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 13 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
            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




            $begingroup$
            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
            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




            $begingroup$
            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
            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




            $begingroup$
            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 12 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
            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




            $begingroup$
            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
            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




            $begingroup$
            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
          13 hours ago




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




          $begingroup$
          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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          Knooppunt Holsloot

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          Knooppunt Holsloot Kenmerken Wegen A37 × N34 Type knooppunt klaverblad Portaal     Verkeer & Vervoer Knooppunt Holsloot is een verkeersknooppunt op de kruising van de autosnelweg A37 en de N34 in Drenthe, tussen Emmen en Coevorden. Oorspronkelijk was deze aansluiting een afritconstructie van het type Haarlemmermeeraansluiting, zoals bij Knooppunt Hooipolder. Sinds 2002 is het knooppunt omgebouwd tot een volledig klaverblad. Als noviteit werd hier voor het eerst gebruikgemaakt van dimbare verlichting in plaats van de normale vaste tijden waarop de verlichting aan en uit gaat, wat een grote kostenbesparing opleverde. [1] Dat Holsloot als volwaardig knooppunt is uitgevoerd, waarbij alle verkeersstromen en afbuigingen ononderbroken kunnen doorgaan, is uitzonderlijk. Het betreft hier namelijk een aansluiting tussen een autosnelweg en niet-snelweg. Een vergelijkbare situatie treft men nabij Hoensbroek bij het knooppunt Ten Esschen.     Aa
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          Altaar (religie)

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          Deel van de serie over kerkelijk gerei waaronder paramenten en liturgisch vaatwerk gebruikt in de liturgie Liturgisch vaatwerk Vasa sacra Miskelk · Pateen Kelklepeltje Ciborie · Monstrans Pyxis · Custodiale Vasa non sacra Ampullen · Wijwatervat Olievaatje Wierookvat · Ablutievat Paramenten Amict · Albe · Baarkleed · Cingel Tuniek · Dalmatiek · Fanon Kazuifel · Manipel · Stola Gremiale · Benedictievelum Mijter Koorkledij Rochet · Superplie Koorkap · Cappa magna Kovel Kelkgerei Bursa · Kelkvelum Ciborievelum Kerklinnen Corporale · Kelkdoekje Lavabodoekje · Palla Altaardwaal Kerkinterieur Altaar · Ambo Biechtstoel · Communiebank Doksaal · Doopvont Faldistorium Godslamp · Hoogaltaar Heilig Kruisaltaar Katheder · Preekstoel Sedilia Tabernakel · Volksaltaar Liturgische boeken Altaarmissaal Benedictionale · Brevier Evangeliarium · Evangelistarium Graduale · Kyriale Lectionarium Psalter · Rituaal Sacramentarium · Volksmissaal Overige Flambou
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          Gregoriusmis

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          Een afbeelding van de Gregoriusmis uit een getijdenboek gemaakt in de zuidelijke Nederlanden tussen 1500 en 1525 Gregoriusmis, Meester van het Akense Altaar, ca. 1500-1525. Collectie Museum Catharijneconvent De Gregoriusmis of de Mis van Sint Gregorius is een motief in katholieke kunst, dat het eerst verscheen in de late Middeleeuwen en dat nog steeds werd gebruikt tijdens de contrareformatie. Paus Gregorius I (ca. 540-604) is afgebeeld, de H. Mis opdragend, terwijl Christus als de Man van Smarten verschijnt op het altaar dat voor Sint Gregorius staat. Volgens de legende had St. Gregorius gebeden om een teken voor een betwijfelaar van de transsubstantiatieleer, waarop de lijdende Christus tijdens het opdragen van de Mis verscheen om hem te overtuigen en waarbij zijn bloed in de altaarkelk liep. Vaak zijn ook de passiewerktuigen afgebeeld. Zie de categorie Mass of Saint Gregory van Wikimedia Commons voor mediabestanden over dit onderwerp. This page is onl
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