address
Lambda Calculus
• The lambda calculus, invented in 1941, is
the foundation of functional programming
• It “…provides simple semantics for
computation with functions so that
properties of functional computation can
be studied.”
http://www.answers.com/topic/lambda-calculus#Motivation
Fundamental Concepts
– Functions don’t have to have a name:
• e.g., (  x . x*x ) is the Square function. Relate to
mathematical notation: f(x) = x*x
• ((  x . x*x )2): apply the function to a value, similar
to f(x) = x2 for x = 2
– Argument names are arbitrary
• (  x . x*x ) same as (  y . y*y )
– A function that has 2 arguments can be rewritten as a function with 1 argument which
produces another function with only one
argument.
• ( x.( y. x*x + y* y))
Lambda Calculus
• Is a general model of computation
– Can express anything that can be expressed by a
Turing machine; in other words, is Turing Complete
• An imperative language works by changing the
data store; a functional language works by
applying functions and returning values
– Lambda calculus models the latter approach
– Some practitioners claim that it is harder to make
mistakes while using this style of programming.
Status of Functions
• In imperative and OO programming,
functions have different (lower) status than
variables.
• In functional programming, functions have
same status as variables; they are first-class
entities.
– They can be passed as arguments in a
call.
– They can transform other functions.
Example
• A function that operates on other functions is
called a functional form. e.g.,
g(f, [x1, x2, … ]) = [f(x1), f(x2), …], becomes
g(Square, [2, 3, 5]) = [4, 9, 25]
• Meaning: function g takes as parameters
another function and a list (sequence) of
values to which the parameter function is
applied.
Programs as Lists
• Lists are written with white-spaceseparated elements enclosed in
parentheses
• Expressions are represented as lists
• Lists are treated as functions with the first
element being the operation and the
remaining elements the arguments.
14.2 Scheme
• A derivative of Lisp
• Our subset:
– omits assignments
– simulates looping via recursion
– simulates blocks via functional composition
• Scheme is Turing complete, even without
while statements – recursion is used for
repetition.
14.2.1 Expressions
• Cambridge prefix notation for all Scheme
expressions:
(f x1 x2 … xn)
•
•
•
•
•
e.g.,
(+ 2 2)
; evaluates to 4
(+(* 5 4 2)(-6 2));means 5*4*2+(6-2)
Note: Scheme comments begin with ;
Cambridge prefix allows operators to have an
arbitrary number of arguments.
The define Function
• Used to define global “variables”; e.g.
– (define f 120)
– define changes its environment but is not
equivalent to an assignment statement. It just
gives a name to a value.
• define can also be used for other
purposes, as we will see.
• setQ and setF are functions that
operate more like assignments; also set!
14.2.2 Expression Evaluation
• Three steps:
– replace names of symbols by their current
bindings.
– Evaluate lists as function calls in Cambridge
prefix, where a list is a set of elements
enclosed in ( ); e.g., (* a 2)
• The first element in the list is always treated as the
function unless you specifically say not to.
– Constants evaluate to themselves.
Examples
5
; evaluates to 5
Given the global definition (define f 120) then
f
; evaluates to 120
(+ f 5)
; evaluates to 125
(+ 5 2 9 13)
; evaluates to 29
#f
; predefined, false
#t
; predefined, true
5, #f, #t are constants, f is a bound
symbol
Lists as Function Calls
• (+ 5 2 9 13) ; evaluates to 29
but
• (+ (5 2 9 13)); an error
Scheme will try to evaluate the second list,
interpreting “5” as a function
• (f); error - f isn’t a function
Preventing Evaluation
(define colors (quote (red
yellow green)))
or
(define colors (‘ (red yellow
green)))
Quoting tells Scheme/LISP that the following
list is not to be evaluated.
Quoted Lists
(define x f)
; defines x as 120 (value of f)
(define x ‘f)
; defines x as the symbol f
(define color ‘red)
; color is defined to be red
(define color red)
; error: no definition of red
14.2.3 Lists
• A list is a series of expressions enclosed in
parentheses.
– Lists represent both functions and data.
– The empty list is written ().
– e.g., (0 2 4 6 8) is
a list of even
numbers.
– Here’s how it’s
stored:
List Node Structure
• Each list node is a record with two fields:
the car and the cdr: car is (a pointer to) the
first field, cdr is (a pointer to) the second.
• Note that in the previous example the cdr
of the last node ( |8|nil| ) is nil.
– This is equivalent to the null pointer in C++
• The nil value can be represented as ( ),
which is also the representation for an
empty list.
“Proper” Lists & Dotted Lists
• Proper lists are assumed to end with the
value ( ) which is implemented by null
reference
• In a dotted list the last cons has some value
other than nil as the value of the cdr field.
– “Dotted” lists are written (0 2 4 6 . 8) The last
node in the previous list would have been |6|8|
Structure of a
List in Scheme
(a)
Figure 14.1
(b)
Notice the difference
between the list with
nil as the last entry,
and the “dotted” list
shown in part b.
