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QUESTION ONE a. Let A be defined as: A = {w:w is a binary string containing...
Question 8, please. 2. Prove: (a) the set of even numbers is countable. (b i=1 3....
Question 8, please.
2. Prove: (a) the set of even numbers is countable. (b i=1 3. The binary relation on pair integers - given by (a,b) - (c,d) iff a.d=cbis an equivalence relation. 4. Given a graph G = (V, E) and two vertices s,t EV, give the algorithm from class to determine a path from s to t in G if it exists. 5. (a) Draw a DFA for the language: ( w w has 010 as a substring)....
Problem 3.3: For a string x € {0,1}*, let af denote the string obtained by changing...
Problem 3.3: For a string x € {0,1}*, let af denote the string obtained by changing all 0's to l's and all l's to O's in x. Given a language L over the alphabet {0,1}, define FLIP-SUBSTR(L) = {uvFw: Uvw E L, U, V, w € {0, 1}*}. Prove that if L is regular, then FLIP-SUBSTR(L) is regular. (For example, (1011)F = 0100. If 1011011 e L, then 1000111 = 10(110) F11 € FLIP-SUBSTR(L). For another example, FLIP-SUBSTR(0*1*) = 0*1*0*1*.)...
6. (12 marks) This question tests your understanding of the equivalence between DFAS and NFAS. Consider...
6. (12 marks) This question tests your understanding of the equivalence between DFAS and NFAS. Consider NFA M (, q2},{0,1}, 6, qı, {q1}) for o defined as: 0 1 {1, 2} Ø 92} {q1, g2}|{1} (a) (4 marks) Draw the state diagrams for M. (b) (2 marks) Based on the construction of Theorem 1.39 in the text, start to build the DFA M' that is equiva- lent to M by identifying the number of DFA states and listing them. (c)...
Question 9 10 pts Select all the statements below which are true: Every dfa is also...
Question 9 10 pts Select all the statements below which are true: Every dfa is also an nfa. A maximum of 1 final state is allowed for a dfa. Alanguage that is accepted by a dfa is a regular language. Each dfa must have a trap state 0 Let M be an nfa, and let w be an input string. If Mends in a non-final state after reading w, then wis rejected. Let = {a,b,c,d}and M be an nfa with...
3. For each of the following languages, . State whether the language is finite or infinite....
3. For each of the following languages, . State whether the language is finite or infinite. . State whether the language is regular or nonregular. . If you claim the language is regular: give a DFA (graphical representation) that recog- nizes the language. . If you claim that the language is not regular, describe the intuition for why this is so. Consider the following languages (a) [8 marks] The language of 8 bit binary strings that begin and end with...
1(a)Draw the state diagram for a DFA for accepting the following language over alphabet {0,1}: {w...
1(a)Draw the state diagram for a DFA for accepting the following language over alphabet {0,1}: {w | the length of w is at least 2 and has the same symbol in its 2nd and last positions} (b)Draw the state diagram for an NFA for accepting the following language over alphabet {0,1} (Use as few states as possible): {w | w is of the form 1*(01 ∪ 10*)*} (c)If A is a language with alphabet Σ, the complement of A is...
Consider the following binary relations R1, R2, and R3 below, each defined over the set of...
Consider the following binary relations R1, R2, and R3 below, each defined over the set of integers between 0 and 4 inclusive and with each tuple (a,b) indicating that a is related to b. R1 = {(0,0), (0,3), (1, 1), (1, 2), (2,0), (2,3), (3, 1), (3,4), (4,0), (4,1)} R2 = {(1, 2), (2, 2), (3,0), (3,2), (4,0), (4,3)} R3 = {(0,0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4,4)} Which of these three relations is an...
3. [20 points] Give short answers to each of the following parts. Each answer should be...
3. [20 points] Give short answers to each of the following parts. Each answer should be at most three sentences. Be sure to define any notation that you use. (a) Explain the difference between a DFA and an NFA. (b) Give a regular expression for the language consisting of strings over the alphabet 2-(0, 1) that contains an even number of 0's and an odd number of 1's and does not contain the substring 01. (c) Give the formal definition...
Consider the NFA N with states labeled q1, q2 and q3, where q1 is the start...
