Iteration and Invariants CoSc 450: Programming Paradigms 03 Tail - - PowerPoint PPT Presentation

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Iteration and Invariants CoSc 450: Programming Paradigms 03 Tail - - PowerPoint PPT Presentation

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CoSc 450: Programming Paradigms 03 Iteration and Invariants CoSc 450: Programming Paradigms 03 Tail recursion: What our author calls iteration is more commonly called tail recursion. A good optimizing compiler can convert a

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CoSc 450: Programming Paradigms

Iteration and Invariants

03

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SLIDE 2

CoSc 450: Programming Paradigms Tail recursion:

  • What our author calls “iteration” is more

commonly called “tail recursion”.

  • A good optimizing compiler can convert a tail

recursive program into an iterative one (as a loop). 03

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SLIDE 3

CoSc 450: Programming Paradigms Tail recursion:

  • What our author calls “iteration” is more

commonly called “tail recursion”.

  • A good optimizing compiler can convert a tail

recursive program into an iterative one (as a loop). 03

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SLIDE 4

CoSc 450: Programming Paradigms Tail recursion:

  • What our author calls “iteration” is more

commonly called “tail recursion”.

  • A good optimizing compiler can convert a tail

recursive program into an iterative one (as a loop). 03

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SLIDE 5

CoSc 450: Programming Paradigms Not tail recursion: 03

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SLIDE 6

CoSc 450: Programming Paradigms Not tail recursion: 03

n! = n ∗ (n − 1)!

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SLIDE 7

CoSc 450: Programming Paradigms Not tail recursion: 03

n! = n ∗ (n − 1)!

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SLIDE 8

CoSc 450: Programming Paradigms Not tail recursion: 03

n! = n ∗ (n − 1)!

Because after the recursive call additional processing must be done before the value is returned

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SLIDE 9

CoSc 450: Programming Paradigms General idea:

  • The function calls a helper function once.
  • The helper function is tail recursive, and calls

itself.

  • The helper function has an extra parameter.
  • The extra processing is done in the parameters
  • f the helper function.

03

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SLIDE 10

CoSc 450: Programming Paradigms General idea:

  • The function calls a helper function once.
  • The helper function is tail recursive, and calls

itself.

  • The helper function has an extra parameter.
  • The extra processing is done in the parameters
  • f the helper function.

03

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SLIDE 11

CoSc 450: Programming Paradigms General idea:

  • The function calls a helper function once.
  • The helper function is tail recursive, and calls

itself.

  • The helper function has an extra parameter.
  • The extra processing is done in the parameters
  • f the helper function.

03

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SLIDE 12

CoSc 450: Programming Paradigms General idea:

  • The function calls a helper function once.
  • The helper function is tail recursive, and calls

itself.

  • The helper function has an extra parameter.
  • The extra processing is done in the parameters
  • f the helper function.

03

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SLIDE 13

CoSc 450: Programming Paradigms General idea:

  • The function calls a helper function once.
  • The helper function is tail recursive, and calls

itself.

  • The helper function has an extra parameter.
  • The extra processing is done in the parameters
  • f the helper function.

03

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SLIDE 14

CoSc 450: Programming Paradigms Not tail recursion: 03

n! = n ∗ (n − 1)!

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SLIDE 15

CoSc 450: Programming Paradigms Not tail recursion: 03

n! = n ∗ (n − 1)!

Helper function, with two parameters a and b that computes a * b!

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SLIDE 16

CoSc 450: Programming Paradigms Not tail recursion: 03

n! = n ∗ (n − 1)!

Helper function, with two parameters a and b that computes a * b!

a · b! = (a · b) · (b − 1)!

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SLIDE 17

CoSc 450: Programming Paradigms Not tail recursion: 03

n! = n ∗ (n − 1)!

Helper function, with two parameters a and b that computes a * b!

a · b! = (a · b) · (b − 1)!

The main function calls the helper function with a value of one for a and n for b.