List Elements Can Be Lists Themselves
This represents
the list (a (b c))
a
nil
b
c
nil
Lists as Binary Trees
• Because each node in a Scheme list
consists of two pointers, internally the lists
are similar to binary trees.
• Recall;
– The links (pointers) are called the 'car' and the
'cdr'.
– A list node (cell) is also called a cons or a
pair.
List Functions
• car: returns the first element (“head”) of a
list
• cdr: returns the tail of the list, which is
itself a list
List Transforming Functions
• Suppose we write
(define evens ‘(0 2 4 6 8)).
Then:
(car evens)
; gives 0
(cdr evens)
; gives (2 4 6 8)
( (null? ‘()) ; gives #t, or true
(cdr(cdr evens)); (4 6 8)
(car‘(6 8)) ; 6 (quoted to stop eval)
List Functions
• cons: used to build lists
– Requires two arguments: an element and a list;
e.g.,
– (cons 8 ( )) ; gives the 1-element list (8)
– (cons 6 (cons 8( ))) ; gives the list (6 8)
– (cons 6 ‘(8)) ; also gives the list (6 8)
– (cons 4(cons 8 9)) ; gives the dotted list
; (4 8 . 9 ) since 9 is not a
; list
List Building Functions
• list takes an arbitrary number of arguments and
creates a list; append takes two lists and
concatenates them:
– (append ‘(1 3 5) evens)
;gives(1 3 5 0 2 4 6 8)
– (append evens (list 3)
; gives (0 2 4 6 8 3)
– (list ‘(1 3 5) evens)
; gives ((1 3 5) (0 2 4 6 8))
– (list 1 2 3 4) ; gives (1 2 3 4)
– (list ‘evens ‘odds) ;gives (evens odds)
Equal
• The function equal? returns true if the
two objects have the same structure and
content.
(equal? 5 ‘(5)) ; gives #f, or false
(equal? 5 5)
; #t
(equal? ‘(1 2 3) ‘(1 (2 3)))
; returns false
14.2.4 – Elementary Values
• Numbers: integers, rationals, and floating point
numbers
• Characters
• Functions
• Symbols
• Strings
• Boolean values #t and #f
• All values except #f and ( ) are interpreted as
true when used as a predicate.
Control Flow
(if test then-part)
(if test then-part else-part)
Example:
(if (< x 0) (- 0 x)) ;returns –x if x<0
(if (< x y) x y)
; returns the smaller of x, y
Control Flow
• The case statement is similar to a Java or
C++ switch statement:
• (case month
(( sep apr jun nov) 30)
((feb) 28)
(else 31) ; optional
)
• All cases take an unquoted list of
constants, except for the else.
Defining Functions
• define is also used to define functions;
according to the following syntax:
(define name (lambda (arguments) body))
or
(define (name arguments) body)
• From the former, one can see that Scheme
is an applied lambda calculus.
Examples
• (define (min x y) (if (< x y) x y))
to return the minimum of x and y
• (define (abs x)(if(< x 0)(- 0 x) x))
to find the absolute value of x
Recursive functions
• (define (factorial n)
(if(< n 1) 1 (* n factorial(-n 1)))
))
• (define (fun1 alist)
(if (null? alist) 0
(+ (car alist)(fun1(cdr alist)))
))
– Suppose ‘alist’ is replaced by ‘evens’
Built-in Functions
• length: returns the number of elements in
a list
– (length ‘(1 2 3 4)) ; returns 4
– (length ‘((1 2 3) (5 6 7) 8)) ; returns 3
• (define (length alist)
(if (null? alist) 0 (+ 1 (length (cdr alist)))
))
Built-in Functions
• member: tests whether an element is in a
list
(member 4 evens);returns (4 6 8)
(member 1 ‘(4 2)); returns ()
• If the element is in the list, return portion of
the list starting with that element, else
return the empty list.
Functions as Parameters
• The mapcar function takes two
parameters: a function fun and a list
alist.
• mapcar applies the function to the list of
values, one at a time, and returns a list of
the results.
Functions as Parameters
(define (mapcar fun alist)
(if (null? alist) ‘()
(cons (fun (car alist))
(mapcar fun (cdr alist)))
))
E.g., if (define (square x) (* x x)) then
(mapcar square ‘(2 3 5 7 9))
returns
(4 9 25 49 81)
14.2.9 Symbolic Differentiation
• Symbolic Differentiation Rules
• Fig 14.2 d (c)  0
c is a constant
dx
d
(x)  1
dx
d
du dv
(u  v)  
u and v are funct ions ofx
dx
dx dx
d
du dv
(u  v)  
dx
dx dx
d
dv
du
(uv)  u  v
dx
dx
dx
 du
d
dv  2
(u /v)  v  u /v
 dx
dx
dx 
(define (diff x expr)
(if (not (list? expr))
(if (equal? x expr) 1 0)
(let ((u (cadr expr)) (v (caddr expr)))
(case (car expr)
((+) (list ‘+ (diff x u) (diff x v)))
((-) (list ‘- (diff x u) (diff x v)))
((*) (list ‘+ (list ‘* u (diff x v))
(list ‘* v (diff x u))))
((/) (list ‘div (list ‘- (list ‘* v (diff x u))
(list ‘* u (diff x v)))
(list ‘* u v)))
))))
Implementation of LISP
and its Dialects
• Usually implemented as an interpreted
language, although there are compiled forms.
