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Chapter 0: Q1E (page 1)

Question: Answer all parts for the following DFA and give reasons for your answers.

a.Is<M,0100>∈ADFA?b. Is<M,011>∈ADFA?c. Is<M>∈ADFA?d. Is<M,0100>∈AREX?e. Is<M>∈EDFA?f. Is<M,M>∈EQDFA?

Short Answer

Expert verified

Answer

  1. Yes
  2. No
  3. No
  4. No
  5. No
  6. Yes

Step by step solution

01

Explain Deterministic Finite Automata 

A Deterministic Finite Automata is the computational model that determines the state of the input symbol provided.

02

Step 2: Is  <M,0100>∈ADFA

a.

Consider the given diagram, assume that the states are named as follows,

M,0100, represents the inputs of the machine. Symbol 0 starts with the state S , the next input 1 takes to the state A . The input symbol 0 moves to the state B and the next input 0 is the last symbol of the string that ends in the state S ,which accepts the input.

Therefore, Yes, M,0100∈ADFA.

03

Step 3: Is  M,011∈ADFA

b.

Consider the given diagram, assume that the states are named as follows,

M,011, represents the inputs of the machine. Symbol 0 starts with the state S , the next input 1 takes to the state A . The input symbol 0 is the last symbol of the string that ends in the state B ,which doesnot accepts the input.

Therefore, No, M,011∉ADFA.

04

Step 4: Is <M>∈ADFA

c.

Consider the given diagram, assume that the states are named as follows,

M, represents the inputs of the machine. The input symbol empty is not accepted by the given automata.

Therefore, No,role="math" M∉ADFA .

05

Step 5: Is  <M,0100>∈AREX

d.

Consider the given diagram, assume that the states are named as follows,

M,0100, represents the inputs of the machine. The input symbol is belongs to DFA not the regular language inputs.

Therefore, No,M,0100∉AREX

06

Step 6: Is  <M>∈EDFA

e.

Consider the given diagram, assume that the states are named as follows,

M, represents the empty input of the machine. The input symbol is belongs to DFA not the regular language inputs.

Therefore, No,M∉EDFA.

07

Step 7: Is  <M,M>∈EQDFA

e.

Consider the given diagram, assume that the states are named as follows,

M,M, represents the same language of the machine. The input symbol belongs to the DFA.

Therefore, Yes,M,M∈EQDFA.

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Most popular questions from this chapter

Let MODEXP={ha,b,c,pi|a,b,c,andpare positive binary integers such that ab≡cmodp}.

Show that MODEXP∈P. (Note that the most obvious algorithm doesn’t run in polynomial time. Hint: Try it first where b is a power of 2 .)

Let CNFk= is a satisfiable cnf-formula where each variable appears in at most k places}.

a. Show thatCNF2?P .

b. Show thatCNF3 isNP-complete.

Question: Each of the following languages is the intersection of two simpler languages. In each part, construct DFAs for the simpler languages, then combine them using the construction discussed in footnote 3 (page 46) to give the state diagram of a DFA for the language given. In all parts,∑=a,b.

a.{w|w\kern1pt has  at  least  three  a's  and  at  least  two  b's}b.w|w\kern1pt has  exactly  two  a's  and  at  least  two  b'sc.w|w\kern1pt has  even number  of  a's  and  one or  two  b'sd.{w|w\kern1pt has  even number  of  a's  and eachais followed by at least  one b}e.w|w\kern1pt starts  with  an  a  and  has at most one  bf.w|w\kern1pt has an odd number ofa's  and ends with a  bg.w|w\kern1pt has even length and an odd number ofa's

Let X=M,wM is a single-tape TM that never modifies the portion of the tape that contains the input w. Is X decidable? Prove your answer.

A Turing machine with left reset is similar to an ordinary Turing machine, but the transition function has the form

δ : Q × Γ−→Q × Γ × {R, RESET}.

If δ(q, a) = (r, b, RESET), when the machine is in state q reading an a, the machine’s head jumps to the left-hand end of the tape after it writes b on the tape and enters state r. Note that these machines do not have the usual ability to move the head one symbol left. Show that Turing machines with left reset recognize the class of Turing-recognizable languages.
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