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CoSc 450: Programming Paradigms Write factorial and factorial-product. 03

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CoSc 450: Programming Paradigms Prove factorial-product is correct. 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

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SLIDE 20

CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Base case Code inspection: (factorial-product a 0) returns a. Math: Therefore, correct in base case.

a · 0! = a

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CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Base case Code inspection: (factorial-product a 0) returns a. Math: Therefore, correct in base case.

a · 0! = a

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CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Base case Code inspection: (factorial-product a 0) returns a. Math: Therefore, correct in base case.

a · 0! = a

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SLIDE 23

CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Base case Code inspection: (factorial-product a 0) returns a. Math: Therefore, correct in base case.

a · 0! = a

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SLIDE 24

CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Inductive case Prove that (factorial-product x b) terminates with value assuming that (factorial-product y (- b 1)) terminates with value as the inductive hypothesis.

x · b! y · (b − 1)!

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SLIDE 25

CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Inductive case Prove that (factorial-product x b) terminates with value assuming that (factorial-product y (- b 1)) terminates with value as the inductive hypothesis.

x · b! y · (b − 1)!

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SLIDE 26

CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Inductive case Prove that (factorial-product x b) terminates with value assuming that (factorial-product y (- b 1)) terminates with value as the inductive hypothesis.

x · b! y · (b − 1)!

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SLIDE 27

CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Inductive case

  • =
  • =
  • (a·b)·(b−1)!

=

  • a·b·(b−1)!

=

  • a·b!
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SLIDE 28

CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Inductive case

  • =
  • =
  • (a·b)·(b−1)!

=

  • a·b·(b−1)!

=

  • a·b!
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SLIDE 29

CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Inductive case

  • =
  • =
  • (a·b)·(b−1)!

=

  • a·b·(b−1)!

=

  • a·b!
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SLIDE 30

CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Inductive case

  • =
  • =
  • (a·b)·(b−1)!

=

  • a·b·(b−1)!

=

  • a·b!
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SLIDE 31

CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Inductive case

  • =
  • =
  • (a·b)·(b−1)!

=

  • a·b·(b−1)!

=

  • a·b!
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SLIDE 32

CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Inductive case

  • =
  • =
  • (a·b)·(b−1)!

=

  • a·b·(b−1)!

=

  • a·b!
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SLIDE 33

CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Inductive case

  • =
  • =
  • (a·b)·(b−1)!

=

  • a·b·(b−1)!

=

  • a·b!
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SLIDE 34

CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Inductive case

  • =
  • =
  • (a·b)·(b−1)!

=

  • a·b·(b−1)!

=

  • a·b!
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SLIDE 35

CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Inductive case

  • =
  • =
  • (a·b)·(b−1)!

=

  • a·b·(b−1)!

=

  • a·b!
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SLIDE 36

CoSc 450: Programming Paradigms 03

(define factorial-product (lambda (a b) ; Returns a*b!, b >= 0. (if (= b 0) a (factorial-product (* a b) (- b 1)))))

Inductive case

  • =
  • =
  • (a·b)·(b−1)!

=

  • a·b·(b−1)!

=

  • a·b!
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SLIDE 37

CoSc 450: Programming Paradigms 03 The power function

> (power 4 5) 1024 (power 4 5) returns 4 to the power 5.

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CoSc 450: Programming Paradigms 03 The power function

> (power 4 5) 1024 (power 4 5) returns 4 to the power 5.

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CoSc 450: Programming Paradigms 03 The power function

> (power 4 5) 1024 (power 4 5) returns 4 to the power 5.

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CoSc 450: Programming Paradigms Not tail recursion: 03

be = b · be−1

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SLIDE 41

CoSc 450: Programming Paradigms Not tail recursion: 03

be = b · be−1

Tail recursion:

(a) · be = (ab) · be−1

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SLIDE 42

CoSc 450: Programming Paradigms Write power and power-product. 03

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CoSc 450: Programming Paradigms 03 Exponentiation is not associative

(34)5 6= 3(45) 320 6= 31024

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CoSc 450: Programming Paradigms 03 Exponentiation is not associative

(34)5 6= 3(45) 320 6= 31024

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CoSc 450: Programming Paradigms 03 Fermat numbers

