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
If n  E, then n + 2  E Induction case
If n  E, then -n  E Induction case
E : 0
, 0+2

4. 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

5. 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, -2, -4, -6, …

6. 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!

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

8. 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

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

10. 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

11. 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

12. 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
Case 3: xy
has kx + ky of each. OK

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

14. In-Class Problem
Problem 1

15. Recursive Data Types
Rooted Binary Trees

16. 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:

17. 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’

18. A Rooted Binary Tree Example

19. Recursive Functions on RBT
nodes(t) → Vertex Set of t
nodes( )::= { },
nodes(makeleft(t))::= nodes(t){ }
Remember, is not supposed to be in Nodes(t)
nodes(makeright(t))::= nodes(t){ }
nodes(makeboth(t1, t2))::=
nodes(t1)nodes(t2){ }

20. Example
= {{ }, { }, { }, { }, { },
{ }, { }, { }, { } }
nodes( )

21. Recursive Functions on RBT
depth(t) → Power Set of t
depth( )::= 1
depth(makeleft(t))::= 1 + depth(t)
depth(makeright(t))::= 1 + depth(t)
depth(makeboth(t1, t2))::= max(depth(t1)+ 1,depth(t2)+ 1)

22. Example
depth( )
= 4

23. Proof by Structural Induction
Lemma: |nodes(t)|  2depth(t)
Proof by Structural Induction
Base Case: t =
|nodes(t)| = 1 = 20 = 2depth(t) OK!

24. Induction Case: makeleft(t)
Assume: |nodes(t)|  2depth(t)
To Prove:|nodes(makeleft(t))|  2depth(makeleft(t))
|nodes(makeleft(t))|
= |nodes(t)| + 1 by def. of makeleft
 2depth(t) + 1 by induction hyp.
 2depth(t) + 2depth(t)
= 2depth(t)+1
= 2depth(makeleft(t)) by def. of depth

25. Induction Case: makeright(t)
Same

26. Induction Case: makeboth(t1, t2)
Assume: |nodes(t1)|  2depth(t1)
|nodes(t2)|  2depth(t2)
To Prove:
|nodes(makeboth(t1, t2))|  2depth(makeboth(t1,t2))
Proof:
VERY SIMILAR

27. In-Class Problem
Problems 2 & 3

28. Examples so far: But what if the objects are all infinite?
Can reduce structural induction to
Strong Induction on SIZE of parts.
But what if the objects are all infinite?
simple example: recursive definition of the set of functions on real numbers from
what is the "size" of the COSINE function?

29. In-Class Problem
Problem 4

30. In-Class Problem
Problem 5

31. Infinitely Wide Trees … …

32. Infinitely Wide Trees

33. Infinitely Wide Trees

34. Infinitely Wide Trees … …

35. In-Class Problem
Problems 6 & 7