Consider the NFA N with states labeled q1, q2 and q3, where q1 is the start state and q2 and q3 are the final (accepting) states. The transition function for N is δ(q1,a) = {q1}, δ(q1,b) = {q1,q2}, δ(q2,a) = {q3}, δ(q2,b)= ∅, δ(q3,a)= ∅, and δ(q3,b)= ∅. Let L be the language recognized by N i.e. L(N). a) Draw the state diagram for N. b) Describe in plain English what's in the language L. c) Via the construction NFA to...
Unless otherwise noted, the alphabet for all questions below is assumed to be Σ (ab). Also...
Unless otherwise noted, the alphabet for all questions below is assumed to be Σ (ab). Also note that all DFA's in your solutions should have one transition for each state in the DFA for each character in the alphabet. 1. (6 marks) This question tests your comfort with "boundary cases" of DFA's. Draw the state diagrams of DFAs recognizing each of the following languages. (a) (2 marks) L = {c) fore the empty string. (b) (2 marks) L (c) (2...
Question 8, please.
2. Prove: (a) the set of even numbers is countable. (b i=1 3. The binary relation on pair integers - given by (a,b) - (c,d) iff a.d=cbis an equivalence relation. 4. Given a graph G = (V, E) and two vertices s,t EV, give the algorithm from class to determine a path from s to t in G if it exists. 5. (a) Draw a DFA for the language: ( w w has 010 as a substring)....
Problem 3.3: For a string x € {0,1}*, let af denote the string obtained by changing all 0's to l's and all l's to O's in x. Given a language L over the alphabet {0,1}, define FLIP-SUBSTR(L) = {uvFw: Uvw E L, U, V, w € {0, 1}*}. Prove that if L is regular, then FLIP-SUBSTR(L) is regular. (For example, (1011)F = 0100. If 1011011 e L, then 1000111 = 10(110) F11 € FLIP-SUBSTR(L). For another example, FLIP-SUBSTR(0*1*) = 0*1*0*1*.)...
6. (12 marks) This question tests your understanding of the equivalence between DFAS and NFAS. Consider NFA M (, q2},{0,1}, 6, qı, {q1}) for o defined as: 0 1 {1, 2} Ø 92} {q1, g2}|{1} (a) (4 marks) Draw the state diagrams for M. (b) (2 marks) Based on the construction of Theorem 1.39 in the text, start to build the DFA M' that is equiva- lent to M by identifying the number of DFA states and listing them. (c)...
Question 9 10 pts Select all the statements below which are true: Every dfa is also an nfa. A maximum of 1 final state is allowed for a dfa. Alanguage that is accepted by a dfa is a regular language. Each dfa must have a trap state 0 Let M be an nfa, and let w be an input string. If Mends in a non-final state after reading w, then wis rejected. Let = {a,b,c,d}and M be an nfa with...
3. For each of the following languages, . State whether the language is finite or infinite. . State whether the language is regular or nonregular. . If you claim the language is regular: give a DFA (graphical representation) that recog- nizes the language. . If you claim that the language is not regular, describe the intuition for why this is so. Consider the following languages (a) [8 marks] The language of 8 bit binary strings that begin and end with...
Consider the following binary relations R1, R2, and R3 below, each defined over the set of integers between 0 and 4 inclusive and with each tuple (a,b) indicating that a is related to b. R1 = {(0,0), (0,3), (1, 1), (1, 2), (2,0), (2,3), (3, 1), (3,4), (4,0), (4,1)} R2 = {(1, 2), (2, 2), (3,0), (3,2), (4,0), (4,3)} R3 = {(0,0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 3), (4,4)} Which of these three relations is an...
3. [20 points] Give short answers to each of the following parts. Each answer should be at most three sentences. Be sure to define any notation that you use. (a) Explain the difference between a DFA and an NFA. (b) Give a regular expression for the language consisting of strings over the alphabet 2-(0, 1) that contains an even number of 0's and an odd number of 1's and does not contain the substring 01. (c) Give the formal definition...
Unless otherwise noted, the alphabet for all questions below is assumed to be Σ (ab). Also note that all DFA's in your solutions should have one transition for each state in the DFA for each character in the alphabet. 1. (6 marks) This question tests your comfort with "boundary cases" of DFA's. Draw the state diagrams of DFAs recognizing each of the following languages. (a) (2 marks) L = {c) fore the empty string. (b) (2 marks) L (c) (2...
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