Class 43: The Limits of Computing

Held Wednesday, April 19, 2000

Overview

Today we consider an important limit of computing: are there things that can't be computed?

Question 43 for discussion today: What metaphors might you use for the halting problem? How might those metaphors help one understand the problem?

Question 44 for discussion Friday: What chapters/topics do you plan to present? When? Please list alternatives.

Notes

• Upcoming events:
  • Thursday night's Africana studies conference
  • Cool talk on bias in standardized testing Thursday at 4:30.
• On Monday, April 24th at 7 p.m. in South Lounge, Katie Koestner, a nationally recognized, outspoken survivor of college date rape and sexual assault prevention specialist, will present a program for students and community members.
• A few of you have suggested a different structure for the last few weeks of the semester: each of you pick a topic (preferably including a chapter in Dewdney or Forester and Morrison) to present. Everyone reads the appropriate chapter. The presenter also reads supplementary materials that she/he finds on the Web and consults with Sarah and me as necessary.

Contents

• Are there uncomputable problems?
• The Halting Problem

Summary

Are there uncomputable problems?

• We've determined that
  • There is a limit to models of computing (Church's thesis)
  • Some problems are difficult to compute effectively (NP-completeness)
• That leads to a new question:
  • Are there things that we can't compute?

The Halting Problem

• The halting problem is a simple problem:
  • Given a program and the input to the program, will the program eventually stop.
• For example, does the following program stop?

  while (x != y) {
    if (x < y) {
      x = (x-1)*(x-1);
      y = sqrt(y);
    }
    else {
      y = (y-1)*(y-1);
      x = sqrt(x);
    }
  } // while
  

• Why is this important? It would be helpful to know if programs stop.
• There are also many other things that it would be helpful to know about programs, such as whether or not two programs are equivalent.
• Surprisingly, it turns out that it is impossible to write a general program that determines if other programs halt.