Unsolvability and Infeasibility

1. Unsolvability and Infeasibility

2. Computability (Solvable) • A problem is computable if it is possible to write a computer program to solve it. • Can all problems be computed? • This question concerned mathematicians even before digital computers were developed. They looked for an algorithm (a finite set of instructions to carry out a task).

3. Turing • Alan Turing developed the concept of a computing machine in the 1930s • A Turing machine, as his model became known, consists of a control unit with a read/write head that can read and write symbols on an infinite tape.

4. Church-Turing Thesis • Any function that can be computed can be computed by a simple Turing Machine. The Turing Machine is as powerful as any algorithm. • Cannot prove thesis

5. Is there an unsolvable problem? • YES • A proof that there is a problem for which there is no algorithm is not • I can’t come up with an algorithm. Therefore there is no algorithm that solves the problem. • I can’t develop a program . Therefore there is no algorithm that solves the problem. • We need some background:

6. Paradox • There is a small town with only one barber. • The barber shaves only those people who do not shave themselves. • All people are shaved. • Who shaves the barber? If… • Assuming that such a barber exists, leads to a contradiction. Thus he can’t exist.

7. Loop • A loopisa set of instructions that is executed repeatedly. x=0 10 times: add 1 to x • What will be the value of x when the loop ends?

8. Loop x=5 while x > 0 subtract 1 from x • What will be the value of x when the loop ends? • How many times will the loop be executed?

9. Loop x=-5 while x ≠ 0 add 1 to x • What will be the value of x when the loop ends? • How many times will the loop be executed?

10. Loop x=0 while x ≠ 0 add 1 to x • What will be the value of x when the loop ends? • How many times will the loop be executed?

11. Loop x=5 while x ≠ 0 add 1 to x • What will be the value of x when the loop ends? • How many times will the loop be executed?

12. Infinite loop • Executes “forever”

13. Halt? • Given the initial value of x, we can predict whether this loop will end or halt. • Given a clock, can you predict whether it will halt? How long will you have to watch it? • Given an arbitrary program, can we predict whether it will end.? • Can we write a program to do this?

14. The Halting problem • Given a program and an input to the program, determine if the program will eventually stop with this input • Running the program is not a solution. Suppose we tried to run the program corresponding to the infinite loop to see if it ends. We would get tired of waiting for the answer, and stop the program. Thus we would still not have a prediction.

15. The Halting problem • Theorem: The Halting Problem is not Computable. • The Halting Problem is important because it proves the existence of an uncomputable/unsolvable problem.

16. Proof of Theorem • See the diagram • Recall that both programs and data are stored in binary. There is no difference in the representation. • Assume there is an algorithm A that solves the Halting Problem. • Write a new program N as follows:

17. N • Given a binary representation of program P • Use algorithm A to determine if P halts on input P • If A says HALTS, N goes into an infinite loop. • If A says LOOPS, then N says HALTS.

18. N • Imagine giving N to itself as data. • If A says that N HALTS, then N LOOPS. • If A says that N LOOPS, then N HALTS. • We have a paradox. A cannot exist. Our only assumption was that there is an algorithm A that solves the halting problem. It was wrong. • The Halting Problem is not computable. It is unsolvable/uncomputable.

19. Complexity • linear time –proportional to the size of the data • Recall • Finding the maximum value in a list of n elements by looking at each element in turn is linear. • Sequential search is linear

20. Complexity • logarithmic time --algorithms that successively cut the amount of data to be processed in half at each step are logarithmic • Finding max by comparing pairs • Finding a value in a list of n sorted elements using the binary search algorithm

21. Complexity • exponential time • Processing all subsets of a set • Trying all moves in chess • factorial time • Processing all permutations • Traveling Salesperson Problem

22. Feasible • A problem is feasible if it can be solved in a reasonable amount of time. • Must consider all algorithms. If at least one is able to do it, the problem is feasible. • A function can be computable, but yet infeasible.

23. Growth

24. Sort a set of n elements (I) The selection sort algorithm • find the largest of the n elements. • find the largest of the remaining n-1 elements. • find the largest of the remaining n-2 elements. … • the smallest element is remaining. • Time: n+ n-1+ ...+ 2+ 1 = n(n+ 1)/2 units quadratic

25. Sort a set of n elements (II) • Exhaustive listing and search • List all permutations (orderings) of the data • Pick the one that is sorted. • Time: n(n-1)(n-2) ... (3)(2)(1) = n! units factorial. Not very efficient!

26. Feasible • Sorting is feasible because there exists a reasonable algorithm. quadratic • Linear Search • Binary Search. logarithmic • Finding maximum. both linear and logarithmic.

27. Infeasible • Playing chess by considering every possible move is not. exponential • Considering every possible seating arrangement for a large group is not. factorial • Checking all possible subsets of a set is not . exponential

28. Feasibility • Logarithmic and linear algorithms are feasible • Exponential and factorial algorithms are infeasible. • An approximate solution may be adequate for an infeasible problem.