Document Open Access Logo

Subsequence Matching and Analysis Problems for Formal Languages

Authors Szilárd Zsolt Fazekas , Tore Koß , Florin Manea , Robert Mercaş , Timo Specht

PDF

File

Thumbnail PDF
PDF
LIPIcs.ISAAC.2024.28.pdf
  • Filesize: 0.77 MB
  • 23 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Stringology
  • String Combinatorics
  • Subsequence
  • Formal Languages
  • Context-Free Languages
  • Context-Sensitive Languages

Metrics

Abstract

In this paper, we study a series of algorithmic problems related to the subsequences occurring in the strings of a given language, under the assumption that this language is succinctly represented by a grammar generating it, or an automaton accepting it. In particular, we focus on the following problems: Given a string w and a language L, does there exist a word of L which has w as subsequence? Do all words of L have w as a subsequence? Given an integer k alongside L, does there exist a word of L which has all strings of length k, over the alphabet of L, as subsequences? Do all words of L have all strings of length k as subsequences? For the last two problems, efficient algorithms were already presented in [Adamson et al., ISAAC 2023] for the case when L is a regular language, and efficient solutions can be easily obtained for the first two problems. We extend that work as follows: we give sufficient conditions on the class of input-languages, under which these problems are decidable; we provide efficient algorithms for all these problems in the case when the input language is context-free; we show that all problems are undecidable for context-sensitive languages. Finally, we provide a series of initial results related to a class of languages that strictly includes the regular languages and is strictly included in the class of context-sensitive languages, but is incomparable to the of class context-free languages; these results deviate significantly from those reported for language-classes from the Chomsky hierarchy.

Cite As Get BibTex

Szilárd Zsolt Fazekas, Tore Koß, Florin Manea, Robert Mercaş, and Timo Specht. Subsequence Matching and Analysis Problems for Formal Languages. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 28:1-28:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.28

Author Details

Szilárd Zsolt Fazekas
  • Akita University, Japan
Tore Koß
  • University of Göttingen, Germany
Florin Manea
  • University of Göttingen, Germany
Robert Mercaş
  • Loughborough University, UK
Timo Specht
  • University of Göttingen, Germany

Funding

  • Fazekas, Szilárd Zsolt: Supported by the JSPS Kakenhi Grant Number 23K10976.
  • Manea, Florin: Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) in the framework of the Heisenberg Programme project number 466789228.
  • Mercaş, Robert: Supported by a Return Fellowship from the Alexander von Humboldt Foundation for a research stay at University of Göttingen, Germany.

References

  1. Amir Abboud, Arturs Backurs, and Virginia Vassilevska Williams. Tight hardness results for LCS and other sequence similarity measures. In IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 59-78, 2015. URL: https://doi.org/10.1109/FOCS.2015.14.
  2. Amir Abboud and Aviad Rubinstein. Fast and deterministic constant factor approximation algorithms for LCS imply new circuit lower bounds. In 9th Innovations in Theoretical Computer Science Conference, ITCS 2018, January 11-14, 2018, Cambridge, MA, USA, pages 35:1-35:14, 2018. URL: https://doi.org/10.4230/LIPIcs.ITCS.2018.35.
  3. Amir Abboud, Virginia Vassilevska Williams, and Oren Weimann. Consequences of faster alignment of sequences. In Automata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Copenhagen, Denmark, July 8-11, 2014, Proceedings, Part I, pages 39-51, 2014. URL: https://doi.org/10.1007/978-3-662-43948-7_4.
  4. Duncan Adamson, Pamela Fleischmann, Annika Huch, Tore Koß, Florin Manea, and Dirk Nowotka. k-universality of regular languages. In Satoru Iwata and Naonori Kakimura, editors, 34th International Symposium on Algorithms and Computation (ISAAC 2023), volume 283 of LIPIcs, pages 4:1-4:21, 2023. Full version: https://arxiv.org/abs/2311.10658. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2023.4.
  5. Duncan Adamson, Maria Kosche, Tore Koß, Florin Manea, and Stefan Siemer. Longest common subsequence with gap constraints. In Anna E. Frid and Robert Mercas, editors, Combinatorics on Words - 14th International Conference, WORDS 2023, Umeå, Sweden, June 12-16, 2023, Proceedings, volume 13899 of Lecture Notes in Computer Science, pages 60-76. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-33180-0_5.
  6. Ashwani Anand and Georg Zetzsche. Priority downward closures. In Guillermo A. Pérez and Jean-François Raskin, editors, 34th International Conference on Concurrency Theory, CONCUR 2023, September 18-23, 2023, Antwerp, Belgium, volume 279 of LIPIcs, pages 39:1-39:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.CONCUR.2023.39.
  7. Alexander Artikis, Alessandro Margara, Martín Ugarte, Stijn Vansummeren, and Matthias Weidlich. Complex event recognition languages: Tutorial. In Proceedings of the 11th ACM International Conference on Distributed and Event-based Systems, DEBS 2017, Barcelona, Spain, June 19-23, 2017, pages 7-10, 2017. URL: https://doi.org/10.1145/3093742.3095106.
  8. Ricardo A. Baeza-Yates. Searching subsequences. Theoretical Computer Science, 78(2):363-376, 1991. URL: https://doi.org/10.1016/0304-3975(91)90358-9.
  9. Laura Barker, Pamela Fleischmann, Katharina Harwardt, Florin Manea, and Dirk Nowotka. Scattered factor-universality of words. In Nataša Jonoska and Dmytro Savchuk, editors, Developments in Language Theory, LNCS, pages 14-28, Cham, 2020. Springer International Publishing. URL: https://doi.org/10.1007/978-3-030-48516-0_2.
  10. Lasse Bergroth, Harri Hakonen, and Timo Raita. A survey of longest common subsequence algorithms. In Pablo de la Fuente, editor, Seventh International Symposium on String Processing and Information Retrieval, SPIRE 2000, A Coruña, Spain, September 27-29, 2000, pages 39-48, , 2000. URL: https://doi.org/10.1109/SPIRE.2000.878178.
  11. Philip Bille, Inge Li Gørtz, Hjalte Wedel Vildhøj, and David Kofoed Wind. String matching with variable length gaps. Theoretical Computer Science, 443:25-34, 2012. URL: https://doi.org/10.1016/j.tcs.2012.03.029.
  12. Karl Bringmann and Bhaskar Ray Chaudhury. Sketching, Streaming, and Fine-Grained Complexity of (Weighted) LCS. In Sumit Ganguly and Paritosh Pandya, editors, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018), volume 122 of LIPIcs, pages 40:1-40:16, 2018. URL: https://doi.org/10.4230/LIPIcs.FSTTCS.2018.40.
  13. Karl Bringmann and Marvin Künnemann. Multivariate fine-grained complexity of longest common subsequence. In Proc. SODA 2018, pages 1216-1235, 2018. URL: https://doi.org/10.1137/1.9781611975031.79.
  14. Sam Buss and Michael Soltys. Unshuffling a square is NP-hard. Journal of Computer and System Sciences, 80(4):766-776, 2014. URL: https://doi.org/10.1016/j.jcss.2013.11.002.
  15. Hiroyuki Chigahara, Szilárd Zsolt Fazekas, and Akihiro Yamamura. One-way jumping finite automata. International Journal of Foundations of Computer Science, 27(3):391-405, 2016. Google Scholar
  16. Vaclav Chvatal and David Sankoff. Longest common subsequences of two random sequences. Journal of Applied Probability, 12(2):306-315, 1975. URL: http://www.jstor.org/stable/3212444.
  17. Maxime Crochemore, Christophe Hancart, and Thierry Lecroq. Algorithms on strings. Cambridge University Press, 2007. Google Scholar
  18. Maxime Crochemore, Borivoj Melichar, and Zdeněk Troníček. Directed acyclic subsequence graph - overview. Journal of Discrete Algorithms, 1(3-4):255-280, 2003. URL: https://doi.org/10.1016/S1570-8667(03)00029-7.
  19. Joel D. Day, Pamela Fleischmann, Maria Kosche, Tore Koß, Florin Manea, and Stefan Siemer. The Edit Distance to k-Subsequence Universality. In Markus Bläser and Benjamin Monmege, editors, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021), volume 187 of LIPIcs, pages 25:1-25:19, 2021. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.25.
  20. Joel D. Day, Maria Kosche, Florin Manea, and Markus L. Schmid. Subsequences with gap constraints: Complexity bounds for matching and analysis problems. In Sang Won Bae and Heejin Park, editors, 33rd International Symposium on Algorithms and Computation, ISAAC 2022, December 19-21, 2022, Seoul, Korea, volume 248 of LIPIcs, pages 64:1-64:18. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.ISAAC.2022.64.
  21. Lukas Fleischer and Manfred Kufleitner. Testing Simon’s congruence. In Igor Potapov, Paul Spirakis, and James Worrell, editors, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018), volume 117 of LIPIcs, pages 62:1-62:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. URL: https://doi.org/10.4230/LIPIcs.MFCS.2018.62.
  22. Pamela Fleischmann, Sungmin Kim, Tore Koß, Florin Manea, Dirk Nowotka, Stefan Siemer, and Max Wiedenhöft. Matching patterns with variables under simon’s congruence. In Olivier Bournez, Enrico Formenti, and Igor Potapov, editors, Reachability Problems - 17th International Conference, RP 2023, Nice, France, October 11-13, 2023, Proceedings, volume 14235 of Lecture Notes in Computer Science, pages 155-170. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-45286-4_12.
  23. Michael L. Fredman and Dan E. Willard. BLASTING through the information theoretic barrier with FUSION TREES. In Harriet Ortiz, editor, Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, May 13-17, 1990, Baltimore, Maryland, USA, pages 1-7. ACM, 1990. URL: https://doi.org/10.1145/100216.100217.
  24. Dominik D. Freydenberger, Pawel Gawrychowski, Juhani Karhumäki, Florin Manea, and Wojciech Rytter. Testing k-binomial equivalence. https://arxiv.org/abs/1509.00622, pages 239-248, 2015. Google Scholar
  25. André Frochaux and Sarah Kleest-Meißner. Puzzling over subsequence-query extensions: Disjunction and generalised gaps. In Benny Kimelfeld, Maria Vanina Martinez, and Renzo Angles, editors, Proceedings of the 15th Alberto Mendelzon International Workshop on Foundations of Data Management (AMW 2023), Santiago de Chile, Chile, May 22-26, 2023, volume 3409 of CEUR Workshop Proceedings. CEUR-WS.org, 2023. URL: https://ceur-ws.org/Vol-3409/paper3.pdf.
  26. Emmanuelle Garel. Minimal separators of two words. In Alberto Apostolico, Maxime Crochemore, Zvi Galil, and Udi Manber, editors, Combinatorial Pattern Matching, 4th Annual Symposium, CPM 93, Padova, Italy, June 2-4, 1993, Proceedings, volume 684 of LNCS, pages 35-53. Springer, 1993. URL: https://doi.org/10.1007/BFB0029795.
  27. Michael R. Garey and David S. Johnson. Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., USA, 1990. Google Scholar
  28. Pawel Gawrychowski, Maria Kosche, Tore Koß, Florin Manea, and Stefan Siemer. Efficiently testing Simon’s congruence. In 38th International Symposium on Theoretical Aspects of Computer Science, STACS 2021, March 16-19, 2021, Saarbrücken, Germany (Virtual Conference), pages 34:1-34:18, 2021. URL: https://doi.org/10.4230/LIPIcs.STACS.2021.34.
  29. Nikos Giatrakos, Elias Alevizos, Alexander Artikis, Antonios Deligiannakis, and Minos N. Garofalakis. Complex event recognition in the big data era: a survey. The VLDB Journal, 29(1):313-352, 2020. URL: https://doi.org/10.1007/s00778-019-00557-w.
  30. Hermann Gruber, Markus Holzer, and Martin Kutrib. More on the size of Higman-Haines sets: Effective constructions. Fundamenta Informaticae, 91(1):105-121, 2009. URL: https://doi.org/10.3233/FI-2009-0035.
  31. Simon Halfon, Philippe Schnoebelen, and Georg Zetzsche. Decidability, complexity, and expressiveness of first-order logic over the subword ordering. In Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '17, pages 1-12, 2017. URL: https://doi.org/10.1109/LICS.2017.8005141.
  32. Jean-Jacques Hébrard. An algorithm for distinguishing efficiently bit-strings by their subsequences. Theoretical Computer Science, 82:35-49, 1991. Google Scholar
  33. Graham Higman. Ordering by divisibility in abstract algebras. Proceedings of the London Mathematical Society, 3(1):326-336, 1952. Google Scholar
  34. Daniel S. Hirschberg. Algorithms for the longest common subsequence problem. Journal of the ACM, 24(4):664-675, 1977. URL: https://doi.org/10.1145/322033.322044.
  35. J. E. Hopcroft and J. D. Ullman. Introduction to Automata Theory, Languages and Computation. Addison-Wesley, 1979. Google Scholar
  36. James W. Hunt and Thomas G. Szymanski. A fast algorithm for computing longest subsequences. Communications of the ACM, 20(5):350-353, 1977. URL: https://doi.org/10.1145/359581.359603.
  37. P. Karandikar, M. Kufleitner, and P. Schnoebelen. On the index of Simon’s congruence for piecewise testability. Information Processing Letters, 115(4):515-519, 2015. URL: https://doi.org/10.1016/J.IPL.2014.11.008.
  38. Prateek Karandikar and Philippe Schnoebelen. The height of piecewise-testable languages and the complexity of the logic of subwords. Logical Methods in Computer Science, 15(2), 2019. URL: https://doi.org/10.23638/LMCS-15(2:6)2019.
  39. Sarah Kleest-Meißner, Rebecca Sattler, Markus L. Schmid, Nicole Schweikardt, and Matthias Weidlich. Discovering event queries from traces: Laying foundations for subsequence-queries with wildcards and gap-size constraints. In Dan Olteanu and Nils Vortmeier, editors, 25th International Conference on Database Theory, ICDT 2022, March 29 to April 1, 2022, Edinburgh, UK (Virtual Conference), volume 220 of LIPIcs, pages 18:1-18:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.ICDT.2022.18.
  40. Sarah Kleest-Meißner, Rebecca Sattler, Markus L. Schmid, Nicole Schweikardt, and Matthias Weidlich. Discovering multi-dimensional subsequence queries from traces - from theory to practice. In Birgitta König-Ries, Stefanie Scherzinger, Wolfgang Lehner, and Gottfried Vossen, editors, Datenbanksysteme für Business, Technologie und Web (BTW 2023), 20. Fachtagung des GI-Fachbereichs ,,Datenbanken und Informationssysteme" (DBIS), 06.-10, März 2023, Dresden, Germany, Proceedings, volume P-331 of LNI, pages 511-533. Gesellschaft für Informatik e.V., 2023. URL: https://doi.org/10.18420/BTW2023-24.
  41. Maria Kosche, Tore Koß, Florin Manea, and Viktoriya Pak. Subsequences in bounded ranges: Matching and analysis problems. In Anthony W. Lin, Georg Zetzsche, and Igor Potapov, editors, Reachability Problems - 16th International Conference, RP 2022, Kaiserslautern, Germany, October 17-21, 2022, Proceedings, volume 13608 of Lecture Notes in Computer Science, pages 140-159. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-19135-0_10.
  42. Maria Kosche, Tore Koß, Florin Manea, and Stefan Siemer. Absent subsequences in words. In Paul C. Bell, Patrick Totzke, and Igor Potapov, editors, Reachability Problems - 15th International Conference, RP 2021, Liverpool, UK, October 25-27, 2021, Proceedings, volume 13035 of LNCS, pages 115-131, 2021. URL: https://doi.org/10.1007/978-3-030-89716-1_8.
  43. Maria Kosche, Tore Koß, Florin Manea, and Stefan Siemer. Absent subsequences in words. Fundam. Informaticae, 189(3-4):199-240, 2022. URL: https://doi.org/10.3233/FI-222159.
  44. Maria Kosche, Tore Koß, Florin Manea, and Stefan Siemer. Combinatorial algorithms for subsequence matching: A survey. In Henning Bordihn, Géza Horváth, and György Vaszil, editors, Proceedings 12th International Workshop on Non-Classical Models of Automata and Applications, NCMA 2022, Debrecen, Hungary, August 26-27, 2022, volume 367 of EPTCS, pages 11-27, , 2022. URL: https://doi.org/10.4204/EPTCS.367.2.
  45. Dietrich Kuske. The subtrace order and counting first-order logic. In Henning Fernau, editor, Computer Science - Theory and Applications, volume 12159 of LNCS, pages 289-302, 2020. URL: https://doi.org/10.1007/978-3-030-50026-9_21.
  46. Dietrich Kuske and Georg Zetzsche. Languages ordered by the subword order. In Mikołaj Bojańczyk and Alex Simpson, editors, Foundations of Software Science and Computation Structures, LNCS, pages 348-364, 2019. URL: https://doi.org/10.1007/978-3-030-17127-8_20.
  47. Martin Lange and Hans Leiß. To CNF or not to CNF? an efficient yet presentable version of the CYK algorithm. Informatica Didact., 8, 2009. URL: http://ddi.cs.uni-potsdam.de/InformaticaDidactica/LangeLeiss2009.
  48. Marie Lejeune, Julien Leroy, and Michel Rigo. Computing the k-binomial complexity of the Thue-Morse word. In Piotrek Hofman and Michał Skrzypczak, editors, Developments in Language Theory, LNCS, pages 278-291, 2019. URL: https://doi.org/10.1007/978-3-030-24886-4_21.
  49. Julien Leroy, Michel Rigo, and Manon Stipulanti. Generalized Pascal triangle for binomial coefficients of words. The Electronic Journal of Combinatorics, 24(1.44):36 pp., 2017. Google Scholar
  50. Chun Li and Jianyong Wang. Efficiently Mining Closed Subsequences with Gap Constraints, pages 313-322. SIAM, 2008. URL: https://doi.org/10.1137/1.9781611972788.28.
  51. Chun Li, Qingyan Yang, Jianyong Wang, and Ming Li. Efficient mining of gap-constrained subsequences and its various applications. ACMTransactions on Knowledge Discovery from Data, 6(1):2:1-2:39, 2012. URL: https://doi.org/10.1145/2133360.2133362.
  52. Markus Lohrey. Algorithmics on SLP-compressed strings: A survey. Groups Complex. Cryptol., 4(2):241-299, 2012. URL: https://doi.org/10.1515/GCC-2012-0016.
  53. Daniel Lokshtanov, Dániel Marx, and Saket Saurabh. Lower bounds based on the exponential time hypothesis. Bull. EATCS, 105:41-72, 2011. URL: http://eatcs.org/beatcs/index.php/beatcs/article/view/92.
  54. David Maier. The complexity of some problems on subsequences and supersequences. Journal of the ACM, 25(2):322-336, April 1978. URL: https://doi.org/10.1145/322063.322075.
  55. Florin Manea, Jonas Richardsen, and Markus L. Schmid. Subsequences with generalised gap constraints: Upper and lower complexity bounds. In Shunsuke Inenaga and Simon J. Puglisi, editors, 35th Annual Symposium on Combinatorial Pattern Matching, CPM 2024, June 25-27, 2024, Fukuoka, Japan, volume 296 of LIPIcs, pages 22:1-22:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPICS.CPM.2024.22.
  56. William J. Masek and Mike Paterson. A faster algorithm computing string edit distances. Journal of Computer and System Sciences, 20(1):18-31, 1980. URL: https://doi.org/10.1016/0022-0000(80)90002-1.
  57. Alexandru Mateescu, Arto Salomaa, and Sheng Yu. Subword histories and Parikh matrices. Journal of Computer and System Sciences, 68(1):1-21, 2004. URL: https://doi.org/10.1016/J.JCSS.2003.04.001.
  58. Alexander Meduna and Petr Zemek. Jumping finite automata. International Journal of Foundations of Computer Science, 23(7):1555-1578, 2012. URL: https://doi.org/10.1142/S0129054112500244.
  59. Victor Mitrana, Andrei Păun, Mihaela Păun, and José-Ramón Sánchez-Couso. Jump complexity of finite automata with translucent letters. Theoretical Computer Science, 992:114450, 2024. URL: https://doi.org/10.1016/J.TCS.2024.114450.
  60. Benedek Nagy and Friedrich Otto. Finite-state acceptors with translucent letters. In A.O. De La Puente G. Bel-Enguix, V. Dahl, editor, BILC 2011: AI Methods for Interdisciplinary Research in Language and Biology, pages 3-13, 2011. Google Scholar
  61. Benedek Nagy and Friedrich Otto. Globally deterministic CD-systems of stateless R-automata with window size 1. International Journal of Computer Mathematics, 90(6):1254-1277, 2013. URL: https://doi.org/10.1080/00207160.2012.688820.
  62. Narao Nakatsu, Yahiko Kambayashi, and Shuzo Yajima. A longest common subsequence algorithm suitable for similar text strings. Acta Informatica, 18:171-179, 1982. URL: https://doi.org/10.1007/BF00264437.
  63. Friedrich Otto. A survey on automata with translucent letters. In Benedek Nagy, editor, Implementation and Application of Automata, pages 21-50, Cham, 2023. URL: https://doi.org/10.1007/978-3-031-40247-0_2.
  64. Rohit Parikh. On context-free languages. Journal of the ACM, 13(4):570-581, 1966. URL: https://doi.org/10.1145/321356.321364.
  65. William E. Riddle. An approach to software system modelling and analysis. Computer Languages, 4(1):49-66, 1979. URL: https://doi.org/10.1016/0096-0551(79)90009-2.
  66. Michel Rigo and Pavel Salimov. Another generalization of abelian equivalence: Binomial complexity of infinite words. Theoretical Computer Science, 601:47-57, 2015. URL: https://doi.org/10.1016/J.TCS.2015.07.025.
  67. Arto Salomaa. Connections between subwords and certain matrix mappings. Theoretical Computer Science, 340(2):188-203, 2005. URL: https://doi.org/10.1016/J.TCS.2005.03.024.
  68. Philippe Schnoebelen and Julien Veron. On arch factorization and subword universality for words and compressed words. In Anna Frid and Robert Mercaş, editors, Combinatorics on Words, volume 13899, pages 274-287, 2023. URL: https://doi.org/10.1007/978-3-031-33180-0_21.
  69. Shinnosuke Seki. Absoluteness of subword inequality is undecidable. Theoretical Computer Science, 418:116-120, 2012. URL: https://doi.org/10.1016/J.TCS.2011.10.017.
  70. Alan C. Shaw. Software descriptions with flow expressions. IEEE Transactions on Software Engineering, 4(3):242-254, 1978. URL: https://doi.org/10.1109/TSE.1978.231501.
  71. Imre Simon. Hierarchies of events with dot-depth one - Ph.D. thesis. University of Waterloo, 1972. Google Scholar
  72. Imre Simon. Piecewise testable events. In H. Brakhage, editor, Automata Theory and Formal Languages, LNCS, pages 214-222, Berlin, Heidelberg, 1975. Springer Berlin Heidelberg. URL: https://doi.org/10.1007/3-540-07407-4_23.
  73. Imre Simon. Words distinguished by their subwords (extended abstract). In Proc. WORDS 2003, volume 27 of TUCS General Publication, pages 6-13, 2003. Google Scholar
  74. Zdeněk Troniček. Common subsequence automaton. In Jean-Marc Champarnaud and Denis Maurel, editors, Conference on Implementation and Application of Automata, volume 2608 of LNCS, pages 270-275, 2002. URL: https://doi.org/10.1007/3-540-44977-9_28.
  75. Jan van Leeuwen. Effective constructions in well-partially- ordered free monoids. Discrete Mathematics, 21(3):237-252, 1978. URL: https://doi.org/10.1016/0012-365X(78)90156-5.
  76. Georg Zetzsche. The Complexity of Downward Closure Comparisons. In Ioannis Chatzigiannakis, Michael Mitzenmacher, Yuval Rabani, and Davide Sangiorgi, editors, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016), volume 55 of LIPIcs, pages 123:1-123:14, 2016. URL: https://doi.org/10.4230/LIPIcs.ICALP.2016.123.
  77. Georg Zetzsche. Separability by piecewise testable languages and downward closures beyond subwords. In Anuj Dawar and Erich Grädel, editors, Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2018, Oxford, UK, July 09-12, 2018, pages 929-938. ACM, 2018. URL: https://doi.org/10.1145/3209108.3209201.
  78. Haopeng Zhang, Yanlei Diao, and Neil Immerman. On complexity and optimization of expensive queries in complex event processing. In International Conference on Management of Data, SIGMOD 2014, Snowbird, UT, USA, June 22-27, 2014, pages 217-228, 2014. URL: https://doi.org/10.1145/2588555.2593671.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact