Question

In Exercises 43–49 find the language recognized by the given nondeterministic finite-state automaton.

Short answer

The result is\({\bf{L(M) = \{ 0,01,11\} }}\).

Step-by-step solution

According to the figure.

Here the given figure contains three states\({{\bf{s}}_{\bf{o}}}{\bf{,}}{{\bf{s}}_{\bf{1}}}{\bf{,}}{{\bf{s}}_{\bf{2}}}\).

If there is an arrow from \({{\bf{s}}_{\bf{i}}}\) to \({{\bf{s}}_{\bf{j}}}\) with label x, then we write down in row \({{\bf{s}}_{\bf{j}}}\)and in the row \({{\bf{s}}_{\bf{i}}}\)and in column x of the following table.

State	0	1
\({{\bf{s}}_{\bf{o}}}\)	\({{\bf{s}}_{\bf{1}}}\),\({{\bf{s}}_{\bf{2}}}\)	\({{\bf{s}}_{\bf{1}}}\)
\({{\bf{s}}_{\bf{1}}}\)		\({{\bf{s}}_{\bf{2}}}\)
\({{\bf{s}}_{\bf{2}}}\)

\({{\bf{s}}_{\bf{o}}}\) is marked as the start state.

Find the final result.

To Move from \({{\bf{s}}_{\bf{o}}}\) the final state \({{\bf{s}}_{\bf{2}}}\)(directly), I require that the input is 0 and thus the bit string 0 is in recognized language.

\(0 \in {\bf{L(M)}}\)

To move from\({{\bf{s}}_{\bf{o}}}\)to \({{\bf{s}}_{\bf{1}}}\), the input can be either 0 or 1. To then move from \({{\bf{s}}_{\bf{1}}}\)to \({{\bf{s}}_{\bf{2}}}\), the second input needs to be 1.Combined , I thus require an input of 01 or 11 to arrive at the final state \({{\bf{s}}_{\bf{2}}}\),which then implies that 01 and 11 are both in the recognized language.

\(01,11 \in {\bf{L(M)}}\)

Therefore, the language generated by the machine is

\({\bf{L(M) = \{ 0,01,11\} }}\)