Class 09: Recursion practice, how recursive programs work Recall the - - PowerPoint PPT Presentation

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Class 09: Recursion practice, how recursive programs work Recall the - - PowerPoint PPT Presentation

Feb 14, 2023 186 likes •755 views

Class 09: Recursion practice, how recursive programs work Recall the list-length procedure (define (list-length alon) (cond [(empty? alon) 0] [(cons? alon) (+ 1 (list-length (rest alon)))])) Why does it correctly compute the length of any

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SLIDE 1

Class 09: Recursion practice, how recursive programs work

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SLIDE 2

Recall the list-length procedure

(define (list-length alon) (cond [(empty? alon) 0] [(cons? alon) (+ 1 (list-length (rest alon)))]))

  • Why does it correctly compute the length of any list?
  • Does it work for lists of length zero?
  • How many of those are there?
  • Does it work for lists of length 1?
  • Yes…
  • Provided that the recursive call computing the length of the rest of the list works
  • The rest of the list has length 0, so its length does get computed correctly
  • Same argument for length 2, 3, 4, …
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SLIDE 3

Recursion diagrams and correct procedures

  • Recursion diagrams prepare for this logical analysis to

work!

  • They ensure that the answer for length-zero lists is

correct

  • They ensure that if the answer for length-(k-1) lists is

correct, then the answer for length-k lists will be as well!

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SLIDE 4

Does it work at a mechanical level?

  • Do the rules of evaluation lead to the results you

expect?

  • Yes (no surprise!)
  • We’ll see this by computing the length of a length-one

list, using the rules.

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SLIDE 5

Warmup: cleaning up Boolean expressions

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SLIDE 6

A recursion problem

  • contains17? takes as input an int list, and returns

true if one of the items is the number 17, and false if the list does not contain a 17.

  • The type-signature is contains17?: (int list) - >

bool

  • With your neighbor, draw recursion diagrams for three

cases:

  • (cons 13 (cons 4 empty))
  • (cons 17 empty)
  • (cons 1 (cons 17 empty))
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SLIDE 7

OI: (cons 13 (cons 4 empty)) RI: (cons 4 empty) RO: false OO: false

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SLIDE 8

OI: (cons 17 empty) RI: empty RO: false OO: true

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SLIDE 9

OI: (cons 1 (cons 17 empty)) RI: (cons 17 empty) RO: true OO: true

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SLIDE 10

OI: (cons 1 (cons 17 empty)) RI: (cons 17 empty) RO: true

If RO is true, OO is true If RO is false, but (first OO) is 17, then OO is true Otherwise OO is false

OO: true

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SLIDE 11

Code

(define (contains17? aloi) (cond [(empty? aloi) false] [(cons? aloi) (if (contains17? (rest aloi)) true (if (= 17 (first aloi)) true false))]))

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SLIDE 12

Code

(define (contains17? aloi) (cond [(empty? aloi) false] [(cons? aloi) (or (contains17? (rest aloi)) (if (= 17 (first aloi)) true false))]))

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SLIDE 13

Code

(define (contains17? aloi) (cond [(empty? aloi) false] [(cons? aloi) (or (contains17? (rest aloi)) (if (= 17 (first aloi)) true false))]))

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SLIDE 14

Code (better)

(define (contains17? aloi) (cond [(empty? aloi) false] [(cons? aloi) (or (contains17? (rest aloi)) (= 17 (first aloi))]))

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SLIDE 15

Code (even better)

(define (contains17? aloi) (cond [(empty? aloi) false] [(cons? aloi) (or (= 17 (first aloi)) (contains17? (rest aloi))]))

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SLIDE 16

Evaluation practice

(already done previously, but left here for reference!)

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SLIDE 17

Color coding

  • Values are in green
  • Environment is a sequence of purple boxes; TLE

shorthand shows just + and – as builtin procs.

