Question

Give context-free grammars that generate the following languages. In all parts, the alphabet $\mathbf{\sum }$is $\left\{0,1\right\}$.

role="math" localid="1660714062992" $$\begin{gathered} \mathit{a}\mathbf{.}\mathbf{}\mathbf{\{}\overline{)\mathbf{w}\mathbf{}}\mathbf{}\mathbf{w}\mathbf{}\mathbf{c}\mathbf{o}\mathbf{n}\mathbf{t}\mathbf{a}\mathbf{i}\mathbf{n}\mathbf{s}\mathbf{}\mathbf{a}\mathbf{t}\mathbf{}\mathbf{l}\mathbf{e}\mathbf{a}\mathbf{s}\mathbf{t}\mathbf{}\mathbf{t}\mathbf{h}\mathbf{r}\mathbf{e}\mathbf{e}\mathbf{}\mathbf{1}\mathbf{s}\mathbf{\}} \\ \mathit{b}\mathbf{.}\mathbf{}\mathbf{\{}\overline{)\mathbf{w}\mathbf{}}\mathbf{}\mathbf{w}\mathbf{}\mathbf{s}\mathbf{t}\mathbf{a}\mathbf{r}\mathbf{t}\mathbf{s}\mathbf{}\mathbf{a}\mathbf{n}\mathbf{d}\mathbf{}\mathbf{e}\mathbf{n}\mathbf{d}\mathbf{s}\mathbf{}\mathbf{w}\mathbf{i}\mathbf{t}\mathbf{h}\mathbf{}\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{}\mathbf{s}\mathbf{a}\mathbf{m}\mathbf{e}\mathbf{}\mathbf{s}\mathbf{y}\mathbf{m}\mathbf{b}\mathbf{o}\mathbf{l}\mathbf{\}} \\ \mathit{c}\mathbf{.}\mathbf{}\mathbf{\{}\overline{)\mathbf{w}\mathbf{}}\mathbf{}\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{}\mathbf{l}\mathbf{e}\mathbf{n}\mathbf{g}\mathbf{t}\mathbf{h}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{w}\mathbf{}\mathbf{i}\mathbf{s}\mathbf{}\mathbf{o}\mathbf{d}\mathbf{d}\mathbf{\}} \\ \mathit{d}\mathbf{.}\mathbf{}\mathbf{\{}\overline{)\mathbf{w}\mathbf{}}\mathbf{}\mathbf{t}\mathbf{h}\mathbf{e}\mathbf{}\mathbf{l}\mathbf{e}\mathbf{n}\mathbf{g}\mathbf{t}\mathbf{h}\mathbf{}\mathbf{o}\mathbf{f}\mathbf{}\mathbf{w}\mathbf{}\mathbf{i}\mathbf{s}\mathbf{}\mathbf{o}\mathbf{d}\mathbf{d}\mathbf{}\mathbf{a}\mathbf{n}\mathbf{d}\mathbf{}\mathbf{i}\mathbf{t}\mathbf{s}\mathbf{}\mathbf{m}\mathbf{i}\mathbf{d}\mathbf{d}\mathbf{l}\mathbf{e}\mathbf{}\mathbf{s}\mathbf{y}\mathbf{m}\mathbf{b}\mathbf{o}\mathbf{l}\mathbf{}\mathbf{i}\mathbf{s}\mathbf{}\mathbf{a}\mathbf{}\mathbf{0}\mathbf{\}} \\ \mathit{e}\mathbf{.}\mathbf{}\mathbf{\{}\overline{)\mathbf{w}\mathbf{}}\mathbf{}\mathbf{w}\mathbf{=}{\mathbf{w}}^{\mathbf{R}}\mathbf{,}\mathbf{}\mathbf{t}\mathbf{h}\mathbf{a}\mathbf{t}\mathbf{}\mathbf{i}\mathbf{s}\mathbf{,}\mathbf{}\mathbf{w}\mathbf{}\mathbf{i}\mathbf{s}\mathbf{}\mathbf{a}\mathbf{}\mathbf{p}\mathbf{a}\mathbf{l}\mathbf{i}\mathbf{n}\mathbf{d}\mathbf{r}\mathbf{o}\mathbf{m}\mathbf{e}\mathbf{\}} \\ \mathit{f}\mathbf{.}\mathbf{}\mathit{T}\mathit{h}\mathit{e}\mathbf{}\mathit{e}\mathit{m}\mathit{p}\mathit{t}\mathit{y}\mathbf{}\mathit{s}\mathit{e}\mathit{t}\mathbf{.} \end{gathered}$$

Short answer

1. $$\begin{gathered} \begin{array}{rcl}& & S\to W1W1W1W \\ W\to 0W\overline{)1W}\overline{)\epsilon }\end{array} \end{gathered}$$
   
2. $$\begin{gathered} \begin{array}{rcl}S& \to & 0W0\overline{)1W1\overline{)0\overline{)1}}}\end{array} \\ W\to 0W\overline{)1W\overline{)\epsilon }} \end{gathered}$$
   
3. $S\to \overline{)0}\overline{)1}0S0\overline{)1S0\overline{)0S1\overline{)1S1}}}$
   
4. $S\to \overline{)0}0S0\overline{)1S0\overline{)0S1\overline{)1S1}}}$
   
5. $S\to \overline{)0}1\overline{)0S0\overline{)1S1\overline{)\epsilon }}}$
   
6. $S\to S$
   

Step-by-step solution

Explain Context-free languages

A context-free grammar consists of productions, variables, and terminals. The language generated by context-free grammar is called a context-free language.

The context-free grammars generate the languages by the productions and the rules.

Give context-free grammars that generate the given language.

a.

Consider context-free language, $\{\overline{)\mathrm{w}}\mathrm{w}\mathrm{contains}\mathrm{at}\mathrm{least}\mathrm{three}1\mathrm{s}\}$. The context-free grammar that produces the above language is as follows,

$$\begin{gathered} \begin{array}{rcl}& & S\to W1W1W1W \\ W\to 0W\overline{)1W}\overline{)\epsilon }\end{array} \end{gathered}$$

In the above grammar, the production S ensures that the string contains at least three 1s.

Therefore, the context-free grammar for the given language has been obtained.

Give context-free grammars that generate the given language.

b.

Consider context-free language,$\{\overline{)\mathrm{w}}\mathrm{w}\mathrm{starts}\mathrm{and}\mathrm{ends}\mathrm{with}\mathrm{the}\mathrm{same}\mathrm{symbol}\}$. The context-free grammar that produces the above language is as follows,

$$\begin{gathered} \begin{array}{rcl}S& \to & 0W0\overline{)1W1\overline{)0\overline{)1}}}\end{array} \\ W\to 0W\overline{)1W\overline{)\epsilon }} \end{gathered}$$

In the above grammar, the production S ensures that the string starts and ends with the same symbol.

Therefore, the context-free grammar for the given language has been obtained.

Give context-free grammars that generate the given language.

c.

Consider context-free language,$\{\overline{)\mathrm{w}}\mathrm{the}\mathrm{length}\mathrm{of}\mathrm{w}\mathrm{is}\mathrm{odd}\}$. The context-free grammar that produces the above language is as follows,

$S\to \overline{)0}\overline{)1}0S0\overline{)1S0\overline{)0S1\overline{)1S1}}}$

In the above grammar, the production S ensures that the length of the string is odd.

Therefore, the context-free grammar for the given language has been obtained.

Give context-free grammars that generate the given language.

d.

Consider context-free language,$\{\overline{)\mathrm{w}}\mathrm{the}\mathrm{length}\mathrm{of}\mathrm{w}\mathrm{is}\mathrm{odd}\mathrm{and}\mathrm{its}\mathrm{middle}\mathrm{symbol}\mathrm{is}\mathrm{a}0\}$. The context-free grammar that produces the above language is as follows,

$S\to \overline{)0}0S0\overline{)1S0\overline{)0S1\overline{)1S1}}}$

In the above grammar, the production S ensures that the length of the string is odd and the middle symbol is 0.

Therefore, the context-free grammar for the given language has been obtained.

Give context-free grammars that generate the given language.

e.

Consider context-free language,$\{\overline{)\mathrm{w}}\mathrm{w}={\mathrm{w}}^{\mathrm{R}},\mathrm{that}\mathrm{is},\mathrm{w}\mathrm{is}\mathrm{a}\mathrm{palindrome}\}$. The context-free grammar that produces the above language is as follows,

$S\to \overline{)0}1\overline{)0S0\overline{)1S1\overline{)\epsilon }}}$

In the above grammar, the production S ensures that the string is palindrome.

Therefore, the context-free grammar for the given language has been obtained.

Give context-free grammars that generate the given language.

f.

Consider context-free language is empty. The context-free grammar that produces the above language is as follows,

$S\to S$

In the above grammar, the production S ensures that the string is empty.

Therefore, the context-free grammar for the given language has been obtained.