Document Open Access Logo

The Complexity of the Diagonal Problem for Recursion Schemes

Author Pawel Parys



PDF
Thumbnail PDF

File

LIPIcs.FSTTCS.2017.45.pdf
  • Filesize: 0.52 MB
  • 14 pages

Document Identifiers

Author Details

Pawel Parys

Cite AsGet BibTex

Pawel Parys. The Complexity of the Diagonal Problem for Recursion Schemes. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 45:1-45:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.FSTTCS.2017.45

Abstract

We consider nondeterministic higher-order recursion schemes as recognizers of languages of finite words or finite trees. We establish the complexity of the diagonal problem for schemes: given a set of letters A and a scheme G, is it the case that for every number n the scheme accepts a word (a tree) in which every letter from A appears at least n times. We prove that this problem is (m-1)-EXPTIME-complete for word-recognizing schemes of order m, and m-EXPTIME-complete for tree-recognizing schemes of order m.
Keywords
  • diagonal problem
  • higher-order recursion schemes
  • intersection types
  • downward closure

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Alfred V. Aho. Indexed grammars - an extension of context-free grammars. J. ACM, 15(4):647-671, 1968. URL: http://dx.doi.org/10.1145/321479.321488.
  2. Kazuyuki Asada and Naoki Kobayashi. Pumping lemma for higher-order languages. In Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, July 10-14, 2017, Warsaw, Poland, volume 80 of LIPIcs, pages 97:1-97:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2017.97.
  3. Luca Breveglieri, Alessandra Cherubini, Claudio Citrini, and Stefano Crespi-Reghizzi. Multi-push-down languages and grammars. Int. J. Found. Comput. Sci., 7(3):253-292, 1996. URL: http://dx.doi.org/10.1142/S0129054196000191.
  4. Christopher H. Broadbent and Naoki Kobayashi. Saturation-based model checking of higher-order recursion schemes. In Simona Ronchi Della Rocca, editor, Computer Science Logic 2013 (CSL 2013), CSL 2013, September 2-5, 2013, Torino, Italy, volume 23 of LIPIcs, pages 129-148. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2013. URL: http://dx.doi.org/10.4230/LIPIcs.CSL.2013.129.
  5. Lorenzo Clemente, Pawel Parys, Sylvain Salvati, and Igor Walukiewicz. Ordered tree-pushdown systems. In Prahladh Harsha and G. Ramalingam, editors, 35th IARCS Annual Conference on Foundation of Software Technology and Theoretical Computer Science, FSTTCS 2015, December 16-18, 2015, Bangalore, India, volume 45 of LIPIcs, pages 163-177. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.FSTTCS.2015.163.
  6. Lorenzo Clemente, Pawel Parys, Sylvain Salvati, and Igor Walukiewicz. The diagonal problem for higher-order recursion schemes is decidable. In Martin Grohe, Eric Koskinen, and Natarajan Shankar, editors, Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '16, New York, NY, USA, July 5-8, 2016, pages 96-105. ACM, 2016. URL: http://dx.doi.org/10.1145/2933575.2934527.
  7. Wojciech Czerwinski, Wim Martens, Lorijn van Rooijen, and Marc Zeitoun. A note on decidable separability by piecewise testable languages. In Adrian Kosowski and Igor Walukiewicz, editors, Fundamentals of Computation Theory - 20th International Symposium, FCT 2015, Gdańsk, Poland, August 17-19, 2015, Proceedings, volume 9210 of Lecture Notes in Computer Science, pages 173-185. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-22177-9_14.
  8. Werner Damm. The IO- and oi-hierarchies. Theor. Comput. Sci., 20:95-207, 1982. URL: http://dx.doi.org/10.1016/0304-3975(82)90009-3.
  9. Matthew Hague, Andrzej S. Murawski, C.-H. Luke Ong, and Olivier Serre. Collapsible pushdown automata and recursion schemes. In Proceedings of the Twenty-Third Annual IEEE Symposium on Logic in Computer Science, LICS 2008, 24-27 June 2008, Pittsburgh, PA, USA, pages 452-461. IEEE Computer Society, 2008. URL: http://dx.doi.org/10.1109/LICS.2008.34.
