A random permitting-context grammar is a context-free grammar $(N, \Sigma, P, S)$ equipped with a function $p : P \rightarrow 2^N$. The rule $A \rightarrow x$ can be applied to $uAw \Rightarrow uxw$ if every symbol in $p(A\rightarrow x)$ appears in $uw$. A forbidding-context grammar is similar, except equipped with a function $f : P \rightarrow 2^N$, and a rule application is allowed if no symbol from $f(A\rightarrow x)$ appears in $uw$.
$\{a^n b^m : 1 \le m \le 2^n\}$ is a random permitting-context language (RPCL), though $\{a^n b^m : m = 2^n\}$ is not (as can be proved by Ewert and Van der Walt's pumping lemma for RPCLs). In a permitting grammar we can allow the nonterminals of some type to double, but we can't force it.
It seems obvious to me that one can't get more than exponential growth, but I can't see how to prove it (the pumping lemma certainly doesn't help, since it will just increase the number of $a$'s). Is there a known result that helps here?
I'm also interested in superexponential growth in forbidding languages (e.g. is $\{a^{2^{2^n}} : n \in \mathbb{N}\}$ an RFCL).
