Scheme Tutorial. Goals Combine several simple ideas into one compound idea to obtain complex ideas Bring two ideas together to obtain relations Seperate

Scheme Tutorial

Goals

• Combine several simple ideas into one compound idea to obtain complex ideas

• Bring two ideas together to obtain relations

• Seperate ideas from all other ideas that accompany them in real existence to obtain general ideas (this is called abstraction)

by John Locke, An Essay Concerning Human Understanding (1690)

Features of LISP

• Recursive Functions of Symbolic Expressions and Their Computation by Machine (John McCarthy, 1960)

• LISP stands for LISt Processing

• An interpreted language (not efficient)

• Second oldest language (after FORTRAN)

• Designed to solve problems in the form of

symbolic differentiation & integration of algebraic expressions

Features of LISP

• LISP’s ability

– to represent procedures as data

– to manipulate programs as data

• LISP is currently a family of dialects (share most of the original features)

• Scheme is a dialect of LISP (small yet powerful)

Characteristics of SCHEME

• Supports functional programming - but not on an exclusive basis

• Functions are first class data objects

• Uses static binding of free names in procedures and functions

• Types are checked and handled at run time - no static type checking

• Parameters are evaluated before being passed - no lazyness

Elements of Language

• Primitive expressions – simple expressions

• Means of combination – compound expressions

• Means of abstraction – compound objects can

be named and manipulated as units

Scheme Expressions - 1

> 395

395

> (+ 137 349)

486

> (/ 10 6)

1.66667


Prefix Notation

take arbitrary number of arguments

> (+ 21 35 12 7)

75

> (* 25 4 12)

1200


Prefix Notation

allow combinations to be nested

> (+ (* 3 5) (- 10 6) )

19

> (* (+ 3 (- (+ 2 1) 5) (/ 4 2)) (* 3 2))

Read – Eval – Print loop

Scheme Pretty-Printing

> (+ (* 3 (+ (* 2 4) (+ 3 5))) (+ (- 10 7) 6))

(+ (* 3

(+ (* 2 4)

(+ 3 5)))

(+ (- 10 7)

6))

> (load “e1.scm”)

Scheme Naming

> (define size 2)

> size

2

> (* 5 size)

10


Compound Procedures

> (define (square x) (* x x))

> (square 3)

9

(define (< name > < formal parameters >) < body > )


Substitution model for procedure application

Conditional Expressions

> (define (abs x)

(cond ((> x 0) x)

((= x 0) 0)

((< x 0) (- x))))

> (define (abs x)

(cond ((< x 0) (- x))

(else x)))

Conditional Expressions(cond (<p1> <e1>) (<p2> <e2>)

: (<pN> <eN>))

p is predicate either true (non-nil) or

false (nil)

e is consequent expression returns the value of it

Conditional Expressions

> (define (abs x)

(if (< x 0)

(- x)

x))

(if <predicate> <consequent> <alternative>)

Logical Operators

Primitive operators : < , > , =

Logical operators : and , or , not

> (define (>= x y)

(or (> x y) (= x y)))

Example: another definition of (>= x y)

Example: 5 < x < 10

Proceduresdifferent than mathematical functions

x = the y such that y 0 and y2 = x

> (define (sqrt x)

(the y (and (>= y 0) (= (square y) x))))

Mathematics – declarative (what is)

Programs – imperative (how to)

Square Roots by Newton’s Method

> (load “e4.scm”) - 2 = 1.4142...

Square Roots by Newton’s Method

break the problem into subproblems

how to tell whether a guess is good enough

how to improve a guess

how to calculate the average of two numbers

etc.

each of the above tasks is a procedure

Square Roots by Newton’s Methodsqrt

sqrt-iter

good-enough?

square abs

improve

average


Procedures – Black Box Abstractions

(define (square x)

(* x x))

(define (square x)

(exp (+ (log x) (log x))))

Procedural Abstractions(define (square x) (* x x))

(define (square y) (* y y))

(define (square variable) (* variable variable))

parameter names that are local to the procedure

bound variables – change throughout the procedure does not change the meaning of the procedure

free variables – if a variable is not bound in the proc

(define (good-enough? guess x)

(< (abs (- (square guess) x)) .001))

Internal DefinitionsEncapsulation – hiding details


Nesting definitions – block structure


Lexical scoping – not necessary to pass x explicitly

to internal procedures, so x becomes free variable

in the internal definitions


Procedures

Procedure :

a pattern for the local evolution of

a computational process.

