recursion - com1022 functional programming and reasoning · 2010-12-07 · recursion com1022...

RecursionCOM1022 Functional Programming and Reasoning

Dr. Hans Georg Schaathun and Prof. Steve Schneider

University of Surrey

Autumn 2010 Week 10

Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week 10 1 / 35

This session

After this session, you shouldunderstand the principle of recursionbe able to use recursion over integers to solve simple problems

ReferenceThompson, Chapter 4


Modular programming

Outline

1 Modular programming

2 Primitive recursion

3 General forms of recursion

4 Iteration

5 Error handling

6 Conclusion


Modular programming

Software development

The key to problem solvingSplit the task into smaller and manageable problems

Software development is the sameEach subproblem gives a function

Many subproblems apply to different tasksWrite the functions such that they can be reused


Modular programming

Planning the development

If I had all the functions in the world, which would I use?

1 This question points to subproblems2 Then move to the key functions – one at a time3 Apply the same question to them

This idea apply in a myriad of waysTodays main topic is a special case

known as recursion


Primitive recursion

Outline




4 Iteration

5 Error handling

6 Conclusion


Primitive recursion

A recursive definition

A recursive function is defined in terms of itself.

Factorial: n! = 1 · 2 · 3 · . . . · nRecursive definition:

0! = 1 (1)n! = n · (n − 1)! for n > 0 (2)

A recursive function is defined in terms of itself.

Why is this not a circular definition?


Primitive recursion

The basic components

The recursive case (n! = (n − 1)! · n)large cases (for n) are reduced to a smaller case (n − 1)

The base case (0! = 1)some small case must be solved explicitely

Without the base case, recursion would never endn,n − 1,n − 2, . . . ,1,0,−1, . . . ,−∞

Without the recursive case, everything would be explicitone line for every n (0,1,2, . . . ,∞)


Primitive recursion

Recursion in Haskell

Haskell supports recursive definitions. Two different styles:

Pattern matching:

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

or by cases:

factorial :: Int -> Intfactorial n| n == 0 = 1| otherwise = n * factorial (n-1)


Primitive recursion

For example

factorial 4= 4 · (factorial 3)

= 4 · 3 · (factorial 2)

= 4 · 3 · 2 · (factorial 1)

= 4 · 3 · 2 · 1 · (factorial 0)

= 4 · 3 · 2 · 1 · 1= 24


Primitive recursion

Why recursion?

Fundamental principle in mathematics and in computerprogramming

Used in all programming paradigmsyou will see it in Java

Recursion tends to make it simple to1 prove correctness2 define and understand algorithms

It often makes computationally efficient algorithms


Primitive recursion

Primitive recursion

What if we knew the value of f (n − 1). How would we definef (n)?

1 f 0 = ...2 f n = ... f (n-1) ...

The recursive case depends onf (n-1)expressions independent of f


Primitive recursion

Integer square root

The integer square root of a number n is the greatest integer whosesquare is less than or equal to n.Examples

intsqrt 15 = 3intsqrt 16 = 4

Exercise: Define intsqrt using primitive recursion.

intsqrt 0 = ???

intsqrt n = ...(intsqrt (n-1))...

If you know intsqrt (n-1)then how can that help you work out intsqrt n ?


Primitive recursion

Cuts and pieces

Exercise: How many pieces can the 2-d plane be separated into withn lines?

pieces 0 = ???

pieces n = ???

If you know pieces (n-1)then how can that help you work out pieces n ?


Primitive recursion

Summing a function

Exercise: Given a function f :: Int -> Int, how do I calculateΣn

i=1f (i) ?

sigma f 0 = ???

sigma f n = ???

If you know sigma f (n-1)then how can that help you work out sigma f n ?

Observe that sigma is a higher-order function. This means that it cantake another function f as an argument.

Question: What is the type of sigma ?


General forms of recursion

Outline




4 Iteration

5 Error handling

6 Conclusion



Many forms of recursion

Sometimes the recursive case does not call n − 1

More generally:

For which values of k would f (k) help us define f (n)?

Sometimes we need f (n − 1) and f (n − 2)

Sometimes we can work with f (n/2)

In general we look for f on some ‘smaller’ arguments



Divisors

How many times can a factor p divide into a number m?

For given p and m, want the largest n such that pn divides m.

