Download Functional Programming Big Picture

Functional Programming
Big Picture
?
What we’ve learned so far:
? Imperative Programming Languages
? Variables,
binding, scoping, reference environment,
etc
?
What’s next:
? Functional Programming Languages
? Semantics of Programming Languages
? Control Flow
? Data Types
? Logic Programming
? Subroutines, Procedures: Control Abstraction
? Data Abstraction and Object-Orientation
Organization of Programming Languages-Cheng (Fall 2004)
Design of Programming Languages
?
Design of imperative languages is based directly
on the von Neumann architecture
? Efficiency is the primary concern, rather than
the suitability of the language for software
development
?
Other designs:
? Functional programming languages
? Based
? Logic
on mathematical functions
programming languages
? Based
on formal logic (specifically, predicate
calculus)
Organization of Programming Languages-Cheng (Fall 2004)
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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
Input
Program
Output
Organization of Programming Languages-Cheng (Fall 2004)
Functional Programming Language
?
Functional programming is a style of
programming in which the primary method of
computation is the application of functions to
arguments
x
Argument
or variable
f(x )
y
Function
Output
Organization of Programming Languages-Cheng (Fall 2004)
Historical Background
1940s:
Alonzo Church and Haskell Curry developed
the lambda calculus, a simple but powerful
mathematical theory of functions.
Organization of Programming Languages-Cheng (Fall 2004)
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2
Historical Background
1960s:
John McCarthy developed Lisp, the first
functional language. Some influences
from the lambda calculus, but still
retained variable assignments.
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Historical Background
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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Historical Background
Mid 1970s:
Robin Milner develops ML, the first of the
modern functional languages, which introduced
type inference and polymorphic types.
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3
Historical Background
Late 1970s - 1980s:
David Turner develops a number of
lazy functional languages leading up
to Miranda, a commercial product.
Organization of Programming Languages-Cheng (Fall 2004)
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Historical Background
1988:
A committee of prominent researchers
publishes the first definition of Haskell,
a standard lazy functional language.
Organization of Programming Languages-Cheng (Fall 2004)
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Historical Background
1999:
The committee publishes the definition
of Haskell 98, providing a long-awaited
stable version of the language.
Organization of Programming Languages-Cheng (Fall 2004)
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4
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)
?
?
?
In mathematics, variables always represent actual
values
?
?
?
domain set: set of all real numbers
range set: set of real numbers from –1 to +1
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
Organization of Programming Languages-Cheng (Fall 2004)
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
Organization of Programming Languages-Cheng (Fall 2004)
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Functional Programming
Summing the integers 1 to 10 in Haskell:
sum [1..10]
The computation method is function application.
Organization of Programming Languages-Cheng (Fall 2004)
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5
Functional Programming
?
Function definition vs function application
? Function definition
? Declaration
describing how a function is to be
computed using formal parameters
? Function
application
?A
call to a declared function using actual
parameters or arguments
?
In mathematics, a distinction is often not clearly
made between a variable and a parameter
? The term independent variable is often used
for both actual and formal parameters
Organization of Programming Languages-Cheng (Fall 2004)
Functional Programming Language
?
The objective of the design of a FPL is to mimic
mathematical functions to the greatest extent
possible
? Imperative language: operations are completed
and results are stored in variables for later use
? Management
of variables is source of complexity for
imperative programming
?
FPL: variables are not necessary
? Without variables, iterative constructs are
impossible
? Repetition must be provided by recursion
rather than iterative constructs
Organization of Programming Languages-Cheng (Fall 2004)
Example: GCD
?
Find the greatest common divisor (GCD) between
12 and 30
? Divisors for 12:
1, 2, 3, 4, 6, 12
? Divisors for 30:
1, 2, 3, 5, 6, 10, 15, 30
? GCD(12,
?
30) is equal to 6
Algorithm:
? u=30, v=12
? u=12, v=6
? u=6, v=0
30 mod 12 = 6
12 mod 6 = 0
since v=0, GCD=u=6
Organization of Programming Languages-Cheng (Fall 2004)
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Computing GCD
int gcd(int u, int v )
{
int temp;
while (v != 0) {
temp = v;
v = u % v;
u = temp;
}
return u;
}
int gcd(int u, int v )
{
if (v==0)
return u;
else
return gcd(v, u % v);
}
* Computing GCD using
iterative construct
* Computing GCD using
recursion
Organization of Programming Languages-Cheng (Fall 2004)
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
Organization of Programming Languages-Cheng (Fall 2004)
Using Imperative PLs to support FPLs
?
