David Evanshttp://www.cs.virginia.edu/~evans

CS200: Computer ScienceUniversity of VirginiaComputer Science

Lecture 6: Conscar cdr sdr wdr

28 January 2004 CS 200 Spring 2004 2

Menu

• Recursion Practice: fibo

• History of Scheme: LISP

• Introducing Lists

28 January 2004 CS 200 Spring 2004 3

Defining Recursive Procedures

1. Be optimistic.– Assume you can solve it.– If you could, how would you solve a bigger

problem.

2. Think of the simplest version of the problem, something you can already solve. (This is the base case.)

3. Combine them to solve the problem.

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Defining fibo

;;; (fibo n) evaluates to the nth Fibonacci

;;; number

(define (fibo n)

(if (or (= n 1) (= n 2))

1 ;;; base case

(+ (fibo (- n 1))

(fibo (- n 2)))))

FIBO (1) = FIBO (2) = 1

FIBO (n) = FIBO (n – 1) + FIBO (n – 2)

for n > 2

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Fibo Results> (fibo 2)1> (fibo 3)2> (fibo 4)3> (fibo 10)55> (fibo 100)Still working after 4 hours…

Why can’t our 100,000x Apollo Guidance Computer calculate (fibo 100)?

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Tracing Fibo> (require-library "trace.ss")> (trace fibo)(fibo)> (fibo 3)|(fibo 3)| (fibo 2)| 1| (fibo 1)| 1|22

This turns tracing on

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> (fibo 5)|(fibo 5)| (fibo 4)| |(fibo 3)| | (fibo 2)| | 1| | (fibo 1)| | 1| |2| |(fibo 2)| |1| 3| (fibo 3)| |(fibo 2)| |1| |(fibo 1)| |1| 2|55

(fibo 5) = (fibo 4) + (fibo 3) (fibo 3) + (fibo 2) + (fibo 2) + (fibo 1) (fibo 2) + (fibo 1) + 1 + 1 + 1 1 + 1 2 + 1 + 2 3 + 2 = 5

To calculate (fibo 5) we calculated:(fibo 4) 1 time(fibo 3) 2 times(fibo 2) 3 times(fibo 1) 2 times = 8 calls to fibo = (fibo 6)

How many calls to calculate (fibo 100)?

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fast-fibo

(define (fast-fibo n) (define (fibo-worker a b count) (if (= count 1) b (fibo-worker (+ a b) a (- count

1)))) (fibo-worker 1 1 n))

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Fast-Fibo Results

> (fast-fibo 1)1> (fast-fibo 10)55> (time (fast-fibo 100))cpu time: 0 real time: 0 gc time: 0354224848179261915075

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Beware the Bunnies!!

;;; The Earth's mass is 6.0 x 10^24 kg> (define mass-of-earth (* 6 (expt 10 24))) ;;; A typical rabbit's mass is 2.5 kilograms> (define mass-of-rabbit 2.5)> (/ (* mass-of-rabbit (fast-fibo 100)) mass-of-earth)0.00014759368674135913> (/ (* mass-of-rabbit (fast-fibo 110)) mass-of-earth)0.018152823441189517> (/ (* mass-of-rabbit (fast-fibo 119)) mass-of-earth)1.379853393132076> (exact->inexact (/ 119 12))9.916666666666666According to Fibonacci’s model, after less than 10 years, rabbits would out-weigh the Earth!

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History of Scheme• Scheme [1975]

– Guy Steele and Gerry Sussman

– Originally “Schemer”

– “Conniver” [1973] and “Planner” [1967]

• Based on LISP – John McCarthy (late 1950s)

• Based on Lambda Calculus– Alonzo Church (1930s)

– Last few lectures in course

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LISP

“Lots of Insipid Silly Parentheses”

“LISt Processing language”

Lists are pretty important – hard to write a useful Scheme program without them.

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Making Lists

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Making a Pair

> (cons 1 2)(1 . 2) 1 2

cons constructs a pair

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Splitting a Pair> (car (cons 1 2))1> (cdr (cons 1 2))2

1 2

cons

car cdr

car extracts first part of a paircdr extracts second part of a pair

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Why “car” and “cdr”?• Original (1950s) LISP on IBM 704

– Stored cons pairs in memory registers– car = “Contents of the Address part of the

Register”– cdr = “Contents of the Decrement part of the

Register” (“could-er”)

• Doesn’t matter unless you have an IBM 704

• Think of them as first and rest(define first car)(define rest cdr)

(The DrScheme “Pretty Big” language already defines these, but they are not part of standard Scheme)

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Implementing cons, car and cdr

Using PS2: (define cons make-point)(define car x-of-point)(define cdr y-of-point)

As we implemented make-point, etc.:(define (cons a b) (lambda (w) (if (w) a

b)))(define (car pair) (pair #t)(define (cdr pair) (pair #f)

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Pairs are fine, but how do we make threesomes?

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Threesome?

(define (threesome a b c) (lambda (w) (if (= w 0) a (if (= w 1) b c))))

(define (first t) (t 0))(define (second t) (t 1))(define (third t) (t 2))Is there a better way of thinking about our triple?

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Triple

A triple is just a pair where one of the parts is a pair!

(define (triple a b c) (cons a (cons b c)))(define (t-first t) (car t))(define (t-second t) (car (cdr t))) (define (t-third t) (cdr (cdr t)))

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Quadruple

A quadruple is a pair where the second part is a triple

(define (quadruple a b c d) (cons a (triple b c d)))(define (q-first q) (car q))(define (q-second q) (t-first (cdr t))) (define (q-third t) (t-second (cdr t)))(define (q-fourth t) (t-third (cdr t)))

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Multuples• A quintuple is a pair where the second part is

a quadruple

• A sextuple is a pair where the second part is a quintuple

• A septuple is a pair where the second part is a sextuple

• An octuple is group of octupi

• A list (any length tuple) is a pair where the second part is a …?

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Lists

List ::= (cons Element List)

A list is a pair where the second part is a list.

One big problem: how do we stop?This only allows infinitely long lists!

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Recursive Transition Networks

ARTICLE ADJECTIVE NOUN endbegin

ORNATE NOUN

ORNATE NOUN ::= ARTICLE ADJECTIVE NOUN

ORNATE NOUN ::= ARTICLE ADJECTIVE ADJECTIVE NOUN

ORNATE NOUN ::= ARTICLE ADJECTIVE ADJECTIVE ADJECTIVE NOUNORNATE NOUN ::= ARTICLE ADJECTIVE ADJECTIVE ADJECTIVE ADJECTIVE NOUN

ORNATE NOUN ::= ARTICLE ADJECTIVE ADJECTIVE ADJECTIVE ADJECTIVE ADJECTIVE NOUN

From Lecture 5

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Recursive Transition Networks

ARTICLE ADJECTIVE NOUN endbegin

ORNATE NOUN

ORNATE NOUN ::= ARTICLE ADJECTIVES NOUN

ADJECTIVES ::= ADJECTIVE ADJECTIVES

ADJECTIVES ::=

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Lists

List ::= (cons Element List)List ::=

A list is either: a pair where the second part is a listor, empty

It’s hard to write this!

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Null

List ::= (cons Element List)List ::=

A list is either: a pair where the second part is a listor, empty (null)

null

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List Examples> null()> (cons 1 null)(1)> (list? null)#t> (list? (cons 1 2))#f> (list? (cons 1 null))#t

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More List Examples

> (list? (cons 1 (cons 2 null)))#t> (car (cons 1 (cons 2 null)))1> (cdr (cons 1 (cons 2 null)))(2)

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Charge

• Next Time: List Recursion

• PS2 Due Monday– Lots of the code we provided uses lists

– But, you are not expected to use lists in your code (but you can if you want)