Fn = 2(2n) + 1

The first few Fermat numbers

n = 0 ⇒ 2 + 1 = 3 n = 1 ⇒ 22 + 1 = 5 n = 2 ⇒ (22)2 + 1 = 17 n = 3 ⇒ ((22)2)2 + 1 = 257

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CoSc 450: Programming Paradigms 03 Fermat numbers

Fn = 2(2n) + 1

The first few Fermat numbers

n = 0 ⇒ 2 + 1 = 3 n = 1 ⇒ 22 + 1 = 5 n = 2 ⇒ (22)2 + 1 = 17 n = 3 ⇒ ((22)2)2 + 1 = 257

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CoSc 450: Programming Paradigms 03 Fermat numbers

Fn = 2(2n) + 1

The first few Fermat numbers

n = 0 ⇒ 2 + 1 = 3 n = 1 ⇒ 22 + 1 = 5 n = 2 ⇒ (22)2 + 1 = 17 n = 3 ⇒ ((22)2)2 + 1 = 257

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SLIDE 48

CoSc 450: Programming Paradigms 03 Fermat numbers

Fn = 2(2n) + 1

The first few Fermat numbers

n = 0 ⇒ 2 + 1 = 3 n = 1 ⇒ 22 + 1 = 5 n = 2 ⇒ (22)2 + 1 = 17 n = 3 ⇒ ((22)2)2 + 1 = 257

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SLIDE 49

CoSc 450: Programming Paradigms 03 Fermat numbers

Fn = 2(2n) + 1

The first few Fermat numbers

n = 0 ⇒ 2 + 1 = 3 n = 1 ⇒ 22 + 1 = 5 n = 2 ⇒ (22)2 + 1 = 17 n = 3 ⇒ ((22)2)2 + 1 = 257

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SLIDE 50

CoSc 450: Programming Paradigms 03 Fermat numbers

Fn = 2(2n) + 1

The first few Fermat numbers

n = 0 ⇒ 2 + 1 = 3 n = 1 ⇒ 22 + 1 = 5 n = 2 ⇒ (22)2 + 1 = 17 n = 3 ⇒ ((22)2)2 + 1 = 257

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CoSc 450: Programming Paradigms 03 Fermat numbers

n = 3 ⇒ ((22)2)2 | {z } +1 = 257

2 repeatedly squared 3 times

(repeatedly-square 2 0) should return (repeatedly-square 2 1) should return (repeatedly-square 2 2) should return

2 22 = 4 (22)2 = 16

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SLIDE 52

CoSc 450: Programming Paradigms 03 Fermat numbers

n = 3 ⇒ ((22)2)2 | {z } +1 = 257

2 repeatedly squared 3 times

(repeatedly-square 2 0) should return (repeatedly-square 2 1) should return (repeatedly-square 2 2) should return

2 22 = 4 (22)2 = 16

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SLIDE 53

CoSc 450: Programming Paradigms 03 Fermat numbers

n = 3 ⇒ ((22)2)2 | {z } +1 = 257

2 repeatedly squared 3 times

(repeatedly-square 2 0) should return (repeatedly-square 2 1) should return (repeatedly-square 2 2) should return

2 22 = 4 (22)2 = 16

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SLIDE 54

CoSc 450: Programming Paradigms 03 Fermat numbers

n = 3 ⇒ ((22)2)2 | {z } +1 = 257

2 repeatedly squared 3 times

(repeatedly-square 2 0) should return (repeatedly-square 2 1) should return (repeatedly-square 2 2) should return

2 22 = 4 (22)2 = 16

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SLIDE 55

CoSc 450: Programming Paradigms 03 Fermat numbers

n = 3 ⇒ ((22)2)2 | {z } +1 = 257

2 repeatedly squared 3 times

(repeatedly-square 2 0) should return (repeatedly-square 2 1) should return (repeatedly-square 2 2) should return

2 22 = 4 (22)2 = 16

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SLIDE 56

CoSc 450: Programming Paradigms 03 Fermat numbers

  • b(2n)

= 2n = 2·2n−1 b2·(2n−1) = xy·z = (xy)z (b2)2n−1 b n b2 n−1

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SLIDE 57

CoSc 450: Programming Paradigms 03 Fermat numbers

  • b(2n)

