Document Open Access Logo

The AC⁰-Complexity of Visibly Pushdown Languages

Authors Stefan Göller, Nathan Grosshans

PDF
Thumbnail PDF

File

LIPIcs.STACS.2024.38.pdf
  • Filesize: 0.87 MB
  • 18 pages

Document Identifiers

Author Details

Stefan Göller
  • School of Electrical Engineering and Computer Science, Universität Kassel, Germany
Nathan Grosshans
  • Independent Scholar, Paris Region, France

Funding

  • Göller, Stefan: The author was supported by the Agence Nationale de la Recherche grant no. ANR-17-CE40-0010.

Cite As Get BibTex

Stefan Göller and Nathan Grosshans. The AC⁰-Complexity of Visibly Pushdown Languages. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 38:1-38:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.STACS.2024.38

Abstract

We study the question of which visibly pushdown languages (VPLs) are in the complexity class AC⁰ and how to effectively decide this question. Our contribution is to introduce a particular subclass of one-turn VPLs, called intermediate VPLs, for which the raised question is entirely unclear: to the best of our knowledge our research community is unaware of containment or non-containment in AC⁰ for any language in our newly introduced class. Our main result states that there is an algorithm that, given a visibly pushdown automaton, correctly outputs exactly one of the following: that its language L is in AC⁰, some m ≥ 2 such that MODₘ (the words over {0,1} having a number of 1’s divisible by m) is constant-depth reducible to L (implying that L is not in AC⁰), or a finite disjoint union of intermediate VPLs that L is constant-depth equivalent to. In the latter of the three cases one can moreover effectively compute k,l ∈ ℕ_{> 0} with k≠l such that the concrete intermediate VPL L(S → ε ∣ ac^{k-1}Sb₁ ∣ ac^{l-1}Sb₂) is constant-depth reducible to the language L. Due to their particular nature we conjecture that either all intermediate VPLs are in AC⁰ or all are not. As a corollary of our main result we obtain that in case the input language is a visibly counter language our algorithm can effectively determine if it is in AC⁰ - hence our main result generalizes a result by Krebs et al. stating that it is decidable if a given visibly counter language is in AC⁰ (when restricted to well-matched words).
For our proofs we revisit so-called Ext-algebras (introduced by Czarnetzki et al.), which are closely related to forest algebras (introduced by Bojańczyk and Walukiewicz), and use Green’s relations.

Subject Classification

ACM Subject Classification
  • Theory of computation → Grammars and context-free languages
  • Theory of computation → Circuit complexity
Keywords
  • Visibly pushdown languages
  • Circuit Complexity
  • AC0

