Consider terminal symbols $0,1,2,3$ and non$-$terminals S$,$ A$,$ B$,$C with S as the start symbol. This ques$-$

tion will involve grammars based on these symbols and the languages they generate.

$($a$)$ If the only productions are S $!0$S$2$ and S $!<=,$ what is the language generated by the grammar? Explain your

answer.

$($b$)$ You want to modify the grammar so as to generate the language consisting of all the strings $0$ m $1$ n $2$ m$+$n $,$ where

m$,$ n $2$ N$.$ You are going to do this by adding the production A $!<=$ and exactly two further productions. What

are the productions, and why does this work? Your answer should describe the role of S and A$.$

$($c$)$ Find a grammar which generates the language consisting of all strings of the form $0$ m $1$ n $2$ p $3$ m$+$n$+$p $,$ where

m$,$ n$,$ p $2$ N$.$ You are not required to use the productions from parts $($a$)$ and $($b$),$ although you may find the

ideas involved in those parts helpful. Once again you should explain why your grammar has the desired

effect, and explain the roles of any non$-$terminals you use