At each step,

the next state of the process is computed from

its current state according to the rules of

interpreting procedures.

Linear Recursion(factorial 6)

(* 6 (factorial 5))

(* 6 (* 5 (factorial 4)))

(* 6 (* 5 (* 4 (factorial 3))))

(* 6 (* 5 (* 4 (* 3 (factorial 2)))))

(* 6 (* 5 (* 4 (* 3 (* 2 (factorial 1))))))

(* 6 (* 5 (* 4 (* 3 (* 2 1)))))

(* 6 (* 5 (* 4 (* 3 2))))

(* 6 (* 5 (* 4 6)))

(* 6 (* 5 24))

(* 6 120)

720 process does grow and shrink

Linear Recursionn! = n * (n-1) * (n-2) ... 2 * 1

n! = n * (n-1)!

n! = n * ( (n-1) * (n-2)! )

n! = n * ( ... ( (n-1) * (n-2) * ... * 1! ) ) ... )

(define (factorial n)

(if (= n 1)

1

(* n (factorial (- n 1)))))

Linear Iteration

(factorial 6)

(fact-iter 1 1 6)

(fact-iter 1 2 6)

(fact-iter 2 3 6)

(fact-iter 6 4 6)

(fact-iter 24 5 6)

(fact-iter 120 6 6)

(fact-iter 720 7 6)

720process does not grow and shrink

Linear IterationProduct, counter = 1

do while counter < n

product = counter * product

counter = counter + 1

(define (factorial n)

(fact-iter 1 1 n))

(define (fact-iter product counter max-count)

(if (> counter max-count)

product

(fact-iter (* counter product)

(+ counter 1)

max-count)))

Tree RecursionFibonacci numbers : 0, 1, 1, 2, 3, 5, 8, 13, 21, ...

0 if n = 0

Fib( n ) = 1 if n = 1

Fib(n-1) + Fib(n-2) otherwise

(define (fib n)

(cond ((= n 0) 0)

((= n 1) 1)

(else (+ (fib (- n 1))

(fib (- n 2))))))

Tree Recursion

Fib(5)

Fib(4) Fib(3)

Fib(3) Fib(2) Fib(2) Fib(1)

Fib(2) Fib(1)

Fib(1) Fib(0)

Fib(1) Fib(0) Fib(1) Fib(0)

Tree Recursion

For Fibonacci numbers,

Use linear iteration instead of tree recursion

(define (fib n)

(fib-iter 1 0 n))

(define (fib-iter a b count)

(if (= count 0)

b

(fib-iter (+ a b) a (- count 1))))

Exponentiation – linear recursion

(define (expt b n)

(if (= n 0)

1

(* b (expt b (- n 1)))))

Exponentiation – linear iteration

(define (expt b n)

(exp-iter b n 1))

(define (exp-iter b counter product)

(if (= counter 0)

product

(exp-iter b

(- counter 1)

(* b product))))

Exponentiation – fast method

(define (fast-exp b n)

(cond ( (= n 0) 1)

( (even? n) (square (fast-exp b (/ n 2))))

(else (* b (fast-exp b (- n 1))))))

(define (even? n)

(= (remainder n 2) 0))

bn = (bn/2)2 if n is even

bn = b * bn-1 if n is odd

Greatest Common Divisor

(define (gcd a b)

(if (= b 0)

a

(gcd b (remainder a b))))

GCD( a , b ) is defined to be the largest integer that evenly divides both a and b.

GCD( a , b ) = GCD( b , r ) where r is the remainder of a / b.