• A LISP interpreter generally consists of:
– a memory manager
– collection of core functions implemented in compiled
form for speed
– a set of support functions implemented with the core
Lisp is an Expression-Oriented
Language.
• Unlike most other languages, no distinction is
made between "expressions" and "statements"
• Code and data are written as expressions.
• When an expression is evaluated, it produces a
value (or list of values), which then can be
embedded into other expressions.
Using Scheme
• In Scheme, processing proceeds by typing
expressions into the Scheme
environment. After each expression is
entered, Scheme
– reads the expression,
– evaluates the expression to determine its
value, and
– prints the resulting value.
– This read-eval-print cycle forms the basis for
all processing within Scheme.
Memory Management in LISP
• Memory is organized into cells (or nodes).
• all cells are the same size, usually 2 words of
memory
• cells form a large pool of available storage,
chained together into the initial free list.
• cells are taken off the free list as needed to
build atoms, lists, etc.
Memory Management in Lisp
• Literal atoms may require several cells to
hold their definition (Remember the list
diagram)
• Each cell must be big enough to hold a
pointer to the next list element, and a
pointer to the current element value
Garbage Collection
• Evaluations in LISP take needed cells off
of the free list. Many results are
temporary and then are of no more use.
• Therefore, must have mechanism for
reclaiming space.
• Lisp was the first language to use
automatic garbage collection extensively
Another Lisp Quote
– Lisp has jokingly been called "the most
intelligent way to misuse a computer". I
think that description is a great compliment
because it transmits the full flavor of
liberation: it has assisted a number of our
most gifted fellow humans in thinking
previously impossible thoughts.
– — Edsger Dijkstra, CACM, 15:10
Other Programming
Paradigms
Event-Driven
Concurrent
Event-Driven P.L. Paradigm
• Event-drive programs do not control the
sequence in which input events occur;
instead, they are written to react to any
reasonable sequence of events.
• Compare to imperative, OO, functional,
and logical paradigms in which the
ordering of input influences the way
program steps are executed
Examples
•
•
•
•
Graphical User Interface (GUI) programs
Online registration systems
Airline reservation systems
Embedded applications for devices such
as cell phones, cars, airplane navigation
systems, robots, home security systems …
Event-driven control
• Imperative programs get input, operate on it and
produce results, either in a loop or in a single
pass. Programmer determines order of input.
• Event-driven programs maintain an event queue
that corresponds to events that occur
asynchronously and are processed as they
happen.
– Event-driven programs tend to run continuously,
waiting for input
Java Event-Processing Features
• Provides certain classes and methods that
can be used to design an interaction
– Specify events, associate an event with an
object, and provide event-handling actions
• Figure 16.3 shows a partial class hierarchy
with subclasses such as button, popup
menu, text area, etc.
• Listeners recognize an event, such as a
mouse click.
Concurrent Programming
• Concurrency occurs when two or more separate
threads or programs execute at the “same” time
– actually, or apparently.
• Example: several objects of the same class (e.g
basketBallPlayer) may model a real-world
situation in which several actual players interact
concurrently
• Example: a web browser downloads a page
from a web site; several threads cooperate by
rendering different parts of the page
History of Concurrency
• Concurrency is implemented in one of two
ways: Concurrent program units execute
– on separate processors at the same time, or
– on a single processor by implementing time
slicing: switching back and forth rapidly
between several separate programs
(processes, threads)
History of Concurrency
• Multiprogramming is an early operating
system technique that time-shares CPU
cycles between several different
processes, all loaded in memory.
– Originally introduced to give the CPU
something to do when a process was waiting
for input
– Later extended by interrupting a running
process to give another process a chance
History of Concurrency
• Concurrency was first implemented at the
operating system level. Later techniques
extended it to application level programs.
• Concurrency at the application level requires
some kind of synchronization to prevent possible
data corruption
– Semaphores
– Monitors – Java threads
– Dedicated concurrent languages: concurrent Pascal,
concurrent Haskell, Cilk (concurrent C), ….
Author Definitions
• A concurrent program is a program designed to
have two or more execution contexts. Such a
program is said to be multithreaded.
– A parallel program is a concurrent program in which
several threads are active simultaneously (implies
multiple processors, but conceptually the same thing)
• A distributed program is a concurrent program
that is designed to executed simultaneously on a
collection of processors that don’t share memory.
– Not the same – cannot share data directly, aren’t
generally considered to be ‘multi-threaded’
Concurrency Issues
• Race conditions / synchronization
• Deadlock
• Interprocess communication