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1 Administrivia Lists, higher order procedures, and symbols Structure and Interpretation of Computer Programs Mike Phillips (mpp) Massachusetts Institute of Technology Project 0 was due today Reminder: Project 1 due at 7pm on Tuesday Mail to If you didn t sign up on Tuesday, let us know Lecture 2 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 1 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 2 / 65 Types Simple data types (+ 5 10) => 15 (+ "hi" 15) => +: expects type <number> as 1st argument, given: "hi"; other arguments were: 15 Addition is not defined for strings Only works for things of type number Scheme checks types for simple built-in functions Everything has a type: Number String Boolean Procedures? Is the type of not the same type as +? Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 3 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 4 / 65

2 What about procedures? Type examples (+ 5 10) => 15 Procedures have their own types, based on arguments and return value number number means takes one number, returns a number (+ "hi" 15) => +: expects type <number> as 1st argument, given: "hi"; other arguments were: 15 What is the type of +? number, number number (mostly) Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 5 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 6 / 65 Type examples More complicated examples Expression: 15 "hi" square > Type of a procedure is a contract... is of type: number string number number number, number boolean If the operands have the specified types, the procedure will result in a value of the specified type Otherwise, its behavior is undefined (lambda (a b c) (if (> a 0) (+ b c) (- b c))) (lambda (p) (if p "hi" "bye")) (lambda (x) (* 3.14 (* 2 5))) number, number, number number boolean string any number Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 7 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 8 / 65

3 Patterns across procedures Summation Procedural abstraction is finding patterns, and making procedures of them: (* 17 17) (* 42 42) (* x x)... (lambda (x) (* x x)) π2 2 8 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 9 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 10 / 65 Summation Complex types (define (sum-integers a b) (+ a (sum-integers (+ 1 a) b)))) (define (sum-squares a b) (+ (square a) (sum-squares (+ 1 a) b)))) (define (pi-sum a b) (+ (/ 1 (square a)) (pi-sum (+ 2 a) b)))) (define (sum term a next b) (+ (term a) (sum term (next a) next b)))) (define (sum term a next b) (+ (term a) (sum term (next a) next b)))) What is the type of this procedure? What type is the output? (number number), number, (number number), number number How many arguments does it have? What is the type of each argument? Higher-order procedures take a procedure as an argument, or return one as a value Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 11 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 12 / 65

4 Higher-order procedures Higher-order procedures b k k=a (define (sum-integers a b) (+ a (sum-integers (+ 1 a) b)))) (define (sum term a next b) (+ (term a) (sum term (next a) next b)))) (define (new-sum-integers a b) (sum (lambda (x) x) a (lambda (x) (+ x 1)) b)) b k=a (define (sum-squares a b) (+ (square a) (sum-squares (+ 1 a) b)))) (define (sum term a next b) (+ (term a) (sum term (next a) next b)))) (define (new-sum-squares a b) (sum square a (lambda (x) (+ x 1)) b)) k 2 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 13 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 14 / 65 Higher-order procedures Returning procedures b k=a k odd 1 k 2 π2 8 (define (pi-sum a b) (+ (/ 1 (square a)) (pi-sum (+ 2 a) b)))) (define (sum term a next b) (+ (term a) (sum term (next a) next b)))) (define (new-pi-sum a b) (sum (lambda (x) (/ 1 (square x))) a (lambda (x) (+ x 2)) b))... takes a procedure as an argument or returns one as a value (define (new-sum-integers a b) (sum (lambda (x) x) a (lambda (x) (+ x 1)) b)) (define (new-sum-squares a b) (sum square a (lambda (x) (+ x 1)) b)) (define (add1 x) (+ x 1)) (define (new-sum-squares a b) (sum square a add1 b)) (define (new-pi-sum a b) (sum (lambda (x) (/ 1 (square x))) a (lambda (x) (+ x 2)) b)) (define (add2 x) (+ x 2)) (define (new-pi-sum a b) (sum (lambda (x) (/ 1 (square x))) a add2 b)) Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 15 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 16 / 65

