Unsolvability and Infeasibility

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).

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.

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

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:

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.

Loop A loop is a 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?

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?

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?

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?

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?

Infinite loop Executes “forever”

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?

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.

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.

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:

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.

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.

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

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

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

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.

Growth nn2n2 2n2n n!

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 = n(n+ 1)/2 units quadratic

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!

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

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

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