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SLIDE 18

(skip to slide 42)

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SLIDE 19

(+ (- 3 4) (+ 2 5)) A B C A: B: C: Value:

+ builtin+

  • builtin-
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SLIDE 20

(+ (- 3 4) (+ 2 5)) A B C A: builtin+ B: C: Value:

+ builtin+

  • builtin-
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SLIDE 21

(+ (- 3 4) (+ 2 5)) A B C A: builtin+ B: C: Value:

  • B: (- 3 4)

D E F D: E: F: value:

+ builtin+

  • builtin-
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SLIDE 22

(+ (- 3 4) (+ 2 5)) A B C A: builtin+ B: C: Value:

  • B: (- 3 4)

D E F D: builtin- E: F: value:

+ builtin+

  • builtin-
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SLIDE 23

(+ (- 3 4) (+ 2 5)) A B C A: builtin+ B: C: Value:

  • B: (- 3 4)

D E F D: builtin- E: 3 F: 4 value:

+ builtin+

  • builtin-
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SLIDE 24

(+ (- 3 4) (+ 2 5)) A B C A: builtin+ B: C: Value:

  • B: (- 3 4)

D E F D: builtin- E: 3 F: 4 value: -1

+ builtin+

  • builtin-
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SLIDE 25

(+ (- 3 4) (+ 2 5)) A B C A: builtin+ B: -1 C: Value:

  • B: (- 3 4)

D E F D: builtin- E: 3 F: 4 value: -1

+ builtin+

  • builtin-
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SLIDE 26

(+ (- 3 4) (+ 2 5)) A B C A: builtin+ B: -1 C: Value:

  • B: (- 3 4) | C: (+ 2 5)

D E F | G H J D: builtin- | G: E: 3 | H: F: 4 | J: value: -1 | value:

+ builtin+

  • builtin-
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SLIDE 27

(+ (- 3 4) (+ 2 5)) A B C A: builtin+ B: -1 C: Value:

  • B: (- 3 4) | C: (+ 2 5)

D E F | G H J D: builtin- | G: builtin+ E: 3 | H: F: 4 | J: value: -1 | value:

+ builtin+

  • builtin-
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SLIDE 28

(+ (- 3 4) (+ 2 5)) A B C A: builtin+ B: -1 C: Value:

  • B: (- 3 4) | C: (+ 2 5)

D E F | G H J D: builtin- | G: builtin+ E: 3 | H: 2 F: 4 | J: value: -1 | value:

+ builtin+

  • builtin-
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SLIDE 29

(+ (- 3 4) (+ 2 5)) A B C A: builtin+ B: -1 C: Value:

  • B: (- 3 4) | C: (+ 2 5)

D E F | G H J D: builtin- | G: builtin+ E: 3 | H: 2 F: 4 | J: 5 value: -1 | value:

+ builtin+

  • builtin-
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SLIDE 30

(+ (- 3 4) (+ 2 5)) A B C A: builtin+ B: -1 C: Value:

  • B: (- 3 4) | C: (+ 2 5)

D E F | G H J D: builtin- | G: builtin+ E: 3 | H: 2 F: 4 | J: 5 value: -1 | value: 7

+ builtin+

  • builtin-
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SLIDE 31

(+ (- 3 4) (+ 2 5)) A B C A: builtin+ B: -1 C: 7 Value:

  • B: (- 3 4) | C: (+ 2 5)

D E F | G H J D: builtin- | G: builtin+ E: 3 | H: 2 F: 4 | J: 5 value: -1 | value: 7

+ builtin+

  • builtin-
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SLIDE 32

(+ (- 3 4) (+ 2 5)) A B C A: builtin+ B: -1 C: 7 Value: 6

  • B: (- 3 4) | C: (+ 2 5)

D E F | G H J D: builtin- | G: builtin+ E: 3 | H: 2 F: 4 | J: 5 value: -1 | value: 7

+ builtin+

  • builtin-
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SLIDE 33

(define (f x) (+ x 1)) (+ (f 1) (f 3)) A B C A: builtin+ B: C: Value:

  • B: (f 1) | C: (f 3)

D E | G H D: closure(x, (+ x 1))| G: E: 1 | H: | | |

+ builtin+

  • builtin-

f closure{(x, (+ x 1))

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SLIDE 34

(define (f x) (+ x 1)) (+ (f 1) (f 3)) A B C A: builtin+ B: C: Value:

  • B: (f 1) | C: (f 3)

D E | G H D: closure(x, (+ x 1))| G: E: 1 | H: Bind x -> 1; evaluate | (+ x 1) | | | |

+ builtin+

  • builtin-

x 1

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SLIDE 35

(define (f x) (+ x 1)) (+ (f 1) (f 3)) A B C A: builtin+ B: C: Value:

  • B: (f 1) | J: (+ x 1)