GCD(206, 40)=GCD(40,6)=GCD(6, 4)=GCD(4, 2)=GCD(2, 0)=2

Higher-Order Procedures

Build abstractions by assigning names to common patterns

(define (cube x) (* x x x))

Procedures that manipulate data

i.e. accept data as argument and return data

What about procedures that manipulate procedures

i.e. accept procedures as argument and return procedures

Higher-Order Procedures

Procedures as Parameters

(define (sum-cubes a b)

(if (> a b)

0

(+ (cube a) (sum-cubes (+ a 1) b))))

(define (sum-integers a b)

(if (> a b)

0

(+ a (sum-integers (+ a 1) b))))

Procedures as Parameters

(define (<name> a b)

(if (> a b)

0

(+ (<term> a) (<name> (<next> a) b))))

Procedures as Parameters(define (sum term a next b)

(if (> a b)

0

(+ (term a)

(sum term (next a) next b))))


(define (inc x)

(+ x 1))

(sum cube a inc b))

Procedures using LambdaLambda – define anonymous

(lambda (x) (+ x 1))

(lambda (<formal-parameters>) <body>)


(sum (lambda (x) (* x x x))

a

(lambda (x) (+ x 1))

b))

Procedures using Lambda

Lambda – define anonymous

(define (plus4 x) (+ x 4))

(define plus4 (lambda (x) (+ x 4))

Lambda Calculus

A lambda expression describes a "nameless" function

Specifies both the parameter(s) and the mapping

Consider this function: cube (x) = x * x * x

Corresponding lambda expr: (x) x * x * x

Can be "applied" to parameter(s) by placing the parameter(s) after the expression

((x) x * x * x)(3)

The above application evaluates to 27

Lambda Calculus

Based on notation (lambda (l) (car (car l)))

l is called a "bound variable"; think of it as a formal parameter name

Lambda expressions can be applied((lambda (l) (car (car l))) '((a b) c

d))

Lambda Examples

> (define x 6)

> (lambda (x) (+ x 1))

> (define inc (lambda (x) (+ x 1)))

> (define same (lambda (x) (x)))

> (if (even? x) inc same)

> ((if (even? x) inc same) 5)

6

Lambda Examples

> ((lambda(x) (+ x 1)) 3)

4

> (define fu-lst (list (lambda (x) (+ x 1)) (lambda (x) (* x 5))))

> fu-lst

(#<procedure> #<procedure>)

> ((second fu-lst) 6)

30

Internal Definitions Internal definitions: the special form, LET

(let ( (x ‘(a b c))

(y ‘(d e f)) )

(cons x y))

* Introduces a list of local names (use define for top-level entities, but use let for internal definitions)

* Each name is given a value

Using Let to Define Local Variables

f(x,y) = x(1 + xy)2 + y(1 – y) + (1 + xy)(1 – y)

a = 1 + xy

b = 1 – y

f(x,y) = xa2 + yb + ab


(define (f x y)

(define a (+ 1 (* x y)))

(define b (– 1 y))

(+ (* x (square a))

(* y b)

(* a b)))


(define (f x y)

(let ((a (+ 1 (* x y)))

(b (– 1 y)))

(+ (* x (square a))

(* y b)

(* a b)))


(define (f x y)

((lambda (a b)

(+ (* x (square a))

(* y b)

(* a b)))

(+ 1 (* x y))

(– 1 y)))


(let ((<var1> <exp1>)

(<var2> <exp2>)

:

(<varN> <expN>))

<body>)

Procedures as Returned ValuesThe derivative of x3 is 3x2

Procedure : derivative

Argument : a function

Return value : another function

Derivative Procedure

If f is a function and dx is some number, then Df of f is the function whose value at any number x is given (limit of dx) by

f(x + dx) – f(x)D f(x) = ---------------------- dx

Procedures as Returned Values

(define (deriv f dx)

(lambda (x)

(/ (- (f (+ x dx)) (f x))

(dx)))

> ((deriv cube .001) 5)

75.015

PairsCompund Structure called Pair

<pair> constructor procedure “cons <head> <rest>”

<head> extractor procedure “car <pair>”