  10. Alexander Kartzow and Pawel Parys. Strictness of the collapsible pushdown hierarchy. In Branislav Rovan, Vladimiro Sassone, and Peter Widmayer, editors, Mathematical Foundations of Computer Science 2012 - 37th International Symposium, MFCS 2012, Bratislava, Slovakia, August 27-31, 2012. Proceedings, volume 7464 of Lecture Notes in Computer Science, pages 566-577. Springer, 2012. URL: http://dx.doi.org/10.1007/978-3-642-32589-2_50.
  11. Naoki Kobayashi. Types and higher-order recursion schemes for verification of higher-order programs. In Zhong Shao and Benjamin C. Pierce, editors, Proceedings of the 36th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL 2009, Savannah, GA, USA, January 21-23, 2009, pages 416-428. ACM, 2009. URL: http://dx.doi.org/10.1145/1480881.1480933.
  12. Naoki Kobayashi. Pumping by typing. In 28th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2013, New Orleans, LA, USA, June 25-28, 2013, pages 398-407. IEEE Computer Society, 2013. URL: http://dx.doi.org/10.1109/LICS.2013.46.
  13. Naoki Kobayashi, Kazuhiro Inaba, and Takeshi Tsukada. Unsafe order-2 tree languages are context-sensitive. In Anca Muscholl, editor, Foundations of Software Science and Computation Structures - 17th International Conference, FOSSACS 2014, Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2014, Grenoble, France, April 5-13, 2014, Proceedings, volume 8412 of Lecture Notes in Computer Science, pages 149-163. Springer, 2014. URL: http://dx.doi.org/10.1007/978-3-642-54830-7_10.
  14. Naoki Kobayashi and C.-H. Luke Ong. A type system equivalent to the modal mu-calculus model checking of higher-order recursion schemes. In Proceedings of the 24th Annual IEEE Symposium on Logic in Computer Science, LICS 2009, 11-14 August 2009, Los Angeles, CA, USA, pages 179-188. IEEE Computer Society, 2009. URL: http://dx.doi.org/10.1109/LICS.2009.29.
  15. Naoki Kobayashi and C.-H. Luke Ong. Complexity of model checking recursion schemes for fragments of the modal mu-calculus. Logical Methods in Computer Science, 7(4), 2011. URL: http://dx.doi.org/10.2168/LMCS-7(4:9)2011.
  16. Gregory M. Kobele and Sylvain Salvati. The IO and OI hierarchies revisited. Inf. Comput., 243:205-221, 2015. URL: http://dx.doi.org/10.1016/j.ic.2014.12.015.
  17. Joseph. B. Kruskal. Well-quasi-ordering, the tree theorem, and Vazsonyi’s conjecture. Transactions of the American Mathematical Society, 95(2):210-225, 1960. URL: http://dx.doi.org/10.2307/1993287.
  18. Pawel Parys. A characterization of lambda-terms transforming numerals. J. Funct. Program., 26:e12, 2016. URL: http://dx.doi.org/10.1017/S0956796816000113.
  19. Pawel Parys. Intersection types and counting. In Naoki Kobayashi, editor, Proceedings Eighth Workshop on Intersection Types and Related Systems, ITRS 2016, Porto, Portugal, 26th June 2016., volume 242 of EPTCS, pages 48-63, 2016. URL: http://dx.doi.org/10.4204/EPTCS.242.6.
  20. Paweł Parys. Homogeneity without loss of generality. Submitted, 2017. Google Scholar
  21. Steven J. Ramsay, Robin P. Neatherway, and C.-H. Luke Ong. A type-directed abstraction refinement approach to higher-order model checking. In Suresh Jagannathan and Peter Sewell, editors, The 41st Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, POPL '14, San Diego, CA, USA, January 20-21, 2014, pages 61-72. ACM, 2014. URL: http://dx.doi.org/10.1145/2535838.2535873.
  22. Sylvain Salvati and Igor Walukiewicz. Simply typed fixpoint calculus and collapsible pushdown automata. Mathematical Structures in Computer Science, 26(7):1304-1350, 2016. URL: http://dx.doi.org/10.1017/S0960129514000590.
  23. Georg Zetzsche. An approach to computing downward closures. In Magnús M. Halldórsson, Kazuo Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Programming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Proceedings, Part II, volume 9135 of Lecture Notes in Computer Science, pages 440-451. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-662-47666-6_35.
  24. 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, July 11-15, 2016, Rome, Italy, volume 55 of LIPIcs, pages 123:1-123:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.ICALP.2016.123.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail