David Evans http://www.cs.virginia.edu/evans Class 33: Learning to Count CS200: Computer Science University of Virginia Computer Science David Evans http://www.cs.virginia.edu/evans
Universal Computation z z z z z z z z z z z z z z z z z z z z ), X, L 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 ), #, R (, #, L 1 2: look for ( Start (, X, R HALT #, 1, - #, 0, - Finite State Machine 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
Making “Primitives” from Only Glue () 23 January 2004 CS 200 Spring 2004
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
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
Finding the Truth if T M N 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
and and or? and x (y. if x y F)) or x (y. if x T y)) 23 January 2004 CS 200 Spring 2004
Lambda Calculus is a Universal Computer? z z z z z z z z z z z z z z z z z z z z ), X, L ), #, R (, #, L 1 2: look for ( 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 Start (, X, R HALT #, 1, - #, 0, - Finite State Machine 23 January 2004 CS 200 Spring 2004
42 forty-two XLII What is 42? cuarenta y dos 23 January 2004 CS 200 Spring 2004
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
Meaning of Numbers pred (succ N) N succ (pred N) succ (pred (succ N)) succ N 23 January 2004 CS 200 Spring 2004
Meaning of Zero zero? zero T zero? (succ zero) F zero? (pred (succ zero)) 23 January 2004 CS 200 Spring 2004
Is this enough? add xy.if (zero? x) y (add (pred x) (succ y)) 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
Can we define lambda terms that behave like zero, zero?, pred and succ? Hint: what if we had cons, car and cdr? 23 January 2004 CS 200 Spring 2004
zero? null? pred cdr succ x . cons F x Numbers are Lists... 23 January 2004 CS 200 Spring 2004
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
cons and car cons x.y.z.zxy car p.p T T x . y. x cons M N = (x.y.z.zxy) M N (y.z.zMy) N z.zMN car p.p T car (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
cdr too! cons xyz.zxy car p.p T cdr p.p F cdr cons M N cdr z.zMN = (p.p F) z.zMN (z.zMN) F FMN N 23 January 2004 CS 200 Spring 2004
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
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
Counting 0 null 1 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
42 = xy. (z. z xy) xy. y xy. (z. z xy) xy. y xy. (z. z xy) xy 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
Arithmetic zero? null? succ x. cons F x pred 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
Lambda Calculus is a Universal Computer z z z z z z z z z z z z z z z z z z z z ), X, L ), #, R (, #, L 1 2: look for ( 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 Start (, X, R HALT #, 1, - #, 0, - Finite State Machine We have this, but we cheated using to make recursive definitions! 23 January 2004 CS 200 Spring 2004
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)) ... This should give you some belief that we might be able to do it. We won’t cover the details of why this works in this class. (CS655 sometimes does.) 23 January 2004 CS 200 Spring 2004
Charge Monday: Review session for Exam 2 Tuesday Office Hours 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 23 January 2004 CS 200 Spring 2004