D E | K L M D: closure(x, (+ x 1))| K: builtin+ E: 1 | L: 1 Bind x -> 1; evaluate | M: 1 J: (+ x 1) | Value: 2 Value: | |

+ builtin+

  • builtin-

x 1

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SLIDE 36

(define (f x) (+ x 1)) (+ (f 1) (f 3)) A B C A: builtin+ B: C: Value:

  • B: (f 1) | J: (+ x 1)

D E | K L M D: closure(x, (+ x 1))| K: builtin+ E: 1 | L: 1 Bind x -> 1; evaluate | M: 1 J: (+ x 1) | Value: 2 Value: 2 | |

+ builtin+

  • builtin-

x 1

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SLIDE 37

(define (f x) (+ x 1)) (+ (f 1) (f 3)) A B C A: builtin+ B: C: Value:

  • B: (f 1) | C: (f 3)

D E | G H D: closure(x, (+ x 1))| G: E: 1 | H: Bind x -> 1; evaluate | (+ x 1) | value: 2 | Drop the new binding |

+ builtin+

  • builtin-
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SLIDE 38

(define (f x) (+ x 1)) (+ (f 1) (f 3)) A B C A: builtin+ B: 2 C: Value:

  • B: (f 1) | C: (f 3)

D E | G H D: closure(x, (+ x 1))| G: closure(x, (+ x 1)) E: 1 | H: 3 Bind x -> 1; evaluate | (+ x 1) | value: 2 | Drop the new binding |

+ builtin+

  • builtin-
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SLIDE 39

(define (f x) (+ x 1)) (+ (f 1) (f 3)) A B C A: builtin+ B: 2 C: Value:

  • B: (f 1) | C: (f 3)

D E | G H D: closure(x, (+ x 1))| G: closure(x, (+ x 1)) E: 1 | H: 3 Bind x -> 1; evaluate | Bind x -> 3; evaluate (+ x 1) | (+ x 1) value: 2 | value: 4 Drop the new binding | Drop the new binding

+ builtin+

  • builtin-

x 3

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SLIDE 40

(define (f x) (+ x 1)) (+ (f 1) (f 3)) A B C A: builtin+ B: 2 C: 4 Value:

  • B: (f 1) | C: (f 3)

D E | G H D: closure(x, (+ x 1))| G: closure(x, (+ x 1)) E: 1 | H: 3 Bind x -> 1; evaluate | Bind x -> 3; evaluate (+ x 1) | (+ x 1) value: 2 | value: 4 Drop the new binding | Drop the new binding

+ builtin+

  • builtin-
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SLIDE 41

(define (f x) (+ x 1)) (+ (f 1) (f 3)) A B C A: builtin+ B: 2 C: 4 Value: 6

  • B: (f 1) | C: (f 3)

D E | G H D: closure(x, (+ x 1))| G: closure(x, (+ x 1)) E: 1 | H: 3 Bind x -> 1; evaluate | Bind x -> 3; evaluate (+ x 1) | (+ x 1) value: 2 | value: 4 Drop the new binding | Drop the new binding

+ builtin+

  • builtin-
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SLIDE 42

(define (f x) (+ x 1)) (f 3)

Formal argument(s)

3

Actual argument(s)

  • During evaluation of a user-defjned procedure, we “temporarily

bind formals to actuals”

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SLIDE 43

T wo examples for len

(len empty) (len (cons 1 empty))

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SLIDE 44

(define (len alon) (cond [(empty? alon) 0] [(cons? alon) (+ 1 (len (rest alon)))])) (len empty) A B A: closure(alon, (cond …)) B: empty-value Bind alon -> empty-value Evaluate the body.

+ builtin+

  • builtin-

len closure(alon, …) alon empty- value

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SLIDE 45

Bind alon -> empty-value Evaluate: (cond P1: [(empty? alon) 0] P2: [(cons? alon) (+ 1 (len (rest alon)))])) P1: condition: (empty? alon) A B A: builtin:empty? B: empty-value [from extension to TLE] value: true

+ builtin+

  • builtin-

len closure(alon, …) n alon empty- value

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SLIDE 46

(define (len alon) (cond [(empty? alon) 0] [(cons? alon) (+ 1 (len (rest alon)))])) (len empty) A B A: closure(alon, (cond …)) B: empty-value Bind alon -> empty-value Evaluate the body. => 0

