Today

- Limits of computation- Complexity analysis examples

The “Hello” Assignment

• Suppose I gave you an assignment to write a computer program which returns “Hello!” and stops. (Assume this is a procedure with no parameters).

• Suppose I promised you full credit for ANY working program which returns “Hello!” and halts, and no credit if it didn’t work.

The “Hello” Assignment

• Can I write a program that automatically grades your homework?

The “Hello” Assignment

• Can I write a program that automatically grades your homework?

• Sure, right?

(define (GradeHello P) (let ((out (P))) (if (eq? out ‘Hello!) 100 0)))

The “Hello” Assignment• No! What if a student submits:

(define (p) (define (q n) (cond ((> n 0) (q n)) (else 'Hello))) (q 10))

• Student’s program runs forever!

•This means the grading program would run forever.

•It would never tell me that the student should fail… • Maybe we can stop the Program after 1 hour and fail the student if the program hasn’t stopped yet.

The “Hello” Assignment• Well, in that case lets submit:

(define (p) (hanoi 1 2 3 50);; hanoi with 50 pegs 'Hello)

• My grading program would say to fail you.

• But technically you deserve to Pass.

The “Hello” Assignment• We keep running in to problems…

• Could it be that writing a computer program to grade such a simple assignment is impossible?

• YES! We’ll prove it…

Is there a limit to the power of computers?

Q: Is there any problem a computer cannot solve?

What is a “problem”?

We need some definition to start with:

And what is a “computer”?

What is a “computer?”A1: Fortran program running on PDP 11.

A3: C++ program running on MAC G4 under OSX at 800MHz.

A4: All of the above

A2: Java program running on a PC with Pentium V under Windows XP at 3GHz.

A5: None of the above

Church-Turing ThesisAll “reasonable” models of programming languagesand computers are equivalent.

Each can emulate the other.

There is nothing essential in Java, C+ + , Cobol, Lambda Calculus, Mathematica, (or even Scheme).

Is there a limit to the power of computers?

Q: Is there any problem a computer cannot solve?

We still have to define what a “problem” means.

We will restrict ourselves to well posedproblems with a yes/no answer.

A solution will be a scheme procedure.

The halting problemThe halting problem H:

• Given a pair of expressions x and y. x is viewed as a Scheme procedure.• H(x,y) is true iff x is a valid procedure and it stops on the input y.

Examplex = (define (f z) (if (= z 1) 1 (+ z (f z)))) , y = 1 H(x,y) is true

x = (define (f z) (if (= z 1) 1 (+ z (f z)))) , y = 2 H(x,y) is false

The halting problem is not solvable

Proof: By contradiction.

Suppose there is a Scheme procedure halt that solves the halting problem. This means that • for all x,y (halt x y) always terminates• for all x,y (halt x y) returns true if and only if the procedure x

stops when applied to argument y.

Now consider the following program D:

(define (D x) (define (loopy) (loopy)) (if (halt x x) (loopy) #t))

The halting problem is not solvable

Now we run D giving it as argument D: (D D)

(define (D x) (define (loopy) (loopy)) (if (halt x x) (loopy) #t))

If D stops on D, then by def of halt procedure

(halt D D) returns #t .then from the code of D, D

enters an infite loop on D,a contradiction.

If D does not stop on D, then by def of halt procedure

(halt D D) returns #f . then from the code of D, D stops

on D, a contradiction.

Conclusion: Both possibilities lead to a contradiction.Therefore a procedure halt that solves the halting problemdoes not exist.

Is there a limit to the power of computers?

Q: Is there any problem a computer cannot solve?

A: YES. There are well posed problems a computer cannot solve!

In fact there are many well posed problems a computer cannot solve! A counting argument: The number of computer programs is countable (since each is a finite string). The number of problems/languages is uncountable. So most problems are not solvable.

But we did not just showed existence, but pointed out aspecific non-computable problem.

Final Exam

~4 Questions• 3 Hours• You may bring any written or printed material (no

laptops..)• You can use every function studied in class \

recitation \ exercise• All covered material except for computability

(first part of this lecture) and parallel computation (previous lecture)

• Good Luck and have a nice vacation!

Complexity Analysis Examples

19

Common Recurrences

T(n) = T(n-1) + (1) (n)

T(n) = T(n-1) + (n) (n2)

T(n) = T(n/2) + (1) (logn)

T(n) = T(n/2) + (n) (n)

T(n) = 2T(n-1) + (1) (2n)

T(n) = 2T(n/2) + (1) (n)

T(n) = 2T(n/2) + (n) (nlogn)

20

)filter

) lambda (positions) (safe? positions)(

) accumulate

append null

) map

) lambda (rest-of-queens)

) map (lambda (new-row)

) adjoin-position

new-row rest-of-queens((

) enumerate-interval 1 board-size(((

) queen-cols (- k 1)((((

(define (mean-record ds)

(let ((records ds))

(let ((n (length records)))

(map (lambda (x) (/ x n))

(accumulate (lambda (x y) (map + x y))

(car records)

(cdr records)))))

(define (onto? n m f) (define (rem-dup s) (if (null? s) s (cons (car s) (filter (lambda(x) (not (eq? (car s) x))) (rem-dup (cdr s)))))) (define (image f n) (rem-dup (map f (integers-between 1 n)))) (eq? m (length (image f n)))

23

Mobile weight

)define (total-weight mobile)

) if (atom? mobile) mobile

) +) total-weight

(

) total-weight

(

(((

)define (atom? x)

) and (not (pair? x)) (not (null? x))((

(branch-structure

(left-branch mobile))(branch-structure

(right-branch mobile))

24

balanced?

)define (balanced? mobile)) or (atom? mobile)

) let ((l (left-branch mobile))) r (right-branch mobile)((

) and =) (

) balanced( ?) balanced((((( ?

(* (branch-length l) (total-weight (branch-structure l))) (* (branch-length r) (total-weight (branch-structure r)))

(branch-structure l) (branch-structure r)

25

Complexity

– Worst case scenario for size n• Need to test all rods• May depend on mobile structure

– Upper bound• Apply total-weight on each sub-mobile• O(n2)

– Lower bound

26

Mobile structures

n

n-1

n-2

n-3

. . .

T(n) = T(n-1) + (n) (for this family of mobiles)

T(n) = (n2)

27

Mobile structures

n/2

T(n) = 2T(n/2) + (n) (for this family of mobiles)

T(n) = (nlogn)

n/2

n/4 n/4 n/4 n/4

n/8 n/8 n/8 n/8 n/8 n/8 n/8 n/8