Size of complement of context-free language












2















Let $L$ be a context-free language, $bar L$ be its complement and $bar L_n$ be the length $n$ words in $bar L_n$.




What is known about $|bar L_n|$?




Note that it is known that $|L_n|$ is either polynomial (e.g., if $L$ is bounded$~$), or grows exponentially.
I wonder if anything similar might be true about $|bar L_n|$.
Warning, $L$ is allowed to be ambiguous!






fl.formal-languages context-free







share|cite|improve this question





























    2















    Let $L$ be a context-free language, $bar L$ be its complement and $bar L_n$ be the length $n$ words in $bar L_n$.




    What is known about $|bar L_n|$?




    Note that it is known that $|L_n|$ is either polynomial (e.g., if $L$ is bounded$~$), or grows exponentially.
    I wonder if anything similar might be true about $|bar L_n|$.
    Warning, $L$ is allowed to be ambiguous!






    fl.formal-languages context-free







    share|cite|improve this question



























      2












      2








      2








      Let $L$ be a context-free language, $bar L$ be its complement and $bar L_n$ be the length $n$ words in $bar L_n$.




      What is known about $|bar L_n|$?




      Note that it is known that $|L_n|$ is either polynomial (e.g., if $L$ is bounded$~$), or grows exponentially.
      I wonder if anything similar might be true about $|bar L_n|$.
      Warning, $L$ is allowed to be ambiguous!






      fl.formal-languages context-free







      share|cite|improve this question
















      Let $L$ be a context-free language, $bar L$ be its complement and $bar L_n$ be the length $n$ words in $bar L_n$.




      What is known about $|bar L_n|$?




      Note that it is known that $|L_n|$ is either polynomial (e.g., if $L$ is bounded$~$), or grows exponentially.
      I wonder if anything similar might be true about $|bar L_n|$.
      Warning, $L$ is allowed to be ambiguous!






      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








      edited 12 hours ago







      domotorp

















      asked 12 hours ago









      domotorpdomotorp

      8,8063078




      8,8063078






















          1 Answer
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          From the proof that determining if a CFL ${L}$ = $Sigma^*$ is undecidable, the set of strings $ID_0#ID_1^R#ID_2#ID_3^R#ldots#ID_t$ where $ID_0,ID_1,ldots,ID_t$ is a list of the configurations of an accepting nondeterministic TM, is the complement of a context-free language. So $|overline{L}_n|$ can basically be any computable function less than exponential.






          share|cite|improve this answer





















          • 1





            Thanks. I fixed it.

            – Lance Fortnow
            3 hours ago











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          From the proof that determining if a CFL ${L}$ = $Sigma^*$ is undecidable, the set of strings $ID_0#ID_1^R#ID_2#ID_3^R#ldots#ID_t$ where $ID_0,ID_1,ldots,ID_t$ is a list of the configurations of an accepting nondeterministic TM, is the complement of a context-free language. So $|overline{L}_n|$ can basically be any computable function less than exponential.






          share|cite|improve this answer





















          • 1





            Thanks. I fixed it.

            – Lance Fortnow
            3 hours ago
















          9














          From the proof that determining if a CFL ${L}$ = $Sigma^*$ is undecidable, the set of strings $ID_0#ID_1^R#ID_2#ID_3^R#ldots#ID_t$ where $ID_0,ID_1,ldots,ID_t$ is a list of the configurations of an accepting nondeterministic TM, is the complement of a context-free language. So $|overline{L}_n|$ can basically be any computable function less than exponential.






          share|cite|improve this answer





















          • 1





            Thanks. I fixed it.

            – Lance Fortnow
            3 hours ago














          9












          9








          9







          From the proof that determining if a CFL ${L}$ = $Sigma^*$ is undecidable, the set of strings $ID_0#ID_1^R#ID_2#ID_3^R#ldots#ID_t$ where $ID_0,ID_1,ldots,ID_t$ is a list of the configurations of an accepting nondeterministic TM, is the complement of a context-free language. So $|overline{L}_n|$ can basically be any computable function less than exponential.






          share|cite|improve this answer















          From the proof that determining if a CFL ${L}$ = $Sigma^*$ is undecidable, the set of strings $ID_0#ID_1^R#ID_2#ID_3^R#ldots#ID_t$ where $ID_0,ID_1,ldots,ID_t$ is a list of the configurations of an accepting nondeterministic TM, is the complement of a context-free language. So $|overline{L}_n|$ can basically be any computable function less than exponential.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 3 hours ago

























          answered 11 hours ago









          Lance FortnowLance Fortnow

          6,8713452




          6,8713452








          • 1





            Thanks. I fixed it.

            – Lance Fortnow
            3 hours ago














          • 1





            Thanks. I fixed it.

            – Lance Fortnow
            3 hours ago








          1




          1





          Thanks. I fixed it.

          – Lance Fortnow
          3 hours ago





          Thanks. I fixed it.

          – Lance Fortnow
          3 hours ago


















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