Introduction to Programming,using Scheme

a.k.a. Bugs, and How to Learn From Them 101

Dinoj Surendran (dinoj@cs, Eckhart basement, Room 2a)

Basic Questions• What is programming? • Writing down an explicit set of instructions that a machine can follow, usually with a task in

mind.

• What is Scheme? • A functional programming language based on Lisp.

• How are my grades calculated? • Homework 60%, Midterm 20%, Final 20%.• 16 homeworks at 4% each. • Extra credit for challenge problems.

• What’s the course webpage?• http://www.cs.uchicago.edu/~dinoj/2002summer115.html

• What are my Office Hours?• Tuesdays and Thursdays 1300 – 1400. I am always accessible by email or via the class

mailing list.

More Basic Questions• What text are we using? Where do I get it?• Structure and Interpretation of Computer Programs, Abelson and Sussman2. The book is online,

see class webpage.

• Can I ask my friends for help with my homework?• No. You can discuss all you want about general concepts, but no more. • Homework suspected to be the product of collaboration receives no credit. Two incidents of this

trouble.

• How much homework will there be? • Lots. Check the class webpage an hour after class. It is due by 9pm of the day of the next class,

and returned to you at the class after that.

• What is the late homework policy?• 20% if handed late, but still on the deadline day, and 50% off if handed in the day after that.

Excuses for not doing it in time: please tell me 24 hours in advance of the deadline.

• What if I can’t finish my homework in time?• Hand it in anyway for partial credit. A short note saying detailing the bug would be good.

Computer Science

• A misnomer – by the same token, Biology should be called Microscope Science or Astronomy Telescope Science

• We study computation, and computers just happen to be the tools we use.

• The purpose of computing is insight, not numbers. -- R. Hamming

• Computer scientists study vision, languages, mathematics, hearing, kinematics, neuroscience, physics, etc in addition to compilers, operating systems and the internet

• Programming is a basic skill, just as calculus is a basic skill for mathematicians. Mathematicians don’t solve calculus problems all the time, but can do so when they have to.

Programming: Errors

• As soon as we started programming, we found to our surprise that it wasn't as easy to get programs right as we had thought. Debugging had to be discovered. I can remember the exact instant when I realized that a large part of my life from then on was going to be spent in finding mistakes in my own programs. -- Maurice Wilkes, 1949

• There are two ways to write error-free programs; only the third works. -- Alan J. Perlis

• I really hate this damned machineI wish that they would sell it.It never does quite what I wantBut only what I tell it. -- A Programmer's Lament

Programming: Erring & Learning

• The road to wisdom?Well its plain and simple to express:Err and err and err again, but less and less and less. -- Piet Hein

• I, myself, have had many failures and I've learned that if you are not failing a lot, you are probably not being as creative as you could be -you aren't stretching your imagination. -- J. Backus

• Learning is never done without errors and defeat. -- Lenin

Programming: Avoiding Errors

• The most important single aspect of software development is to be clear about what you are trying to build. -- Bjarne Stroustrup

• ...well over half of the time you spend working on a project (on the order of 70 percent) is spent thinking, and no tool, no matter how advanced, can think for you. Consequently, even if a tool did everything except the thinking for you -- if it wrote 100 percent of the code, wrote 100 percent of the documentation, … -- the best you could hope for would be a 30 percent improvement in productivity. In order to do better than that, you have to change the way you think.

-- Fred Brook [paraphrased]

Programming: Avoiding Errors II

• Haste is of the devil. Slowness is of God. -- H L Mencken

• You're bound to be unhappy if you optimize everything. -- Donald Knuth

• Programming can be fun, so can cryptography; however they should not be combined.