5 Returning procedures Returning procedures (define (add1 x) (+ x 1)) (define (add2 x) (+ x 2)) (define incrementby (lambda (n)... )) (define add1 (incrementby 1)) (define add2 (incrementby 2)) (define add37.5 (incrementby 37.5)) type of incrementby: number (number number) (define incrementby ; type: num -> (num->num) (lambda (n) (lambda (x) (+ x n)))) ( incrementby 2 ) ( (lambda (n) (lambda (x) (+ x n))) 2 ) (lambda (x) (+ x 2)) ( (incrementby 2) 4) ((lambda (x) (+ x 2)) 4) (+ 4 2) 6 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 17 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 18 / 65 Procedural abstraction Procedural abstraction (define sqrt (lambda (x) (try 1 x)) (define try (lambda (guess x) (if (good-enough? guess x) guess (try (improve guess x) x)))) (define good-enough? (lambda (guess x) (< (abs (- (square guess) x)) 0.001))) (define improve (lambda (guess x) (average guess (/ x guess)))) (define average (lambda (a b) (/ (+ a b) 2))) (define sqrt (lambda (x) (define try (lambda (guess x) (if (good-enough? guess x) guess (try (improve guess x) x)))) (define good-enough? (lambda (guess x) (< (abs (- (square guess) x)) 0.001))) (define improve (lambda (guess x) (average guess (/ x guess)))) (try 1 x)) (define average (lambda (a b) (/ (+ a b) 2))) Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 19 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 20 / 65

6 Summary of types Compound data A type is a set of values Every value has a type Procedure types (types which include ) indicate: Number of arguments required Type of each argument Type of the return value They provide a mathematical theory for reasoning efficiently about programs Useful for preventing some common types of errors Basis for many analysis and optimization algorithms Need a way of (procedure for) gluing data elements together into a unit that can be treated as a simple data element Need ways of (procedures for) getting the pieces back out Need a contract between glue and unglue Ideally want this gluing to have the property of closure: The result obtained by creating a compound data structure can itself be treated as a primitive object and thus be input to the creation of another compound object. Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 21 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 22 / 65 Pairs (cons cells) Pairs are tasty (cons <a> <b>) <p> Where <a> and <b> are expressions that map to <a-val> and <b-val> Returns a pair <p> whose car-part is <a-val> and whose cdr-part is <b-val> (car <p>) <a-val> (cdr <p>) <b-val> (define p1 (cons 4 (+ 3 2))) (car p1) ; -> 4 (cdr p1) ; -> 5 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 23 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 24 / 65

7 Pairs are a data abstraction Pair abstraction Constructor (cons A B) Pair<A,B> Accessors (car Pair<A,B>) A (cdr Pair<A,B>) B Contract (car (cons A B)) A (cdr (cons A B)) B Operations (pair? Q) returns #t if Q evaluates to a pair, #f otherwise Abstraction barrier Once we build a pair, we can treat it as if it were a primitive Pairs have the property of closure we can use a pair anywhere we would expect to use a primitive data element: (cons (cons 1 2) 3) Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 25 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 26 / 65 Building data abstractions Building data abstractions (define (make-point x y) (cons x y)) (define (point-x point) (car point)) (define (point-y point) (cdr point)) (define p1 (make-point 2 3)) (define p2 (make-point 4 1)) (define make-point cons) (define point-x car) (define point-y cdr) (define p1 (make-point 2 3)) (define p2 (make-point 4 1)) What type is make-point? number, number Point Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 27 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 28 / 65

8 Building on earlier abstraction Using data abstractions ;;; Point abstraction (define (make-point x y) (cons x y)) (define (point-x point) (car point)) (define (point-y point) (cdr point)) (define p1 (make-point 2 3)) (define p2 (make-point 4 1)) ;;; Segment abstraction (define (make-seg pt1 pt2) (cons pt1 pt2)) (define (start-point seg) (car seg)) (define (end-point seg) (cdr seg)) (define s1 (make-seg p1 p2)) (define p1 (make-point 2 3)) (define p2 (make-point 4 1)) (define s1 (make-seg p1 p2)) (define (stretch-point pt scale) (make-point (* scale (point-x pt)) (* scale (point-y pt)))) (stretch-point p1 2) -> (4. 6) p1 -> (2. 3) Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 29 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 30 / 65 Using data abstractions Using data abstractions (define p1 (make-point 2 3)) (define p2 (make-point 4 1)) (define s1 (make-seg p1 p2)) (define (stretch-point pt scale) (make-point (* scale (point-x pt)) (* scale (point-y pt)))) What type is stretch-point? Point, number Point (define p1 (make-point 2 3)) (define p2 (make-point 4 1)) (define s1 (make-seg p1 p2)) (define (stretch-seg seg scale) (make-seg (stretch-point (start-point seg) scale) (stretch-point (end-point seg) scale))) (define (seg-length seg) (sqrt (+ (square (- (point-x (start-point seg)) (point-x (end-point seg)))) (square (- (point-y (start-point seg)) (point-y (end-point seg))))))) Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 31 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 32 / 65