Limitations
? Imperative languages often have restrictions
on the types of values returned by the function
? Example:
in Fortran, only scalar values can be
returned
? Most
of them cannot return functions
limits the kinds of functional forms that can be
provided
? This
? Functional
side-effects
? Occurs
when the function changes one of its
parameters or a global variable
Organization of Programming Languages-Cheng (Fall 2004)
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Characteristics of Pure FPL
?
Pure FP languages tend to
? Have no side-effects
? Have no assignment statements
? Often have no variables!
? Be built on a small, concise framework
? Have a simple, uniform syntax
? Be implemented via interpreters rather than
compilers
Organization of Programming Languages-Cheng (Fall 2004)
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
functions
? Functions are first-class values
Organization of Programming Languages-Cheng (Fall 2004)
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 * x
? Nameless 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
Organization of Programming Languages-Cheng (Fall 2004)
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Lambda Expressions
?
Function application
? Lambda 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
Organization of Programming Languages-Cheng (Fall 2004)
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))
Organization of Programming Languages-Cheng (Fall 2004)
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)
Organization of Programming Languages-Cheng (Fall 2004)
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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))
Organization of Programming Languages-Cheng (Fall 2004)
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
Organization of Programming Languages-Cheng (Fall 2004)
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
?
?
Organization of Programming Languages-Cheng (Fall 2004)
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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
Organization of Programming Languages-Cheng (Fall 2004)
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 entities
? They can be the values of expressions and
elements of lists
? They can be assigned to variables and passed
as parameters
Organization of Programming Languages-Cheng (Fall 2004)
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
Organization of Programming Languages-Cheng (Fall 2004)
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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)
Organization of Programming Languages-Cheng (Fall 2004)
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)
Organization of Programming Languages-Cheng (Fall 2004)
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?
Organization of Programming Languages-Cheng (Fall 2004)
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Lambda Expressions
?
Form is based on ? notation
? (LAMBDA (L) (CAR (CDR L)))
? L is called a bound variable
? Lambda expressions can be applied
? ((LAMBDA(L)
(CAR (CDR L))) '((A B) C D))
Organization of Programming Languages-Cheng (Fall 2004)
Constructing Functions
?
DEFINE - Two forms:
? To bind a symbol to an expression
? (DEFINE
? (DEFINE
? To
pi 3.141593)
two_pi (* 2 pi))
bind names to lambda expressions
? (DEFINE
? then
(cube x) (* x x x))
you can use: (cube x)
Organization of Programming Languages-Cheng (Fall 2004)
Evaluation Process
?
Evaluation process (for normal functions):
? Parameters are evaluated, in no particular
order
? The values of the parameters are substituted
into the function body
? The function body is evaluated
? The value of the last expression in the body is
the value of the function
Organization of Programming Languages-Cheng (Fall 2004)
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Control Flow
?
Selection- the special form, IF
?
(IF
predicate
else_exp)
?
(IF
0 )
then_exp
(<> count 0)
(/ sum count)
(DEFINE (factorial n)
( IF ( = n 0) 1 (*
) )
)
)
n
(factorial(-
n 1)
Organization of Programming Languages-Cheng (Fall 2004)
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
Organization of Programming Languages-Cheng (Fall 2004)
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 )))
))
Organization of Programming Languages-Cheng (Fall 2004)
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Example Scheme Function
?
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 '())
))
Organization of Programming Languages-Cheng (Fall 2004)
Example Scheme Function
?
equal
? returns #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 '())
))
Organization of Programming Languages-Cheng (Fall 2004)
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)))
))
Organization of Programming Languages-Cheng (Fall 2004)
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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))))
Organization of ))
Programming Languages-Cheng (Fall 2004)
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
Organization of Programming Languages-Cheng (Fall 2004)
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
Organization of Programming Languages-Cheng (Fall 2004)
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Scheme: Output Functions
?
Output Utility Functions:
? (DISPLAY expression)
? Example:
(DISPLAY (car ‘(A B C)))
? (NEWLINE)
Organization of Programming Languages-Cheng (Fall 2004)
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