Sept 2004 SDP-MSc Slide 1 Section 6 Introduction to Functional Programming.

Sept 2004 SDP-MSc Slide

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Section 6

Introduction to Functional Programming


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Section 6 - References

About Haskell

http://www.haskell.org/aboutHaskell.html

Why Functional Programming Matters

http://www.md.chalmers.se/~rjmh/Papers/whyfp.pdf


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What is a Functional Language?

• Functional programming is style of programming in which the primary method of computation is the application of functions to arguments;

• A functional language is one that supports and encourages the functional style.

Opinions differ, and it is difficult to give a precise definition, but generally speaking:


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Historical Background1920s - 1940s:

Alonzo Church and Haskell Curry develop the lambda calculus, a simple but powerful mathematical theory of functions. 4


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Historical Background1960s:

John McCarthy develops 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 Background1999:

The definition of Haskell 98 published, providing a long-awaited stable version of the language.

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Principles- Functional Programs

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•No Side effects

•No variable assignment

•A Function has no effect other than calculating its value


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Key Features- Functional Programs

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•Higher Order Functions

•Lazy Evaluation

•Currying

•Lambda Expressions

•List Processing


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Functional Programs

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•Program consists only of Functions

•Main program is a function, receives input as parameters

•May in turn be defined as other functions


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Functional Programs

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using System;class Fun1{ public static int sum(int a, int b){

return a+b;}

public static void Main(string[] args) { int x=Convert.ToInt32(args[0]); int y=Convert.ToInt32(args[1]); Console.WriteLine("Sum of values:{0}", sum(x,y));}

}


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- No side effects

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class Fun2{ public static int res; // global variable

public static void sum(int a, int b){ res= a+b;} // side effect

public static void Main(string[] args) { int x=Convert.ToInt32(args[0]); int y=Convert.ToInt32(args[1]); sum(x,y); Console.WriteLine("Sum of values:{0}", res);

}

}


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Function result only depends on input parameters

Always returns the same result, whenever called

Order of function calls irrelevant:

f(x)=2*x + 1 => f(2)=5 always

Next: A counter example

Functional Programming -


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class Fun3{ public static int a,b; // global variables

public static int sum(){ return a+b;}

public static void Main(string[] args) { a=Convert.ToInt32(args[0]); b=Convert.ToInt32(args[1]); Console.WriteLine("Sum of values:{0}", sum()); a=9; b=8; Console.WriteLine("Sum of values:{0}", sum());

}

}


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Sample call produces different results


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No Side-effects - Summary

•No global variables

•No variable assignment

•Function just calculates a value based on input parameters

Next: Another counter example


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class Fact1{

public static int factorial(int n){ int res=1;

for (int i=1;i<=n;i++) res*=i; // var assignment return res;}

public static void Main(string[] args) { int x=Convert.ToInt32(args[0]); Console.WriteLine("Factorial= {0}", factorial(x));

}

}


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Recursion

Most Functional Programming Languages use recursion

- avoids variable assignment

- facilitates Lazy Evaluation


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class Fact2{ public static int factorial(int n){

if (n==0) return 1; else return n * factorial(n-1); }

public static void Main(string[] args) { int x=Convert.ToInt32(args[0]);Console.WriteLine("Factorial= {0}", factorial(x)); }

}


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class SumRange{ public static int sumr(int s, int e){

int res=0; for(int i=s;i<=e;i++) res=res+i; //assignment return res;}

public static void Main(string[] args) {int x=Convert.ToInt32(args[0]);int y=Convert.ToInt32(args[1]);Console.WriteLine("SumRange= "+ sumr(x,y)); }

}


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Next: Recursive Version


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class SumRange { public static int sumr(int s, int e){

if (s==e) return s; else return s+ sumr(s+1,e);}

public static void Main(string[] args) {int x=Convert.ToInt32(args[0]);int y=Convert.ToInt32(args[1]);Console.WriteLine("SumRange= {0}", sumr(x,y));