-- Kreitzberg and Shneiderman

• Programs must be written for people to read, and only incidentally for machines to execute. -- SICP

General Advice

• I hear and I forget; I see and I remember; I do and I understand. -- Anonymous Chinese Proverb

• He who asks is a fool for five minutes; he who does not ask remains a fool forever. -- ACP

• Never hesitate to ask a lesser person. -- Confucius • The first step in fixing a broken program is getting

it to fail repeatedly (on the simplest example possible). -- T. Duff

• To arrive at the simple is difficult. -- R. Elisha

Now (finally!) to Scheme…

Expressions

• Everything in Scheme is an expression, and has a value. For example, these are primitives, and are their own value:5#\c‘babalas “braai”4.1223+2i#t, #f

• The value is sometimes undefined, like (/ 1 0)

Combinations

• These are more complicated expressions:(+ 4 1 49) ( sqrt (* 2 8) )( string-append “the doh of” “Homer”)(= (/ 18 3) (- 8 2) )(* 3+3i 4+4i )

• These evaluate to 54, 4, “the doh ofHomer”, #t and 0+24i respectively.

• Note the prefix notation! Everything is of the form (function arg1 … argn)

More fun with prefix notation

Evaluate (* (/ 4 1) (- 4 2 9) (+ (/ 6 3) (sqrt 9)))- evaluate each argument separately:

(/ 4 1) 4/1 = 4

(- 4 2 9) 4-2-9 = -7

(+ (/ 6 3) (sqrt 9) ) (+ 2 3) 5- then put them together

(* 4 –7 5) -140

How Expressions are Evaluated

• To evaluate (operator arg1 arg2 … argn)– Evaluate all arguments (RECURSIVE!)– Apply operator to the value of each argument

• e.g.(+ (* (- 5 3) (sqrt 9)) (/ (* 20 2) (+ 3 2)))

(+ (* 2 3 ) (/ 10 5)

(+ 6 2 )

8

FunctionsProgramming languages like C, Java make a huge

distinction between functions and data. Not Scheme.Think of them as ‘pipes’ and ‘balls’, both of which

are objects.(lambda (x) (* x 2)) defines a function that doubles its argument(lambda (x y z) (+ x y z) ) defines a function that takes three arguments and returns

their sum.(lambda () 0) defines a function that takes no arguments and returns zero.(lamba (x) 0) defines a function taking one argument and returning zero.

How to use functions: place them at the start of a compound expression with the correct number of arguments, e.g.

( (lambda (x) (* x 2)) 3 ) 6( (lambda (x y z) (+ x y z)) 100 28 4) 132

Parentheses

We are used to using parentheses just to make expressions clearer. In Scheme, they have a specific meaning:

“What is inside me is a function and its arguments, and I’m going to apply the function to the arguments, and return the result.”

Therefore, expressions like (5) or (‘foo 3) are meaningless since 5 and ‘foo are not procedures.

Similarly, (+ 3 4) is meaningful, ((+ 3 4)) is not.

Data Abstraction

• Make it, name it, forget about it, use it• e.g. Incrementer: -- “make it” (lambda (x) (+ x 1)) -- “name it” (define inc (lambda (x) (+ x 1))) -- “forget about it” – you no longer need to remember

how inc was implemented, just know what it does -- “use it” (inc 312) 313• both data and functions can be abstracted, e.g. (define

quayle-iq 50) means that you can say (inc quayle-iq) and receive 51

Why is Data Abstraction a Good Thing?

• Don’t need to know how something is implemented in order to use it – imagine a world where you needed to know how your car worked to get your driver’s license (it would be a world with a lot of illegal drivers).

• Can change the implementation of something without changing the way it is used – your car engine can be replaced with a new one without you having to change the way you drive your car.

• Code reuse – you don’t have to implement everything from scratch – if you know you need new brakes, and know the specs used by your car, you can get them from someone else without having to make them yourself.

