Number of words of length n in a context-free language











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15
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Denote by $w_n$ the number of words of length $n$ in a (possibly ambiguous) context-free language.




What is known about $w_n$?




I'm sure this has been studied a lot, but I couldn't find anything at all on it.






fl.formal-languages context-free







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  • 3




    There is a quasi-polynomial time randoimized algorithm to approximate $w_n$ to within a $(1+epsilon)$ approximation. sciencedirect.com/science/article/pii/S0890540197926213
    – Chandra Chekuri
    yesterday








  • 1




    For unambiguous CFLs, the classic Chomsky–Schützenberger enumeration theorem should be of interest.
    – Martin Berger
    yesterday















up vote
15
down vote

favorite
3












Denote by $w_n$ the number of words of length $n$ in a (possibly ambiguous) context-free language.




What is known about $w_n$?




I'm sure this has been studied a lot, but I couldn't find anything at all on it.






fl.formal-languages context-free







share|cite|improve this question


















  • 3




    There is a quasi-polynomial time randoimized algorithm to approximate $w_n$ to within a $(1+epsilon)$ approximation. sciencedirect.com/science/article/pii/S0890540197926213
    – Chandra Chekuri
    yesterday








  • 1




    For unambiguous CFLs, the classic Chomsky–Schützenberger enumeration theorem should be of interest.
    – Martin Berger
    yesterday













up vote
15
down vote

favorite
3









up vote
15
down vote

favorite
3






3





Denote by $w_n$ the number of words of length $n$ in a (possibly ambiguous) context-free language.




What is known about $w_n$?




I'm sure this has been studied a lot, but I couldn't find anything at all on it.






fl.formal-languages context-free







share|cite|improve this question













Denote by $w_n$ the number of words of length $n$ in a (possibly ambiguous) context-free language.




What is known about $w_n$?




I'm sure this has been studied a lot, but I couldn't find anything at all on it.






fl.formal-languages context-free




fl.formal-languages context-free






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked yesterday









domotorp

8,5143075




8,5143075








  • 3




    There is a quasi-polynomial time randoimized algorithm to approximate $w_n$ to within a $(1+epsilon)$ approximation. sciencedirect.com/science/article/pii/S0890540197926213
    – Chandra Chekuri
    yesterday








  • 1




    For unambiguous CFLs, the classic Chomsky–Schützenberger enumeration theorem should be of interest.
    – Martin Berger
    yesterday














  • 3




    There is a quasi-polynomial time randoimized algorithm to approximate $w_n$ to within a $(1+epsilon)$ approximation. sciencedirect.com/science/article/pii/S0890540197926213
    – Chandra Chekuri
    yesterday








  • 1




    For unambiguous CFLs, the classic Chomsky–Schützenberger enumeration theorem should be of interest.
    – Martin Berger
    yesterday








3




3




There is a quasi-polynomial time randoimized algorithm to approximate $w_n$ to within a $(1+epsilon)$ approximation. sciencedirect.com/science/article/pii/S0890540197926213
– Chandra Chekuri
yesterday






There is a quasi-polynomial time randoimized algorithm to approximate $w_n$ to within a $(1+epsilon)$ approximation. sciencedirect.com/science/article/pii/S0890540197926213
– Chandra Chekuri
yesterday






1




1




For unambiguous CFLs, the classic Chomsky–Schützenberger enumeration theorem should be of interest.
– Martin Berger
yesterday




For unambiguous CFLs, the classic Chomsky–Schützenberger enumeration theorem should be of interest.
– Martin Berger
yesterday










1 Answer
1






active

oldest

votes

















up vote
21
down vote



accepted










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 $w_nle p(n)$ for all $n$

  • Or there exists a $c>1$, so that $w_nge c^n$ 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:




Pawel Gawrychowski, Dalia Krieger, Narad Rampersad, Jeffrey Shallit:

"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







share|cite|improve this answer



















  • 2




    Very interesting connection: The term growth rate is a well known one in group theory and heavily studied. However virtually free groups have exponential growth rate and we know by Muller and Schupp (1983) that word problems of virtually free groups are deterministic context-free. Do you know if there is further work about the growth rate of deterministic context-free languages?
    – datell
    yesterday











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up vote
21
down vote



accepted










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 $w_nle p(n)$ for all $n$

  • Or there exists a $c>1$, so that $w_nge c^n$ 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:




Pawel Gawrychowski, Dalia Krieger, Narad Rampersad, Jeffrey Shallit:

"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







share|cite|improve this answer



















  • 2




    Very interesting connection: The term growth rate is a well known one in group theory and heavily studied. However virtually free groups have exponential growth rate and we know by Muller and Schupp (1983) that word problems of virtually free groups are deterministic context-free. Do you know if there is further work about the growth rate of deterministic context-free languages?
    – datell
    yesterday















up vote
21
down vote



accepted










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 $w_nle p(n)$ for all $n$

  • Or there exists a $c>1$, so that $w_nge c^n$ 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:




Pawel Gawrychowski, Dalia Krieger, Narad Rampersad, Jeffrey Shallit:

"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







share|cite|improve this answer



















  • 2




    Very interesting connection: The term growth rate is a well known one in group theory and heavily studied. However virtually free groups have exponential growth rate and we know by Muller and Schupp (1983) that word problems of virtually free groups are deterministic context-free. Do you know if there is further work about the growth rate of deterministic context-free languages?
    – datell
    yesterday













up vote
21
down vote



accepted







up vote
21
down vote



accepted






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 $w_nle p(n)$ for all $n$

  • Or there exists a $c>1$, so that $w_nge c^n$ 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:




Pawel Gawrychowski, Dalia Krieger, Narad Rampersad, Jeffrey Shallit:

"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







share|cite|improve this answer














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 $w_nle p(n)$ for all $n$

  • Or there exists a $c>1$, so that $w_nge c^n$ 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:




Pawel Gawrychowski, Dalia Krieger, Narad Rampersad, Jeffrey Shallit:

"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








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edited yesterday

























answered yesterday









Gamow

3,50431129




3,50431129








  • 2




    Very interesting connection: The term growth rate is a well known one in group theory and heavily studied. However virtually free groups have exponential growth rate and we know by Muller and Schupp (1983) that word problems of virtually free groups are deterministic context-free. Do you know if there is further work about the growth rate of deterministic context-free languages?
    – datell
    yesterday














  • 2




    Very interesting connection: The term growth rate is a well known one in group theory and heavily studied. However virtually free groups have exponential growth rate and we know by Muller and Schupp (1983) that word problems of virtually free groups are deterministic context-free. Do you know if there is further work about the growth rate of deterministic context-free languages?
    – datell
    yesterday








2




2




Very interesting connection: The term growth rate is a well known one in group theory and heavily studied. However virtually free groups have exponential growth rate and we know by Muller and Schupp (1983) that word problems of virtually free groups are deterministic context-free. Do you know if there is further work about the growth rate of deterministic context-free languages?
– datell
yesterday




Very interesting connection: The term growth rate is a well known one in group theory and heavily studied. However virtually free groups have exponential growth rate and we know by Muller and Schupp (1983) that word problems of virtually free groups are deterministic context-free. Do you know if there is further work about the growth rate of deterministic context-free languages?
– datell
yesterday


















 

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