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## David Evans cs.virginia/~evans

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**Lecture 6: Cons**car cdr sdr wdr David Evans http://www.cs.virginia.edu/~evans CS200: Computer Science University of Virginia Computer Science**Menu**• Recursion Practice: fibo • History of Scheme: LISP • Introducing Lists CS 200 Spring 2004**Defining Recursive Procedures**• Be optimistic. • Assume you can solve it. • If you could, how would you solve a bigger problem. • Think of the simplest version of the problem, something you can already solve. (This is the base case.) • Combine them to solve the problem. CS 200 Spring 2004**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 CS 200 Spring 2004**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)? CS 200 Spring 2004**Tracing Fibo**> (require-library "trace.ss") > (trace fibo) (fibo) > (fibo 3) |(fibo 3) | (fibo 2) | 1 | (fibo 1) | 1 |2 2 This turns tracing on CS 200 Spring 2004**> (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 |5 5 (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)? CS 200 Spring 2004**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)) CS 200 Spring 2004**Fast-Fibo Results**> (fast-fibo 1) 1 > (fast-fibo 10) 55 > (time (fast-fibo 100)) cpu time: 0 real time: 0 gc time: 0 354224848179261915075 CS 200 Spring 2004**;;; 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)) 9.916666666666666 According to Fibonacci’s model, after less than 10 years, rabbits would out-weigh the Earth! Beware the Bunnies!! CS 200 Spring 2004**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 CS 200 Spring 2004**LISP**“Lots of Insipid Silly Parentheses” “LISt Processing language” Lists are pretty important – hard to write a useful Scheme program without them. CS 200 Spring 2004**Making Lists**CS 200 Spring 2004**Making a Pair**> (cons 1 2) (1 . 2) 1 2 consconstructs a pair CS 200 Spring 2004**Splitting a Pair**cons > (car (cons 1 2)) 1 > (cdr (cons 1 2)) 2 1 2 car cdr car extracts first part of a pair cdr extracts second part of a pair CS 200 Spring 2004**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) CS 200 Spring 2004**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) CS 200 Spring 2004**Pairs are fine, but how do we make threesomes?**CS 200 Spring 2004**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? CS 200 Spring 2004**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))) CS 200 Spring 2004**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))) CS 200 Spring 2004**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 …? CS 200 Spring 2004**Lists**List ::= (consElementList) 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! CS 200 Spring 2004**From Lecture 5**Recursive Transition Networks ORNATE NOUN end begin NOUN ARTICLE ADJECTIVE ORNATE NOUN ::= ARTICLE ADJECTIVE NOUN ORNATE NOUN ::= ARTICLE ADJECTIVE ADJECTIVE NOUN ORNATE NOUN ::= ARTICLE ADJECTIVE ADJECTIVE ADJECTIVE NOUN ORNATE NOUN ::= ARTICLE ADJECTIVE ADJECTIVE ADJECTIVE ADJECTIVE NOUN ORNATE NOUN ::= ARTICLE ADJECTIVE ADJECTIVE ADJECTIVE ADJECTIVE ADJECTIVE NOUN CS 200 Spring 2004**Recursive Transition Networks**ORNATE NOUN end begin NOUN ARTICLE ADJECTIVE ORNATE NOUN ::= ARTICLE ADJECTIVES NOUN ADJECTIVES ::= ADJECTIVE ADJECTIVES ADJECTIVES ::= CS 200 Spring 2004**Lists**List ::= (consElementList) List ::= It’s hard to write this! A list is either: a pair where the second part is a list or, empty CS 200 Spring 2004**Null**List ::= (consElementList) List ::= null A list is either: a pair where the second part is a list or, empty (null) CS 200 Spring 2004**List Examples**> null () > (cons 1 null) (1) > (list? null) #t > (list? (cons 1 2)) #f > (list? (cons 1 null)) #t CS 200 Spring 2004**More List Examples**> (list? (cons 1 (cons 2 null))) #t > (car (cons 1 (cons 2 null))) 1 > (cdr (cons 1 (cons 2 null))) (2) CS 200 Spring 2004**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) CS 200 Spring 2004