= 2n = 2·2n−1 b2·(2n−1) = xy·z = (xy)z (b2)2n−1 b n b2 n−1

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SLIDE 58

CoSc 450: Programming Paradigms 03 Fermat numbers

  • b(2n)

= 2n = 2·2n−1 b2·(2n−1) = xy·z = (xy)z (b2)2n−1 b n b2 n−1

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SLIDE 59

CoSc 450: Programming Paradigms 03 Fermat numbers

  • b(2n)

= 2n = 2·2n−1 b2·(2n−1) = xy·z = (xy)z (b2)2n−1 b n b2 n−1

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SLIDE 60

CoSc 450: Programming Paradigms 03 Fermat numbers

  • b(2n)

= 2n = 2·2n−1 b2·(2n−1) = xy·z = (xy)z (b2)2n−1 b n b2 n−1

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SLIDE 61

CoSc 450: Programming Paradigms 03 Fermat numbers

  • b(2n)

= 2n = 2·2n−1 b2·(2n−1) = xy·z = (xy)z (b2)2n−1 b n b2 n−1

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SLIDE 62

CoSc 450: Programming Paradigms Write fermat-number and repeatedly-square. 03

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CoSc 450: Programming Paradigms 03 Perfect number

  • The sum of its divisors is twice the number, or
  • The number is equal to the sum of its divisors
  • ther than itself.
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CoSc 450: Programming Paradigms 03 Perfect number

  • The sum of its divisors is twice the number, or
  • The number is equal to the sum of its divisors
  • ther than itself.

2·6 = 1+2+3+6 6 = 1+2+3

  • ⇒
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CoSc 450: Programming Paradigms 03 Perfect number

  • The sum of its divisors is twice the number, or
  • The number is equal to the sum of its divisors
  • ther than itself.

2·6 = 1+2+3+6 6 = 1+2+3

  • ⇒

There are no known odd perfect numbers.

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SLIDE 66

CoSc 450: Programming Paradigms 03 Suppose you have a function sum-of-divisors such that (sum-of-divisors 4) returns 1+2+4, (sum-of-divisors 5) returns 1+5, (sum-of-divisors 6) returns 1+2+3+6, etc.

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SLIDE 67

CoSc 450: Programming Paradigms 03 Suppose you have a function sum-of-divisors such that (sum-of-divisors 4) returns 1+2+4, (sum-of-divisors 5) returns 1+5, (sum-of-divisors 6) returns 1+2+3+6, etc. Write perfect? such that (perfect? 4) returns #f, (perfect? 5) returns #f, (perfect? 6) returns #t, etc.

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SLIDE 68

CoSc 450: Programming Paradigms 03 The shape of sum-of-divisors

(define sum-of-divisors (lambda (n) (define sum-from-plus ; sum of all divisors of n which are (lambda (low addend) ; >= low, plus addend (if (> low n) addend ; no divisors of n are greater than n (sum-from-plus (+ low 1) (if (divides? low n) (+ addend low) addend))))) (sum-from-plus 1 0)))

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SLIDE 69

CoSc 450: Programming Paradigms 03 The shape of sum-of-divisors

(define sum-of-divisors (lambda (n) (define sum-from-plus ; sum of all divisors of n which are (lambda (low addend) ; >= low, plus addend (if (> low n) addend ; no divisors of n are greater than n (sum-from-plus (+ low 1) (if (divides? low n) (+ addend low) addend))))) (sum-from-plus 1 0)))

sum-from-plus is defined

inside sum-of-divisors. So, it has access to n.