} }


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class Arr1{ public static int add(int[] a, int first){

if (a.Length==0)return 0; if (first==a.Length-1) return a[first]; else return a[first] + add(a,first+1);

}

public static void Main() { int[] v={3,4,5,6,5}; Console.WriteLine("Sum=: {0}", add(v,0));

}}

Add elements in array


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class Arr2{ public static int count_target(int[] a,int first,int target){

if (a.Length==0)return 0; if (first>a.Length-1) return 0;

else if (a[first]==target)

return 1 + count_target(a,first+1,target); else return 0 + count_target(a,first+1,target);

}:

Count occurrences of target in array


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public static void Main(string[] args) { int[] v={3,4,5,6,5}; int t=Convert.ToInt32(args[0]); Console.WriteLine("No of occurences: {0}",

count_target(v,0,t)); }}


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Q1

Write a C# application which contains a recursive function

static bool search_target(int[] a,int first,int target){

which return ‘true’ if ‘target’ is an element of array ‘a’ otherwise returns ‘false’

e.g. search_target([2,3,4], 0, 3) returns true

Exercises:


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Q2


static bool gr_than_tar(int[] a,int first,int target){

which return ‘true’ if all elements of array ‘a’ are > target otherwise returns ‘false’

e.g. gr_than_tar([2,3,4], 0, 1) returns true

gr_than_tar([2,3,4], 0, 2) returns false

Exercises:


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Q3


static int largest(int[] a, int first){

which return the maximum element in the Array

e.g. largest([2,3,4,1], 0) returns 4


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Functional Programming

Two other key features

- List Processing

- Polymorphic Types


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int Add(object) - adds to end

bool Contains(object)

int indexOf(object)

insert( int index, object value );

=========================================public virtual object this[ int index ] {get; set;} // accessor

e.g object el=alist[2]; // gets 3rd element alist[1]=“abc”;==========================================

ArrayList - Collection Class


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public virtual int Count {get;} // number of elements

int size= alist.Count===========================================void RemoveAt(int) - remove el at given index

setElementAt(Object, int)

Sort()

public virtual int Capacity {get; set;} // sets,gets max size // 16 by defaultAlternative:ArrayList text = new ArrayList(newCapactity)


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using System;using System.Collections;

class Vec1{

public static void Main() { ArrayList a=new ArrayList(); a.Add("a"); a.Add("b"); a.Add("c"); for(int i=0; i<a.Count;i++) Console.WriteLine(" {0}“, a[i]);

}}


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class Vec2{

public static void Main() { ArrayList a=new ArrayList(); a.Add(1); a.Add(2); a.Add(3); for(int i=0; i<a.Count;i++) Console.WriteLine(" {0}", a[i]);

}}


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class Vec3{ public static int total(ArrayList vec){

int res=0; for(int i=0; i<vec.Count;i++) { res+= (int)vec[i]; } return res;}

public static void Main() { ArrayList v=new ArrayList(); v.Add(1); v.Add(2); v.Add(3); Console.WriteLine("Sum of Elements: {0}",

total(v));} }


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Next Fuctional Version


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class Vec4{ public static int total(ArrayList vec, int first) { if (vec.Count == 0) return 0; if (first == vec.Count - 1) return (int)vec[first]; else return (int)vec[first] + total(vec, first + 1); }


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: public static void Main(string[] args) { ArrayList v = new ArrayList(); v.Add(Convert.ToInt32(args[0])); v.Add(Convert.ToInt32(args[1])); v.Add(Convert.ToInt32(args[2])); Console.WriteLine("Sum of Elements: {0}",total(v,0)); Console.Read(); }}


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Now to search vector [3,4,5] for a value


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class Vec5{public static bool our_search(ArrayList vec,int first,int target) { if (vec.Count == 0) return false; if (first > vec.Count - 1) return false; else if ((int)vec[first]== target) return true; else return our_search(vec, first + 1,target); }


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public static void Main(string[] args) { ArrayList v = new ArrayList(); v.Add(3); v.Add(4); v.Add(5); int t = Convert.ToInt32(args[0]); Console.WriteLine("Element found: {0}", our_search(v, 0, t)); Console.Read(); }}


