Functional Programming
Universitatea Politehnica Bucuresti
2008-2009
Adina Magda Florea
http://aimas.cs.pub.ro/fp_09
L1 – Course content
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Introduction, programming paradigms, basic
concepts
Introduction to Scheme
Expressions, types, and functions
Lists
Programming techniques
Name binding, recursion, iterations, and
continuations
Introduction to Haskell
Problem solving paradigms in FP
Course materials
Scheme resources
Almost all you want to know and find about Scheme
http://www.schemers.org/
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Development and execution environment
http://www.drscheme.org/
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Books and research articles on Scheme
http://library.readscheme.org/page2.html
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R. Kent Dybvig
The Scheme Programming Language, Second Edition – online book
http://www.scheme.com/tspl2d/index.html
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Haskell
http://www.haskell.org/
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Haskell language design
http://haskell.readscheme.org/lang_sem.html
Requirements
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Laboratory: min 6
Laboratory assignments
Homeworks
Final exam
Grading
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Laboratory assignments and homewwork
50%
Final exam 50%
Lecture No. 1
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Introduction to FP
Mathematical functions
LISP
Introduction to Scheme
1. Introduction to FP
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The design of the imperative languages is based
directly on the von Neumann architecture
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Efficiency is the primary concern, rather than the
suitability of the language for software development
The design of the functional languages is based
on mathematical functions
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A solid theoretical basis that is also closer to the
user, but relatively unconcerned with the
architecture of the machines on which programs will
run
1.1 Principles of FP
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treats computation as evaluation of mathematical
functions (and avoids state)
data and programs are represented in the same
way
functions as first-class values
– higher-order functions: functions that operate on, or
create, other functions
– functions as components of data structures
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lamda calculus provides a theoretical framework
for describing functions and their evaluation
it is a mathematical abstraction rather than a
programming language
1.2 History
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lambda calculus (Church, 1932)
simply typed lambda calculus (Church, 1940)
lambda calculus as prog. lang. (McCarthy(?), 1960,
Landin 1965)
polymorphic types (Girard, Reynolds, early 70s)
algebraic types ( Burstall & Landin, 1969)
type inference (Hindley, 1969, Milner, mid 70s)
lazy evaluation (Wadsworth, early 70s)
Equational definitions Miranda 80s
Type classes Haskell 1990s
1.3 Varieties of FP languages
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typed (ML, Haskell) vs untyped (Scheme,
Erlang)
Pure vs Impure
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impure have state and imperative features
pure have no side effects, “referential
transparency”
Strict vs Lazy evaluation
1.4 Declarative style of programming
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Declarative Style of programming - emphasis is
placed on describing what a program should do rather
than prescribing how it should do it.
Functional programming - good illustration of the
declarative style of programming.
A program is viewed as a function from input to
output.
Logic programming – another paradigm
A program is viewed as a collection of logical rules
and facts (a knowledge-based system). Using logical
reasoning, the computer system can derive new facts
from existing ones.
1.5 Functional style of programming
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A computing system is viewed as a function
which takes input and delivers output.
The function transforms the input into output .
Functions are the basic building blocks from
which programs are constructed.
The definition of each function specifies what
the function does.
It describes the relationship between the input
and the output of the function.
Examples
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Describing a game as a function
Text processing
Text processing: translation
Compiler
1.6 Why functional programming
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Functional programming languages are carefully
designed to support problem solving.
There are many features in these languages which help
the user to design clear, concise, abstract, modular,
correct and reusable solutions to problems.
The functional Style of Programming allows the
formulation of solutions to problems to be as easy,
clear, and intuitive as possible.
Since any functional program is typically built by
combining well understood simpler functions, the
functional style naturally enforces modularity.
Why functional programming
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Programs are easy to write because the system relieves the
user from dealing with many tedious implementation
considerations such as memory management, variable
declaration, etc .
Programs are concise (typically about 1/10 of the size of a
program in non-FPL)
Programs are easy to understand because functional
programs have nice mathematical properties (unlike imperative
programs) .