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SLIDE 70

CoSc 450: Programming Paradigms 03 The shape of sum-of-divisors

(define sum-of-divisors (lambda (n) (define sum-from-plus ; sum of all divisors of n which are (lambda (low addend) ; >= low, plus addend (if (> low n) addend ; no divisors of n are greater than n (sum-from-plus (+ low 1) (if (divides? low n) (+ addend low) addend))))) (sum-from-plus 1 0)))

sum-from-plus is defined

inside sum-of-divisors. So, it has access to n.

low addend

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SLIDE 71

CoSc 450: Programming Paradigms 03 The shape of sum-of-divisors

(define sum-of-divisors (lambda (n) (define sum-from-plus ; sum of all divisors of n which are (lambda (low addend) ; >= low, plus addend (if (> low n) addend ; no divisors of n are greater than n (sum-from-plus (+ low 1) (if (divides? low n) (+ addend low) addend))))) (sum-from-plus 1 0)))

sum-from-plus returns the sum of all the divisors

  • f n that are greater than or equal to low

plus the addend.

low addend

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SLIDE 72

CoSc 450: Programming Paradigms 03

sum-from-plus returns the sum of all the divisors

  • f n that are greater than or equal to low

plus the addend.

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SLIDE 73

CoSc 450: Programming Paradigms 03

sum-from-plus returns the sum of all the divisors

  • f n that are greater than or equal to low

plus the addend. Is 12 perfect?

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SLIDE 74

CoSc 450: Programming Paradigms 03

sum-from-plus returns the sum of all the divisors

  • f n that are greater than or equal to low

plus the addend. Is 12 perfect? 1 2 3 4 5 6 7 8 9 10 11 12

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SLIDE 75

CoSc 450: Programming Paradigms 03

sum-from-plus returns the sum of all the divisors

  • f n that are greater than or equal to low

plus the addend. Is 12 perfect? 1 2 3 4 5 6 7 8 9 10 11 12

n = 12

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SLIDE 76

CoSc 450: Programming Paradigms 03

sum-from-plus returns the sum of all the divisors

  • f n that are greater than or equal to low

plus the addend. Is 12 perfect? 1 2 3 4 5 6 7 8 9 10 11 12

low = 6 n = 12

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SLIDE 77

CoSc 450: Programming Paradigms 03

sum-from-plus returns the sum of all the divisors

  • f n that are greater than or equal to low

plus the addend. Is 12 perfect? 1 2 3 4 5 6 7 8 9 10 11 12

addend = 1+2+3+4 low = 6 n = 12

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SLIDE 78

CoSc 450: Programming Paradigms 03

sum-from-plus returns the sum of all the divisors

  • f n that are greater than or equal to low

plus the addend. Is 12 perfect? 1 2 3 4 5 6 7 8 9 10 11 12

addend = 1+2+3+4 low = 6 n = 12 (sum-from-plus 6 10)

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SLIDE 79

CoSc 450: Programming Paradigms 03

sum-of-divisors

(define sum-of-divisors (lambda (n) (define sum-from-plus ; sum of all divisors of n which are (lambda (low addend) ; >= low, plus addend (if (> low n) addend ; no divisors of n are greater than n (sum-from-plus (+ low 1) (if (divides? low n) (+ addend low) addend))))) (sum-from-plus 1 0))) low addend

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SLIDE 80

CoSc 450: Programming Paradigms 03

(define sum-of-divisors (lambda (n) (define sum-from-plus ; sum of all divisors of n which are (lambda (low addend) ; >= low, plus addend (if (> low n) addend ; no divisors of n are greater than n (sum-from-plus (+ low 1) (if (divides? low n) (+ addend low) addend))))) (sum-from-plus 1 0))) low addend

sum-of-divisors

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SLIDE 81

CoSc 450: Programming Paradigms 03

(define sum-of-divisors (lambda (n) (define sum-from-plus ; sum of all divisors of n which are (lambda (low addend) ; >= low, plus addend (if (> low n) addend ; no divisors of n are greater than n (sum-from-plus (+ low 1) (if (divides? low n) (+ addend low) addend))))) (sum-from-plus 1 0))) low addend

sum-of-divisors

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SLIDE 82

CoSc 450: Programming Paradigms 03

(define sum-of-divisors (lambda (n) (define sum-from-plus ; sum of all divisors of n which are (lambda (low addend) ; >= low, plus addend (if (> low n) addend ; no divisors of n are greater than n (sum-from-plus (+ low 1) (if (divides? low n) (+ addend low) addend))))) (sum-from-plus 1 0))) low addend

sum-of-divisors

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SLIDE 83

CoSc 450: Programming Paradigms 03

(define sum-of-divisors (lambda (n) (define sum-from-plus ; sum of all divisors of n which are (lambda (low addend) ; >= low, plus addend (if (> low n) addend ; no divisors of n are greater than n (sum-from-plus (+ low 1) (if (divides? low n) (+ addend low) addend))))) (sum-from-plus 1 0))) low addend

sum-of-divisors

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SLIDE 84

CoSc 450: Programming Paradigms 03

(define sum-of-divisors (lambda (n) (define sum-from-plus ; sum of all divisors of n which are (lambda (low addend) ; >= low, plus addend (if (> low n) addend ; no divisors of n are greater than n (sum-from-plus (+ low 1) (if (divides? low n) (+ addend low) addend))))) (sum-from-plus 1 0))) low addend

sum-of-divisors

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SLIDE 85

CoSc 450: Programming Paradigms 03

(define sum-of-divisors (lambda (n) (define sum-from-plus ; sum of all divisors of n which are (lambda (low addend) ; >= low, plus addend (if (> low n) addend ; no divisors of n are greater than n (sum-from-plus (+ low 1) (if (divides? low n) (+ addend low) addend))))) (sum-from-plus 1 0))) low addend

sum-of-divisors

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SLIDE 86

CoSc 450: Programming Paradigms 03

(define sum-of-divisors (lambda (n) (define sum-from-plus ; sum of all divisors of n which are (lambda (low addend) ; >= low, plus addend (if (> low n) addend ; no divisors of n are greater than n (sum-from-plus (+ low 1) (if (divides? low n) (+ addend low) addend))))) (sum-from-plus 1 0))) low addend

sum-of-divisors

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SLIDE 87

CoSc 450: Programming Paradigms 03 Hallmarks of pure functional programming

  • A function returns a value.
  • There is no call by reference.
  • All parameters are called by value.
  • There are no loops.
  • Repetition is achieved by recursion.
  • There is no assignment statement.
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SLIDE 88

CoSc 450: Programming Paradigms 03 The Golden Ratio

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SLIDE 89

CoSc 450: Programming Paradigms 03 The Golden Ratio

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SLIDE 90

CoSc 450: Programming Paradigms 03 The Golden Ratio A A

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SLIDE 91

CoSc 450: Programming Paradigms 03 The Golden Ratio A B A

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SLIDE 92

CoSc 450: Programming Paradigms 03 The Golden Ratio A B A A B = A + B A = 1 + B A = 1 + 1 A/B φ = A/B φ = 1 + 1 φ

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SLIDE 93

CoSc 450: Programming Paradigms 03 The Golden Ratio A B A A B = A + B A = 1 + B A = 1 + 1 A/B φ = A/B φ = 1 + 1 φ

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SLIDE 94

CoSc 450: Programming Paradigms 03 The Golden Ratio A B A A B = A + B A = 1 + B A = 1 + 1 A/B φ = A/B φ = 1 + 1 φ

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SLIDE 95

CoSc 450: Programming Paradigms 03 The Golden Ratio A B A A B = A + B A = 1 + B A = 1 + 1 A/B φ = A/B φ = 1 + 1 φ

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SLIDE 96

CoSc 450: Programming Paradigms 03 The Golden Ratio A B A A B = A + B A = 1 + B A = 1 + 1 A/B φ = A/B φ = 1 + 1 φ

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SLIDE 97

CoSc 450: Programming Paradigms 03 Successive approximations of the Golden Ratio

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SLIDE 98

CoSc 450: Programming Paradigms 03 Successive approximations of the Golden Ratio

φ0 = 1 φ1 = 1 + 1 φ0 = 2 φ2 = 1 + 1 φ1 = 3 2 φ3 = 1 + 1 φ2 = 5 3 φ4 = 1 + 1 φ3 = 8 5

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SLIDE 99

CoSc 450: Programming Paradigms 03 Successive approximations of the Golden Ratio

φ0 = 1 φ1 = 1 + 1 φ0 = 2 φ2 = 1 + 1 φ1 = 3 2 φ3 = 1 + 1 φ2 = 5 3 φ4 = 1 + 1 φ3 = 8 5

