# CMPSCI 401: Theory of Computation

### David Mix Barrington

### Spring, 2009

# Homework Assignment #4

#### Posted Monday 30 March 2009

#### 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}
A_{TM} and that A_{TM} ≤_{m} 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 A_{TM} is TR-complete.
- Problem D-3 (10 extra credit) Of the six languages proved undecidable
in Section 5.1: HALT
_{TM}, E_{TM}, REGULAR_{TM},
EQ_{TM}, E_{LBA}, and ALL_{CFG}, 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.

Last modified 30 March 2009