Chapter 4: 17P (page 212)

Prove that $E Q_{D F A}$ is decidable by testing the two DFAs on all strings up to a certain size. Calculate a size that works.

Short Answer

Expert verified

The string length must be less than or equal to n.m, where $A (w^{n}) \neq B (w^{m}) .$

Step by step solution

Explain decidability

A problem is decidable if all the instances of that problem are accepted. A Turing machine is used to decide a problem.

Calculate a size that works

Construct a Turing machine that takes the input w, with the size of n.m for $A (w) \neq B (w) .$

By contradiction, the first string provides a different output of the DFAs A and B which is $w'$ . The length l of $|w'|$ is greater than n.m.

The sequence of the states of A and B are $a_{0}, a_{1}, a_{2}, \dots a_{l}$ and $b_{0}, b_{1}, b_{2}, \dots b_{l},$ which describe the transition $w' .$

The value of l is greater than n.m, if the above sequences are provided as input then $a_{i} = a_{j}, b_{i} = b_{j}, i < j$ are also present.

Therefore, the string w has the length less than equal to n.m such that $A (w^{n}) \neq B (w^{m})$ .

Therefore, the string length must be less than or equal to n.m , where $A (w^{n}) \neq B (w^{m})$ .

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Prove that $E Q_{D F A}$ is decidable by testing the two DFAs on all strings up to a certain size. Calculate a size that works.

Short Answer

Step by step solution

Explain decidability

Calculate a size that works

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Prove that EQDFA is decidable by testing the two DFAs on all strings up to a certain size. Calculate a size that works.

Short Answer

Step by step solution

Explain decidability

Calculate a size that works

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Computer Science Textbooks

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Prove that $E Q_{D F A}$ is decidable by testing the two DFAs on all strings up to a certain size. Calculate a size that works.