Functional programs are referentially transparent , that is, if a
variable is set to be a certain value in a program; this value
cannot be changed again. That is, there is no assignment but
only a true mathematical equality.
Why functional programming
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Programs are easy to reason about because
functional programs are just mathematical
functions;
hence, we can prove or disprove claims about
our programs using familiar mathematical
methods and ordinary proof techniques (such as
those encountered in high school Algebra).
For example we can always replace the left
hand side of a function definition by the
corresponding right hand side.
1.7 Examples of FP languages
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Lisp (1960, the first functional language….dinosaur, has no
type system)
Hope (1970s an equational fp language)
ML (1970s introduced Polymorphic typing systems)
Scheme (1975, static scoping)
Miranda (1980s equational definitions, polymorphic typing
Haskell (introduced in 1990, all the benefits of above +
facilities for programming in the large.)
Erlang (1995 - a general-purpose concurrent programming
language and runtime system, introduced by Ericsson)
The sequential subset of Erlang is a functional language, with
dynamic typing.
2. Mathematical functions
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Def: A mathematical function is a mapping of
members of one set, called the domain set, to
another set, called the range set
A lambda expression specifies the parameter(s)
and the mapping of a function in the following
form
(x) x * x * x
for the function cube (x) = x * x * x
Mathematical functions
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Lambda expressions describe nameless
functions
Lambda expressions are applied to parameter(s)
by placing the parameter(s) after the expression
e.g. ((x) x * x * x)(3)
which evaluates to 27
Mathematical functions
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Functional Forms
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Def: A higher-order function, or functional form, is
one that either takes functions as parameters or
yields a function as its result, or both
Functional forms
1. Function Composition
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A functional form that takes two functions as
parameters and yields a function whose value is the
first actual parameter function applied to the
application of the second
Form: h  f ° g
which means h (x)  f ( g ( x))
For f (x)  x * x * x and g (x)  x + 3,
h  f ° g yields (x + 3)* (x + 3)* (x + 3)
Functional forms
2. Construction
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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
Form: [f, g]
For f (x)  x * x * x and g (x)  x + 3,
[f, g] (4) yields (64, 7)
Functional forms
3. Apply-to-all
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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
Form: 
For h (x)  x * x * x
( h, (3, 2, 4)) yields (27, 8, 64)
3. LISP
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Lambda notation is used to specify functions and
function definitions. Function applications and data
have the same form.
e.g., 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
The first LISP interpreter appeared only as a
demonstration of the universality of the computational
capabilities of the notation
4. Introduction to Scheme
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A mid-1970s dialect of LISP, designed to be a
cleaner, more modern, and simpler version than
the contemporary dialects of LISP
Invented by Guy Lewis Steele Jr. and Gerald
Jay Sussman
Emerged from MIT
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Originally called Schemer
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Shortened to Scheme because of a 6 character limitation
on file names
Introduction to Scheme
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Designed to have very few regular constructs
which compose well to support a variety of
programming styles
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Functional, object-oriented, and imperative
All data type are equal
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What one can do to one data type, one can do to all
data types
Introduction to Scheme
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Uses only static scoping
Functions are first-class entities
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They can be the values of expressions and elements
of lists
They can be assigned to variables and passed as
parameters
Introduction to Scheme
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Primitive Functions
1. Arithmetic: +, -, *, /, ABS,
SQRT, REMAINDER, MIN, MAX
e.g., (+ 5 2) yields 7
Introduction to Scheme
2. QUOTE -takes one parameter; returns the
parameter without evaluation