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Counting Els in [3,4,5,2]


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class Vec6{public static int our_count(ArrayList vec, int first, int target) { if (vec.Count == 0) return 0; if (first > vec.Count - 1) return 0; else if ((int)vec[first]== target) return 1+our_count(vec,first+1,target); else return 0+our_count(vec,first+1,target); }


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:

public static void Main(string[] args) { ArrayList v = new ArrayList(); v.Add(3); v.Add(4); v.Add(5); v.Add(3); int t = Convert.ToInt32(args[0]); Console.WriteLine("No of Element found: {0}", our_count(v, 0, t)); Console.Read(); }}


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Q3

Write a java application which contains a recursive function

static boolean gr_than_tar(Vector v,int first,int target){

which return ‘true’ if all elements of array ‘a’ are > target otherwise returns ‘false’

e.g. gr_than_tar([2,3,4], 0, 1) returns true

gr_than_tar([2,3,4], 0, 2) returns false

Exercises:


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Q4


static int largest(int[] a, int first){

which return the maximum element in the Array



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Q5. Rewrite Q4 but this time using a Vector


static int largest(Vector v,int first){

which return the maximum element in the Vector



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Functional Programming -other key features

•Higher Order Functions•Lazy Evaluation•Currying•Lambda expressions (Nameless functions)

Need a pure Functional Language to implement these fully

we will briefly look at Haskell


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What is Hugs?• Hugs is an interactive Haskell system, and

is the most widely used implementation of Haskell;

• Hugs is available on the web from:

www.haskell.org/hugs

Download has type ‘.msi’ may have to download a mircosoft application to unzip this also


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•From File Manager

•Change to folder

c:/Program Files/Hugs98/lib

•double click on “Prelude.hs

Starting Hugs


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Starting Hugs% hugs

__ __ __ __ ____ ___ _________________________________________|| || || || || || ||__ Hugs 98: Based on the Haskell 98 standard||___|| ||__|| ||__|| __|| Copyright (c) 1994-1999||---|| ___|| World Wide Web: http://haskell.org/hugs|| || Report bugs to: [email protected]|| || Version: February 2000 _________________________________________

Haskell 98 mode: Restart with command line option -98 to enable extensions

Reading file "/usr/local/share/hugs/lib/Prelude.hs": Hugs session for:/usr/local/share/hugs/lib/Prelude.hsType :? for help

Prelude>


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The Hugs > prompt means that the Hugs system is ready to evaluate an expression.

For example:> 5+2*311

> (5+2)*321

> sqrt (3^2 + 4^2)5.0


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The Standard PreludeThe library file Prelude.hs provides a large number of standard functions. In addition to the familiar numeric functions such as + and *, the library also provides many useful functions on lists.

• Calculating the length of a list:

> length [1,2,3,4]4


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> reverse [1,2,3,4][4,3,2,1]

> [1,2,3] ++ [4,5,6][1,2,3,4,5,6]

> head [1,2,3,4]1

> tail [1,2,3,4][2,3,4]

> last [1,2,3,4]4


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double x = x + x

quadruple x = double (double x)

When developing a Haskell script, it is useful to keep two windows open, one running an editor for the script, and the other running Hugs.

Start an editor, type in the following two function definitions, and save the script as test.hs

Save in: c:/Program Files/Hugs98/lib

From Hugs prompt type > :load “test.hs”


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> quadruple 1040

> double 24

Now both Prelude.hs and test.hs are loaded, and functions from both scripts can be used:

increment :: Int -> Intincrement n = (n+1)


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larger :: Int -> Int -> Intlarger a b = if a > b then a else b

Larger


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> :r

If we modify file “test.hs”

must reload


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factorial :: Int -> Intfactorial n = if n==0 then 1 else n * factorial (n - 1)


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search :: Int -> [Int] -> Boolsearch t [] = Falsesearch t (x:xs) = if x == t then True else search t xs

> search 5 [3,4,5,6]True

> search 8 [3,4,5,6]False


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count_els :: [Int] -> Intcount_els [] = 0count_els (x:xs) = 1 + count_els xs