Examplespowerdiv 2 8 = 3

powerdiv 3 8 = 0

powerdiv 3 18 = 2

powerdiv 3 1 = 0

powerdiv 2 1000 = 3

Define powerdiv p m in terms of powerdiv on ‘smaller’ arguments.

What’s the base case?What’s the recursive case?[note that powerdiv p (m-1) does not help towards powerdivp m]



Split-and-Conquer

qn = q · q · . . . · q (3)

How do you compute this efficiently?Primitive recursion can be used here: but n − 1 multiplications isquite expensive

q0 = 1, (4)

q1 = q, (5)

qn =

{(qbn/2c)2, if n is even(qbn/2c)2 · q, if n is odd

(6)



A Haskell example

myExp q 0 = 1myExp q 1 = qmyExp q n| n ‘mod‘ 2 == 0 = h*h| n ‘mod‘ 2 == 1 = h*h*qwhere h = myExp q (n ‘div‘ 2)

where allows you to define a variable for internal use in thedefinition



Split-and-Conquer

This is a split-and-conquer algorithmeach recursive step halves the problemreaches base case in 2 log2 n steps.

2 log2 n << n − 1 (except for small n)



Fibonacci numbers

Fibonacci sequence

1,1,2,3,5,8,13,21,34,55,89,144,233, . . .

Each number is the sum of the two preceeding numbersHow do we write this in recursive form?For example

fib 1 = 1fib 2 = 1fib n = fib (n-1) + fib (n-2)

Each step requires two preceeding numbers



Efficiency

fib n = fib (n-1) + fib (n-2)fib (n-2) is calculated twice

once for fib (n-1) and once for fib n

fib (n-3) is calculated three timesonce for fib (n-2) and twice (!) for fib (n-1)

The compiler/interpreter may or may not optimisei.e. remember and reuse previous calculation

This is explored as an exercise.


Iteration

Outline




4 Iteration

5 Error handling

6 Conclusion


Iteration

Space considerations

Recursion can be expensive in terms of space.

Each recursive call must keep track of the context it was called in, untilthe final computation once the base case is reached.

Recall:

factorial 4= 4 · (factorial 3)

= 4 · 3 · (factorial 2)

= 4 · 3 · 2 · (factorial 1)

= 4 · 3 · 2 · 1 · (factorial 0)

= 4 · 3 · 2 · 1 · 1= 24


Iteration

Iteration

If a recursive call has no surrounding context, then the spacerequirements will not grow.

A recursive definition of this form is called an iterative definition.

??? But f n = f (n-1) is not much use ???

Generally more arguments are required:f m n = f (...m...n...) (n-1)


Iteration

Factorial revisited

partfact tot 0 = tot

partfact tot n = partfact (tot * n) (n-1)

factorial n = partfact 1 n

factorial 4

= partfact 1 4

= partfact 4 3

= partfact 12 2

= partfact 24 1

= partfact 24 0

= 24


Iteration

Highest Common Factor

The highest common factor of m and n — the highest number thatdivides into both of them.Examples

hcf 3 8 = 1

hcf 4 6 = 2

hcf 4 12 = 4

hcf 91 299 = 13

hcf 5 5 = 5

Question: Give an iterative definition of hcf m n:Base case ?Recursive call (on ‘smaller’ arguments) ?

[think about which of the examples above are a base case —where the answer is immediate ?]


Iteration

hcf base case — worked example


Iteration

hcf inductive step — worked example


Error handling

Outline




4 Iteration

5 Error handling

6 Conclusion


Error handling

Erroneous callsFactorial

factorial (-2)

What happens?

(−2)(−3)(−4)(−5) . . . (−∞)You never reach any base case

n! is undefined for n < 0factorial (-2) is an error


Error handling

Declaring an error

The error clause

fac :: Int -> Intfac n| n < 0 = error "Undefined for negative numbers"| n == 0 = 1| n > 0 = n * fac (n-1)

The error clausehalts the programissues the given error message


Conclusion

Outline




4 Iteration

5 Error handling

6 Conclusion


Conclusion

Concluding remarks

Recursion is a fundamental method in function definitionsgo away and practice using it

(Pure) Functional languages do not have loopsrecursion is used insteadeven when loops are available, recursion may be easier to read

We will later return to recursion on listsand recursion will be used in many later exercises


recursion - com1022 functional programming and reasoning · 2010-12-07 · recursion com1022...

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