Document Open Access Logo

Lower Bounds for Graph-Walking Automata

Authors Olga Martynova , Alexander Okhotin



PDF
Thumbnail PDF

File

LIPIcs.STACS.2021.52.pdf
  • Filesize: 0.94 MB
  • 13 pages

Document Identifiers

Author Details

Olga Martynova
  • Department of Mathematics and Computer Science, St. Petersburg State University, Russia
Alexander Okhotin
  • Department of Mathematics and Computer Science, St. Petersburg State University, Russia

Cite AsGet BibTex

Olga Martynova and Alexander Okhotin. Lower Bounds for Graph-Walking Automata. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 52:1-52:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.STACS.2021.52

Abstract

Graph-walking automata (GWA) traverse graphs by moving between the nodes following the edges, using a finite-state control to decide where to go next. It is known that every GWA can be transformed to a GWA that halts on every input, to a GWA returning to the initial node in order to accept, as well as to a reversible GWA. This paper establishes lower bounds on the state blow-up of these transformations: it is shown that making an n-state GWA traversing k-ary graphs return to the initial node requires at least 2(n-1)(k-3) states in the worst case; the same lower bound holds for the transformation to halting automata. Automata satisfying both properties at once must have at least 4(n-1)(k-3) states. A reversible automaton must have at least 4(n-1)(k-3)-1 states. These bounds are asymptotically tight to the upper bounds proved using the methods from the literature.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Models of computation
Keywords
  • Finite automata
  • graph-walking automata
  • halting
  • reversibility

Metrics

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

References

  1. Mikolaj Bojanczyk and Thomas Colcombet. Tree-walking automata cannot be determinized. Theor. Comput. Sci., 350(2-3):164-173, 2006. URL: https://doi.org/10.1016/j.tcs.2005.10.031.
  2. Lothar Budach. Automata and labyrinths. Mathematische Nachrichten, 86(1):195-282, 1978. URL: https://doi.org/10.1002/mana.19780860120.
  3. Yann Disser, Jan Hackfeld, and Max Klimm. Undirected graph exploration with ⊝(log log n) pebbles. In Robert Krauthgamer, editor, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 25-39. SIAM, 2016. URL: https://doi.org/10.1137/1.9781611974331.ch3.
  4. Amr Elmasry, Torben Hagerup, and Frank Kammer. Space-efficient basic graph algorithms. 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 288-301. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL: https://doi.org/10.4230/LIPIcs.STACS.2015.288.
  5. Pierre Fraigniaud, David Ilcinkas, Guy Peer, Andrzej Pelc, and David Peleg. Graph exploration by a finite automaton. Theor. Comput. Sci., 345(2-3):331-344, 2005. URL: https://doi.org/10.1016/j.tcs.2005.07.014.
  6. Viliam Geffert, Carlo Mereghetti, and Giovanni Pighizzini. Complementing two-way finite automata. Inf. Comput., 205(8):1173-1187, 2007. URL: https://doi.org/10.1016/j.ic.2007.01.008.
  7. Attila Kondacs and John Watrous. On the power of quantum finite state automata. In 38th Annual Symposium on Foundations of Computer Science, FOCS '97, Miami Beach, Florida, USA, October 19-22, 1997, pages 66-75. IEEE Computer Society, 1997. URL: https://doi.org/10.1109/SFCS.1997.646094.
  8. Michal Kunc and Alexander Okhotin. Reversible two-way finite automata over a unary alphabet. Technical Report 1024, Turku Centre for Computer Science, 2011. Google Scholar
  9. Michal Kunc and Alexander Okhotin. Reversibility of computations in graph-walking automata. Inf. Comput., 275:104631, 2020. URL: https://doi.org/10.1016/j.ic.2020.104631.
  10. Rolf Landauer. Irreversibility and heat generation in the computing process. IBM J. Res. Dev., 5(3):183-191, 1961. URL: https://doi.org/10.1147/rd.53.0183.
  11. Klaus-Jörn Lange, Pierre McKenzie, and Alain Tapp. Reversible space equals deterministic space. J. Comput. Syst. Sci., 60(2):354-367, 2000. URL: https://doi.org/10.1006/jcss.1999.1672.
  12. Kenichi Morita. A deterministic two-way multi-head finite automaton can be converted into a reversible one with the same number of heads. In Robert Glück and Tetsuo Yokoyama, editors, Reversible Computation, 4th International Workshop, RC 2012, Copenhagen, Denmark, July 2-3, 2012. Revised Papers, volume 7581 of Lecture Notes in Computer Science, pages 29-43. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-36315-3_3.
  13. Anca Muscholl, Mathias Samuelides, and Luc Segoufin. Complementing deterministic tree-walking automata. Inf. Process. Lett., 99(1):33-39, 2006. URL: https://doi.org/10.1016/j.ipl.2005.09.017.
  14. Alexander Okhotin. Graph-walking automata: From whence they come, and whither they are bound. In Michal Hospodár and Galina Jirásková, editors, Implementation and Application of Automata - 24th International Conference, CIAA 2019, Košice, Slovakia, July 22-25, 2019, Proceedings, volume 11601 of Lecture Notes in Computer Science, pages 10-29. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-23679-3_2.
  15. Michael Sipser. Lower bounds on the size of sweeping automata. J. Comput. Syst. Sci., 21(2):195-202, 1980. URL: https://doi.org/10.1016/0022-0000(80)90034-3.
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