I have a language which is made out of and has a grammar only operating on well parenthesized words or words of one symbol. I want to know if the problem of belonging to this language is decidable.
Example. Consider the following grammar: $$a \to a(aa) \\ a \to b \\a (a(a(ab))) \to b$$ I'll denote by $L(x)$ the language generated by the above grammar and rule $S \to x$.
Trivially, $b \in L(a)$, using rule 2.
Also $b \in L(a(aa))$ by the sequence of productions $a(aa) \to a(a(a(aa))) \to a(a(a(ab))) \to b$.
However $b \notin L(aa)$ because after applying rule 1 any number of times and then rule 2, symbol $a$ will be repeated $2n + 1$ times and rule 3 can only remove $4m$ of them.
Note, the grammar is actually dealing with trees. Parentheses denote a subtree and symbols denote leaves. Hopefully the notation is clear enough. The language seems much weaker than RE but stronger than context free.