Determining a language with a Turing Machine

Question

How can I build a Turing Machine that determines the following language?

$$L_{E - DFA} = \{\langle A \rangle | \text{$A$ is a $DFA$ and $L(A) = \varnothing$}\}$$

Thanks alot

Answer 1

Hint: Given a description of the DFA $A$, the Turing machine checks to see if any accepting state is reachable from the starting state using the transition function.

Answer 2

Hint: Let $A$ be a deterministic finite automaton. There exists an automaton $A^*$ such that $L(A) = L(A^*)$ and for all automata $B$ with $L(A)=L(B)$, $B$ has at least as many states as $A^*$. This $A^*$ is called the minimal automaton for $L(A)$. Now, if $L(A) = \emptyset$, can you guess what is $A^*$? Can you see how to compute $A^*$ from the description of $A$?