1. and Structural Induction
Mathematics for Computer Science MIT 6.042J/18.062J
Recursive Definitions
and
Structural Induction
Copyright © Albert Meyer, 2002.
Prof. Albert Meyer & Dr. Radhika Nagpal

2. Build something from a simpler version of the same thing:
Recursive Definitions
Build something from a simpler version of the same thing:
Base case(s) that don’t depend on anything else
Induction (Construction) case(s) that depend on simpler cases

3. Example Definition: set E
Define set E  Z, recursively:
0  E Base Case

4. Example Definition: set E
Define set E  Z, recursively:
0  E Base Case
E : 0

5. Example Definition: set E
Define set E  Z, recursively:
0  E Base Case
If n  E, then n + 2  E Induction case
If n  E, then -n  E Induction case
E : 0
, 0+2

6. Example Definition: set E
Define set E  Z, recursively:
0  E
If n  E, then n + 2  E
If n  E, then -n  E
E : 0, 2, 2+2
, (2+2)+2

7. Example Definition: set E
Define set E  Z, recursively:
0  E
If n  E, then n + 2  E
If n  E, then -n  E
E : 0, 2, 4, 6, … All even numbers
Also, , -4, -6, …

8. Example Definition: set E
So, E contains the even integers

9. Example Definition: set E
So, E contains the even integers
Anything Else?

10. Example Definition: set E
So, E contains the even integers
Anything Else?
No!

11. Recursive Definition: Extremal Clause
So, E contains the even integers
Anything Else?
No!
0  E
If n  E, then n + 2  E
If n  E, then -n  E

12. Recursive Definition: Extremal Clause
So, E contains the even integers
Anything Else?
No!
0  E
If n  E, then n + 2  E
If n  E, then -n  E
Implicit part of recursive definition:
THAT’S ALL!

13. Example Definition: set E
So E is exactly:
The Even Integers

14. Define set of strings, S  {a, b}*
Another Example
Define set of strings, S  {a, b}*
l  S, (the empty string)
If x, y  S, then the following are in S:
axb
bxa xy

15. Proof by Structural Induction on S
Lemma:
Every x in S has an equal number of a’s and b’s.

16. Structural Induction on S
Proof: Let
EQ ::= {strings with equal # of a’s and b’s}
P(s) ::= s  S  s  EQ

17. Structural Induction on S
Proof: Let
EQ ::= {strings with equal # of a’s and b’s}
P(s) ::= s  S  s  EQ
Base Case: s = l.
0 a’s and 0 b’s. OK

18. Structural Induction on S
Inductive Step:
Assume: P(x) and P(y)
To Prove: P(axb), P(bxa), and P(xy)

19. Structural Induction on S
Inductive Step:
Assume: P(x) and P(y)
To Prove: P(axb), P(bxa), and P(xy)
Case 1: axb
has k + 1 a’s and b’s
if x has k of each. OK

20. Structural Induction on S
Inductive Step:
Assume: P(x) and P(y)
To Prove: P(axb), P(bxa), and P(xy)
Case 2: bxa SAME

21. Structural Induction on S
Inductive Step:
Assume: P(x) and P(y)
To Prove: P(axb), P(bxa), and P(xy)
Case 3: xy
has kx + ky of each. OK

22. Theorem: S = EQ Show by strong induction on length of string in EQ.
Proof in Notes 5.

23. In-Class Problem
Problem 1

24. Recursive Data Types
Rooted Binary Trees

25. Defined recursively as follows: A single node r is in T:
Rooted Binary Trees
Defined recursively as follows:
A single node r is in T:

26. t t t t’ Rooted Binary Trees
If t, t’ are RBTs, then the following are RBTs
Makeleft(t) t
Makeright(t) t
Makeboth(t,t’) t t’

27. A Rooted Binary Tree Example