Example 1.1

• A function that returns the circumference of a circle, when given its radius

• (define pi 3.14159)• (define circum

(lambda (r) (* 2 r pi)))

• Alternatively (define (circum x) (* 2 x pi) )• Note how name of parameter (x or r) is irrelevant• Usage: (circum 5) 31.4159

Example 1.2

• Function which greets when you give it a name• (define hello

(lambda (name) (string-append “hello ” name “ you slimy

worm”) ) )

• (define (hello name) (string-append “hello ” name “ you slimy worm”) )

• Usage: (hello “don”) “hello don you slimy worm”

Example 1.3

• (define (piggy x) (* 2 x) )• (define (gonzo x y) (- x y) )• (define (bunsen x y) (/ x y) )• (define (kermit x y)

(bunsen (piggy x) (gonzo x y) ) )Usage: (gonzo 100 2) 98(bunsen 100 0) “aaargh!!”(piggy (bunsen 6 2)) (piggy 3) 6(kermit 5 2) (bunsen (piggy 5) (gonzo 5 2) )

(bunsen 10 3) 10/3

The Substitution Model

• How expressions are evaluated– If the expression is a primitive, it is its own

value– If it is a compound expression -- not a special

form -- like (function arg1 … argn), then• Evaluate function, arg1, … argn

• Apply function to arguments, by replacing the parameters in the body of the function definition with the arguments, and then evaluating this

Example 1.4(Substitution Model)

• (define piggy (lambda (x) (* 2 x)) )• (define gonzo (lambda (x y) (- x y)) )• Evaluate (piggy (gonzo 3 5))

-- evaluate piggy: (lambda (x) (* 2 x))-- evaluate (gonzo 3 5) -- evaluate gonzo: (lambda (x y) (- x y) ) -- evaluate 3 5: these are primitives so are themselves. -- apply (lambda (x y) (- x y)) to 3 5: eval (- 3 5) -2 -- (gonzo 3 5) has been evaluated to get –2-- apply (lambda (x) (* 2 x)) to –2: eval (* 2 –2) -4

Substitution Model : fine print

• This is an idealized version of how things work• EXCEPTION: If the function is AND or OR, and

the first argument evaluates to #f or #t respectively, the second argument is not evaluated.

• Therefore (or (= 2 2) (/ 1 0) ) will not produce an error since (/ 1 0) will not be evaluated

• Suppose we defined our own OR function(define myor (lambda (x y) (or x y)) )(define (myor x y) (or x y) )

• (myor (= 2 2) (/ 1 0))will produce an error. WHY?

Conditional Statements

• The general form of a conditional:(cond (<p1> <e1>)

(<p2> <e2>) ... (<pn> <en>))

• For example, this function returns the max of a,b(lambda (a b)

(cond ( (> a b) a)( (< a b) b)( (= a b) a)

))

Conditional Statements (else)

• For convenience, the last condition can be “else”, which covers every other condition

(lambda (a b)

(cond ((> a b) a)

( else b)

)

)

Conditional Statements (if)

• A common occurrence is that there are only two conditions:

(if condition dothis dothat)

• For example

(lambda (a b) (if (> a b) a b) )

(lambda (x) ( (if (> x 0) + -) x) )

implement the MAX and ABS function

Example 1.5• The IF statement can be replaced by the COND

statement.

(define (myif condition dothis dothat)

(cond

(condition dothis)

(else dothat)

)

)

Example 1.5

• Consider the following function to return (sin x)/x(define (lambda x)

(myif (= x 0) 1 (/ (sin x) x)

))

• This crashes when given the input x=0. • The same function, with MYIF replaced by IF, does not

crash. WHY?• Moral: IF can be replaced by COND, but only

internally.

Example 1.6

• Define a procedure that displays the roots of ax2+bx+c = 0. If there are no roots, it displays “no roots”. You can assume a is nonzero.

