1. Functional Programming Languages
Chapter 15
Functional Programming Languages

2. Chapter 15 Introduction Mathematical Functions
Fundamentals of Functional Programming
Languages
The First Functional Programming Language:
LISP
Introduction to Scheme
COMMON LISP ML
Haskell
Applications of Functional Languages
Comparison of Functional and Imperative

3. Introduction
The design of the imperative languages is based on the von Neumann architecture
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
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

4. Mathematical Functions
Definition:
A mathematical function is a mapping
of members of one set, called the domain set,
to another set, called the range set
Mathematical functions can be specified in a number of ways.
A lambda expression is one of those ways

5. Lambda Expressions A lambda expression specifies the parameter(s)
and the mapping of a function
Example: for the function cube(x) = x*x*x
(x) x * x * x
Lambda expressions describe nameless functions
Lambda expressions are applied to parameter(s) by placing the parameter(s) after the expression
Example: ((x) x * x * x)(2) evaluates to 8

6. A higher-order function, or functional form,
Functional Forms
Definition:
A higher-order function, or functional form,
is one that either takes functions as parameters or returns a function as its result, or both.
We will use two functional forms:
1. functional composition
2. apply-to-all

7. Function Composition Definition:
Function composition is a functional form
that takes two functions as parameters
and yields a function whose value
is the first actual parameter function
applied to the result
of the second actual parameter function
Form: h  f ° g
which means h (x)  f ( g (x) )
For f (x)  x + 2 and g (x)  3 * x,
h  f ° g yields (3 * x) + 2

8. Apply-to-all Definition: Apply-to-all is 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
( h, (2, 3, 4) ) yields (4, 9, 16)

9. Fundamentals of Functional Programming Languages
The objective of the design of an FPL
is to mimic mathematical functions
to the greatest extent possible
In an FPL,
variables are not necessary,
as is the case in mathematics

10. Fundamentals of Functional Programming Languages
The basic process of computation is fundamentally different in a FPL than in an imperative language
In an imperative language, operations are done and the results are stored in variables for later use
Management of variables is a constant concern and source of complexity for imperative programming

11. Referential Transparency
In an FPL,
the evaluation of a function
always produces the same result
given the same parameters

12. The First Functional Programming Language: LISP
Data object types:
originally only atoms and lists
List form:
parenthesized collections of sublists and/or atoms
e.g., (A B (C D) E)
Lambda notation is used to specify functions and function definitions
Function applications, and data all have the same form

13. LISP Lists vs. Functions
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 parmeters, B and C

14. LISP Interpretation The first LISP interpreter
appeared only as a demonstration
of the universality
of the computational capabilities
of the notation
Originally, LISP was a typeless language
LISP lists are stored internally
as single-linked lists

15. Origins of Scheme A mid-1970s dialect of LISP
designed to be cleaner and simpler
than contemporary dialects of LISP
Uses only static scoping
Functions are first-class entities
They can be the values of expressions and elements of lists
They can be assigned to variables and passed as parameters

16. Primitive Functions Arithmetic: +, -, *, /,
ABS, SQRT, REMAINDER, MIN, MAX
e.g., (+ 5 2) yields 7

17. '(A B) is equivalent to (QUOTE (A B))
Primitive Functions
QUOTE
takes one parameter
returns the parameter without evaluation
used to avoid parameter evaluation
the Scheme interpreter, named EVAL, always evaluates parameters to function applications before applying the function.
abbreviated with a prefix apostrophe
'(A B) is equivalent to (QUOTE (A B))

18. Function Definition: LAMBDA
Lambda Expressions
Form is based on  notation
e.g., (LAMBDA (x) (* x x)
x is called a bound variable
Lambda expressions can be applied
e.g., ((LAMBDA (x) (* x x)) 7)

19. Special Form Function: DEFINE
A Function for Constructing Functions DEFINE - Two forms:
1. To bind a symbol to an expression
e.g.: (DEFINE pi )
Example use:
(DEFINE two_pi (* 2 pi))
2. To bind names to lambda expressions
e.g.: (DEFINE (square x) (* x x))
(square 5)

20. Evaluation Process 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

21. Output Functions
(DISPLAY expression)
(NEWLINE)

22. Numeric Predicate Functions
#T is true
() is false
=, <>, >, <, >=, <=
EVEN?, ODD?, ZERO?, NEGATIVE?

23. Numeric Predicate Functions
(define (quadratic-solutions a b c)
(display
(/ (+ (- 0 b) (sqrt (- (square b)
(* 4 a c)))) (* 2 a)))
(newline)
(/ (- (- 0 b) (sqrt (- (square b)
(define (square dum) (* dum dum) )

