I am having challenges (in two phases) with creating a CFG.Derive the CFG for the following language Show parse trees for the strings cacab and aacabbb obtained from the grammar designed above.I am getting a bit mixed up by the exercise especially because my CFG appears not to produce a parse tree.Here is the language:$$    L = \{a^n (ca)^m b^{n+1} \mid m \ge 0 , n \ge 0 \}$$So far my grammar looks as follows:$$ \begin{align}    S &\to Ab \mid Bb \mid Cb \mid b \\    A &\to aA \mid \epsilon \\    B &\to caB \mid \epsilon \\    C &\to bC \mid \epsilon \\\end{align} $$
Derive a Context Free Grammar from a language
context free;formal grammars;tree grammars
You're on the right track: $A$ recognizes $a^*$ (i.e. $\{a^n \mid n \ge 0\}$), $B$ recognizes $(ca)^*$ and $C$ recognizes $b^*$.You need to assemble them correctly. $Ab | Bb | Cb | b$ recognizes words that are made of something recognized by $A$ followed by $b$, or of something recognized by $B$ followed by $b$, etc. A word in $L$ consists of a part that's recognized by $A$ followed by a part that's recognized by $B$, etc. So the pieces need to be concatenated together: $S_1 \to ABCb$.$S_1$ recognizes $\{a^n (ca)^m b^{p+1} \mid m,n,p \ge 0\}$. This is too much: $L$ consists only of the words for which $n = p$ in this decomposition. Instead of defining $A$ and $C$ separately, you need to relate them. Whenever you put an $a$ on the left of the $B$ part, put a $b$ on the right.$$ \begin{align}  S_2 &\to B \\  S_2 &\to a S_2 b \\\end{align} $$This constraints the number of $a$'s on the left to be equal to the number of $b$'s on the right. The words in $L$ actually have one more $b$ on the right. One way to add that $b$ is to append it at the end:$$ S \to S_2 b $$Another approach is to tack on the $b$ to the end of the $B$ part. $S$ and $S'$ recognize the same language, with slightly different parse trees.$$ \begin{align}  S' &\to B b \\  S' &\to a S' b \\\end{align} $$To construct the parse trees, work out where each rule is applied to recognize the sample words.