<rest> extractor procedure “cdr <pair>”

(cadr <arg>) = (car (cdr <arg>))

(cons 1 2)

1

2 Box & Pointer Representation

Pairs (continued)(cons (cons 1 2) (cons 3 4))

1 2

3 4

(cons (cons 1 (cons 2 3)) 4)

1 2 3

4

Hierarchical Data

Pairs enable us to represent hierarchical data

hierarchical data – data made up of parts

Data structures such as sequences and trees

Data Structures

Lists

(list <a1> <a2> ... <aN>) is equal to

(cons <a1> (cons <a2> (cons ... (cons <aN> nil))...)

List Operations – append, delete, list, search, nth, len

Sets > (load “e8.scm”)

Trees > (load “e9.scm”)

Symbols and Quote

(define a 1)

(define b 2)

(list a b) (1 2)

(list ‘a ‘b) (a b)

(car ‘(a b c)) a

(cdr ‘(a b c)) (b c)

Data Abstraction

from Primitive Data to Compund Data

Real numbers – Rational numbers

Operations on primitive data : +, -, *, /

Operations on compound data : +rat, -rat, *rat, /rat

Generic operators for all numbers : add, sub, mul, div

Rational Numbers

(define (make-rat n d) (cons n d))

(define (numer x) (car x))

(define (denom x) (cdr x))

(define (+rat x y)

(make-rat (+ (* (numer x) (denom y))

(* (denom x) (numer y))

(* (denom x) (denom y))))

Use of Complex Numbers

Operations on compound data : +rat, -rat, *rat, /rat

+c -c *c /c

Complex arithmetic package

Rectangular representation Polar representation

List structure and primitive machine arithmetic

Use of Numbers

Generic operators for all numbers : add, sub, mul, div

add sub mul div

Generic arithmetic package

List structure and primitive machine arithmetic

+rat –rat *rat /rat +c –c *c /c + – * /

Rectangular Polar

Complex arithmetic Real arithmetic

Rational arithmetic

Complex Arithmetic - 1

z = x + i y where i2 = -1

Real coordinate is x

Imaginary coordinate is y

(define (make-rect x y) (cons x y))

(define (real-part z) (car z))

(define (imag-part z) (cdr z))

Im

Re

ry

xA

z = x + i y = r eiA

Complex Arithmetic - 2z = x + i y = r eiA

Magnitude is r

Angle is A

(define (make-polar r a)

(cons (* r (cos a)) (* r (sin a))))

(define (magnitude z)

(sqrt (+ (square (car z)) (square (cdr z)))))

(define (angle z)

(atan (cdr z) (car z)))


(define (+c z1 z2)

(make-rect (+ (real-part z1) (real-part z2))

(+ (imag-part z1) (imag-part z2))))

(define (*c z1 z2)

(make-polar (* (magnitude z1) (magnitude z2))

(+ (angle z1) (angle z2))))


We may choose to implement complex numbers in polar form instead of rectangular form.

(define (make-polar r a) (cons r a))

(define (make-rect x y)

(cons (sqrt (+ (square x) (square y))) (atan y x)))

The discipline of data abstraction ensures that the

implementation of complex-number operators is

independent of which representation we choose.

Manifest Types

A data object that has a type that can be recognized

and tested is said to have manifest type.

(define (attach-type type contents)

(cons type contents))

For complex numbers,

we have two types rectangular & polar

Manifest Types

(define (type datum)

(if (not (atom? datum))

(car datum)

(error “Bad typed datum – Type ” datum)))

(define (contents datum)

(if (not (atom? datum))

(cdr datum)

(error “Bad typed datum – Contents ” datum)))

Complex Numbers

(define (make-rect x y)

(attach-type ‘rect (cons x y)))

(define (make-polar r a)

(attach-type ‘polar (cons r a)))

(define (rect? z)

(eq? (type z) ‘rect))

(define (polar? z)

(eq? (type z) ‘polar))

Complex Numbers(define (real-part z)

(cond ( (rect? z)

(real-part-rect (contents z)))

( (polar? z)

