91.502 Homework Assignments

(Unless otherwise stated, homework is due next Monday after the assignment is posted.)

 

HW1 (posted 9/12)

1.      1.1.3 (e) (That is, problem 3(e) from Exercise 1.1)

2.      1.1.4 (c)

3.      1.1.5 (b)

4.      1.1.6

5.      1.1.8 (Hint: Use Arden’s Lemma and variable elimination in linear algebra)

6.      1.2.2 (c)

7.      1.2.3 (b)

8.      1.2.4

9.      1.2.5 (optional; Hint: Read Example 1.20 first)

 

HW2 (posted 9/19)

1.      2.1.2

2.      2.2.1 (a), (b), (c), (g), (h)

3.      2.2.2 (b)

4.      2.3.3 (a), (b), (d)

5.      2.4.1 (b), (d)

 

HW3 (posted 9/26)

1.      2.5.1 (a)

2.      2.5.4 (Figure 2.42 (b))

3.      2.6.3 (c), (g) (Hint: See 2.6.2 (b) for definition of A Ĺ B and read Example 2.39 for ideas)

4.      2.7.2 (c) (Hint: Read Example 2.54 first)

5.      2.7.4 (a)

6.      2.7.5

 

HW4 (posted 10/4, due 10/10 in class)

1.      2.8.1 (a), (c), (f)

2.      2.8.2 (b), (d) (Note: #0(x) denotes the number of 0’s in x)

3.      2.8.5

 

HW5 (posted 10/11)

1.      3.1.3

2.      3.1.4 (a)

3.      3.1.8

4.      3.2.2 (a), (c)

5.      3.3.1 (a)

6.      3.3.2 (a)

7.      3.4.3 (a), (b)

 

HW6 (posted 10/17; Note: this assignment is due 10/30 in class, a week after the midterm exam)

1.      3.5.1 (b)

2.      3.6.2 (a), (c), (g), (h)

3.      4.1.2

 

HW7 (posted 10/31/06)

1.      4.2.1 (a)

2.      4.2.3 (a) (Hint: First describe your construction in English + math notations, then convert it into a state diagram)

3.      4.3.3 (Note: You only need to describe your construction in English with math notations. Do not try to construct a state diagram)

4.      4.5.2

5.      4.5.3 (a)

 

HW8 (posted 11/07/06)

1.      Encode the DTM given in Problem 4.1.2 into a binary string using the standard ASCII code. Use q_i to denote qi and use ((p,a) (q,b,D)) to denote δ(p,a) = (q,b,D), where D is either L or R. For simplicity, use two-digit hexadecimal representation of ASCII code.

2.      5.2.1 (a), (b)

3.      Use the dovetailing technique to show that the language B defined in Example 5.11 is r.e.

4.      5.2.3 (Note that the up-arrow means “undefined” and XA is the characteristics function of A; namely, XA(x) = 1 if x is in A, and XA(x) = 0 otherwise.)

5.      5.3.3 (a)

 

HW9 (posted 11/14/06)

1.      5.3.2

2.      5.3.7

3.      Show that if A is reducible to B via f and B is reducible to C via g, then A is reducible to C. That is, you need to show why g(f(x)) is a reduction from A to C in the proof of Proposition 5.24.

4.      Let A = {<M,x> | M is a TM and halts on some y > x}. Show that A is not decidable. (Hint: Reduce the un-decidable Turing machine halting problem H = {<M,x> | M is a TM and halts on x} to A.)

 

HW10 (posted 11/21/06. Note: the number of problems is reduced for the Thanksgiving week)

1.      Let NEQ = {<M1, M2> | M1 and M2 are TMs and L(M1) ≠ L(M2)}. Show that NEQ is not r.e.

2.      Show why TOT’ (i.e., the complementation of TOT), FIN, and REC defined in class are not r.e.

3.      Let A = {<M, x> | M is a TM and M on input x outputs xR}. Show that A is r.e. but not decidable.

 

HW11 (posted 11/28/06)

1.      6.1.3

2.      6.1.5

3.      6.2.3

4.      6.2.7

 

HW12 (posted 12/05/06; this is the last assignment)

1.      6.3.8

2.      6.4.4

3.      Let M be a 2-tape NTM with time bound t(n). Show that there exists a one-tape NTM M' with time bound O(t2(n)) such that L(M) = L(M').

4.      Provide proofs of Proposition 7.19 on page 340.

5.      Show that if an NP-complete problem is in P, then P = NP.