CMPSCI 601: Theory of Computation

Offered through the PEEAS distance learning program

Notes on Homework Assignment #2

David Mix Barrington

Posted 19 July 2003

Given the misunderstandings shown on your solutions to HW#2, I wanted to make some general comments on solving this type of problem before you attack similar problems on the midterm. You should also review my specific comments on your own HW#2 and review the HW#2 solutions carefully.

• Types of variables: Many of your answers quickly devolved into nonsense because you made statements that were not valid for the type of variables to which they applied. For example, you might refer to a language accepting an input, or a string being an element of a machine, where in fact machines operate on strings and decide whether those machines are in languages.

  You should be familiar with typing from programming and even more familiar with it from your work with the LPL software. The computer will simply not accept statements that do not type-check, and you should train yourself likewise so you don't make such statements in your answers. A single error of confusing one type with another can get you off the track forever. Once you are no longer addressing the correct question, what you say is no longer likely to help you much.

  One particular place to be concerned with typing is with the input and output types of any machine or algorithm. For example, the language K is defined as {n: n in L(M_n)}. Thus to say that an object x is in K or not, x must be a number. And an alleged algorithm for K must take a number as input and return a boolean. A reduction from K to some other language A must take a number as its input and return an output appropriate for membership or non-membership in A.

• Definitions: We think about the concepts of this course both rigorously, using definitions and proof, and informally, using intuition. Always remember that your intuitions are suspect and can only be trusted when they can be backed up rigorously.

  For example, if a language A is r.e. then by definition there exists a Turing machine M such that A = L(M). You may use this machine to show that other things are computable, but you may not use it to say that anything is uncomputable. There will in general be many different machines with the same language, and saying that one machine does not do something tells you nothing about what another machine might or might not do.

• Types of Questions:

  Here are some standard types of questions and how to answer them:

  • "Upper Bounds": To show that a language is solvable within certain resources, for example that it is recursive, primitive recursive, or in P, you must show the existence of an algorithm. Then you must argue that your algorithm follows whatever resource constraints you are claiming for it. For example, to show that A is recursive it suffices to show that some always-halting Turing machine decides A. To show that A is primitive recursive, you must show that some Bloop program decides A. And to show that A is in P, you must show that there is an algorithm that decides whether a string w is in A using time polynomial in the length of w.
  • To show that a language A is r.e., you must show the existence of a Turing machine whose language is A, meaning that this machine outputs 1 iff its input is in A. You can do this directly or indirectly, by reducing A to some other r.e. language. Note that showing A to be recursive also shows it to be r.e., and showing it r.e. says nothing in itself about whether A is recursive.
  • To show that a language A is r.e.-complete, you must show that A is r.e. and then show that any other r.e. language reduces to A. Sometimes you do this directly as in the proof that K is r.e.-complete (the halting problem proof). But more often you reduce K or some other known r.e.-complete language to A, which shows that A is either r.e.-complete or even harder.
  • To show that a language is not r.e. at all, you can show that a known non-r.e. language reduces to it. The easiest example of a language that is not r.e. is one whose complement is r.e.-complete. For example, K-bar is not r.e., and if K-bar reduces to A, then A is not r.e. at all.

• For each of these types of problems, you should consider whether you currently believe there are other valid ways to solve them (such as the ones I saw on HW#2) and determine whether these ways are actually valid. (I'm glad to answer questions about this by email.)

Last modified 20 July 2003