(real-part-polar (contents z)))))

(define (imag-part z)

(cond ( (rect? z)

(imag-part-rect (contents z)))

( (polar? z)

(imag-part-polar (contents z)))))

Complex Numbers

(define (real-part-rect z) (car z))

(define (imag-part-rect z) (cdr z))

Let Expressions

(let ((a 4) (b -3))

(let ((a-squared (* a a))

(b-squared (* b b)))

(+ a-squared b-squared)))

25

Let Expressions

(let ((x 1))

(let ((x (+ x 1)))

(+ x x)))

4

Let Expressions

Shadowing may be avoided by choosing different names for variables. The expression above could be rewritten so that the variable bound by the inner let is new-x.

(let ((x 1))

(let ((new-x (+ x 1)))

(+ new-x new-x)))

4

Lambda Expressions

((lambda (x) (+ x x)) (* 3 4)) ⇒ 24

Because procedures are objects, we can establish a procedure as the value of a variable and use the procedure more than once.

(let ((double (lambda (x) (+ x x))))

(list (double (* 3 4))

(double (/ 99 11))

(double (- 2 7)))) (24 18 -10⇒ )

Lambda Expressions

(let ((double-any (lambda (f x) (f x x))))

(list (double-any + 13)

(double-any cons 'a))) (26 (a . a))⇒

Lambda Expressions

(define double-any

(lambda (f x)

(f x x)))

The variable double-any now has the same status as cons or the name of any other primitive procedure. We can use double-any as if it were a primitive procedure.

(double-any + 10) 20⇒

(double-any cons 'a) (a . a)⇒

Lambda Expressions

(map abs '(1 -2 3 -4 5 -6)) (1 2 3 4 5 6)⇒

(map cons '(a b c) '(1 2 3)) ((a . 1) (b . 2) (c . 3))⇒

(map (lambda (x) (* x x)) '(1 -3 -5 7)) (1 9 25 49)⇒

Lambda Expressions

(define map1

(lambda (p ls)

(if (null? ls)

'()

(cons (p (car ls))

(map1 p (cdr ls))))))

(map1 abs '(1 -2 3 -4 5 -6)) (1 2 3 4 5 6)⇒

Lambda Expressions

(let ((x 'a))

(let ((f (lambda (y) (list x y))))

(f 'b))) (a b)⇒

The occurrence of x within the lambda expression refers to the x outside the lambda that is bound by the outer let expression.

The variable x is said to occur free in the lambda expression or to be a free variable of the lambda expression.

The variable y does not occur free in the lambda expression since it is bound by the lambda expression.

Free & Bound Variables> (occurs-free? ’x ’x)

#t

> (occurs-free? ’x ’y)

#f

> (occurs-free? ’x ’(lambda (x) (x y)))

#f

> (occurs-free? ’x ’(lambda (y) (x y)))

#t

> (occurs-free? ’x ’((lambda (x) x) (x y)))

#t

> (occurs-free? ’x ’(lambda (y) (lambda (z) (x (y z)))))

#t

Free & Bound VariablesWe can summarize these cases in the rules:

• If the expression e is a variable, then the variable x occurs free in e if and only if x is the same as e.

• If the expression e is of the form (lambda (y) e), then the variable x occurs free in e if and only if y is different from x and x occurs free in e.

• If the expression e is of the form (e1 e2), then x occurs free in e if and only if it occurs free in e1 or e2. Here, we use “or” to mean inclusive or, meaning that this includes the possibility that x occurs free in both e1 and e2. We will generally use “or” in this sense.

Free & Bound Variables

(define occurs-free?

(lambda (var exp)

(cond

((symbol? exp) (eqv? var exp))

((eqv? (car exp) ’lambda)

(and (not (eqv? var (car (cadr exp))))

(occurs-free? var (caddr exp))))

(else

(or

(occurs-free? var (car exp))

(occurs-free? var (cadr exp)))))))

Documents

Scheme Tutorial. Goals Combine several simple ideas into one compound idea to obtain complex ideas Bring two ideas together to obtain relations Seperate