9 Using data abstractions Abstractions have two communities (define p1 (make-point 2 3)) (define p2 (make-point 4 1)) (define s1 (make-seg p1 p2)) (define (stretch-point pt scale) (make-point (* scale (point-x pt)) (* scale (point-y pt)))) (stretch-point p1 2) -> (4. 6) p1 -> (2. 3) Builders (define (make-point x y) (cons x y)) (define (point-x point) (car point)) Users (* scale (point-x pt)) Frequently the same person Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 33 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 34 / 65 Pairs are a data abstraction Rational number abstraction Constructor (cons A B) Pair<A,B> Accessors (car Pair<A,B>) A (cdr Pair<A,B>) B Contract (car (cons A B)) A (cdr (cons A B)) B Operations (pair? Q) returns #t if Q evaluates to a pair, #f otherwise Abstraction barrier A rational number is a ratio n d Addition: Multiplication: a b + c d = ad + bc bd = = a b c d = ac bd = 2 9 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 35 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 36 / 65

10 Rational number abstraction Rational number abstraction Constructor ; make-rat: integer, integer -> Rat (make-rat <n> <d>) -> <r> Accessors ; numer, denom: Rat -> integer (numer <r>) (denom <r>) Contract (numer (make-rat <n> <d>)) = <n> (denom (make-rat <n> <d>)) = <d> Operations (+rat x y) (*rat x y) Abstraction barrier Constructor Accessors Contract Operations Abstraction barrier Implementation ; Rat = Pair<integer, integer> (define (make-rat n d) (cons n d)) (define (numer r) (car r)) (define (denom r) (cdr r)) Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 37 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 38 / 65 Additional operators Using our system ; What is the type of +rat? Rat, Rat -> Rat (define (+rat x y) (make-rat (+ (* (numer x) (denom y)) (* (numer y) (denom x))) (* (denom x) (denom y)))) ; The type of *rat: Rat, Rat -> Rat (define (*rat x y) (make-rat (* (numer x) (numer y)) (* (denom x) (denom y)))) (define one-half (make-rat 1 2)) (define three-fourths (make-rat 3 4)) (define new (+rat one-half three-fourths)) (numer new) ;? (denom new) ;? We get 10 8, not the simplified 5 4 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 39 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 40 / 65

11 Rationalizing implementation Rationalizing implementation (define (gcd a b) (if (= b 0) a (gcd b (remainder a b)))) (define (make-rat n d) (cons n d)) (define (numer r) (/ (car r) (gcd (car r) (cdr r)))) (define (denom r) (/ (cdr r) (gcd (car r) (cdr r)))) Remove common factors when accessed (define (gcd a b) (if (= b 0) a (gcd b (remainder a b)))) (define (make-rat n d) (cons (/ n (gcd n d)) (/ d (gcd n d)))) (define (numer r) (car r)) (define (denom r) (cdr r)) Remove common factors when created Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 41 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 42 / 65 Grouping together larger collections Conventional interfaces lists We want to group a set of rational numbers (cons r1 r2)... A list is a type that can hold an arbitrary number of ordered items. Formally, a list is a sequence of pairs with the following properties: The car-part of a pair holds an item The cdr-part of a pair holds the rest of the list The list is terminated by the empty list: () Lists are closed under cons and cdr Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 43 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 44 / 65

12 Lists and pairs as pictures Lists (cons <el1> <el2>) el2 (list <el1> <el2>... <eln>) el1 Sequences of cons cells Better, and safer, to abstract: (define first car) (define rest cdr) (define adjoin cons) el1 el2 eln... but we don t for lists and pairs (list ) ; -> ( ) (null? <z>) ; -> #t if <z> evaluates to empty list Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 45 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 46 / 65 cons ing up lists cdr ing down lists (define 1thru4 (list )) (define 2thru7 (list )) (define (enumerate from to) (if (> from to) () (cons from (enumerate (+ 1 from) to)))) (define (length lst) (if (null? lst) 0 (+ 1 (length (cdr lst))))) (define (append list1 list2) (if (null? list1) list2 (cons (car list1) (append (cdr list1) list2)))) Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 47 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 48 / 65

13 Transforming lists Map (define (square-list lst) (if (null? lst) () (cons (square (car lst)) (square-list (cdr lst))))) (define (double-list lst) (if (null? lst) () (cons (* 2 (car lst)) (double-list (cdr lst))))) (define (map proc lst) (if (null? lst) () (cons (proc (car lst)) (map proc (cdr lst))))) (define (map proc lst) (if (null? lst) () (cons (proc (car lst)) (map proc (cdr lst))))) What is the type of map? (A B), List<A> List<B> Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 49 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 50 / 65 Choosing just part of a list Data Types in Scheme (define (filter pred lst) (cond ((null? lst) ()) ((pred (car lst)) (cons (car lst) (filter pred (cdr lst)))) (else (filter pred (cdr lst))))) (filter even? (list )) ;-> (2 4 6) Conventional Numbers: 29, 35, 1.34, 1.2e5 Characters and Strings: #\a "this is a string" Booleans: #t, #f Vectors: #(1 2 3 "hi" 3.7) Scheme-specific Procedures: value of +, result of evaluating (lambda (x) x) Pairs and lists: (42. 8), ( ) Symbols: pi, +, x, foo, hello-world What is the type of filter? (A Boolean), List<A> List<A> Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 51 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 52 / 65