24. Control Flow IF Selection - the special form, IF
(IF predicate then_exp else_exp)
(IF (<> count 0)
(/ sum count) )
Example:
(DEFINE (somefunction n)
(IF (= n 0) 1
(* n (somefunction (- n 1)))

25. Control Flow COND Multiple Selection-the special form, COND (COND
(predicate_1 expr {expr})
...
(predicate_n expr {expr})
(ELSE expr {expr}) )
Example:
(DEFINE (compare x y)
((> x y) (DISPLAY "x greater than y"))
((< x y) (DISPLAY "y greater than x"))
(ELSE (DISPLAY "x and y are equal")) ))

26. List Functions CONS and LIST
takes two parameters,
the first can be either an atom or a list
the second is a list
returns a new list that includes
the first parameter as its first element, and
the second parameter as the rest of the list
Example: (CONS 'A '(B C)) returns (A B C)
LIST
takes any number of parameters
returns a list with the parameters as elements

27. List Functions CAR and CDR
takes a list parameter
returns the first element of that list
Example: (CAR '(A B C)) returns A
(CAR '((A B) C D)) returns (A B)
CDR
returns the list after removing its first element
Example: (CDR '(A B C)) returns (B C)
(CDR '((A B) C D)) returns (C D)

28. Predicate Function EQ? EQ? takes two parameters
it returns #T if both parameters are atoms and the two are the same
Example: (EQ? 'A 'A) yields #T
(EQ? 'A 'B) yields ()
Note that if EQ? is called with list parameters, the result is undefined
Also EQ? does not work for numeric atoms

29. Predicate Functions LIST? and NULL?
takes one parameter;
returns
#T if the parameter is a list
() otherwise
NULL?
#T if the parameter is the empty list
Note: NULL? returns #T if the parameter is()

30. Example Scheme Function 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))) ))

31. Example Scheme Function 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 '()) ))

32. Example Scheme Function 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 '()) ))

33. Example Scheme Function 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))) ))

34. Example Scheme Function LET
Syntax:
(LET (
(name_1 expression_1)
(name_2 expression_2)
...
(name_n expression_n))
body )
Semantics:
Evaluate all expressions
Bind the values to the names
Evaluate the body

35. LET Example (DEFINE (quadratic_roots a b c) (LET ( ( root_prt_ovr_2a
( / (SQRT (- (* b b)(* 4 a c)))(* 2 a)) )
( mins_b_ovr_2a
( / (- 0 b) (* 2 a)) ) )
(DISPLAY (+ mins_b_ovr_2a root_prt_ovr_2a))
(NEWLINE)
(DISPLAY (- mins_b_ovr_2a root_prt_ovr_2a))

36. An Introduction to Scheme
List Processing
QUOTE takes: an s-expression
returns: the input without evaluation
(abbreviated by apostrophe prefix operator)
CONS takes: an s-expression and a list
returns: a list composed of the two inputs
LIST takes: any number of parameters
returns: a list with the input parameters as
elements

37. An Introduction to Scheme
Control Flow
1. Selection - the special form, IF
(IF predicate then_exp else_exp)
2. Multiple Selection - the special form, COND
(COND
(predicate_1 expr {expr})
(predicate_2 expr {expr})
...
(predicate_n expr {expr})
(ELSE expr {expr}) )

38. Scheme Functional Forms
Composition
The previous examples have used it
(CDR (CDR ‘(A B C))) returns (C)
(CAR (CDR ‘(A B C))) returns B
Most lisp languages abbreviate this:
(CADR ‘(A B C)) returns B
Frequently, this expands farther: (CADDDR ‘(A B C D E)) returns D

39. Scheme Functional Forms
Apply to All - one form in Scheme is mapcar
takes: a function and a list
returns: a list of results of function applied
to all elements of the list;
Applies the given function to all elements of the given list;
(DEFINE (mapcar fun lis)
(COND
((NULL? lis) '())
(ELSE (CONS (fun (CAR lis))
(mapcar fun (CDR lis)))) ))

40. Functions That Build Code
Frequently, it is useful to define a function that builds code and evaluates it
It is possible in Scheme to define a function that
builds Scheme code, and
requests its interpretation
This is possible because the interpreter is a user-available function, EVAL

41. Functions That Build Code
(DEFINE (adder lis)
(COND ((NULL? lis) 0)
(ELSE (EVAL (CONS '+ lis))) )
The function adder is defined to do the job.
The parameter lis is a list of numbers.
First CONS prefixes the atom + onto lis.
Note that + is quoted to prevent evaluation.
The new list is given to EVAL for evaluation.

