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CSci 4011: Formal Languages and Automata Theory
Fall 2007, University of Minnesota
Homework 6

Posted: Nov 14, 2008
Due: Before class on Nov 25, 2008

Note: This deadline is a little tight. However, there is a Thanksgiving break that begins on Nov 25 and I want to get the last homework out before this.

Problem 1

Show that EQCFG is undecidable. This is Exercise 5.1 from the book.

Problem 2

If A &lem B and B is a regular language, does this imply that A is a regular language? Why or why not?

This is Exercise 5.4 from the book.

Problem 3

Let
T = { <M> | M is a Turing machine that accepts wR whenever it accepts w}
Show that T is undecidable using
  1. a direct argument and

  2. Rice's theorem.
Note that there are two parts to this question and you need to do both for full credit.

Problem 4

Consider the problem of determining whether a Turing machine M on an input w ever attempts to move its head left from the leftmost tape cell.
  1. Formulate this problem as a language question whose decidability or undecidability we might try to show. Call the language you describe NOLEFTTM.

  2. Show that NOLEFTTM is undecidable.

Problem 5

Consider the problem of determininng whether or not a Turing machine M on input w ever attempts to move its head left at any point during its computation on w.
  1. Formulate this problem as a language question.

  2. Show that this question is in fact decidable.

Problem 6

Let
J = { w | either w = 0x for some x &isin ATM or w = 1y for some y &isin ATM.
Show that neither J nor J is Turing-recognizable.

This is Problem 5.24 from the book.

Problem 7

On page 205, second line, the author asks you to find a match that is independent of the acceptance of w by M if we view P' as an instance of the PCP instead of the MPCP. Respond to this question by exhibiting the match.

Problem 8 (Extra Credit)

Do Problem 5.21 in the book. This is a little long so I won't type it out. If you do not have the U.S. version of the second edition of the book, look for the one that is about showing AMBIGCFG is undecidable. Check with Zach, Myung-Hwan or me to make sure you have the right problem.

Last updated on Nov 14, 2008 by gopalan@cs.umn.edu