代做CS3240 Assignment 1代写数据结构语言程序

日期：2024-09-01 01:44

CS3240 Assignment 1

1 [20 points]

Assume the alphabet Σ = {0, 1}. Consider the Deterministic Finite Automaton (DFA) M

depicted in Fig. 1 and answer the following questions.

Figure 1: The DFA in Question 1.

1. Given a string w = 001, will DFA M accept w?

2. Given a string w = 111, will DFA M accept w?

3. Given a string w = 101, will DFA M accept w?

4. Please use accurate natural language to describe the language L(M ) of DFA M , i.e., which binary strings in Σ^∗ can be accepted by M . You can also describe L(M ) using math or computer science style. notation.

2 [20 points]

Assume the alphabet Σ = {0, 1}. Consider the Nondeterministic Finite Automaton (NFA)

N depicted in Fig. 2 and answer the following questions.

1. Given a string w = 00, will NFA N accept w?

2. Given a string w = 101, will NFA N accept w?

3. Given a string w = 111, will NFA N accept w?

4. Does there exist a computation path such that the NFA N will “hang” after running on string w = 000?

5. Please use accurate natural language to describe the language L(N ) of NFA N , i.e., which binary strings in Σ^∗ can be accepted by N . You can also describe L(N ) using math or computer science style. notation.

Figure 2: The NFA N in Question 2.

3 [20 points]

Let alphabet Σ = {0, 1} and L = {100} ⊆ Σ^∗, i.e., language L only contains one string 100. Please design a DFA whose language is L.

You can provide details of this DFA by using the formal definition or insert a picture of the state diagram. Please use a comfortable and clear picture if you want to have a handwritten state diagram.

4 [20 points]

Assume the alphabet Σ = {0, 1}. Please answer the following questions based on theo- rems and conclusions you have learned in lectures (and textbooks).

1. Let L = {w ∈ Σ^∗|w has an odd number of 1’s and contains 101 as a subtring.}. Is L

a regular language? Please justify your answer.

2. Let language L contain every binary string w of Σ^∗ that satisfies following properties,

(1) w has both 101 and 11 as substrings, (2) the occurrence of 101 is before the oc- currence of 11. For example, 1010011 ∈ L, 1100101 ∈/ L since the occurrence of 101 is after the occurrence of 11. Is L a regular language? Please justify your answer.

3. Assume L ⊆ Σ^∗ is a regular language. Is that possible that a subset L^′ of L (L^′ ⊆ L) is not a regular language?

5 [20 points]

Assume the alphabet Σ = {0, 1}. Please provide a regular expression for each of the following language.

1. L = {w ∈ Σ^∗|w ̸= ϵ}.

2. L = {w ∈ Σ^∗| both the first and the last character of w is 0.}.

3. L = {w ∈ Σ^∗|w contains 111 as a substring.}.

6 [20 points]

Please determine truth values of following statements.

1. For any valid regular expression with respect to an alphabet Σ, we can construct a DFA and an NFA such that the language of each of those two machines is exactly the language described by the regular expression.

2. For any valid DFA, we can have a regular expression that exactly describes the lan- guage of this DFA.

3. Any language consisting of a finite number of strings is a regular language.

4. Assume A and B are regular languages over the same alphabet Σ, then the language

A \ B = {w ∈ Σ^∗|w ∈ A, w ∈/ B} is also a regular language.

5. Assume the alphabet Σ = {0, 1}. A = {0ⁿ1ⁿ|n ≥ 0} is not a regular language and

B = {0^m1ⁿ|m, n ≥ 0} is a regular language.