42. COMMON LISP
A combination of many of the features of the popular dialects of LISP around in the early 1980s
Includes:
records
arrays
complex numbers
character strings
powerful i/o capabilities
packages with access control
imperative features like those of Scheme
iterative control statements

43. ML A static-scoped functional language
(types determined at compile time)
Syntax is closer to Pascal than to LISP
Usually uses type inferencing to determine the types of undeclared variables
Also has type declarations
Strongly typed and has no type coercions
(Scheme is essentially typeless)
Includes exception handling and a module facility for implementing abstract data types
Supports lists and list processing

44. ML Uses val instead of DEFINE to bind a name to a value
Function declaration syntax and example:
fun <name> (<parameters>) = <body>;
fun cube (x: int) = x * x * x;
The default numeric type is int
Example of patterned parameters in functions:
fun fact(0) = 1
| fact(n: int): int = n * fact(n –1);
List notation: brackets and commas
All elements of a list must be the same type

45. Haskell Similarities to ML
(syntax, static scoped, strongly typed, type inferencing)
Differences from ML (and most other functional languages)
Haskell is purely functional
(no variables, no assignment statements, and no side effects of any kind)
Most Important Features
Uses lazy evaluation (evaluate no subexpression until the value is needed)
Has list comprehensions, which allow it to deal with infinite lists

46. Function Definitions with Parameter Forms
Haskell
Function Definitions with Parameter Forms
Examples
1. Factorial
fact 0 = 1
fact n = n * fact(n – 1)
2. Fibonacci numbers
fib 0 = 1
fib 1 = 1
fib (n + 2) = fib (n + 1) + fib n
3. Factorial using guards
fact n
| n == 0 = 1
| n > 0 = n * fact(n – 1)

47. Haskell List Operations
List notation: brackets and commas
directions = [“north”,“south”,“east”,“west”]
Length: #
#directions is 4
Arithmetic series with the .. operator
[2, 4..10] is the same as [2, 4, 6, 8, 10]
Catenation with the ++ operator
[1, 3] ++ [5, 7] results in [1, 3, 5, 7]
CONS, CAR, CDR via the colon operator
(as in Prolog)
1:[3, 5, 7] results in [1, 3, 5, 7]

48. Haskell List Operations
Examples
product [ ] = 1
product (a:x) = a * product x
fact n = product [1..n]
quadratic_root a b c =
[part1 – part2, part1 + part2]
where
part1 = -b/(2.0*a)
part2 = sqrt(b^2–4.0*a*c)/(2.0*a)

49. Haskell List Comprehensions
Set Notation
A list of the squares of the first 20 positive integers is given by this function:
[ n * n | n <- [1..20] ]
This function computes all of the factors of its given parameter:
factors n = [ i | i [ 1..n div 2 ], n mod i == 0 ]
Quicksort:
sort [ ] = [ ]
sort (h:t) = sort [ b | b <- t; b <= h ]
++ [ h ]
++ sort [ b | b <- t; b > h ]

50. Haskell Lazy Evaluation
Only compute those that are necessary positives = [ 0.. ]
The member function could be written as
member [ ] b = False
member (a:x) b = (a == b) || member x b
Determining if 16 is a square number (Problem?)
squares = [ n * n | n <- [ 0.. ] ]
member squares 16
The following version will always work
member2 (m:x) n
| m < n = member2 x n
| m == n = True
| otherwise = False

51. Applications of Functional Languages
APL is used for throw-away programs
LISP is used for artificial intelligence
Knowledge representation
Machine learning
Natural language processing
Modeling of speech and vision
Scheme is used to teach introductory programming at a significant number of universities

52. Fundamentals of Functional Programming Languages
The basic process of computation is fundamentally different in a FPL than in an imperative language
Imperative Languages
variables are necessary to mimic machine operation
operations are done and the results are stored in variables for later use
repetition done with loops made with control variables
FPLs
mathematics has unknowns, not variables
repetition done by recursion
referential transparency
FPLs are simpler than imperative languages

53. Comparing Functional Languages and Imperative Languages
Efficient execution
Complex semantics
Complex syntax
Concurrency is programmer designed
Functional Languages
Simple semantics
Simple syntax
Inefficient execution
Programs can automatically be made concurrent

54. Summary Functional programming languages
control program execution using
function application
conditional expressions
recursion
functional forms
do not use imperative features
variables
assignments
have advantages over imperative alternatives, but lower efficiency prevents widespread use

55. Summary
LISP began as a purely functional language and later included imperative features
Scheme is a relatively simple dialect of LISP that uses static scoping exclusively
COMMON LISP is a large LISP-based language
ML is a static-scoped and strongly typed functional language which includes many features desired by programmers
Haskell is a lazy functional language supporting infinite lists and set comprehension.