+ builtin+

  • builtin-

len closure(alon, …) alon empty- value

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SLIDE 47

(define (len alon) (cond [(empty? alon) 0] [(cons? alon) (+ 1 (len (rest alon)))])) (len empty) A B A: closure(alon, (cond …)) B: empty-value Bind alon -> empty-value Evaluate the body. => 0 Remove the extra binding(s)

+ builtin+

  • builtin-

len closure(alon, …) alon empty- value

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SLIDE 48

(define (len alon) (cond [(empty? alon) 0] [(cons? alon) (+ 1 (len (rest alon)))])) (len (cons 1 empty)) A B A: closure(alon, (cond …)) B: (cons 1 empty-value) Bind alon -> (cons 1 empty-value) Evaluate the body.

+ builtin+

  • builtin-

len closure(alon, …) alon (cons 1 empty-value)

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SLIDE 49

Bind alon -> (cons 1 empty) Evaluate: (cond P1: [(empty? alon) 0] P2: [(cons? alon) (+ 1 (len (rest alon)))])) P1: condition: (empty? alon) A B A: builtin:empty? B: (cons 1 empty)[from extension to TLE] value: false P2: condition: (cons? alon) A B A: builtin:cons? B: (cons 1 empty) [from TLE extension] value: true Cond-expression: evaluate (+ 1 (len (rest alon))) Value: ???

+ builtin+

  • builtin-

len closure(alon, …) alon (cons 1 empty-value)

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SLIDE 50

evaluate (+ 1 (len (rest alon))) A B C A: builtin:+ B: 1 C: Value:

+ builtin+

  • builtin-

len closure(alon, …) alon (cons 1 empty-value)

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SLIDE 51

evaluate (+ 1 (len (rest alon))) A B C A: builtin:+ B: 1 C: Value:

  • (len (rest alon))

D E D: closure(alon, ...) E: empty-value [Why?] Value: 0 (see prev. slides!)

+ builtin+

  • builtin-

len closure(alon, …) alon (cons 1 empty- value) + builtin+

  • builtin-

len closure(alon, …) alon (cons 1 empty- value) alon empty-value

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SLIDE 52

Bind alon -> empty-value Evaluate: (cond P1: [(empty? alon) 0] P2: [(cons? alon) (+ 1 (len (rest alon)))])) P1: condition: (empty? alon) A B A: builtin:empty? B: empty-value [from extension to TLE] value: true ...

+ builtin+

  • builtin-

len closure(alon, …) alon (cons 1 empty- value) alon empty- value

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SLIDE 53

evaluate (+ 1 (len (rest alon))) A B C A: builtin:+ B: 1 C: 0 Value: 1

  • (len (rest alon))

D E D: closure(alon, ...) E: empty-value Value: 0 (see prev. slides!)

+ builtin+

  • builtin-

len closure(alon, …) alon (cons 1 empty-value) + builtin+

  • builtin-

len closure(alon, …) alon (cons 1 empty- value) alon empty-value

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SLIDE 54

(define (len alon) (cond [(empty? alon) 0] [(cons? alon) (+ 1 (len (rest alon)))])) (len (cons 1 empty)) A B A: closure(alon, (cond …)) B: (cons 1 empty) Bind alon -> (cons 1 empty) Evaluate the body. (We got 1) Remove the temporary bindings Value: 1 (yay!)

+ builtin+

  • builtin-

len closure(alon, …) alon (cons 1 empty-value)

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SLIDE 55

New topic: operation counting

  • T
  • measure whether a procedure we write is “fast” or

”slow”

  • Look at how it performs as the size of the input grows
  • For us: ”size” is almost always “the length of the input list”
  • Use a mathematical thing called a “recurrence relation” to

express this

  • “Solve” the recurrence to get a more compact and

understandable answer

  • Answers will use the idea of one function being

“eventually greater than” another

  • Answers will not be perfect!
  • ….but they’ll be good enough to tell us the information we

almost always need

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SLIDE 56

“Operations”

  • Binding a name to a value
  • Looking up the value associated to a name
  • Computing the value of a bool, number, or string
  • Evaluating cons, fjrst, rest, empty, empty?, cons?, =, >,

<, …

  • Apply “or” or “and” to two items
  • Arithmetic operations, all the builtins we’ve seen so far
  • T

esting whether a Boolean is T/F