## Homework Assignments

### Homework 1, Due Tuesday 02/09

1. Exercise 1.4, part f, page 83.

2. Exercise 1.5, part f, page 84.

3. Exercise 1.6, part j, page 84.
Further Instructions: Argue using a formal mathematical proof that your DFA recognizes the language given in the problem.

4. Problem 1.36, page 89.
Further Instructions: Present an abstract definition of your DFA. That is, specify the input-output relation of the transition function δ mathematically.

5. Problem 1.37, page 89.
Note: Assume the input x ∈ {0,1}*. This bit pattern x represents a number in base 2 (binary). We want that number to be divisible by n. The most significant bit of x will be read by the DFA first.

Hint: The C2 case is trivial, since we can detect divisibility by 2 just by looking at the least significant bit. First show that C3 is regular. Then argue that your method generalizes to any integer n ≥ 3.

### Homework 2, Due Tuesday 02/16 02/23

1. Exercise 1.7, part c, page 84.

2. Exercise 1.7, part e, page 84.

3. Problem 1.40, part b only, page 89.
Note: you are being asked to show that if A is regular then NOEXTEND(A) is also regular.

4. Problem 1.43, page 90.

### Homework 3, Due Tuesday 03/02

1. Exercise 1.16, part a, page 86.

2. Exercise 1.18, parts f, h & j, page 86.
I.e., give regular expressions for the languages specified in 1.6.f, 1.6.h and 1.6.j on page 84. Briefly explain the main parts of your expression.

3. Exercise 1.21, part a, page 86.

4. Problem 1.39, page 89.
Further Instructions: First write down what you need to show in order to prove the claim.

### Homework 4, Due Tuesday 03/09

1. Exercise 1.46, parts a, c & d, page 90.

2. Exercise 1.57, page 92.

3. Exercise 2.4, part c, page 128.
Further Instructions: briefly document your grammar. That is, for each variable in your grammar, explain briefly which strings are derived from that variable.

4. Exercise 2.6, part b, page 129.
Further Instructions: briefly document your grammar. That is, for each variable in your grammar, explain briefly which strings are derived from that variable.

### Homework 5, Due Tuesday 03/23

1. Give a high-level description of a pushdown automaton (PDA) for the following language and then draw the transition diagram for your PDA.
L2 = { ai bj ck | ij or jk }
Note: a high-level English description of your PDA has comments like "push X whenever a 0 is read."

2. Problem 2.20, page 130.

3. Problem 2.22, page 130.

4. Problem 2.26, page 130.

### Homework 6, Due Tuesday 03/30

1. Use the pumping lemma for regular languages to show that the following language is not regular:
A = { x ∈ {0,1}* | x has exactly three 0's, the length of x is odd and the middle symbol of x is 0. }

2. Problem 2.30, part a, page 131.

3. Problem 2.30, part d, page 131.

4. Problem 2.31, page 131.

### Homework 7, Due Tuesday 04/06

1. Problem 3.11, page 161.

2. Problem 3.12, page 161.

3. Problem 3.13, page 161.

4. Problem 3.19, page 162.
Hint: you may assume that Problem 3.18 has been solved.

### Homework 8, Due Tuesday 04/13

1. Problem 4.10, page 183.

2. Problem 4.16, page 184.

3. Problem 4.19, page 184.

4. Problem 4.27, page 184.

### Homework 9, Due Tuesday 04/20

1. Exercise 5.2, page 211.

2. Problem 5.14 & 5.15, page 212.

3. Problem 5.22, page 212.

4. Problem 5.23, page 212.

### Homework 10, Due Tuesday 04/27

1. Problem 5.23, page 212.

1. Problem 5.24, page 212.

2. Problem 5.25, page 212.

3. A set is cofinite if it is the complement of a finite set. Let
COF = { < M > | M is a TM and L(M) is cofinite}
FIN = { < M > | M is a TM and L(M) is finite}
Prove that FIN ≤m COF. That is FIN reduces to COF via a many-one reduction (a.k.a. a "mapping" reduction).

### Homework 11, Due Tuesday 05/04

1. Express the following language using first-order quantifiers (∃ and ∀ over natural numbers or strings) and a decidable predicate:
EQUIVTM = { ⟨ M1, M2 ⟩ | L(M1) = L(M2) }

2. Show that K17 defined below is ≤m-complete for the Turing recognizable languages.
K17 = { ⟨ M ⟩ | L(M) contains at least 17 strings }

Note: as a consequece K17 is undecidable.

3. Let EQ17 be as defined below. Show that the complement of ATMm-reduces to EQ17.

EQ17 = { ⟨ M ⟩ | L(M) contains exactly 17 strings }

Note: as a consequece EQ17 is not Turing-recognizable.

4. Problem 6.6, page 242.

### Homework 12, Due Tuesday 05/11 Thursday 05/13

1. Problem 7.21, page 296.

2. Problem 7.27, page 297.

3. Problem 7.45, page 300.