Question

Answer each part for the following context-free grammar G

$\begin{array}{l}\mathbf{R}\mathbf{\to }\mathbf{XRX}\mathbf{|}\mathbf{S}\\ \mathbf{S}\mathbf{\to }\mathbf{aTb}\mathbf{|}\mathbf{bTa}\\ \mathbf{T}\mathbf{\to }\mathbf{XTX}\mathbf{\left|}\mathbf{X}\mathbf{\right|}\mathbf{\epsilon }\\ \mathbf{X}\mathbf{\to }\mathbf{a}\mathbf{|}\mathbf{b}\end{array}$

1. What are the variables of $\mathbf{G}$?
2. What are the terminals of $\mathbf{G}$?
3. Which is the start variable of $\mathbf{G}$?
4. Give three strings in $\mathbf{L}\mathbf{\left(}\mathbf{G}\mathbf{\right)}$.
5. Give three strings not in$\mathbf{L}\mathbf{\left(}\mathbf{G}\mathbf{\right)}$ .
6. True or False: $\mathbf{T}\mathbf{\Rightarrow }\mathbf{aba}$.
7. True or False: $\mathbf{T}\stackrel{\mathbf{\ast }}{\mathbf{\Rightarrow }}\mathbf{aba}$.
8. True or False:$\mathbf{T}\mathbf{\Rightarrow }\mathbf{T}$ .
9. True or False: $\mathbf{T}\stackrel{\mathbf{\ast }}{\mathbf{\Rightarrow }}\mathbf{T}$.
10. True or False:$\mathbf{XXX}\stackrel{\mathbf{\ast }}{\mathbf{\Rightarrow }}\mathbf{aba}$ .
11. True or False: $\mathbf{X}\stackrel{\mathbf{\ast }}{\mathbf{\Rightarrow }}\mathbf{aba}$
12. True or False:role="math" localid="1660812124187" $\mathbf{T}\stackrel{\mathbf{\ast }}{\mathbf{\Rightarrow }}\mathbf{XX}$.
13. True or False: $\mathbf{T}\stackrel{\mathbf{\ast }}{\mathbf{\Rightarrow }}\mathbf{XXX}$.
14. True or False: $\mathbf{S}\stackrel{\mathbf{\ast }}{\mathbf{\Rightarrow }}\mathbf{\epsilon }$.
15. Give a description in English of $\mathbf{L}\mathbf{\left(}\mathbf{G}\mathbf{\right)}$ .

Short answer

1. $\mathrm{V}=\{\mathrm{R},\mathrm{X},\mathrm{S},\mathrm{T}\}$
   
2. $\{\mathrm{a},\mathrm{b}\}$
   
3. $\mathrm{R}$
4. Three strings in$\mathrm{L}\left(\mathrm{G}\right)$ are$\mathrm{ab}$ ,$\mathrm{ba}$ , and $\mathrm{aab}$.
5. Three strings are not in$\mathrm{L}\left(\mathrm{G}\right)$ are$\mathrm{a}$ , $\mathrm{b}$, and $\mathrm{\epsilon }$.
6. False.
7. True.
8. False.
9. True.
10. True.
11. False.
12. True.
13. True.
14. False.
15. $\mathrm{L}\left(\mathrm{G}\right)$consists of all strings over $\mathrm{a}$and$\mathrm{b}$ that are not palindromes.

Step-by-step solution

Define the context-free Grammar

Context-free grammar is a notation used for describing languages. It is more powerful than finite automata or Regular expressions.

Determine the variable of G

(a)

The capital letters are called variables because the value of that alphabet changes by replacing the productions.

Thus, the variables in the given grammar are,$\mathrm{V}=\{\mathrm{R},\mathrm{X},\mathrm{S},\mathrm{T}\}$.

Therefore, the variables are:$\mathrm{V}=\{\mathrm{R},\mathrm{X},\mathrm{S},\mathrm{T}\}$

Determine the terminals of G.

(b)

The letters which are represented in small are called terminals. Thus, the terminals in the given grammar are, $\{\mathrm{a},\mathrm{b}\}$

Hence, the terminals are$\{\mathrm{a},\mathrm{b}\}$

Determine the start variable of G

(c)

The special nonterminal symbol which appears in the initial string generated by the given grammar.

Hence, the start variable is $\mathrm{R}$.

Determine the three strings in L(G) .

(d)

The three strings in language$\mathrm{L}\left(\mathrm{G}\right)$of grammar $\mathrm{G}$are as follows,

1. ab

Proof:

Consider the production $\mathrm{R}\to \mathrm{S}$,Replace $\mathrm{S}$with $\mathrm{a}\mathrm{Tb}$from the production $\mathrm{R}\to \mathrm{aTb}$.Replace $\mathrm{T}$with $\epsilon$ from the production $\mathrm{R}\to \mathrm{a\epsilon b}$. The resulting string is $\mathrm{R}\to \mathrm{ab}$.

