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Problem 10

Let \(I N F I N I T E_{\mathrm{DFA}}=\\{\langle A\rangle \mid A\) is a DFA and \(L(A)\) is an infinite language \(\\} .\) Show that INFINITE DFA is decidable.

Expert verified

The INFINITE_DFA is decidable by following these steps: (1) Simplify the DFA by removing unreachable states, (2) identify loops in the simplified DFA using a cycle detection algorithm, and (3) check if any identified loops lead to an accepting state. If there exists a loop leading to an accepting state, L(A) is infinite; otherwise, L(A) is finite.

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Chapter 4

Let \(E=\\{\langle M\rangle \mid M\) is a DFA that accepts some string with more 1 s than \(0 s\\}\). Show that \(E\) is decidable. (Hint: Theorems about CFLs are helpful here.)

Chapter 4

Say that an NFA is ambiguous if it accepts some string along two different computation branches. Let \(A M B I G_{N F A}=\\{\langle N\rangle \mid N\) is an ambiguous NFA \(\\} .\) Show that \(A M B I G_{\text {NFA }}\) is decidable. (Suggestion: One elegant way to solve this problem is to construct a suitable DFA and then run \(E_{D F A}\) on it.)

Chapter 4

Prove that the class of decidable languages is not closed under homomorphism.

Chapter 4

Consider the problem of determining whether a DFA and a regular expression are equivalent. Express this problem as a language and show that it is decidable.

Chapter 4

Let \(A=\\{\langle M\rangle \mid M\) is a DFA that doesn't accept any string containing an odd number of \(1 \mathrm{~s}\\}\). Show that \(A\) is decidable.

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