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A Tale of Conjunctive Grammars

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11088))

Abstract

Conjunctive grammars are an extension of ordinary (“context-free”) grammars with a conjunction operator, which can be used in any rules to specify a substring that satisfies several syntactic conditions simultaneously. This family has been systematically studied since the turn of the century, and is a subject of current studies. This paper gives an overview of the current state of the art in the research on conjunctive grammars.

Supported by the Russian Science Foundation, project 18-11-00100.

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References

  1. Aizikowitz, T., Kaminski, M.: Conjunctive grammars and alternating pushdown automata. Acta Informatica 50(3), 175–197 (2013). https://doi.org/10.1007/978-3-540-69937-8_6

    Article  MathSciNet  MATH  Google Scholar 

  2. Aizikowitz, T., Kaminski, M.: Linear conjunctive grammars and one-turn synchronized alternating pushdown automata. Int. J. Found. Comput. Sci. 25(6), 781–802 (2014). https://doi.org/10.1142/S0129054114500336

    Article  MathSciNet  MATH  Google Scholar 

  3. Aizikowitz, T., Kaminski, M.: LR(0) conjunctive grammars and deterministic synchronized alternating pushdown automata. J. Comput. Syst. Sci. 82(8), 1329–1359 (2016). https://doi.org/10.1016/j.jcss.2016.05.008

    Article  MathSciNet  MATH  Google Scholar 

  4. Ajdukiewicz, K.: Die syntaktische Konnexität. In: Ajdukiewicz, K., Ingarden, R., Twardowski, K. (eds.) Studia Philosophica, vol. 1, pp. 1–27 (1935)

    Google Scholar 

  5. Autebert, J., Berstel, J., Boasson, L.: Context-free languages and pushdown automata. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, pp. 111–174. Springer, Heidelberg (1997). https://doi.org/10.1007/978-3-642-59136-5_3

    Chapter  Google Scholar 

  6. Bakinova, E., Basharin, A., Batmanov, I., Lyubort, K., Okhotin, A., Sazhneva, E.: Formal languages over GF(2). In: Klein, S.T., Martín-Vide, C., Shapira, D. (eds.) LATA 2018. LNCS, vol. 10792, pp. 68–79. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-77313-1_5

    Chapter  Google Scholar 

  7. Bar-Hillel, Y., Gaifman, H., Shamir, E.: On categorial and phrase structure grammars. Bull. Res. Counc. Isr. 9F, 155–166 (1960)

    MathSciNet  MATH  Google Scholar 

  8. Barash, M.: Programming language specification by a grammar with contexts. In: NCMA (2013)

    Google Scholar 

  9. Barash, M., Okhotin, A.: An extension of context-free grammars with one-sided context specifications. Inf. Comput. 237, 268–293 (2014). https://doi.org/10.1016/j.ic.2014.03.003

    Article  MathSciNet  MATH  Google Scholar 

  10. Barash, M., Okhotin, A.: Two-sided context specifications in formal grammars. Theor. Comput. Sci. 591, 134–153 (2015). https://doi.org/10.1016/j.tcs.2015.05.004

    Article  MathSciNet  MATH  Google Scholar 

  11. Barash, M., Okhotin, A.: Linear grammars with one-sided contexts and their automaton representation. RAIRO Informatique Théorique et Applications 49(2), 153–178 (2015). http://dx.doi.org/10.1051/ita/2015004

  12. Barash, M., Okhotin, A.: Generalized LR parsing algorithm for grammars with one-sided contexts. Theory Comput. Syst. 61(2), 581–605 (2017). https://doi.org/10.1007/s00224-016-9683-3

    Article  MathSciNet  MATH  Google Scholar 

  13. Barash, M., Okhotin, A.: Linear-space recognition for grammars with contexts. Theor. Comput. Sci. 719, 73–85 (2018). https://doi.org/10.1016/j.tcs.2017.11.006

    Article  MathSciNet  MATH  Google Scholar 

  14. Boullier, P.: A cubic time extension of context-free grammars. Grammars 3(2–3), 111–131 (2000). https://doi.org/10.1023/A:1009907814595

    Article  MathSciNet  MATH  Google Scholar 

  15. Chomsky, N.: Three models for the description of language. IRE Trans. Inf. Theory 2(3), 113–124 (1956). https://doi.org/10.1109/TIT.1956.1056813

    Article  MATH  Google Scholar 

  16. Clark, A., Eyraud, R., Habrard, A.: Using contextual representations to efficiently learn context-free languages. J. Mach. Learn. Res. 11, 2707–2744 (2010). http://www.jmlr.org/papers/v11/clark10a.html

  17. Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. J. ACM 9, 350–371 (1962). https://doi.org/10.1145/321127.321132

    Article  MathSciNet  MATH  Google Scholar 

  18. Greibach, S.A.: A new normal-form theorem for context-free phrase structure grammars. J. ACM 12, 42–52 (1965). https://doi.org/10.1145/321250.321254

    Article  MathSciNet  MATH  Google Scholar 

  19. Greibach, S.A., Shi, W., Simonson, S.: Single tree grammars. In: Ullman, J.D. (ed.) Theoretical Studies in Computer Science, pp. 73–99. Academic Press, Cambridge (1992)

    Chapter  Google Scholar 

  20. Heilbrunner, S., Schmitz, L.: An efficient recognizer for the Boolean closure of context-free languages. Theor. Comput. Sci. 80, 53–75 (1991). https://doi.org/10.1016/0304-3975(91)90205-G

    Article  MathSciNet  MATH  Google Scholar 

  21. Hellings, J.: Conjunctive context-free path queries. In: 17th International Conference on Database Theory, ICDT 2014, Athens, Greece, 24–28 March 2014, pp. 119–130 (2014)

    Google Scholar 

  22. Ibarra, O.H., Kim, S.M.: Characterizations and computational complexity of systolic trellis automata. Theor. Comput. Sci. 29, 123–153 (1984). https://doi.org/10.1016/0304-3975(84)90015-X

    Article  MathSciNet  MATH  Google Scholar 

  23. Jeż, A.: Conjunctive grammars can generate non-regular unary languages. Int. J. Found. Comput. Sci. 19(3), 597–615 (2008). https://doi.org/10.1142/S012905410800584X

    Article  MATH  Google Scholar 

  24. Jeż, A., Okhotin, A.: Conjunctive grammars over a unary alphabet: undecidability and unbounded growth. Theory Comput. Syst. 46(1), 27–58 (2010). https://doi.org/10.1007/s00224-008-9139-5

    Article  MathSciNet  MATH  Google Scholar 

  25. Jeż, A., Okhotin, A.: Complexity of equations over sets of natural numbers. Theory Comput. Syst. 48(2), 319–342 (2011). https://doi.org/10.1007/s00224-009-9246-y

    Article  MathSciNet  MATH  Google Scholar 

  26. Jeż, A., Okhotin, A.: One-nonterminal conjunctive grammars over a unary alphabet. Theory Comput. Syst. 49(2), 319–342 (2011). https://doi.org/10.1007/s00224-011-9319-6

    Article  MathSciNet  MATH  Google Scholar 

  27. Jeż, A., Okhotin, A.: On the number of nonterminal symbols in unambiguous conjunctive grammars. In: Kutrib, M., Moreira, N., Reis, R. (eds.) DCFS 2012. LNCS, vol. 7386, pp. 183–195. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31623-4_14

    Chapter  MATH  Google Scholar 

  28. Jeż, A., Okhotin, A.: Computational completeness of equations over sets of natural numbers. Inf. Comput. 237, 56–94 (2014). https://doi.org/10.1016/j.ic.2014.05.001

    Article  MathSciNet  MATH  Google Scholar 

  29. Jeż, A., Okhotin, A.: Unambiguous conjunctive grammars over a one-symbol alphabet. Theor. Comput. Sci. 665, 13–39 (2017). https://doi.org/10.1016/j.tcs.2016.12.009

    Article  MathSciNet  MATH  Google Scholar 

  30. Kanazawa, M.: The Lambek calculus enriched with additional connectives. J. Log. Lang. Inf. 1, 141–171 (1992). https://doi.org/10.1007/BF00171695

    Article  MathSciNet  MATH  Google Scholar 

  31. Kanchan Devi, K., Arumugam, S.: Probabilistic conjunctive grammar. In: Arumugam, S., Bagga, J., Beineke, L.W., Panda, B.S. (eds.) ICTCSDM 2016. LNCS, vol. 10398, pp. 119–127. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-64419-6_16

    Chapter  Google Scholar 

  32. Knuth, D.E.: On the translation of languages from left to right. Inf. Control 8(6), 607–639 (1965). https://doi.org/10.1016/S0019-9958(65)90426-2

    Article  MathSciNet  MATH  Google Scholar 

  33. Kountouriotis, V., Nomikos, C., Rondogiannis, P.: Well-founded semantics for Boolean grammars. Inf. Comput. 207(9), 945–967 (2009). https://doi.org/10.1016/j.ic.2009.05.002

    Article  MathSciNet  MATH  Google Scholar 

  34. Kuznetsov, S.: Conjunctive grammars in greibach normal form and the lambek calculus with additive connectives. In: Morrill, G., Nederhof, M.-J. (eds.) FG 2012-2013. LNCS, vol. 8036, pp. 242–249. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39998-5_15

    Chapter  Google Scholar 

  35. Kuznetsov, S., Okhotin, A.: Conjunctive categorial grammars. In: Proceedings of the 15th Meeting on the Mathematics of Language, MOL 2017, London, UK, 13–14 July 2017, pp. 141–151. ACL (2017)

    Google Scholar 

  36. Lang, B.: Deterministic techniques for efficient non-deterministic parsers. In: Loeckx, J. (ed.) ICALP 1974. LNCS, vol. 14, pp. 255–269. Springer, Heidelberg (1974). https://doi.org/10.1007/3-540-06841-4_65

    Chapter  Google Scholar 

  37. Lange, M.: Alternating context-free languages and linear time \(\mu \)-calculus with sequential composition. Electron. Notes Theor. Comput. Sci. 68(2), 70–86 (2002). https://doi.org/10.1016/S1571-0661(05)80365-2

    Article  MATH  Google Scholar 

  38. Latta, M., Wall, R.: Intersective context-free languages. In: Martin-Vide, C. (ed.) 9th Congress on Natural and Formal Languages, Reus, Spain, 20–22 December 1993, pp. 15–43 (1993)

    Google Scholar 

  39. Lehtinen, T., Okhotin, A.: Boolean grammars and GSM mappings. Int. J. Found. Comput. Sci. 21(5), 799–815 (2010). https://doi.org/10.1142/S0129054110007568

    Article  MathSciNet  MATH  Google Scholar 

  40. Lehtinen, T., Okhotin, A.: On language equations \(XXK = XXL\) and \(XM = N\) over a unary alphabet. In: Gao, Y., Lu, H., Seki, S., Yu, S. (eds.) DLT 2010. LNCS, vol. 6224, pp. 291–302. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14455-4_27

    Chapter  MATH  Google Scholar 

  41. Okhotin, A.: Conjunctive grammars. J. Autom. Lang. Comb. 6(4), 519–535 (2001)

    MathSciNet  MATH  Google Scholar 

  42. Okhotin, A.: Conjunctive grammars and systems of language equations. Program. Comput. Soft. 28(5), 243–249 (2002). https://doi.org/10.1023/A:1020213411126

    Article  MathSciNet  MATH  Google Scholar 

  43. Okhotin, A.: Top-down parsing of conjunctive languages. Grammars 5(1), 21–40 (2002). https://doi.org/10.1023/A:1014219530875

    Article  MATH  Google Scholar 

  44. Okhotin, A.: LR parsing for conjunctive grammars. Grammars 5(2), 81–124 (2002). https://doi.org/10.1023/A:1016329527130

    Article  MathSciNet  MATH  Google Scholar 

  45. Okhotin, A.: Boolean grammars. Inf. Comput. 194(1), 19–48 (2004). https://doi.org/10.1016/j.ic.2004.03.006

    Article  MathSciNet  MATH  Google Scholar 

  46. Okhotin, A.: On the equivalence of linear conjunctive grammars to trellis automata. Informatique Théorique et Applications 38(1), 69–88 (2004). https://doi.org/10.1051/ita:2004004

    Article  MathSciNet  MATH  Google Scholar 

  47. Okhotin, A.: On the number of nonterminals in linear conjunctive grammars. Theor. Comput. Sci. 320(2–3), 419–448 (2004). https://doi.org/10.1016/j.tcs.2004.03.002

    Article  MathSciNet  MATH  Google Scholar 

  48. Okhotin, A.: The dual of concatenation. Theor. Comput. Sci. 345(2–3), 425–447 (2005). https://doi.org/10.1016/j.tcs.2005.07.019

    Article  MathSciNet  MATH  Google Scholar 

  49. Okhotin, A.: On the existence of a Boolean grammar for a simple programming language. In: Automata and Formal Languages, Proceedings of AFL 2005, Dobogókő, Hungary, 17–20 May 2005

    Google Scholar 

  50. Okhotin, A.: Generalized LR parsing algorithm for Boolean grammars. Int. J. Found. Comput. Sci. 17(3), 629–664 (2006). https://doi.org/10.1142/S0129054106004029

    Article  MathSciNet  MATH  Google Scholar 

  51. Okhotin, A.: Recursive descent parsing for Boolean grammars. Acta Informatica 44(3–4), 167–189 (2007). https://doi.org/10.1007/s00236-007-0045-0

    Article  MathSciNet  MATH  Google Scholar 

  52. Okhotin, A.: Notes on dual concatenation. Int. J. Found. Comput. Sci. 18(6), 1361–1370 (2007). https://doi.org/10.1142/S0129054107005406

    Article  MathSciNet  MATH  Google Scholar 

  53. Okhotin, A.: Unambiguous Boolean grammars. Inf. Comput. 206, 1234–1247 (2008). https://doi.org/10.1016/j.ic.2008.03.023

    Article  MathSciNet  MATH  Google Scholar 

  54. Okhotin, A.: Expressive power of LL(\(k\)) Boolean grammars. Theor. Comput. Sci. 412(39), 5132–5155 (2011). https://doi.org/10.1016/j.tcs.2011.05.013

    Article  MathSciNet  MATH  Google Scholar 

  55. Okhotin, A.: Conjunctive and Boolean grammars: the true general case of the context-free grammars. Comput. Sci. Rev. 9, 27–59 (2013). https://doi.org/10.1016/j.cosrev.2013.06.001

    Article  MATH  Google Scholar 

  56. Okhotin, A.: Parsing by matrix multiplication generalized to Boolean grammars. Theor. Comput. Sci. 516, 101–120 (2014). https://doi.org/10.1016/j.tcs.2013.09.011

    Article  MathSciNet  MATH  Google Scholar 

  57. Okhotin, A.: Improved normal form for grammars with one-sided contexts. Theor. Comput. Sci. 588, 52–72 (2015). https://doi.org/10.1016/j.tcs.2015.03.041

    Article  MathSciNet  MATH  Google Scholar 

  58. Okhotin, A.: The hardest language for conjunctive grammars. In: Kulikov, A.S., Woeginger, G.J. (eds.) CSR 2016. LNCS, vol. 9691, pp. 340–351. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-34171-2_24

    Chapter  Google Scholar 

  59. Okhotin, A.: Underlying principles and recurring ideas of formal grammars. In: Klein, S.T., Martín-Vide, C., Shapira, D. (eds.) LATA 2018. LNCS, vol. 10792, pp. 36–59. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-77313-1_3

    Chapter  MATH  Google Scholar 

  60. Okhotin, A., Reitwießner, C.: Conjunctive grammars with restricted disjunction. Theor. Comput. Sci. 411(26–28), 2559–2571 (2010). https://doi.org/10.1016/j.tcs.2010.03.015

    Article  MathSciNet  MATH  Google Scholar 

  61. Okhotin, A., Reitwießner, C.: Parsing Boolean grammars over a one-letter alphabet using online convolution. Theor. Comput. Sci. 457, 149–157 (2012). https://doi.org/10.1016/j.tcs.2012.06.032

    Article  MathSciNet  MATH  Google Scholar 

  62. Okhotin, A., Rondogiannis, P.: On the expressive power of univariate equations over sets of natural numbers. Inf. Comput. 212, 1–14 (2012). https://doi.org/10.1016/j.ic.2012.01.004

    Article  MathSciNet  MATH  Google Scholar 

  63. Rabkin, M.: Recognizing two-sided contexts in cubic time. In: Hirsch, E.A., Kuznetsov, S.O., Pin, J.É., Vereshchagin, N.K. (eds.) CSR 2014. LNCS, vol. 8476, pp. 314–324. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-06686-8_24

    Chapter  Google Scholar 

  64. Rosenkrantz, D.J., Stearns, R.E.: Properties of deterministic top-down grammars. Inf. Control 17, 226–256 (1970). https://doi.org/10.1016/S0019-9958(70)90446-8

    Article  MathSciNet  MATH  Google Scholar 

  65. Rounds, W.C.: LFP: a logic for linguistic descriptions and an analysis of its complexity. Comput. Linguist. 14(4), 1–9 (1988)

    Google Scholar 

  66. Szabari, A.: Alternujúce Zásobníkové Automaty (Alternating Pushdown Automata), in Slovak, diploma work (M.Sc. thesis), University of Košice (Czechoslovakia), 45 pp. (1991)

    Google Scholar 

  67. Terrier, V.: On real-time one-way cellular array. Theor. Comput. Sci. 141(1–2), 331–335 (1995). https://doi.org/10.1016/0304-3975(94)00212-2

    Article  MathSciNet  MATH  Google Scholar 

  68. Terrier, V.: Some computational limits of trellis automata. In: Dennunzio, A., Formenti, E., Manzoni, L., Porreca, A.E. (eds.) AUTOMATA 2017. LNCS, vol. 10248, pp. 176–186. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-58631-1_14

    Chapter  MATH  Google Scholar 

  69. Tomita, M.: An efficient augmented context-free parsing algorithm. Comput. Linguist. 13(1), 31–46 (1987)

    Google Scholar 

  70. Valiant, L.G.: General context-free recognition in less than cubic time. J. Comput. Syst. Sci. 10(2), 308–314 (1975). https://doi.org/10.1016/S0022-0000(75)80046-8

    Article  MathSciNet  MATH  Google Scholar 

  71. Wotschke, D.: The Boolean closures of the deterministic and nondeterministic context-free languages. In: Brauer, W. (ed.) GI Gesellschaft für Informatik e. V. LNCS, vol. 1, pp. 113–121. Springer, Heidelberg (1973). https://doi.org/10.1007/978-3-662-41148-3_11

    Chapter  Google Scholar 

  72. Yoshinaka, R.: Learning conjunctive grammars and contextual binary feature grammars. In: Dediu, A.-H., Formenti, E., Martín-Vide, C., Truthe, B. (eds.) LATA 2015. LNCS, vol. 8977, pp. 623–635. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-15579-1_49

    Chapter  MATH  Google Scholar 

  73. Zhang, Q., Su, Z.: Context-sensitive data-dependence analysis via linear conjunctive language reachability. In: Principles of Programming Languages (POPL 2017), pp. 344–358 (2017). http://dx.doi.org/10.1145/3009837.3009848

  74. Zier-Vogel, R., Domaratzki, M.: RNA pseudoknot prediction through stochastic conjunctive grammars. In: The Nature of Computation: Logic, Algorithms, Applications (CiE 2013, Milan, Italy, 1–5 July 2013), Informal Proceedings, pp. 80–89 (2013)

    Google Scholar 

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Okhotin, A. (2018). A Tale of Conjunctive Grammars. In: Hoshi, M., Seki, S. (eds) Developments in Language Theory. DLT 2018. Lecture Notes in Computer Science(), vol 11088. Springer, Cham. https://doi.org/10.1007/978-3-319-98654-8_4

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