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Every context-free language has either polynomial growth or exponential growth. In the notation of the question poser:

• Either there is a polynomial $$p$$ so that $$w_n\le p(n)$$ for all $$n$$
• Or there exists a $$c>1$$, so that $$w_n\ge c^n$$ for infinitely many $$n$$.

This has been shown for instance in:

Roberto Incitti:
"The growth function of context-free languages"
Theoretical Computer Science 255 (2001), Pages 601-605

Martin R. Bridson, Robert H. Gilman:
"Context-Free Languages of Sub-exponential Growth"
Journal of Computer and System Sciences 64 (2002), Pages 308-310

And for a given context-free grammar, one can decide in polynomial time whether the generated language has polynomial or exponential growth:

"Finding the Growth Rate of a Regular or Context-Free Language in Polynomial Time.
International Journal of Foundations of Computer Science 21 (2010), Pages 597-618

Gamow
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Every context-free language has either polynomial growth or exponential growth. In the notation of the question poser:

• Either there is a polynomial $$p$$ so that $$w_n\le p(n)$$ for all $$n$$
• Or there exists a $$c>1$$, so that $$w_n\ge c^n$$ for infinitely many $$n$$.

This has been shown for instance in:

Roberto Incitti:
"The growth function of context-free languages"
Theoretical Computer Science 255, (2001), Pages 601-605

Martin R. Bridson, Robert H. Gilman:
"Context-Free Languages of Sub-exponential Growth"
Journal of Computer and System Sciences 64, (2002), Pages 308-310

And for a given context-free grammar, one can decide in polynomial time whether the generated language has polynomial or exponential growth:

"Finding the Growth Rate of a Regular or Context-Free Language in Polynomial Time.
International Journal of Foundations of Computer Science 21 (2010), Pages 597-618

Gamow
• 5.7k
• 6
• 21
• 38

Every context-free language has either polynomial growth or polynomialexponential growth: In. In the notation of the question poser:

• Either there is a polynomial $$p$$ so that $$w_n\le p(n)$$ for all $$n$$
• Or there exists a $$c>1$$, so that $$w_n\ge c^n$$ for infinitely many $$n$$.

This has been shown for instance in:

Roberto Incitti:
"The growth function of context-free languages"
Theoretical Computer Science 255, (2001), Pages 601-605

Martin R. Bridson, Robert H. Gilman:
"Context-Free Languages of Sub-exponential Growth"
Journal of Computer and System Sciences 64, (2002), Pages 308-310

Gamow
• 5.7k
• 6
• 21
• 38