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_up(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 1 st 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