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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 evaluation when
it is not appropriate
QUOTE can be abbreviated with the apostrophe
prefix operator
e.g., '(A B) is equivalent to (QUOTE (A B))
Introduction to Scheme
3. CAR takes a list parameter; returns the first
element of that list
e.g., (CAR '(A B C)) yields A
(CAR '((A B) C D)) yields (A B)
4. CDR takes a list parameter; returns the list
after removing its first element
e.g., (CDR '(A B C)) yields (B C)
(CDR '((A B) C D)) yields (C D)
Introduction to Scheme
5. CONS takes two parameters, the first of which
can be either an atom or a list and the second of
which is a list; returns a new list that includes
the first parameter as its first element and the
second parameter as the remainder of its result
e.g., (CONS 'A '(B C)) returns (A B C)
Introduction to Scheme
6. LIST - takes any number of parameters;
returns a list with the parameters as elements
Introduction to Scheme
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Lambda Expressions
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Form is based on  notation
e.g., (LAMBDA (L) (CAR (CAR L)))
L is called a bound variable
 Lambda expressions can be applied
e.g.,
((LAMBDA (L) (CAR (CAR L))) '((A B) C D))
Introduction to Scheme
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A Function for Constructing Functions
DEFINE - Two forms:
1. To bind a symbol to an expression
e.g.,
(DEFINE pi 3.141593)
(DEFINE two_pi (* 2 pi))
Introduction to Scheme
2. To bind names to lambda expressions
e.g.,
(DEFINE (cube x) (* x x x))
Example use:
(cube 4)
Introduction to Scheme
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Evaluation process (for normal functions):
1. Parameters are evaluated, in no particular order
2. The values of the parameters are substituted into
the function body
3. The function body is evaluated
4. The value of the last expression in the body is the
value of the function
(Special forms use a different evaluation process)
Introduction to Scheme
Example:
(DEFINE (square x) (* x x))
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(DEFINE (hypotenuse side1 side1)
(SQRT (+ (square side1)
(square side2)))
)
Introduction to Scheme
Example:
(define
(list-sum lst)
(cond
((null? lst) 0)
((pair? (car lst))
(+(list-sum (car lst)) (list-sum
(cdr lst))))
(else
(+ (car lst) (list-sum (cdr
lst))))))
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Some conventions
Naming Conventions
 A predicate is a procedure that always returns a
boolean value (#t or #f). By convention,
predicates usually have names that end in `?'.
 A mutation procedure is a procedure that alters
a data structure. By convention, mutation
procedures usually have names that end in `!'.
Cases
Uppercase and Lowercase
 Scheme doesn't distinguish uppercase and
lowercase forms of a letter except within
character and string constants; in other words,
Scheme is case-insensitive.
 For example, Foo is the same identifier as
FOO, but 'a' and 'A' are different characters.
Predicate functions
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Predicate Functions: (#t is true and #f or
()is false)
1. EQ? takes two symbolic parameters; it returns #T
if both parameters are atoms and the two are the
same
e.g., (EQ? 'A 'A) yields #t
(EQ? 'A '(A B)) yields ()
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Note that if EQ? is called with list parameters, the
result is not reliable
Also, EQ? does not work for numeric atoms
Predicate functions
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Predicate Functions:
2. LIST? takes one parameter; it returns #T if the
parameter is a list; otherwise()
3. NULL? takes one parameter; it returns #T if the
parameter is the empty list; otherwise()
Note that NULL? returns #T if the parameter is()
4. Numeric Predicate Functions
=, <>, >, <, >=, <=, EVEN?, ODD?,
ZERO?, NEGATIVE?
Control flow
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Control Flow
1. Selection- the special form, IF
(IF predicate then_exp
else_exp)
e.g.,
(IF (<> count 0)
(/ sum count)
0
)
Control flow
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Control Flow
2. Multiple Selection - the special form, COND
General form:
(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
(DEFINE (compare x y)
(COND
((> x y) (DISPLAY “x is greater
than y”))
((< x y) (DISPLAY “y is greater
than x”))
(ELSE (DISPLAY “x and y are
equal”))
)
)
Example
1. member - takes an atom and a simple list; returns #T
if the atom is in the list; () otherwise
(DEFINE (member atm lis)
(COND
((NULL? lis) '())
((EQ? atm (CAR lis)) #T)
((ELSE (member atm (CDR lis)))
))
Example
2. equalsimp - takes two simple lists as parameters;
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 '())
))
Example
3. equal - takes two general lists as parameters; returns #T if the
two 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 '())
))
Example
4. 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)))
))