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Functional Programming Languages (FPL)
- Definitions
- Applications
- Examples
- FPL Characteristics:
- Lambda calculus (LC)
- Functions in FPLs
- Modern functional languages
- Scheme overview
- 8.1. Get your own Scheme from MIT...........................
- 8.2. General overview...................................................
- 8.3. Data Typing...........................................................
- 8.4. Comments..............................................................
- 8.5. Recursion Instead of Iteration................................
- 8.6. Evaluation..............................................................
- 8.7. Storing and using Scheme code.............................
- 8.8. Variables................................................................
- 8.9. Data types...............................................................
- 8.10. Arithmetic functions...........................................
- 8.11. Selection functions..............................................
- 8.12. Iteration................................................................
- 8.13. Defining functions..............................................
- ML
- Haskell
1. Definitions
Functional programming languages were originally developed
specifically to handle symbolic computation and list-
processing applications.
In FPLs the programmer is concerned only with functionality,
not with memory-related variable storage and assignment
sequences.
FPL can be categorized into two types;
PURE functional languages, which support only the
functional paradigm (Haskell), and
Impure functional languages that can also be used for
writing imperative-style programs (LISP).
2. Applications
AI is the main application domain for functional
programming, covering topics such as:
expert systems
knowledge representation
machine learning
natural language processing
modelling speech and vision
value, there are no variables to manipulate and hence no
possibility for side effects.
Programs are constructed by composing function
applications - the values produced by one or more functions
become the parameters to another.
For reasons of efficiency (because the underlying machine
is, in fact, imperative) most functional languages provide
some imperative-style capabilities, including variables with
assignment, sequences of statements, and imperative style
loop structures.
Note that the functional paradigm can also be used with
some imperative languages - e.g. C has both a conditional
expression and support for recursion - so the factorial
function code be coded in functional style in C (or C++ or
Java) as follows:
int fact(int x)
{ return (x == 0)? 1 : x * fact(x - 1); }
Three primary components:
A set of data object: A single, high-level
data structure like a list
A set of built-in functions for object
manipulation: Building, deconstructing,
and accessing lists
A set of functional forms for building
new functions: Composition, reduction,
etc.
5. Lambda calculus (LC)
A method of modeling the computational aspects of
functions
It helps us understand the elements and semantics of
functional programming languages independent of
syntax
LC expressions are of three forms:
e1 : A single identifier (such as x , or 3 )
e2 : A function definition of the form
( x.e)
The expression e , with x being a
bound variable
e is the body of the function, x
is a parameter
e may be any of the three types
of expressions
square( x ) would be written as
( x.xx)*
e3 : A function application of the form
e1 e
Meaning e1 applied e
square applied to 2 would be
(( x.xx) 2)*
Free and Bound Variables:
A variable appearing in a function F is
said to be free if it is not bound in F
Bound variables are like formal
parameters, and act like local variables
r1 : Renaming
x i
.e x j
.[x j
/x i
]e , where x j
is not
free in e
We can replace all occurrences of the
name of a bound variable with
another name without changing the
meaning
r2 : Application
( x.e1)e2 [e2/x]e
Replace the bound variable with the
argument to the application
r3 : Redundant function elimination
x.(e x) e , if x is not free in e
An expression that can no longer be
reduced is said to be in normal
form:
- ( x.( y.x + y) 5)(( y.y * y) 6) =
- ( x.x + 5)(( y.y * y) 6) =
- ( x.x + 5)(6 * 6) =
6. Functions in FPLs
In a functional language, the basic unit of computation is the
FUNCTION.
The function definitions typically include a name for the
function, its associated parameter list, and the expressions
used to carry out the computation.
A function computes a single value based on 0 or more
parameters.
Though the parameters of a function look like
variables in an imperative language, they are
different in that they are not subject to having
their value changed by assignment - i.e. they
retain their initial value throughout the
computation of the function.
Pure functional languages don't need an
assignment statement.
Function construction : given one or more functions as
parameters, as well as a list of other parameters,
construction essentially calls each function and passes it the
list of "other" parameters.
Function composition : applying one function to the result
of another. E.g. square_root(absolute_value(-3))
Apply-to-all functions : takes a single function as a
parameter along with list of operand values. It then applies
the function to each parameter, and returns a list containing
the results of each call.
Example:
IPL vs. FPL
Note that in imperative programming we concern ourselves
with both the computation sequence and maintaining the
program state (i.e. the collection of current data values).
Unlike IPLs, purely functional languages (no variables and
hence no assignments) have no equivalent concept of state:
the programmer focuses strictly on defining the desired
functionality.
Iteration is not accomplished by loop statements, but rather
by conditional recursion.
Functional programmers are concerned only with
functionality. This comes at a direct cost in terms of
efficiency, since the code is still translated into something
running on Von Neuman architecture.
8. Scheme overview
8.1. Get your own Scheme from MIT
swissnet.ai.mit.edu/projects/scheme/index.html
8.2. General overview
Scheme is a functional programming language
Scheme is a small derivative of LISP:
LIS t P rocessing
Dynamic typing and dynamic scooping
Scheme introduced static scooping
Data Objects
An expression is either an atom or a list
An atom is a string of characters
A
Austria
; and finally for (2+3)-(2*2)
; we'll start the statements to be evaluated
; on the next line
; Value: 5
; Value: 11
; Value: 1
8.5. Recursion Instead of Iteration
Since we are expressing the entire computation as a
composition of functions into a single function,
recursion is usually used rather than iteration
Example:
; the first line is the header for the Fibonacci
function:
(define Fibonacci (lambda (n)
; next is the termination case
( if (< n 3) 1
; and the recursive cal
(+ (Fibonacci (- n 1)) (Fibonacci (- n 2))))))
(Fibonacci 6)
; Value: 8
8.6. Evaluation
The functional approach sometimes requires us to
take a "bottom-up" view of the problem: creating
functions to compute the lowest layer of values,
then other functions taking those as operands.
Example: Design a code to compute (a + b + c) / (x
Compute the numerator and denominator
separately,
; for the numerator
(+ a b c)
; for the denominator
(+ x y z)
and then decide how to apply division with those
two functions as operands, i.e.:
(/ (+ a b c) (+ x y z))
8.7. Storing and using Scheme code
The load function is available to load a Scheme
program stores in a an text file, e.g.:
(load "myfile.txt")
; Loading "myfile.txt" -- done
8.9. Data types
Literals are described as self-evaluating , in that
evaluating the literal returns the value they
represent. (E.g. evaluating 3 returns the
integer value 3.)
The primitive types are:
characters
strings (in double-quotes)
Booleans:
True: #t
False: The empty set for false or
#f (see example below).
Integers
rational numbers
real numbers
complex numbers.
List: There is also a composite data type,
called the list, which is a fundamental part
of Scheme. Lists are considered in detail in a
later section.
Numbers
There are integers, rationals, reals, and complex
numbers.
In general, Scheme will return as exact an answer as it
can (i.e. it will give an exact integer or rational over a
real approximation).
Examples:
Let's see the results of some basic arithmetic:
(/ 3.2 1.6)
; Value: 2.
(/ 16 10)
; Value: 8/
Suppose we were to try some comparisons:
(< 2 3)
; Value: #t
(< 4 3)
; Value: ()
8.10. Arithmetic functions
There are many built-in arithmetic functions. Some of
the commonly used ones include:
max, min
quotient, modulo, remainder
ceiling, floor, abs, magnitude, round, truncate
gcd, lcm
exp, log, sqrt
sin, cos, tan
There are also a number of comparison
operators returning Boolean values
Selection in a functional language still controls the choice
between different computations, but is expressed by
returning the results of functions representing the different
computations.
The two major Boolean control operations are:
IF
COND.
IF:
For example, suppose if x is less than 0 we
want to return y - x: (if (< x 0) (- y x))
Now suppose that if x is less than 0 we want
to return 0, otherwise we want to return the
value x - 1:
(if (< x 0) 0
(- x 1))
COND statement is somewhat like the C switch
statement, allowing a series of conditions to test for (with
corresponding functions to evaluate and return) and a
default case:
(cond ((= x y) 0)
((> x y) 1)
(else -1)
Lists
Lists are the main composite data type in Scheme.
Lists are composed of a series of elements, enclosed
in brackets.
Implementation note: the typical implementation
format for lists is to represent each element in a list
using two pointers:
One points to the actual implementation of
the element (hence allowing us to use
anything we like as a list element, the
pointer can refer to a primitive data element,
a list, a string, etc)
The other points to the next element in the
list
Example:
(a b c d) has the four elements a, b, c, and d.
The empty list is denoted ()
Examples of lists include
'(a) ; a list with a single element
'(a b c) ; a list with three elements
'() ; an empty, or null, list
'((a b)) ; a list with a single element, which
happens to be another list
