$$\,\,\,\,\,\,\,\,{\rm I}.\,\,\,\,\,\,\,$$ For an unrestricted grammar $$G$$ and a string $$W,$$ whether $$w \in L\left( G \right)$$

$$\,\,\,\,\,\,{\rm II}.\,\,\,\,\,\,\,$$ Given a Turing machine $$M,$$ whether $$L(M)$$ is regular

$$\,\,\,\,{\rm III}.\,\,\,\,\,\,\,$$ Given two grammars $${G_1}$$ and $${G_2}$$, whether $$L\left( {{G_1}} \right) = L\left( {{G_2}} \right)$$

$$\,\,\,\,{\rm IV}.\,\,\,\,\,\,\,$$ Given an $$NFA$$ $$N,$$ whether there is a deterministic $$PDA$$ $$P$$ such that $$N$$

$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$\,\,\,\,\,\,\,\\,\,\,$$and $$P$$ accept the same language.

Which one of the following statements is correct?

$$\,\,\,\,\,\,\,\,\,\,\,\,$$ $${L_1} = \left\{ {\left\langle M \right\rangle |M} \right.$$ takes at least $$2016$$ steps on some input $$\left. \, \right\},$$

$$\,\,\,\,\,\,\,\,\,\,\,\,$$ $${L_2} = \left\{ {\left\langle M \right\rangle |M} \right.$$ takes at least $$2016$$ steps on all inputs $$\left. \, \right\}$$ and

$$\,\,\,\,\,\,\,\,\,\,\,\,$$ $${L_3} = \left\{ {\left\langle M \right\rangle |M} \right.$$ accepts $$\left. \varepsilon \right\},$$

where for each Turing machine $${M,\left\langle M \right\rangle }$$ denotes a specific encoding of $$M.$$ Which one of the following is

**TRUE**?

**TRUE**?

Let $$L = \left\{ { < M > \left| M \right.} \right.$$ is a Turing machine that accepts a string of length $$\left. {2014} \right\}.$$ Then, $$L$$ is