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

CS200: Computer ScienceUniversity of VirginiaComputer Science

Class 33:Learning to Count

23 January 2004 CS 200 Spring 2004 2

Universal Computationz z z z z z z z z z z z z z z zz z z z

1

Start

HALT

), X, L

2: look for (

#, 1, -

), #, R

(, #, L

(, X, R

#, 0, -

Finite State Machine

Read/Write Infinite TapeMutable Lists

Finite State MachineNumbers to keep track of state

ProcessingWay of making decisions (if)Way to keep going

To prove Lambda Calculus is as powerful as a UTM, we must show we can make everything we need to simulate any TM.

23 January 2004 CS 200 Spring 2004 3

Making “Primitives”from Only Glue ()

23 January 2004 CS 200 Spring 2004 4

In search of the truth?

• What does true mean?

• True is something that when used as the first operand of if, makes the value of the if the value of its second operand:

if T M N M

23 January 2004 CS 200 Spring 2004 5

Don’t search for T, search for if

T x (y. x) xy. x

F x ( y. y))if pca . pca

23 January 2004 CS 200 Spring 2004 6

Finding the Truth

T x . (y. x)

F x . (y. y)

if p . (c . (a . pca)))

if T M N ((pca . pca) (xy. x)) M N

(ca . (x.(y. x)) ca)) M N

(x.(y. x)) M N

(y. M )) N M

Is the if necessary?

23 January 2004 CS 200 Spring 2004 7

and and or?

and x (y. if x y F))

or x (y. if x T y))

23 January 2004 CS 200 Spring 2004 8

Lambda Calculus is a Universal Computer?

z z z z z z z z z z z z z z z zz z z z

1

Start

HALT

), X, L

2: look for (

#, 1, -

), #, R

(, #, L

(, X, R

#, 0, -

Finite State Machine

• Read/Write Infinite Tape? Mutable Lists• Finite State Machine? Numbers to keep track of state• Processing Way of making decisions (if)? Way to keep going

23 January 2004 CS 200 Spring 2004 9

What is 42?

42forty-two

XLIIcuarenta y dos

23 January 2004 CS 200 Spring 2004 10

Meaning of Numbers

• “42-ness” is something who’s successor is “43-ness”

• “42-ness” is something who’s predecessor is “41-ness”

• “Zero” is special. It has a successor “one-ness”, but no predecessor.

23 January 2004 CS 200 Spring 2004 11

Meaning of Numberspred (succ N)

Nsucc (pred N)

Nsucc (pred (succ N))

succ N

23 January 2004 CS 200 Spring 2004 12

Meaning of Zerozero? zero

T zero? (succ zero)

F zero? (pred (succ zero))

T

23 January 2004 CS 200 Spring 2004 13

Is this enough?

• Can we define add with pred, succ, zero? and zero?

add xy.if (zero? x) y

(add (pred x) (succ y))

23 January 2004 CS 200 Spring 2004 14

Can we define lambda terms that behave likezero, zero?, pred and succ?

Hint: what if we had cons, car and cdr?

23 January 2004 CS 200 Spring 2004 15

Numbers are Lists...

zero? null?pred cdr succ x . cons F x

23 January 2004 CS 200 Spring 2004 16

Making Pairs

Remember PS2…

(define (make-point x y) (lambda (selector) (if selector x y)))

(define (x-of-point point) (point #t)) (define (y-of-point point) (point #f))

23 January 2004 CS 200 Spring 2004 17

cons and carcons x.y.z.zxy

cons M N = (x.y.z.zxy) M N (y.z.zMy) N

z.zMN

car p.p Tcar (cons M N) car (z.zMN) (p.p T) (z.zMN)

(z.zMN) T

TMN

(x . y. x) MN

(y. M)N

M

T x . y. x

23 January 2004 CS 200 Spring 2004 18

cdr too!cons xyz.zxy

car p.p Tcdr p.p F

cdr cons M Ncdr z.zMN = (p.p F) z.zMN

(z.zMN) F

FMN

N

23 January 2004 CS 200 Spring 2004 19

Null and null?

null x.T

null? x.(x y.z.F)

null? null x.(x y.z.F) (x. T)

(x. T)(y.z.F)

T

23 January 2004 CS 200 Spring 2004 20

Null and null?

null x.T

null? x.(x y.z.F)

null? (cons M N) x.(x y.z.F) z.zMN (z.z MN)(y.z.F)

(y.z.F) MN

F

23 January 2004 CS 200 Spring 2004 21

Counting

0 null1 cons F 0

2 cons F 1

3 cons F 2

...

succ x.cons F x

pred x.cdr x

23 January 2004 CS 200 Spring 2004 22

42 = xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y xy.(z.z xy) xy. y x.T

23 January 2004 CS 200 Spring 2004 23

Arithmeticzero? null?succ x. cons F xpred x.x F

pred 1 = (x.x F) cons F null (cons F null) F

(xyz.zxy F null) F

(z.z F null) F

F F null

null 0

23 January 2004 CS 200 Spring 2004 24

Lambda Calculus is a Universal Computer

z z z z z z z z z z z z z z z zz z z z

1

Start

HALT

), X, L

2: look for (

#, 1, -

), #, R

(, #, L

(, X, R

#, 0, -

Finite State Machine

• Read/Write Infinite Tape Mutable Lists• Finite State Machine Numbers to keep track of state• Processing Way of making decisions (if) Way to keep going

We have this, butwe cheated using to make recursive definitions!

23 January 2004 CS 200 Spring 2004 25

Way to Keep Going

( f. (( x.f (xx)) ( x. f (xx)))) (z.z)

(x.(z.z)(xx)) ( x. (z.z)(xx))

(z.z) ( x.(z.z)(xx)) ( x.(z.z)(xx))

(x.(z.z)(xx)) ( x.(z.z)(xx))

(z.z) ( x.(z.z)(xx)) ( x.(z.z)(xx))

(x.(z.z)(xx)) ( x.(z.z)(xx))

...

This should give you some belief that we mightbe able to do it. We won’t cover the details of whythis works in this class. (CS655 sometimes does.)

23 January 2004 CS 200 Spring 2004 26

Charge

• Monday: Review session for Exam 2– Exam 2 covers through today– There will definitely be a question that

requires understanding the answer to question 5 on the PS7 comments!

• Tuesday Office Hours– 1-2pm and 4-5pm

• Exam 2 out Wednesday