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