You must finish this exam within 24 hours of when you first read this. You can consult the textbook for the course and any notes or assignments from the course; you are not permitted to consult any other sources. You are not permitted to discuss anything about the test with anyone except the instructor until after the final deadline of Friday at 9:00.

  1. The outer-thirds operator is defined as follows: for every string xyz such that |x| = |y| = |z|, OT(xyz) = xz. Extend OT to languages in the standard way. Is the class of regular languages closed under outer-thirds?
  2. Stan and Cartman have started their own business selling computing machines. They have introduced a new machine, the CHEF 3000 onto the market. The CHEF 3000 comes with a control, an input device, and a counter. Having learned design techniques from Bill Gates, their machine has one oddity: every third increment operation (starting with the third one) increments the counter by two instead of one.

    3. The sales literature for the CHEF 3000 says "For an additional $1000 we will include a patch to the system which will fix the increment problem."

    Explain why buying the patch would be a waste of money.

  3. Prove that if a finite acceptor, M, has 17 states, and it accepts a string of length 21, then the language accepted by M is infinite.
  4. Let L = { (a È b)* such that number of a’s equals number of b’s}

If L is regular, then since the set of regular languages is closed under finite transduction, we can proceed by building the following finite state transducer:

This yields the language L’ = {(a)* such that the number of a’s is even}

Since there is a DFA for L’, notably:

 

We conclude that L is regular. What is wrong with this argument?