A finite state machine with the following state table has a single input $$X$$ and a single out $$Z$$.

If the initial state is unknown, then the shortest input sequence to reach the final state $$C$$ is here, since initial make unknown $$m$$ $$10$$ input we can each final state $$C$$ with shortest path.

Consider the grammar with the following productions.
$$S \to a\,\alpha \,\,b\left| {\,\,b\,\alpha } \right.\,c\,\left| {aB} \right.$$
$$S \to \alpha S\,\left| b \right.$$
$$S \to \alpha \,bb\,\left| {ab} \right.$$
$$S\alpha \to bdb\,\left| b \right.$$
the above grammar is