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

… Single node is a Rooted Tree. If t1, t2, …, tn are RTs, so is: t1 t2 Rooted Trees Single node is a Rooted Tree. If t1, t2, …, tn are RTs, so is: … t1 t2 tn

Rooted Trees In scheme: (((())()())((())())((()))(()(()())))

Structural vs Ordinary Induction Structural Induction on Rooted Trees reduces to Strong Induction on SIZE of subtrees. But what is the "size" of the COSINE function? Data objects may be infinite: The 18.01 Functions

Structural vs Ordinary Induction For 18.01F, replace structural induction by induction on number of steps to construct an object . Function no. of steps x, sin(x), cos(x), ex : 1 log(x) [inverse,ex] 2 cos(log(x)) [cos,log,compose] 4 x ·cos(log(x)) [id,times,] 6

Structural vs Ordinary Induction derivation tree for x ·cos(log(x)) * x cos(log x) x  cos(log x) cos inv log ex

Structural vs Ordinary Induction But Structural Induction also applies to Infinite Trees

… … … … … Infinitely wide Game Tree for Problem 1 (0,0) (0,1) (1,0) (1,1) (0,2) (2,0) (1,2) (2,1) … … (0,0) (0,0) … … (0,0) (0,1) (0,2) (0,1) (0,2) Infinitely wide

In-Class Problem Problem 1

… … Single node is an FP tree If t1, t2, …, tn are FPs, so is: t1 t2 Finite-Path Trees Single node is an FP tree If t1, t2, …, tn are FPs, so is: … … t1 t2 tn May have infinitely many subtrees

Finite-Path Trees with Rooted Trees are Finite-Path Trees with finitely many nodes

Finite-Path Trees … …

Finite-Path Trees … …

Finite-Path Trees: Unboundedly Deep

Finite-Path Trees: Wide & Deep

But every downward path in an FP eventually stops Finite-Path Trees But every downward path in an FP eventually stops

Proof by structural induction on def. of FP Finite-Path Trees Proof by structural induction on def. of FP Base Case (t is one node): Has only a 0 length path

Proof by structural induction on def. of FP Finite-Path Trees Proof by structural induction on def. of FP Induction Step (t has subtrees): Assume in subtrees all down paths are finite. Now path from root of t goes to root of subtree. Rest of the path in subtree must be finite by induction. QED

In-Class Problem Problem 2

Mathematics for Computer Science MIT 6.042J/18.062J Sums, Products & Asymptotics Copyright © Albert Meyer, 2002. Prof. Albert Meyer & Dr. Radhika Nagpal

C. F. Gauss Picture source: http://www-groups.dcs.st-and.ac.uk/~history/PictDisplay/Gauss.html

Sum for children 8900 + 9003 + 9106 + 9209 + 9312 + 9415 + … 9930 + … 10,445 + 10,960 + … +11,372

Sum for children Nine-year old Gauss (so the story goes) saw that each number was 103 greater than the previous one.

Sum for children 8900 + (8900 + 103) + (8900 + 2(103)) + (8900 + 3(103)) + … + (8900 + 24(103)) = 8900(25) + 103(1 + 2 + 3 + … + 24)

Arithmetic Series A ::= 1 + 2 + … + (n-1) + n

A ::= 1 + 2 + … + (n-1) + n A = 1 + 2 + … + (n-1) + n Arithmetic Series A ::= 1 + 2 + … + (n-1) + n A = 1 + 2 + … + (n-1) + n

A ::= 1 + 2 + … + (n-1) + n A = n + (n-1) + … + 2 + 1 Arithmetic Series A ::= 1 + 2 + … + (n-1) + n A = n + (n-1) + … + 2 + 1

2A =(n+1)+(n+1) + … + (n+1) + (n+1) Arithmetic Series A ::= 1 + 2 + … + (n-1) + n A = ------------------------------------------------ n + (n-1) + … + 2 + 1 2A =(n+1)+(n+1) + … + (n+1) + (n+1)

2A =(n+1)+(n+1) + … + (n+1) + (n+1) Arithmetic Series A ::= 1 + 2 + … + (n-1) + n A = ------------------------------------------------ n + (n-1) + … + 2 + 1 2A =(n+1)+(n+1) + … + (n+1) + (n+1) = n(n+1)

Arithmetic Series So

Geometric Series

Geometric Series

Geometric Series

Geometric Series 1 - xn+1

Geometric Series n+1

The future value of $$. I will promise to pay you $100 Annuities The future value of $$. I will promise to pay you $100 in exactly one year, if you will pay me $X now.

X = $100/1.03 ≈ $97.09 My bank will pay me 3% interest. Annuities My bank will pay me 3% interest. I can’t lose if you pay me: X = $100/1.03 ≈ $97.09

97.09¢ today is worth $1.00 in a year Annuities 97.09¢ today is worth $1.00 in a year $1.00 in a year is worth $1/1.03 today $n in a year is worth $nr today, where r = 1/1.03

$n in two years is worth $nr2 today $n in k years is worth $nrk today Annuities $n in two years is worth $nr2 today $n in k years is worth $nrk today

I will pay you $100/year for 10 years If you will pay me $Y now. Annuities I will pay you $100/year for 10 years If you will pay me $Y now. I can’t lose if you pay me 100r + 100r2 + 100r3 + … + 100r10 =100r(1+ r + … + r9) = 100r(1-r10)/(1-r) = $853.02

In-Class Problem Problems 3 & 4