QuestionGiven the language $L_n = \{ a^n \}$ for a natural number $n \geq 2$. Is there a symbol minimal context-free grammar $G$ that generates $L_n$ and contains a rule of the form $A \rightarrow aB$ where $A$ and $B$ are non-terminals?RemarkSymbol minimal means that there is no other context-free grammar that generates $L_n$ with fewer characters (symbols) for all rules. For instance, the number of symbols of the rules in $\{ S \rightarrow a, S \rightarrow aS \}$ is 7 (these rules are also minimal for $\{ a \}^+$).Known ResultsFacts about G (a symbol minimal context-free language for $L_n$) that I have already proved:The right hand side of the rules of $G$ consist of at least 2 characters (non-terminals and/or terminals) (otherwise its easy to find a grammar that has fewer symbols).For each non-terminal of $G$ there exists exactly one rule (can be shown by contradiction).There are no loops in the derivations of $G$ (because $L_n$ is finite).There is an upper bound for the number of symbols: we can trivially generate $L_n$ with the rules $\{ S \rightarrow a^n \}$. We count the number of symbols and get $n + 2$  thats an upper bound.I wrote a program that finds symbol minimal grammars for a specific $n$ by creating all possible grammars generating $L_n$ (there are only a finite number because we have an upper bound). Here are the results for some small $n$:$n = 2$: $\{ S \rightarrow aa \}$, $4$ symbols$n = 3$: $\{ S \rightarrow aaa \}$, $5$ symbols$n = 7$: $\{ S \rightarrow aaaaaaa \}$, $9$ symbols$n = 8$: $\{ S \rightarrow a^8\}$, $\{ S \rightarrow AA, A \rightarrow a^4 \}$, $\{ S \rightarrow AAAA, A \rightarrow aa \}$, $10$ symbols$n = 9$: $\{ S \rightarrow AAA, A \rightarrow a^3 \}$, $10$ symbols$n = 10$: $\{ S \rightarrow AA, A \rightarrow a^5 \}$, $\{ S \rightarrow A^5, A \rightarrow aa \}$, $\{ S \rightarrow aA^3, A \rightarrow a^3 \}$, $11$ symbols$n = 11$: $\{ S \rightarrow aAA, A \rightarrow a^5 \}$, $\{ S \rightarrow aA^5, A \rightarrow aa \}$, $\{ S \rightarrow aaA^3, A \rightarrow a^3 \}$, $12$ symbols$n = 12$: $\{ S \rightarrow AAA, A \rightarrow a^4 \}$, $\{ S \rightarrow A^4, A \rightarrow aaa \}$, $11$ symbols$n = 13$: ,  $12$ symbols$n = 14$: ,  $13$ symbols$n = 15$: ,  $12$ symbols$n = 16$: ,  $12$ symbolsThere are nice ups and downs in the number of symbols, right? I have seen grammars that contain rules like $A \rightarrow aB$ but they were not symbol minimal 
Investigation of Symbol Minimal Context-Free Grammars for the Language $a^n$
fl.formal languages;optimization;context free