Therefore, the string is present in .

2. ba

Proof:

Consider the production$\mathrm{R}\to \mathrm{S}$,Replace$\mathrm{S}$ with $\mathrm{bTa}$from the production $\mathrm{R}\to \mathrm{bTa}$. Replace $\mathrm{T}$with $\epsilon$from the production $\mathrm{R}\to \mathrm{b\epsilon a}$.The resulting string is$\mathrm{R}\to \mathrm{ba}$ .

Therefore, the string is present in.

3. aab

Proof:

Consider the production$\mathrm{R}\to \mathrm{S}$,Replace$\mathrm{S}$ with$\mathrm{aTb}$ from the production$\mathrm{R}\to \mathrm{aTb}$ .Replace $\mathrm{T}$with $\mathrm{x}$from the production $\mathrm{R}\to \mathrm{aXb}$. Replace$\mathrm{x}$ with$\mathrm{a}$ from the production$\mathrm{X}\to \mathrm{a}$ .The resulting string is $\mathrm{R}\to \mathrm{aab}$.

Hence, the three strings are$\mathrm{ab},\mathrm{ba}$ and$\mathrm{aab}$.

Determine the three strings that are not in L(G) . 

(e)

The three strings not in L(G) are

$\mathrm{aaa}$,$\mathrm{bab}$ , $\mathrm{bbb}$

Proof:

Substitute the production rules in variables, starting from the starts symbol. The above four strings cannot be obtained.

Hence, the strings $\mathrm{aaa}$, $\mathrm{bab}$,and$\mathrm{bbb}$ arenot in L (G).

Determine if  T⇒aba is True or False 

(f)

$\mathrm{T}\Rightarrow \mathrm{aba}$

False because there is no rule which translates $T$directly to $\mathrm{aba}$.

Hence, the given statement is false.

Determine if T⇒∗aba is True or False

(g)

$\mathrm{T}\stackrel{\ast }{\Rightarrow }\mathrm{aba}$, can be produced by the production $\mathrm{T}\to \mathrm{XTX}$. Thus, the given statement isTrue.

Hence,$\mathrm{T}\stackrel{\ast }{\Rightarrow }\mathrm{aba}$ is true.

Determine If T⇒T is True or False

(h)

The grammar has no rule $\mathrm{T}\Rightarrow \mathrm{T}$.

Hence,$\mathrm{T}\Rightarrow \mathrm{T}$ is false.

Determine if T⇒∗T is True or False

(i)

The production $\mathrm{T}\stackrel{\ast }{\Rightarrow }\mathrm{T}$can be derived from the given grammar. Thus, the given statement is True.

Hence,$\mathrm{T}\stackrel{\ast }{\Rightarrow }\mathrm{T}$ is True.

Determine if XXX⇒∗aba is True or False

(j)

The given production is true because each $\mathrm{X}$can be substituted with a or b.

Hence,$\mathrm{XXX}\stackrel{\ast }{\Rightarrow }\mathrm{aba}$ is true.

Determine if X⇒∗aba  is True or False. 

(k)

The given rule is false since $\mathrm{X}$can only become a or b.

Hence,$\mathrm{X}\stackrel{\ast }{\Rightarrow }\mathrm{aba}$ is false.

Determine if T⇒∗XX is True or False.

(l)

The given rule is true because as$\mathrm{T}$ can be substituted with an empty string $\in$.

Hence, $\mathrm{T}\stackrel{\ast }{\Rightarrow }\mathrm{XX}$is true.

Determine if T⇒XTX⇒XXX is True or False.

(m)

The given rule is true, since $\mathrm{T}\Rightarrow \mathrm{XTX}\Rightarrow \mathrm{XXX}$can be substituted with $\mathrm{X}$. $\mathrm{X}\stackrel{\ast }{\Rightarrow }\mathrm{aba}$.

Hence, $\mathrm{T}\stackrel{\ast }{\Rightarrow }\mathrm{XXX}$is true.

Determine if S⇒∗∈  is True or False.

(n)

False because any string derived from $\mathrm{S}$has at least one occurrence of a and b.

Hence, role="math" localid="1660815034121" $\mathrm{S}\stackrel{*}{\Rightarrow }\in$is false.

Write description of language in English 

(o)

Since from $\mathrm{R}$get any number of ${\mathrm{X}}^{\text{'}}$on both sides, $\mathrm{L}\left(\mathrm{G}\right)={\{\mathrm{a},\mathrm{b})}^{\ast }\backslash ({\mathrm{a}}^{\ast }\cup {\mathrm{b}}^{\ast })$. It produces strings of the form ${\mathrm{X}}^{\mathrm{n}}{\mathrm{SX}}^{\mathrm{n}}$for value of $\mathrm{n}$greater than or equal to 0.

Therefore, $\mathrm{L}\left(\mathrm{G}\right)$ consists of all strings over$\mathrm{a}$ and $\mathrm{b}$that are not palindromes.