CSCI 301 L31 Worksheet

Lecture 31 - Exercises

Part A - Decidability of Machines

Determine whether each of the following languages are decidable:

1. \(L_{DFA} = \{<M_{DFA}, w> : M_{DFA} \text{ accepts }w\}\)
2. \(L_{NFA}\)
3. \(L_{PDA}\)
4. \(L_{TM}\)

Part B - Reductions

1. Prove or disprove that the following language is decidable: \[ \begin{align*} L_{HELLO} = \{\langle H \rangle : &\text{ $H$ is a Turing machine that prints ``HELLO''}\\ &\text{somewhere on its first tape when run with an empty input tape}\} \end{align*} \]

2. Prove or disprove the following language is decidable: \[ E_{DFA} = \{\langle M \rangle : M \text{ is a DFA accepting the language } \varnothing \} \] Hint: Try an approach that doesn’t involve a reduction.

3. Prove or disprove the following language is decidable: \[ EQ_{DFA} = \{\langle M_1, M_2 \rangle : M_1 \text{ and $M_2$ are DFAs accepting the same language} \} \]

   Hint: Try reducing this to \(E_{DFA}\) from #2.

4. (Bonus) Prove or disprove the following language is decidable: \[ EQ_{TM} = \{\langle M_1, M_2 \rangle : M_1 \text{ and $M_2$ are turing machines accepting the same language} \} \]