> count_els [3,4,5,6]4


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> count_occur 4 [3,4,5,6,4]2

count_occur :: Int -> [Int] -> Intcount_occur t [] = 0count_occur t (x:xs) = if x==t then 1 + count_occur t xs else 0 + count_occur t xs


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add_els :: [Int] -> Intadd_els [] = 0add_els (x:xs) = x + add_els xs

> add_els [3,4,5,6]18


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mylast :: [Int] -> Intmylast (x:xs) = if xs ==[] then x else mylast xs add_fl :: [Int] -> Intadd_fl [] = 0add_fl (x:xs) = x + mylast xs

> add_fl[3,4,5,6]9


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Q1a. Define function ‘decrement n’

decrement 10 9

Q1b smaller 8 6 6

Q1c equal 8 6 equal 8 8 6 True

Q1d length [2,3,4,4] 4

Q1e product [ 2, 3,4 ] 24i.e. 2 * 3 * 4

Huskell


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Q2. Compete the following function count_small :: [Int]->Int

count_small [4, 5, 16, 2, 6, 22] return 4count_small [ ] returns 0

Returns the number of items in the list less than 10


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Q3. Compete the following function max :: [Int]->Int

max [4,5,6,2,6,1] returns 6max [ ] returns 0


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Q4. Compete the following function gr_than_tar :: [Int]->Int-> Bool

gr_than_tar [4,5,6,2,6] 1 returns Truegr_than_tar [4,5,6,2,6] 2 returns False

i.e Are all els in Array greater than target

Huskell Q2


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Higher Order Functions

A Function which takes another function as parameter

Function can also be returned as a result

I.e Functions treated as Objects

increment :: Int -> Intincrement n = (n+1)


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The Map FunctionThe higher-order library function called map applies a function to every element of a list.

map :: (a b) [a] [b]

For example: > map increment [1,2,3,4]

[2,3,4,5]


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The map function can be defined as

map1 :: (a -> b) -> [a] -> [b]map1 f [] = []

map1 f (x:xs) = f x : map1 f xs

> map1 increment [1,2,3,4]

[2,3,4,5]


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Write a function which : - takes a function & list of Ints as paramemets - applies function to first & last element - return result as sum of these two wlaues

>sum_fl increment [3,4,5,6] 11 i.e 4 + 7

sum_fl :: (Int -> Int) -> [Int] -> Intsum_fl f (x:xs) = f x + f (last xs)


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small :: Int -> Boolsmall n = if n<10 then True else False

myfilter :: (a -> Bool) -> [a] -> [a]myfilter p [] = []myfilter p (x:xs) = if p x == True then x : myfilter p xs else myfilter p xs


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Q5. Define a function ‘greater’

greater :: Int -> Int ->Bool

greater 2 3 False

greater 3 2True


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Q6. Now define a function

myfilter2 :: (Int -> Int -> Bool) -> [Int] -> Int -> [Int]

myfilter greater [3,4,5,6,2,7] 3 [4,5,6,7]

I.e returns all values in List greater than 3


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Q7. Define a function ‘large’

Outline a Haskell function ‘large ’ which takes an integer as parameter and returns True if the value is greater than 9

Examples: large 16 returns True large 4 returns False


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Q8. This question concerns Higher Order Functions in Haskell.

Outline a Haskell function ‘count_qualify’ which takes 2 parameters -a function (like large ) which takes an integer and returns a Boolean -a list of integers

Our new higher function will apply the supplied function to each

element of the list in turn.

The overall function will count and return the number of elements

in the list satisfying the inner function

Examples: count_qualify large [15, 6, 17, 5, 18] returns 3 count_qualify large [5, 6, 17, 5, 8] returns 1


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Useful Hugs CommandsCommand Meaning

:l name load script name:r reload current script:e name edit script name:e edit current script:t expr show type of expr:? show all commands:q quit

Sept 2004 SDP-MSc Slide 1 Section 6 Introduction to Functional Programming.

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