14 Symbols How do we refer to Symbols? So far, we ve seen them as the names of variables (define foo (+ bar 2)) But, in Scheme, all data types are first class, so we should be able to: Pass symbols as arguments to procedures Return them as values of procedures Associate them as values of variables Store them in data structures For example: (chocolate caffeine sugar) Evaluation rule for symbols Value of a symbol is the value it is associated with in the environment. We associate symbols with values using the special form define (define pi ) (* pi 2 r) But how do we get to the symbol itself? (define baz pi)?? baz chocolate caffeine sugar Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 53 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 54 / 65 Referring to Symbols New special form: quote Say your favorite color Say your favorite color In the first case, we want the meaning associated with the expression In the second, we want the expression itself We use the concept of quotation in Scheme to distinguish between these two cases We want a way to tell the evaluator: I want the following object as whatever it is, not as an expression to be evaluated (quote foo) foo (define baz (quote pi)) undefined baz pi (+ pi baz) ERROR +: expects type <number> as 2nd argument, given: pi; other arguments were: (list (quote foo) (quote bar) (quote baz)) (foo bar baz) Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 55 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 56 / 65

15 Syntactic sugar Making list structures with symbols The Reader (part of the Read-Eval-Print Loop, REPL) knows a short-cut When it sees pi it acts just like it had read (quote pi) The latter is what is actually evaluated Examples: pi pi "Hello world" "Hello world" (1 2 3) (1 2 3) (list (quote brains) (quote caffeine) (quote sugar)) ; -> (brains caffeine sugar) (list brains caffeine sugar) ; -> (brains caffeine sugar) (brains caffeine sugar) ; -> (brains caffeine sugar) (define x 42) (define y (x y z)) (list (list foo bar) (list x y) (list baz quux squee)) ; -> ((foo bar) (42 (x y z)) (baz quux squee)) ((foo bar) (x y) (bar quux squee)) ; -> ((foo bar) (x y) (bar quux squee)) Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 57 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 58 / 65 Confusing examples Operations on symbols (define x 20) (+ x 3) ; -> 23 (+ x 3) ; -> (+ x 3) (list (quote +) x 3) ; -> (+ 20 3) (list + x 3) ; -> (+ 20 3) (list + x 3) ; -> (#<procedure:+> 20 3) symbol? has type anytype boolean, returns #t for symbols (symbol? (quote foo)) #t (symbol? foo) #t (symbol? 4) #f (symbol? (1 2 3)) #f (symbol? foo) It depends on what value foo is bound to eq? tests the equality of symbols Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 59 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 60 / 65

16 An aside: Testing for equality (= 4 10) ; -> #f (= 4 4) ; -> #t (equal? 4 4) ; -> #t (equal? (/ 1 2) 0.5) ; -> #f (eq? 4 4) ; -> #t (eq? (expt 2 70) (expt 2 70)) ; -> #f eq? tests if two things are exactly the same object in memory. Not for strings or numbers. = tests the equality of numbers equal? tests if two things print the same symbols, numbers, strings, lists of those, lists of lists (= "foo" "foo") ; -> Error! (eq? "foo" "foo") ; -> #f (equal? "foo" "foo") ; -> #t (eq? (1 2) (1 2)) ; -> #f (equal? (1 2) (1 2)) ; -> #t (define a (1 2)) (define b (1 2)) (eq? a b) ; -> #f (define a b) (eq? a b) ; -> #t Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 61 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 62 / 65 Tagged data Benefits of tagged data Attaching a symbol to all data values that indicates the type Can now determine if something is the type you expect (define (make-point x y) (list point x y)) (define (make-rat n d) (list rat x y)) (define (point? thing) (and (pair? thing) (eq? (car thing) point))) (define (rat? thing) (and (pair? thing) (eq? (car thing) rat))) Data-directed programming - decide what to do based on type (define (stretch thing scale) (if (point? thing) (stretch-point thing scale) (stretch-seg thing scale))) Defensive programming - Determine if something is the type you expect, give a better error (define (stretch-point pt) (if (not (point? pt)) (error "stretch-point passed a non-point:" pt) ;;...carry on )) Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 63 / 65 Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 64 / 65

17 Recitation time! Mike Phillips (MIT) Lists, higher order procedures, and symbols Lecture 2 65 / 65

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