Mathematics for Computer Science MIT 6 042 J18

Mathematics for Computer Science MIT 6. 042 J/18. 062 J Recursive Definitions and Structural Induction Copyright © Albert Meyer, 2002. Prof. Albert Meyer & Dr. Radhika Nagpal March 6, 2002 1

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 March 6, 2002 2

Example Definition: set E Define set E Z, recursively: • 0 E Base Case • If n E, then n + 2 E • If n E, then -n E E: Induction case 0, 0+2 March 6, 2002 3

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 March 6, 2002 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: Also, 0, 2, 4, 6, … All even numbers -2, -4, -6, … March 6, 2002 5

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! March 6, 2002 6

Example Definition: set E So E is exactly: The Even Integers March 6, 2002 7

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 March 6, 2002 8

Proof by Structural Induction on S Lemma: Every x in S has an equal number of a’s and b’s. March 6, 2002 9

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 March 6, 2002 10

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 March 6, 2002 11

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 March 6, 2002 12

Theorem: S = EQ Show by strong induction on length of string in EQ. Proof in Notes 5. March 6, 2002 13

In-Class Problem 1 March 6, 2002 14

Recursive Data Types Rooted Binary Trees March 6, 2002 15

Rooted Binary Trees Defined recursively as follows: • A single node r is in T: March 6, 2002 16

Rooted Binary Trees If t, t’ are RBTs, then the following are RBTs Makeleft(t) t Makeright(t) Makeboth(t, t’) t t’ t March 6, 2002 17

A Rooted Binary Tree Example March 6, 2002 18

Recursive Functions on RBT nodes(t) → Vertex Set of t }, nodes(makeleft(t)): : = nodes(t) { } nodes( Remember, ): : = { is not supposed to be in Nodes(t) nodes(makeright(t)): : = nodes(t) { } nodes(makeboth(t 1, t 2)): : = nodes(t 1) nodes(t 2) { } March 6, 2002 19

Example nodes( ) = {{ }, { }, { }, { } } March 6, 2002 20

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(t 1, t 2)): : = max(depth(t 1)+ 1, depth(t 2)+ 1) March 6, 2002 21

Example depth( ) =4 March 6, 2002 22

Lemma: |nodes(t)| 2 depth(t) Proof by Structural Induction Base Case: t = |nodes(t)| = 1 = 20 = 2 depth(t) OK! March 6, 2002 23

Induction Case: makeleft(t) Assume: |nodes(t)| 2 depth(t) To Prove: |nodes(makeleft(t))| 2 depth(makeleft(t)) |nodes(makeleft(t))| = |nodes(t)| + 1 2 depth(t) + 2 depth(t) by def. of makeleft by induction hyp. = 2 depth(t)+1 = 2 depth(makeleft(t)) by def. of depth March 6, 2002 24

Induction Case: makeright(t) Same March 6, 2002 25

Induction Case: makeboth(t 1, t 2) Assume: |nodes(t 1)| 2 depth(t ) |nodes(t 2)| 2 depth(t ) 1 2 To Prove: |nodes(makeboth(t 1, t 2))| 2 depth(makeboth(t , t )) 1 2 Proof: VERY SIMILAR March 6, 2002 26

In-Class Problems 2 & 3 March 6, 2002 27

Examples so far: 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 18. 01. what is the "size" of the COSINE function? March 6, 2002 28

In-Class Problem 4 March 6, 2002 29

In-Class Problem 5 March 6, 2002 30

Infinitely Wide Trees … … March 6, 2002 31

Infinitely Wide Trees March 6, 2002 32

Infinitely Wide Trees March 6, 2002 33

Infinitely Wide Trees … … March 6, 2002 34

In-Class Problems 6 & 7 March 6, 2002 35