Question

I believe it is a language that starts with an a and ends with a b, but I’m not sure.

G = (V,∑,P,S) where V={A,B,S,T}, ∑={a,b},  
P = { S → ε | T | AB, T → aTb | ε, A → aA | Aaa | ε, B → bB | ε }

Answer 1

Look at the start. For the first substitution, there are three possibilities. If it’s S -> ε, you’re finished. So the language contains the empty word ε. If the first substitution is S -> T, what can happen with T? Simple, T can become either ε or aTb, no other nonterminals can ever become involved, so that production produces the sublanguage (I’ll leave that for you to work out, it’s homework). Finally, what if the first substitution is S -> AB? Looking at the productions starting with A resp. B, the right hand sides con only ever contain one nonterminal, the one you started with and substituting A may add one or two a, substituting B either ends (ε) or adds another b. So which combinations of a and b are reachable? Does ba belong to the language? Why resp. why not?

Hint: S -> AB -> Aε = A -> aA -> aε = a is valid, so a nonempty word need not end with b.