Question

Question 6. Give a brief description of the language generated by the following production rules. S → abc

S → aXbc

Xb → bX

Xc → Y bcc

bY → Y b

aY → aa

aY → aaX

Answer 1

The language produced by the given grammar is $\huge \mathbf{\{a^nb^nc^n, n\geq2\}}$

A few examples of strings belonging to the language is and so on.

aabbcc, aaabbbccc

How to derive this language from the production rules ?

Let me label the production rules :

1. S  $\huge \rightarrow$ aXbc

2. Xb $\huge \rightarrow$ bx

3. Xc $\huge \rightarrow$ Y bcc

4. bY $\huge \rightarrow$ Yb

5. aY $\huge \rightarrow$ aa

6. aY $\huge \rightarrow$ aaX

Rule 1 simply produces allows for aXbc. Rule 2 propagates X to the end of $\huge b^n$ . Rule 3 produces an extra "bc" between $\huge b^n$ and $\huge c^n$ part of the string along with an additional Y. Rule 4 propagates Y to the end of $\huge a^n$ which is the start of $\huge b^n$ . Both Rule 5 and 6 add an extra a between $\huge a^n$ and $\huge b^n$ . Rule 5 gives us the option to stop whereas Rule 6 allows to generate the next larger string. Each Y is generated from an X and this allows the number of bc and a added at appropriate places to be equal. Hence, the Grammar produces the language : $\huge \mathbf{\{a^nb^nc^n, n\geq2\}}$

I am also adding a derivation below to show how the the first string aabbcc is generated by the language :

$\huge {\color{DarkGreen} S} \xrightarrow{ 1} a{\color{DarkGreen} Xb}c \xrightarrow{ 2} ab{\color{DarkGreen} Xc} \xrightarrow{ 3} a{\color{DarkGreen} bY}bcc\xrightarrow{ 4} {\color{DarkGreen} aY}bbcc \xrightarrow{ 5} {\color{Red} aabbcc}$