Question

Let CFG $G$be thefollowing grammar.

$$\begin{gathered} S\to aSb\left|bY\right|Ya \\ Y\to bY\left|aY\right|\epsilon \end{gathered}$$

Give a simple description of $L\left(G\right)$in English. Use that description to give a CFG for$L\left(G\right)$ , the complement of $L\left(G\right)$.

Short answer

$$\begin{gathered} S\to aSb\left|bY\right|Ya \\ Y\to bY\left|aY\right|\epsilon \end{gathered}$$

The description of context free grammar for $L\left(G\right)$, the complement of $L\left(G\right)$ is given below.

Step-by-step solution

Define context free grammar.

The context free languages are used to describe the context free grammar. It consists of a finite set of grammar rules that includes non-terminal, terminal symbols, start symbol and set of production rules.

The context free language is generated by context free grammar.

These languages are accepted by Pushdown Automata. These are the superset of regular languages.

Consider context-free languages $L_1$described as $G_1=(V_1,S,R_1,S_1).$

Consider context-free language $L_2$ described as $G_2=(V_2,S,R_2,S_2).$

Give the description of L(G) language.

Clearly, $Y$generates $(a\cup b)*$.

$S$, then, generates strings like a $n(a\cup b)*abnandanb(a\cup b)*bn$.

Thus we can get strings like $a_i{b}_{j}wherei>j,$and we can also get strings like $a_i{b}_j$where $i>j,$but cannot get $a_i{b}_j$where $i\ne j,$. Furthermore, we can generate any string beginning with a b or ending with an a, and every string beginning with a and ending with b that is not of the form $a_i{b}_j$. This, then, is exactly the complement of the language $\left\{a^n{b}^n|n⩾0\right\}.$

A grammar for the complement of this language is $\left\{a^n{b}^n|n⩾0\right\}.$

$S\to aSb|\in .$

The language $L\left(G\right)$ is

$$\begin{gathered} S\to aSb\left|bY\right|Ya \\ Y\to bY\left|aY\right|\epsilon \end{gathered}$$

So, $L\left(G\right)$contains a string of $a\text{'}sandb\text{'}s$such that the strings satisfy the following conditions:

1. The string starts with $na\text{'}sandmb\text{'}s$, where$nandm$ can be zero.

2. The string has any number of $a\text{'}sandb\text{'}s$followed by an $a$.

3. $ab$ followed by any number of $a\text{'}sandb\text{'}s$.

Using the description to give a context free grammar for $L\left(G\right)$, the complement of

$L\left(G\right)$is written as $L\text{'}\left(G\right)$

$L\text{'}\left(G\right)$are elements not in $L\left(G\right)$and they are,

$L\text{'}\left(G\right)=a^{n}b(a?b)*b^n{a}^n\left(ab\right)*a{b}^n$ .