# Minimal context-free Grammar for a special one-letter Language

For natural numbers $n \geq 5$, $m \geq 2^{n-2} + 1$ the following context-free language is given: $$L_{n,m} = \{ a^i | 2 \leq i \leq m \} \setminus \{a^{2^i}|2 \leq i \leq n-2\}$$ Find and prove a minimal context-free grammar according to the number of rules that generates $L_{n,m}$.

I think this cannot be done with a fewer number of rules as shown below: $$P = \{S \rightarrow a^{2^{i-1} + 1} A^{2^{i-1} - 2} | 2 \leq i \leq n-2 \} \cup \\ \{S \rightarrow a^{2^{n-2}+1}A^{m - (2^{n-2}+1)}, A \rightarrow a, A \rightarrow \epsilon\}$$

Note: For the "complement" language $$\{a^{2^i}|1 \leq i \leq n\}$$ it’s known that a minimal context-free grammar needs exactly $n$ rules (one rule for each word $a^{2^i}$).

Any suggestions to prove or disprove my conjecture?

• Unless I am missing something this does not look like a research level question. Instead it looks like something out of my undergrad automata theory class. I could be wrong though. – Woot4Moo May 15 '12 at 14:44
• This concrete language is important for me. It would help me to find the range of descriptional complexity (minimal number of rules of a context-free grammer that generates the language) a context-free languages can have under certain operations (like union, concatenation). Because we consider the minimal number of rules of context-free grammars this is not really part of theory of automata. – Ronny May 16 '12 at 9:52
• I see. I will make an attempt at this over the weekend and if anything comes of it I will make a post. – Woot4Moo May 18 '12 at 15:46
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• It’s no homework. It’s for a PhD thesis. This language could be from general interest for description complexity. Thus it’s relevant for theoretical computer science. – Ronny Jun 26 '12 at 10:09