Metrics

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

Related Versions

References

  1. Rajeev Alur, Viraj Kumar, P. Madhusudan, and Mahesh Viswanathan. Congruences for visibly pushdown languages. In Luís Caires, Giuseppe F. Italiano, Luís Monteiro, Catuscia Palamidessi, and Moti Yung, editors, Automata, Languages and Programming, 32nd International Colloquium, ICALP 2005, Lisbon, Portugal, July 11-15, 2005, Proceedings, volume 3580 of Lecture Notes in Computer Science, pages 1102-1114. Springer, 2005. URL: https://doi.org/10.1007/11523468_89.
  2. Rajeev Alur and P. Madhusudan. Visibly pushdown languages. In László Babai, editor, Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, USA, June 13-16, 2004, pages 202-211. ACM, 2004. URL: https://doi.org/10.1145/1007352.1007390.
  3. Vince Bárány, Christof Löding, and Olivier Serre. Regularity problems for visibly pushdown languages. In Bruno Durand and Wolfgang Thomas, editors, STACS 2006, 23rd Annual Symposium on Theoretical Aspects of Computer Science, Marseille, France, February 23-25, 2006, Proceedings, volume 3884 of Lecture Notes in Computer Science, pages 420-431. Springer, 2006. URL: https://doi.org/10.1007/11672142_34.
  4. David A. Mix Barrington. Bounded-Width Polynomial-Size Branching Programs Recognize Exactly Those Languages in NC¹. J. Comput. Syst. Sci., 38(1):150-164, 1989. URL: https://doi.org/10.1016/0022-0000(89)90037-8.
  5. David A. Mix Barrington, Kevin J. Compton, Howard Straubing, and Denis Thérien. Regular languages in nc¹. J. Comput. Syst. Sci., 44(3):478-499, 1992. URL: https://doi.org/10.1016/0022-0000(92)90014-A.
  6. Richard Beigel. The polynomial method in circuit complexity. In Proceedings of the Eigth Annual Structure in Complexity Theory Conference, San Diego, CA, USA, May 18-21, 1993, pages 82-95. IEEE Computer Society, 1993. URL: https://doi.org/10.1109/SCT.1993.336538.
  7. Mikolaj Bojanczyk and Igor Walukiewicz. Forest algebras. In Jörg Flum, Erich Grädel, and Thomas Wilke, editors, Logic and Automata: History and Perspectives [in Honor of Wolfgang Thomas], volume 2 of Texts in Logic and Games, pages 107-132. Amsterdam University Press, 2008. Google Scholar
  8. Ashok K. Chandra, Larry J. Stockmeyer, and Uzi Vishkin. Constant depth reducibility. SIAM J. Comput., 13(2):423-439, 1984. URL: https://doi.org/10.1137/0213028.
  9. Silke Czarnetzki, Andreas Krebs, and Klaus-Jörn Lange. Visibly pushdown languages and free profinite algebras. CoRR, abs/1810.12731, 2018. URL: https://arxiv.org/abs/1810.12731.
  10. Patrick W. Dymond. Input-driven languages are in log n depth. Inf. Process. Lett., 26(5):247-250, 1988. URL: https://doi.org/10.1016/0020-0190(88)90148-2.
  11. Javier Esparza, Michael Luttenberger, and Maximilian Schlund. A brief history of strahler numbers. In Adrian-Horia Dediu, Carlos Martín-Vide, José Luis Sierra-Rodríguez, and Bianca Truthe, editors, Language and Automata Theory and Applications - 8th International Conference, LATA 2014, Madrid, Spain, March 10-14, 2014. Proceedings, volume 8370 of Lecture Notes in Computer Science, pages 1-13. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-04921-2_1.
  12. Stefan Göller and Nathan Grosshans. The ac^0-complexity of visibly pushdown languages. CoRR, abs/2302.13116, 2023. URL: https://doi.org/10.48550/ARXIV.2302.13116.
  13. Yuri Gurevich and Harry R. Lewis. A logic for constant-depth circuits. Inf. Control., 61(1):65-74, 1984. URL: https://doi.org/10.1016/S0019-9958(84)80062-5.
  14. Michael A. Harrison. Introduction to Formal Language Theory. Addison-Wesley, 1978. Google Scholar
  15. Johan Håstad. Almost optimal lower bounds for small depth circuits. In Juris Hartmanis, editor, Proceedings of the 18th Annual ACM Symposium on Theory of Computing, May 28-30, 1986, Berkeley, California, USA, pages 6-20. ACM, 1986. URL: https://doi.org/10.1145/12130.12132.
  16. Neil Immerman. Languages that capture complexity classes. SIAM J. Comput., 16(4):760-778, 1987. URL: https://doi.org/10.1137/0216051.
  17. Neil Immerman. Descriptive complexity. Graduate texts in computer science. Springer, 1999. URL: https://doi.org/10.1007/978-1-4612-0539-5.
  18. Stasys Jukna. Boolean Function Complexity - Advances and Frontiers, volume 27 of Algorithms and combinatorics. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-24508-4.
  19. Andreas Krebs, Klaus-Jörn Lange, and Michael Ludwig. Visibly counter languages and constant depth circuits. In Ernst W. Mayr and Nicolas Ollinger, editors, 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, March 4-7, 2015, Garching, Germany, volume 30 of LIPIcs, pages 594-607. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.STACS.2015.594.
  20. Michael Ludwig. Tree-Structured Problems and Parallel Computation. PhD thesis, University of Tübingen, Germany, 2019. URL: https://publikationen.uni-tuebingen.de/xmlui/handle/10900/85960/.
  21. Kurt Mehlhorn. Pebbling Moutain Ranges and its Application of DCFL-Recognition. In J. W. de Bakker and Jan van Leeuwen, editors, Automata, Languages and Programming, 7th Colloquium, Noordweijkerhout, The Netherlands, July 14-18, 1980, Proceedings, volume 85 of Lecture Notes in Computer Science, pages 422-435. Springer, 1980. URL: https://doi.org/10.1007/3-540-10003-2_89.
  22. Marcel Paul Schützenberger. On finite monoids having only trivial subgroups. Inf. Control., 8(2):190-194, 1965. URL: https://doi.org/10.1016/S0019-9958(65)90108-7.
  23. Howard Straubing. Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser Boston, 1994. URL: https://doi.org/10.1007/978-1-4612-0289-9.
  24. Heribert Vollmer. Introduction to Circuit Complexity - A Uniform Approach. Texts in Theoretical Computer Science. An EATCS Series. Springer, 1999. URL: https://doi.org/10.1007/978-3-662-03927-4.
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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact