Languages that we cannot (dis)prove to be Context-Free












11












$begingroup$


I'm looking for languages which are "probably not Context-Free" but we are not able to (dis)prove it using known standard techniques.




Is there a recent survey on the subject or an open problem section from a recent conference ?



Probably there are not many languages which are not known to be CF, so if you know one you can also post it as an answer.




The examples I found are:




  • the well known language of Primitive words $Q = { w mid w neq u^i (|u| > 1) }$ (there's a whole nice recent book on it: Context-Free Languages and Primitive Words)

  • the Base-k representations of the co-domain of a polynomial (see question "Base-k representations of the co-domain of a polynomial - is it context-free?" on cstheory)


Note: as showed by Aryeh in his answer you can build a whole class of such languages if you "link" a language to an unknown conjecture about the (non)finiteness or (non)emptiness of some sets (e.g. $L_{Goldbach} = { 1^{2n} mid 2n$ cannot be expressed as a sum of two primes$}$). I'm not quite interested in such examples.






reference-request big-list context-free







share|cite|improve this question











$endgroup$

















    11












    $begingroup$


    I'm looking for languages which are "probably not Context-Free" but we are not able to (dis)prove it using known standard techniques.




    Is there a recent survey on the subject or an open problem section from a recent conference ?



    Probably there are not many languages which are not known to be CF, so if you know one you can also post it as an answer.




    The examples I found are:




    • the well known language of Primitive words $Q = { w mid w neq u^i (|u| > 1) }$ (there's a whole nice recent book on it: Context-Free Languages and Primitive Words)

    • the Base-k representations of the co-domain of a polynomial (see question "Base-k representations of the co-domain of a polynomial - is it context-free?" on cstheory)


    Note: as showed by Aryeh in his answer you can build a whole class of such languages if you "link" a language to an unknown conjecture about the (non)finiteness or (non)emptiness of some sets (e.g. $L_{Goldbach} = { 1^{2n} mid 2n$ cannot be expressed as a sum of two primes$}$). I'm not quite interested in such examples.






    reference-request big-list context-free







    share|cite|improve this question











    $endgroup$















      11












      11








      11


      4



      $begingroup$


      I'm looking for languages which are "probably not Context-Free" but we are not able to (dis)prove it using known standard techniques.




      Is there a recent survey on the subject or an open problem section from a recent conference ?



      Probably there are not many languages which are not known to be CF, so if you know one you can also post it as an answer.




      The examples I found are:




      • the well known language of Primitive words $Q = { w mid w neq u^i (|u| > 1) }$ (there's a whole nice recent book on it: Context-Free Languages and Primitive Words)

      • the Base-k representations of the co-domain of a polynomial (see question "Base-k representations of the co-domain of a polynomial - is it context-free?" on cstheory)


      Note: as showed by Aryeh in his answer you can build a whole class of such languages if you "link" a language to an unknown conjecture about the (non)finiteness or (non)emptiness of some sets (e.g. $L_{Goldbach} = { 1^{2n} mid 2n$ cannot be expressed as a sum of two primes$}$). I'm not quite interested in such examples.






      reference-request big-list context-free







      share|cite|improve this question











      $endgroup$




      I'm looking for languages which are "probably not Context-Free" but we are not able to (dis)prove it using known standard techniques.




      Is there a recent survey on the subject or an open problem section from a recent conference ?



      Probably there are not many languages which are not known to be CF, so if you know one you can also post it as an answer.




      The examples I found are:




      • the well known language of Primitive words $Q = { w mid w neq u^i (|u| > 1) }$ (there's a whole nice recent book on it: Context-Free Languages and Primitive Words)

      • the Base-k representations of the co-domain of a polynomial (see question "Base-k representations of the co-domain of a polynomial - is it context-free?" on cstheory)


      Note: as showed by Aryeh in his answer you can build a whole class of such languages if you "link" a language to an unknown conjecture about the (non)finiteness or (non)emptiness of some sets (e.g. $L_{Goldbach} = { 1^{2n} mid 2n$ cannot be expressed as a sum of two primes$}$). I'm not quite interested in such examples.






      reference-request big-list context-free




      reference-request big-list context-free






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 14 hours ago







      Marzio De Biasi

















      asked 17 hours ago









      Marzio De BiasiMarzio De Biasi

      18.5k243113




      18.5k243113






















          2 Answers
          2






          active

          oldest

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          7












          $begingroup$

          How about the language $L_{TP}$ of twin primes? I.e., all pairs of natural numbers $(p,p')$ (represented, say, in unary), such that $p,p'$ are both prime and $p'=p+2$? If twin primes conjecture is true, then $L_{TP}$ is not context-free; otherwise, it's finite.



          Edit: Let me give a quick proof sketch that the twin primes conjecture implies that $L_{TP}$ is not context-free. Associate to any language $L$ its length sequence $0le a_1le a_2leldots$, where the integer $ell$ appears in the sequence iff there is a word of length $ell$ in $L$. It is a consequence of the pumping lemma(s) that for $L$ that are regular or CFL, the length sequence satisfies the bounded differences property: there is an $R>0$ such that $a_{n+1}-a_nle R$ for all $n$. It is an easy and well-known fact in number theory that the primes do not have bounded differences. Finally, any infinite subsequence of a sequence violating the bounded differences property itself must violate it.






          share|cite|improve this answer











          $endgroup$









          • 2




            $begingroup$
            Nice, thanks! But I'm not quite interested in languages that are linked to unknown conjectures about the (non)finiteness of some sets. BTW if those conjectures are true the resulting language is also regular :-)
            $endgroup$
            – Marzio De Biasi
            14 hours ago










          • $begingroup$
            If there are infinitely many twin primes, how do you see that $L_{TP}$ is regular?
            $endgroup$
            – Aryeh
            14 hours ago










          • $begingroup$
            I mean if (dis)proving a conjecture, results in a finite set. In your case - twin primes - if the conjecture is false.
            $endgroup$
            – Marzio De Biasi
            13 hours ago






          • 1




            $begingroup$
            If there are infinitely many twin primes, how do you show that $L_{TP}$ is not context-free?
            $endgroup$
            – Emil Jeřábek
            13 hours ago






          • 1




            $begingroup$
            Oh, sorry, I didn’t notice you represent the numbers in unary. Then it is clear. (I believe that proving this for binary representation would require a considerable progress on the twin primes conjecture.)
            $endgroup$
            – Emil Jeřábek
            12 hours ago





















          6












          $begingroup$

          Another good one is the complement of the set $S$ of contiguous subwords (aka "factors") of the Thue-Morse sequence ${bf t} = 0110100110010110 cdots $. To give some context, Jean Berstel proved that the complement of the set $T$ of prefixes of the Thue-Morse word is context-free (and actually something more general than that). But the corresponding result for subwords is still open.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Great, thanks! If you saw it stated somewhere (perhaps in one of your many papers on the Thue-Morse sequence? ;-) you can add the reference (even if stated in the iterated morphism form).
            $endgroup$
            – Marzio De Biasi
            9 hours ago












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          2 Answers
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          2 Answers
          2






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          active

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          7












          $begingroup$

          How about the language $L_{TP}$ of twin primes? I.e., all pairs of natural numbers $(p,p')$ (represented, say, in unary), such that $p,p'$ are both prime and $p'=p+2$? If twin primes conjecture is true, then $L_{TP}$ is not context-free; otherwise, it's finite.



          Edit: Let me give a quick proof sketch that the twin primes conjecture implies that $L_{TP}$ is not context-free. Associate to any language $L$ its length sequence $0le a_1le a_2leldots$, where the integer $ell$ appears in the sequence iff there is a word of length $ell$ in $L$. It is a consequence of the pumping lemma(s) that for $L$ that are regular or CFL, the length sequence satisfies the bounded differences property: there is an $R>0$ such that $a_{n+1}-a_nle R$ for all $n$. It is an easy and well-known fact in number theory that the primes do not have bounded differences. Finally, any infinite subsequence of a sequence violating the bounded differences property itself must violate it.






          share|cite|improve this answer











          $endgroup$









          • 2




            $begingroup$
            Nice, thanks! But I'm not quite interested in languages that are linked to unknown conjectures about the (non)finiteness of some sets. BTW if those conjectures are true the resulting language is also regular :-)
            $endgroup$
            – Marzio De Biasi
            14 hours ago










          • $begingroup$
            If there are infinitely many twin primes, how do you see that $L_{TP}$ is regular?
            $endgroup$
            – Aryeh
            14 hours ago










          • $begingroup$
            I mean if (dis)proving a conjecture, results in a finite set. In your case - twin primes - if the conjecture is false.
            $endgroup$
            – Marzio De Biasi
            13 hours ago






          • 1




            $begingroup$
            If there are infinitely many twin primes, how do you show that $L_{TP}$ is not context-free?
            $endgroup$
            – Emil Jeřábek
            13 hours ago






          • 1




            $begingroup$
            Oh, sorry, I didn’t notice you represent the numbers in unary. Then it is clear. (I believe that proving this for binary representation would require a considerable progress on the twin primes conjecture.)
            $endgroup$
            – Emil Jeřábek
            12 hours ago


















          7












          $begingroup$

          How about the language $L_{TP}$ of twin primes? I.e., all pairs of natural numbers $(p,p')$ (represented, say, in unary), such that $p,p'$ are both prime and $p'=p+2$? If twin primes conjecture is true, then $L_{TP}$ is not context-free; otherwise, it's finite.



          Edit: Let me give a quick proof sketch that the twin primes conjecture implies that $L_{TP}$ is not context-free. Associate to any language $L$ its length sequence $0le a_1le a_2leldots$, where the integer $ell$ appears in the sequence iff there is a word of length $ell$ in $L$. It is a consequence of the pumping lemma(s) that for $L$ that are regular or CFL, the length sequence satisfies the bounded differences property: there is an $R>0$ such that $a_{n+1}-a_nle R$ for all $n$. It is an easy and well-known fact in number theory that the primes do not have bounded differences. Finally, any infinite subsequence of a sequence violating the bounded differences property itself must violate it.






          share|cite|improve this answer











          $endgroup$









          • 2




            $begingroup$
            Nice, thanks! But I'm not quite interested in languages that are linked to unknown conjectures about the (non)finiteness of some sets. BTW if those conjectures are true the resulting language is also regular :-)
            $endgroup$
            – Marzio De Biasi
            14 hours ago










          • $begingroup$
            If there are infinitely many twin primes, how do you see that $L_{TP}$ is regular?
            $endgroup$
            – Aryeh
            14 hours ago










          • $begingroup$
            I mean if (dis)proving a conjecture, results in a finite set. In your case - twin primes - if the conjecture is false.
            $endgroup$
            – Marzio De Biasi
            13 hours ago






          • 1




            $begingroup$
            If there are infinitely many twin primes, how do you show that $L_{TP}$ is not context-free?
            $endgroup$
            – Emil Jeřábek
            13 hours ago






          • 1




            $begingroup$
            Oh, sorry, I didn’t notice you represent the numbers in unary. Then it is clear. (I believe that proving this for binary representation would require a considerable progress on the twin primes conjecture.)
            $endgroup$
            – Emil Jeřábek
            12 hours ago
















          7












          7








          7





          $begingroup$

          How about the language $L_{TP}$ of twin primes? I.e., all pairs of natural numbers $(p,p')$ (represented, say, in unary), such that $p,p'$ are both prime and $p'=p+2$? If twin primes conjecture is true, then $L_{TP}$ is not context-free; otherwise, it's finite.



          Edit: Let me give a quick proof sketch that the twin primes conjecture implies that $L_{TP}$ is not context-free. Associate to any language $L$ its length sequence $0le a_1le a_2leldots$, where the integer $ell$ appears in the sequence iff there is a word of length $ell$ in $L$. It is a consequence of the pumping lemma(s) that for $L$ that are regular or CFL, the length sequence satisfies the bounded differences property: there is an $R>0$ such that $a_{n+1}-a_nle R$ for all $n$. It is an easy and well-known fact in number theory that the primes do not have bounded differences. Finally, any infinite subsequence of a sequence violating the bounded differences property itself must violate it.






          share|cite|improve this answer











          $endgroup$



          How about the language $L_{TP}$ of twin primes? I.e., all pairs of natural numbers $(p,p')$ (represented, say, in unary), such that $p,p'$ are both prime and $p'=p+2$? If twin primes conjecture is true, then $L_{TP}$ is not context-free; otherwise, it's finite.



          Edit: Let me give a quick proof sketch that the twin primes conjecture implies that $L_{TP}$ is not context-free. Associate to any language $L$ its length sequence $0le a_1le a_2leldots$, where the integer $ell$ appears in the sequence iff there is a word of length $ell$ in $L$. It is a consequence of the pumping lemma(s) that for $L$ that are regular or CFL, the length sequence satisfies the bounded differences property: there is an $R>0$ such that $a_{n+1}-a_nle R$ for all $n$. It is an easy and well-known fact in number theory that the primes do not have bounded differences. Finally, any infinite subsequence of a sequence violating the bounded differences property itself must violate it.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 13 hours ago

























          answered 15 hours ago









          AryehAryeh

          5,84411840




          5,84411840








          • 2




            $begingroup$
            Nice, thanks! But I'm not quite interested in languages that are linked to unknown conjectures about the (non)finiteness of some sets. BTW if those conjectures are true the resulting language is also regular :-)
            $endgroup$
            – Marzio De Biasi
            14 hours ago










          • $begingroup$
            If there are infinitely many twin primes, how do you see that $L_{TP}$ is regular?
            $endgroup$
            – Aryeh
            14 hours ago










          • $begingroup$
            I mean if (dis)proving a conjecture, results in a finite set. In your case - twin primes - if the conjecture is false.
            $endgroup$
            – Marzio De Biasi
            13 hours ago






          • 1




            $begingroup$
            If there are infinitely many twin primes, how do you show that $L_{TP}$ is not context-free?
            $endgroup$
            – Emil Jeřábek
            13 hours ago






          • 1




            $begingroup$
            Oh, sorry, I didn’t notice you represent the numbers in unary. Then it is clear. (I believe that proving this for binary representation would require a considerable progress on the twin primes conjecture.)
            $endgroup$
            – Emil Jeřábek
            12 hours ago
















          • 2




            $begingroup$
            Nice, thanks! But I'm not quite interested in languages that are linked to unknown conjectures about the (non)finiteness of some sets. BTW if those conjectures are true the resulting language is also regular :-)
            $endgroup$
            – Marzio De Biasi
            14 hours ago










          • $begingroup$
            If there are infinitely many twin primes, how do you see that $L_{TP}$ is regular?
            $endgroup$
            – Aryeh
            14 hours ago










          • $begingroup$
            I mean if (dis)proving a conjecture, results in a finite set. In your case - twin primes - if the conjecture is false.
            $endgroup$
            – Marzio De Biasi
            13 hours ago






          • 1




            $begingroup$
            If there are infinitely many twin primes, how do you show that $L_{TP}$ is not context-free?
            $endgroup$
            – Emil Jeřábek
            13 hours ago






          • 1




            $begingroup$
            Oh, sorry, I didn’t notice you represent the numbers in unary. Then it is clear. (I believe that proving this for binary representation would require a considerable progress on the twin primes conjecture.)
            $endgroup$
            – Emil Jeřábek
            12 hours ago










          2




          2




          $begingroup$
          Nice, thanks! But I'm not quite interested in languages that are linked to unknown conjectures about the (non)finiteness of some sets. BTW if those conjectures are true the resulting language is also regular :-)
          $endgroup$
          – Marzio De Biasi
          14 hours ago




          $begingroup$
          Nice, thanks! But I'm not quite interested in languages that are linked to unknown conjectures about the (non)finiteness of some sets. BTW if those conjectures are true the resulting language is also regular :-)
          $endgroup$
          – Marzio De Biasi
          14 hours ago












          $begingroup$
          If there are infinitely many twin primes, how do you see that $L_{TP}$ is regular?
          $endgroup$
          – Aryeh
          14 hours ago




          $begingroup$
          If there are infinitely many twin primes, how do you see that $L_{TP}$ is regular?
          $endgroup$
          – Aryeh
          14 hours ago












          $begingroup$
          I mean if (dis)proving a conjecture, results in a finite set. In your case - twin primes - if the conjecture is false.
          $endgroup$
          – Marzio De Biasi
          13 hours ago




          $begingroup$
          I mean if (dis)proving a conjecture, results in a finite set. In your case - twin primes - if the conjecture is false.
          $endgroup$
          – Marzio De Biasi
          13 hours ago




          1




          1




          $begingroup$
          If there are infinitely many twin primes, how do you show that $L_{TP}$ is not context-free?
          $endgroup$
          – Emil Jeřábek
          13 hours ago




          $begingroup$
          If there are infinitely many twin primes, how do you show that $L_{TP}$ is not context-free?
          $endgroup$
          – Emil Jeřábek
          13 hours ago




          1




          1




          $begingroup$
          Oh, sorry, I didn’t notice you represent the numbers in unary. Then it is clear. (I believe that proving this for binary representation would require a considerable progress on the twin primes conjecture.)
          $endgroup$
          – Emil Jeřábek
          12 hours ago






          $begingroup$
          Oh, sorry, I didn’t notice you represent the numbers in unary. Then it is clear. (I believe that proving this for binary representation would require a considerable progress on the twin primes conjecture.)
          $endgroup$
          – Emil Jeřábek
          12 hours ago













          6












          $begingroup$

          Another good one is the complement of the set $S$ of contiguous subwords (aka "factors") of the Thue-Morse sequence ${bf t} = 0110100110010110 cdots $. To give some context, Jean Berstel proved that the complement of the set $T$ of prefixes of the Thue-Morse word is context-free (and actually something more general than that). But the corresponding result for subwords is still open.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Great, thanks! If you saw it stated somewhere (perhaps in one of your many papers on the Thue-Morse sequence? ;-) you can add the reference (even if stated in the iterated morphism form).
            $endgroup$
            – Marzio De Biasi
            9 hours ago
















          6












          $begingroup$

          Another good one is the complement of the set $S$ of contiguous subwords (aka "factors") of the Thue-Morse sequence ${bf t} = 0110100110010110 cdots $. To give some context, Jean Berstel proved that the complement of the set $T$ of prefixes of the Thue-Morse word is context-free (and actually something more general than that). But the corresponding result for subwords is still open.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Great, thanks! If you saw it stated somewhere (perhaps in one of your many papers on the Thue-Morse sequence? ;-) you can add the reference (even if stated in the iterated morphism form).
            $endgroup$
            – Marzio De Biasi
            9 hours ago














          6












          6








          6





          $begingroup$

          Another good one is the complement of the set $S$ of contiguous subwords (aka "factors") of the Thue-Morse sequence ${bf t} = 0110100110010110 cdots $. To give some context, Jean Berstel proved that the complement of the set $T$ of prefixes of the Thue-Morse word is context-free (and actually something more general than that). But the corresponding result for subwords is still open.






          share|cite|improve this answer









          $endgroup$



          Another good one is the complement of the set $S$ of contiguous subwords (aka "factors") of the Thue-Morse sequence ${bf t} = 0110100110010110 cdots $. To give some context, Jean Berstel proved that the complement of the set $T$ of prefixes of the Thue-Morse word is context-free (and actually something more general than that). But the corresponding result for subwords is still open.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 10 hours ago









          Jeffrey ShallitJeffrey Shallit

          6,4832635




          6,4832635












          • $begingroup$
            Great, thanks! If you saw it stated somewhere (perhaps in one of your many papers on the Thue-Morse sequence? ;-) you can add the reference (even if stated in the iterated morphism form).
            $endgroup$
            – Marzio De Biasi
            9 hours ago


















          • $begingroup$
            Great, thanks! If you saw it stated somewhere (perhaps in one of your many papers on the Thue-Morse sequence? ;-) you can add the reference (even if stated in the iterated morphism form).
            $endgroup$
            – Marzio De Biasi
            9 hours ago
















          $begingroup$
          Great, thanks! If you saw it stated somewhere (perhaps in one of your many papers on the Thue-Morse sequence? ;-) you can add the reference (even if stated in the iterated morphism form).
          $endgroup$
          – Marzio De Biasi
          9 hours ago




          $begingroup$
          Great, thanks! If you saw it stated somewhere (perhaps in one of your many papers on the Thue-Morse sequence? ;-) you can add the reference (even if stated in the iterated morphism form).
          $endgroup$
          – Marzio De Biasi
          9 hours ago


















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          Statuo de Libereco

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          Koordinatoj: 40°41′21″N 74°02′40″W   /   40.68917°N , 74.04444°U  / 40.68917; -74.04444  ( Statuo de Libereco ) Statuo de Libereco angle: Statue of Liberty statuo Statuo de Libereco de antaŭe Oficiala nomo: Liberty Enlightening the World Lando  Usono Ŝtato Novjorkio Regiono Novjorko Insulo Insulo de Libereco Situo Statuo de Libereco  - koordinatoj 40°41′21″N 74°02′40″W   /   40.68917°N , 74.04444°U  / 40.68917; -74.04444  ( Statuo de Libereco ) Alteco de statuo 46 m  - alteco kun soklo 93 m Pezo 205 000 kg (451 948 lb) Materialoj bronzo, ŝtalo Proponinto Gustave Eiffel  - skulptisto Frédéric Auguste Bartholdi  - proponinto de soklo Richard Morris Hunt  - kapo laŭ modelo de Isabella Eugenie Boyer Transdonita 4-an de julio 1886 Por publiko alirebla por publiko Horzono OT (UTC-5)  - somera tempo OT (UTC-4) Loko
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          Tanganjiko

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          Lago Tanganjiko Situo centra Afriko Ĉefaj fontoj Kalambo, Lufuko, Malagarasi kaj Ruzizi Ĉefaj elfluoj Lukugo Setlejoj Bujumbura, Kalemie Datenoj Koordinatoj 6°  S , 30°  O -6.3827777777778 29.615 782 Koordinatoj: 6°  S , 30°  O DEC Supermara alteco 782 m Surfaca areo 32 893 km² f5 Maksimuma longo 673 km f6 Maksimuma larĝo 72 km f7 Akva volumeno 18.900 km³ dep1 f8 Maksimuma profundo 1470 m f10 Averaĝa profundo 570 m f11 v   •   d   •   r Mapo de la lago Tanganjiko estas lago en orienta Afriko, inter Burundo, Kongo Kinŝasa, Tanzanio kaj Zambio. Ĝi apartenas al la Granda Rifto kaj estas unu el la Afrikaj Grandaj Lagoj. Tanzanio (46%) kaj DRK (40%) havas la majoritaton de la lago. Ĝi havas longon de 673 km, areon de ĉ. 32.900 km² kaj maksimuman profundon taksatan je 1470 metroj; pro tiu ĉi granda profundo ankaŭ ĝia volumeno estas sufiĉe granda, 18.900 km³. Oni konsi
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          Liste der Baudenkmäler in Enneberg

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          Die Liste der Baudenkmäler in Enneberg (ladinisch Mareo , italienisch Marebbe ) enthält die 36 als Baudenkmäler ausgewiesenen Objekte auf dem Gebiet der Gemeinde Enneberg in Südtirol. Basis ist das im Internet einsehbare offizielle Verzeichnis der Baudenkmäler in Südtirol. Dabei kann es sich beispielsweise um Sakralbauten, Wohnhäuser, Bauernhöfe und Adelsansitze handeln. Die Reihenfolge in dieser Liste orientiert sich an der Bezeichnung, alternativ ist sie auch nach der Adresse oder dem Datum der Unterschutzstellung sortierbar. Liste | Foto   Bezeichnung Standort Eintragung Beschreibung Metadaten ja Altes Gasthaus Janesch ID: 14567 46° 42′ 54″ N, 11° 52′ 42″ O 76882557800 0 8. Sep. 1986 (BLR-LAB 4974) Ländliches Wohnhaus/Bauernhof KG:  Welschellen Bauparzelle: 391 ja Asch (Prack) ID: 14540 46° 43′ 12″ N, 11° 54′ 35″ O 75496485300 20. Nov. 1950 (MD) Ansitz KG:  Enneberg Bauparzelle: 38 ja Biccia mit Stade
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