# Homework Assignment #4

#### Due on paper in class, Friday 10 April 2009

There are twelve questions for 100 total points plus 10 extra credit. All but two are from the textbook, Introduction to the Theory of Computation by Michael Sipser (second edition). The number in parentheses following each problem is its individual point value.

Students are responsible for understanding and following the academic honesty policies indicated on this page.

• Problem 4.15 (10)

• Problem 4.16 (10)

• Problem D-1 (10) Define the language BTHP (the blank-tape halting problem) to be the set {(M): the Turing machine M halts (either accepts or rejects) when started on input ε (the empty string)} Prove that BTHP is undecidable by any valid method. Prove both that BTHP ≤m ATM and that ATMm BTHP.

• Problem 4.17 (10) I think this is starred because it's in Chapter 4, but becomes much easier with ideas from Chapter 5. (Hint: Look at computation histories as in Definition 5.5.)

• Problem 4.19 (10)

• Problem 4.27 (10) (Hint: Use the CFL Pumping Lemma.)

• Problem 5.9 (5)

• Problem 5.15 (10) (Hint: You may quote solutions from HW#3. Solved Problems 5.10 and 5.11 may also be helpful.)

• Problem 5.23 (10)

• Problem 5.33 (10)

• Problem D-2 (5) A language X is defined to be TR-complete if (1) X is TR, and (2) for any language Y, if Y is TR then Y ≤m X. Prove that ATM is TR-complete.

• Problem D-3 (10 extra credit) Of the six languages proved undecidable in Section 5.1: HALTTM, ETM, REGULARTM, EQTM, ELBA, and ALLCFG, which are TR-complete? Justify your answer, noting that many of the reductions you may need are shown to exist already in Section 5.3.