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SLIDE 100

CoSc 450: Programming Paradigms 03 Successive approximations of the Golden Ratio

φ0 = 1 φ1 = 1 + 1 φ0 = 2 φ2 = 1 + 1 φ1 = 3 2 φ3 = 1 + 1 φ2 = 5 3 φ4 = 1 + 1 φ3 = 8 5

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SLIDE 101

CoSc 450: Programming Paradigms 03 Successive approximations of the Golden Ratio

φ0 = 1 φ1 = 1 + 1 φ0 = 2 φ2 = 1 + 1 φ1 = 3 2 φ3 = 1 + 1 φ2 = 5 3 φ4 = 1 + 1 φ3 = 8 5

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SLIDE 102

CoSc 450: Programming Paradigms 03 Successive approximations of the Golden Ratio

φ0 = 1 φ1 = 1 + 1 φ0 = 2 φ2 = 1 + 1 φ1 = 3 2 φ3 = 1 + 1 φ2 = 5 3 φ4 = 1 + 1 φ3 = 8 5

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SLIDE 103

CoSc 450: Programming Paradigms 03 Successive approximations of the Golden Ratio (phi 0) returns 1, (phi 1) returns 2, (phi 2) returns 3/2, (phi 3) returns 5/3, (phi 4) returns 8/5, etc. Write phi

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SLIDE 104

CoSc 450: Programming Paradigms 03 The Josephus Problem Rule: Everybody stands in a circle. Starting with the first, kill every third person. Which two people remain alive?

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CoSc 450: Programming Paradigms 03 The Josephus Problem 1 5 7 3 2 4 8 6

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CoSc 450: Programming Paradigms 03 The Josephus Problem 1 5 7 3 2 4 8 6

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SLIDE 107

CoSc 450: Programming Paradigms 03 The Josephus Problem 1 5 7 3 2 4 8 6

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CoSc 450: Programming Paradigms 03 The Josephus Problem 1 5 7 3 2 4 8 6

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SLIDE 109

CoSc 450: Programming Paradigms 03 The Josephus Problem 1 5 7 3 2 4 8 6

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SLIDE 110

CoSc 450: Programming Paradigms 03 The Josephus Problem 1 5 7 3 2 4 8 6

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SLIDE 111

CoSc 450: Programming Paradigms 03 The Josephus Problem 1 5 7 3 2 4 8 6

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SLIDE 112

CoSc 450: Programming Paradigms 03 The Josephus Problem 1 5 7 3 2 4 8 6

(survives? 4 8) returns #t (survives? 7 8) returns #t (survives? k 8) returns #f

for all other values of k

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SLIDE 113

CoSc 450: Programming Paradigms 03 The Josephus Problem 1 5 7 3 2 4 8 6

(survives? 4 8) returns #t (survives? 7 8) returns #t (survives? k 8) returns #f

for all other values of k

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SLIDE 114

CoSc 450: Programming Paradigms 03 The Josephus Problem 1 5 7 3 2 4 8 6

(survives? 4 8) returns #t (survives? 7 8) returns #t (survives? k 8) returns #f

for all other values of k

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SLIDE 115

CoSc 450: Programming Paradigms 03 The Josephus Problem 1 5 7 3 2 4 8 6

(survives? 4 8) returns #t (survives? 7 8) returns #t (survives? k 8) returns #f

for all other values of k

Exercise for the student: Write survives?

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SLIDE 116

CoSc 450: Programming Paradigms 03 The Josephus Problem 1 5 7 3 2 4 8 6

Hint: Parentheses are the new numbers after the first person is killed.

(survives? 6 8)

is recursively the same as

(survives? 3 7)

(1) (2) (3) (4) (5) (6) (7)

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SLIDE 117

CoSc 450: Programming Paradigms 03 The Josephus Problem 1 5 7 3 2 4 8 6

Hint: Parentheses are the new numbers after the first person is killed.

(survives? 6 8)

is recursively the same as

(survives? 3 7)

(1) (2) (3) (4) (5) (6) (7)