• Solution:– calculate b2-4ac– If this is negative, display “no roots”– Otherwise, display (-b +/- sqrt(b2-4ac) ) / 2a

Example 1.6

• (define (discrim a b c) (- (* b b) (* 4 a c) ) )• (define (solvequadratic a b c) ; assumes a nonzero

(if (< (discrim a b c) 0)(begin (display “no roots”) (newline) )(begin (display (/ (+ (- b) (sqrt (discrim a b c))) (* 2 a))) (newline) (display (/ (- (- b) (sqrt (discrim a b c))) (* 2 a))) (newline)

) ))

Example 1.6

• (define (solvequadratic a b c) (define (discrim a b c) (- (* b b) (* 4 a c) ) ) (if (< (discrim a b c) 0)

(begin (display “no roots”) (newline) )(begin (display (/ (+ (- b) (sqrt (discrim a b c))) (* 2 a))) (newline) (display (/ (- (- b) (sqrt (discrim a b c))) (* 2 a))) (newline)

) ))

Example 1.6

• (define (solvequadratic a b c) (define discrim (- (* b b) (* 4 a c) ) ) (if (< discrim 0)

(begin (display “no roots”) (newline) )(begin (display (/ (+ (- b) (sqrt discrim)) (* 2 a))) (newline) (display (/ (- (- b) (sqrt discrim)) (* 2 a)))

(newline) ) ))

Example 1.6

• Improved solution: store intermediate variables. (More on the LET statement in future lectures.) (define (solvequadratic a b c)

(define discrim (- (* b b) (* 4 a c) ) )(if (< discrim 0) (begin (display "no roots") (newline) ) (let* ((denom (* 2 a))

(part1 (/ (- b) denom) )(part2 (/ (sqrt discrim) denom) ))

(begin (display (- part1 part2) ) (newline)

(display (+ part1 part2) ) (newline)

))))

Example 1.7• Rewrite previous example so that it deals with all possible

cases of a, b, c. (define (solvegeneralquad a b c)

(if (= a 0) (if (= b 0) (if (= c 0) (begin (display "any x is solution") (newline) )

(begin (display "no solutions") (newline) ) ) (begin (display (/ (- c) b) ) (newline) ) )

(solvequadratic a b c) )

)

Example 1.8

Finding square roots with Newton’s Method: begin x:= 1

while x not close enough to sqrt(A)x = ½ (x + A/x)

end

x A/x ½(x + A/x)

1 2/1 = 2 ½ (2 + 1) = 1.5   1.5 2/1.5 = 1.3333 ½ (1.3333 + 1.5) =1.4167   1.4167 2/1.4167 = 1.4118 ½ (1.4167 + 1.4118)

=1.4142

Example 1.8

Implementing a procedure like this will become standard to you. For now, you have to think a bit. The basic idea is to keep trying new guesses:

(define (trythis x A)(if (goodenough x A)

x(trythis (newguess x A) A)

))

You now need to figure out what (goodenough x A) and (newguess x A) should be.

Example 1.8

The algorithm clearly states how we should find a new guess given the previous guess:

(define (newguess x A) (/ (+ x (/ A x)) 2) )

An unsophisticated way of deciding whether a guess x is good enough is checking if |x2 – A| < 0.001

(define (goodenough x A)

(< (abs (- (* x x) A)) 0.001)

)

Example 1.8

We have now defined three separate parts of our program:(define (trythis x A)

(if (goodenough x A)x(trythis (newguess x A) A)))

(define (newguess x A) (/ (+ x (/ A x)) 2) )(define (goodenough x A) (< (abs (- (* x x) A)) 0.001) )

How do we put all this together? We need to make the call (trythis 1 A).

Example 1.8

(define (NewtonRoot A)

(define (trythis x A)

(if (goodenough x A)

x

(trythis (newguess x A) A)))

(define (newguess x A) (/ (+ x (/ A x)) 2) )

(define (goodenough x A) (< (abs (- (* x x) A)) 0.001) )

(trythis 1.0 A)

)

Summary

• Computer Science isn’t all about programming• Programming is all about making errors and learning from them.

And also about thinking.• In Scheme, every expression has a value.• Parentheses mean (evaluate me).• Scheme makes no distinction between functions and data. • Functions are defined using lambda.• The Substitution Model gives an idea as to how Scheme

evaluates expressions.• Data abstraction means naming data and functions and then

using them via these names• Conditional statements are implemented using if and cond.