This session. Recursion. Planning the development. Software development. COM1022 Functional Programming and Reasoning

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1 This session Recursion COM1022 Functional Programming and Reasoning Dr. Hans Georg Schaathun and Prof. Steve Schneider University of Surrey After this session, you should understand the principle of recursion be able to use recursion over integers to solve simple problems Reference Thompson, Chapter 4 Autumn 2010 Week 10 Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week 10 1 / 35 Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week 10 2 / 35 Modular programming Software development Modular programming Planning the development The key to problem solving Split the task into smaller and manageable problems Software development is the same Each subproblem gives a function Many subproblems apply to different tasks Write the functions such that they can be reused If I had all the functions in the world, which would I use? 1 This question points to subproblems 2 Then move to the key functions one at a time 3 Apply the same question to them This idea apply in a myriad of ways Todays main topic is a special case known as recursion Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week 10 4 / 35 Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week 10 5 / 35

2 A recursive definition The basic components A recursive function is defined in terms of itself. Factorial: n! = n Recursive 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? 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 end n, n 1, n 2,..., 1, 0, 1,..., Without the recursive case, everything would be explicit one line for every n (0, 1, 2,..., ) Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week 10 7 / 35 Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week 10 8 / 35 Recursion in Haskell For example Haskell supports recursive definitions. Two different styles: Pattern matching: factorial :: Int -> Int factorial 0 = 1 factorial n = n * factorial (n-1) or by cases: factorial :: Int -> Int factorial n n == 0 = 1 otherwise = n * factorial (n-1) factorial 4 = 4 (factorial 3) = 4 3 (factorial 2) = (factorial 1) = (factorial 0) = = 24 Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week 10 9 / 35 Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35

3 Why recursion? Fundamental principle in mathematics and in computer programming Used in all programming paradigms you will see it in Java Recursion tends to make it simple to 1 prove correctness 2 define and understand algorithms It often makes computationally efficient algorithms What if we knew the value of f (n 1). How would we define f (n)? 1 f 0 =... 2 f n =... f (n-1)... The recursive case depends on f (n-1) expressions independent of f Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35 Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35 Integer square root Cuts and pieces The integer square root of a number n is the greatest integer whose square is less than or equal to n. Examples intsqrt 15 = 3 intsqrt 16 = 4 Exercise: Define intsqrt using primitive recursion. intsqrt 0 =??? intsqrt n =...(intsqrt (n-1))... Exercise: How many pieces can the 2-d plane be separated into with n lines? pieces 0 =??? pieces n =??? If you know pieces (n-1) then how can that help you work out pieces n? If you know intsqrt (n-1) then how can that help you work out intsqrt n? Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35 Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35

4 Summing a function Many forms of recursion Exercise: Given a function f :: Int -> Int, how do I calculate Σ n i=1 f (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 can take another function f as an argument. Question: What is the type of sigma? 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 Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35 Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35 Divisors Split-and-Conquer How many times can a factor p divide into a number m? For given p and m, want the largest n such that p n divides m. Examples powerdiv 2 8 = 3 powerdiv 3 8 = 0 powerdiv 3 18 = 2 powerdiv 3 1 = 0 powerdiv = 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 powerdiv p m] Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35 q n = q q... q (3) How do you compute this efficiently? can be used here: but n 1 multiplications is quite expensive q 0 = 1, (4) q 1 = q, (5) { q n (q n/2 ) 2, if n is even = (q n/2 ) 2 (6) q, if n is odd Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35

5 A Haskell example Split-and-Conquer myexp q 0 = 1 myexp q 1 = q myexp q n n mod 2 == 0 = h*h n mod 2 == 1 = h*h*q where h = myexp q (n div 2) This is a split-and-conquer algorithm each recursive step halves the problem reaches base case in 2 log 2 n steps. 2 log 2 n << n 1 (except for small n) where allows you to define a variable for internal use in the definition Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35 Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35 Fibonacci numbers Efficiency Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,... Each number is the sum of the two preceeding numbers How do we write this in recursive form? For example fib 1 = 1 fib 2 = 1 fib n = fib (n-1) + fib (n-2) Each step requires two preceeding numbers 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 times once for fib (n-2) and twice (!) for fib (n-1) The compiler/interpreter may or may not optimise i.e. remember and reuse previous calculation This is explored as an exercise. Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35 Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35

6 Space considerations Recursion can be expensive in terms of space. Each recursive call must keep track of the context it was called in, until the final computation once the base case is reached. Recall: factorial 4 = 4 (factorial 3) = 4 3 (factorial 2) = (factorial 1) = (factorial 0) = = 24 If a recursive call has no surrounding context, then the space requirements 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) Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35 Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35 Factorial revisited Highest Common Factor 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 Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35 The highest common factor of m and n the highest number that divides into both of them. Examples hcf 3 8 = 1 hcf 4 6 = 2 hcf 4 12 = 4 hcf = 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?] Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35

7 hcf base case worked example hcf inductive step worked example Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35 Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35 Erroneous calls Factorial Error handling Declaring an error Error handling The error clause factorial (-2) What happens? ( 2)( 3)( 4)( 5)... ( ) You never reach any base case n! is undefined for n < 0 factorial (-2) is an error fac :: Int -> Int fac n n < 0 = error "Undefined for negative numbers" n == 0 = 1 n > 0 = n * fac (n-1) The error clause halts the program issues the given error message Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35 Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35

8 Concluding remarks Conclusion Recursion is a fundamental method in function definitions go away and practice using it (Pure) Functional languages do not have loops recursion is used instead even when loops are available, recursion may be easier to read We will later return to recursion on lists and recursion will be used in many later exercises Dr. Hans Georg Schaathun and Prof. Steve Schneider Recursion Autumn 2010 Week / 35