Recursion

1. Recursion

2. Definitions I • A recursive definition is a definition in which the thing being defined occurs as part of its own definition • Example: A list consists of: • An open parenthesis, "(" • Zero or more atoms or lists, and • A close parenthesis, ")"

3. Definitions II • Indirect recursion is when a thing is defined in terms of other things, but those other things are defined in terms of the first thing • Example: A list is: • An open parenthesis, • Zero or more S-expressions, and • A close parenthesis • An S-expression is an atom or a list

4. Understanding recursion • The usual way to teach recursion is to “trace through” a recursion, seeing what it does at each level • This may be a good way to understand how recursion works... • ...but it's a terrible way to try to use recursion • There is a better way

5. Base cases and recursive cases • Every valid recursive definition consists of two parts: • One or more base cases, where you compute the answer directly, without recursion • One or more recursive cases, where you do part of the work, and recur with a simpler problem

6. Information hiding • function spread (int A[], int size) { int max, min; sort(A, size); min = A[0]; max = A[size - 1]; return max - min;} • Can you understand this function without looking at sort?

7. Stepping through called functions • Functions should do something simple and understandable • When you try to understand a function, you should not have to step through the code of the functions that it calls • When you try to understand a recursive function, you should not have to step through the code of the functions it calls

8. We have small heads • It's hard enough to understand one level of one function at a time • It's almost impossible to keep track of many levels of the same function all at once • But you can understand one level of one function at a time... • ...and that's all you need to understand in order to use recursion well

9. The four rules • Do the base cases first • Recur only with a simpler case • Don't use global or reference variables • Don't look down

10. Do the base cases first • Every recursive function must have some things it can do without recursion • These are the simple, or base, cases • Test for these cases, and do them first • This is just writing ordinary, nonrecursive code

11. Recur only with a simpler case • If the problem isn't simple enough to be a base case, break it into two parts: • A simpler problem of the same kind (for example, a smaller number, or a shorter list) • Extra work not solved by the simpler problem • Combine the results of the recursion and the extra work into a complete solution • “Simpler” means “more like a base case”

12. Example 1: member • Is value X a member of list L? • boolean member(X, L) { • if (L is the empty list) return false; // this is a base case • if (X equals the first element in L) return true; // another base case • return member(X, L - first element);// simpler because more like empty list}

13. MEMBER in Lisp (DEFUN MEMBER (X L) (COND; base case: L is empty ((NULL L) NIL); base case: X = (CAR L) ((EQ X (CAR L)) T); recur with a case that is simpler ; because it’s more like 1st base case (T (MEMBER X (CDR L))) ) )

14. Example 2: double • Double every element of a list of numbers • function double(L) { • if (L is the empty list) return the empty list; // base case • else { L2 = double (L - first element); // recur • D = 2 * first element in L; // extra work • return (list made by adding D to L2); // combine }}

15. DOUBLE in Lisp (DEFUN DOUBLE (L) (COND; if L is the empty list, return the empty list ((NULL L) ()) ; otherwise, double the first number and ; cons it to the double of the rest of the list (T (CONS (* 2 (CAR L)) (DOUBLE (CDR L))) ) )

16. It's OK to use locals variables and parameters passed by value • A function has its own copy of • local variables • parameters passed by value • Each level of a recursive function has its own copy of these variables and parameters • Changing them at one level does not change them at other levels • One level can't interfere with another level

17. It's bad to use global variables or parameters passed by reference • There is only one copy of a global variable • If a parameter is passed by reference, there is only one copy of it • If such a variable is changed by a recursive function, it's changed at all levels • The various levels interfere with one another • This can get very confusing • Don't let this happen to you!

18. Don't look down • When you write or debug a recursive function, think about this level only • Wherever there is a recursive call, assume that it works correctly • If you can get this level correct, you will automatically get all levels correct • You really can't understand more than one level at a time, so don't even try

19. MEMBER again • (DEFUN MEMBER (X L) (COND ((NULL L) NIL) • This says: if list L is empty, then X isn’t an element of L • Is this a true statement? • ((EQ X (CAR L)) T) • This says: if X = the first element in L, then it’s in L • Is this a true statement? • (T (MEMBER X (CDR L))) ) ) • This says: if X isn’t the first element of L, then X is in L if and only if X is in the tail of L • Is this a true statement? • Did we cover all possible cases? • Did we recur only with simpler cases? • Did we change any global variables? • We’re done!

20. Reprise • Do the base cases first • Recur only with a simpler case • Don't use global or reference variables • Don't look down

21. The End