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Graph-Walking Automata: From Whence They Come, and Whither They are Bound

  • Alexander OkhotinEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11601)

Abstract

Graph-walking automata are finite automata walking on graphs given as an input; tree-walking automata and two-way finite automata are their well-known special cases. Graph-walking automata can be regarded both as a model of navigation in an unknown environment, and as a generic computing device, with the graph as the model of its memory. This paper presents the known results on these automata, ranging from their limitations in traversing graphs, studied already in the 1970s, to the recent work on the logical reversibility of their computations.

References

  1. 1.
    Aho, A.V., Ullman, J.D.: Translations on a context free grammar. Inf. Control 19(5), 439–475 (1971)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Albers, S., Henzinger, M.R.: Exploring unknown environments. SIAM J. Comput. 29(4), 1164–1188 (2000).  https://doi.org/10.1137/S009753979732428XMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aleliunas, R., Karp, R.M., Lipton, R.J., Lovász, L., Rackoff, C.: Random walks, universal traversal sequences, and the complexity of maze problems. In: Proceedings of 20th Annual Symposium on Foundations of Computer Science, FOCS 1979, pp. 218–223. IEEE Computer Society (1979).  https://doi.org/10.1109/SFCS.1979.34
  4. 4.
    Asano, T., et al.: Depth-first search using \(O(n)\) bits. In: Ahn, H.-K., Shin, C.-S. (eds.) ISAAC 2014. LNCS, vol. 8889, pp. 553–564. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-13075-0_44CrossRefGoogle Scholar
  5. 5.
    Bennett, C.H.: The thermodynamics of computation–a review. Int. J. Theor. Phys. 21(12), 905–940 (1982).  https://doi.org/10.1007/BF02084158CrossRefGoogle Scholar
  6. 6.
    Blum, M., Kozen, D.: On the power of the compass (or, why mazes are easier to search than graphs). In: Proceedings of 19th Annual Symposium on Foundations of Computer Science, FOCS 1978, pp. 132–142. IEEE Computer Society (1978).  https://doi.org/10.1109/SFCS.1978.30
  7. 7.
    Bojańczyk, M., Colcombet, T.: Tree-walking automata cannot be determinized. Theor. Comput. Sci. 350(2–3), 164–173 (2006).  https://doi.org/10.1016/j.tcs.2005.10.031MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bojańczyk, M., Colcombet, T.: Tree-walking automata do not recognize all regular languages. SIAM J. Comput. 38(2), 658–701 (2008).  https://doi.org/10.1137/050645427MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Budach, L.: Automata and labyrinths. Math. Nachr. 86(1), 195–282 (1978).  https://doi.org/10.1002/mana.19780860120MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Disser, Y., Hackfeld, J., Klimm, M.: Undirected graph exploration with \(O(\log \log n)\) pebbles. In: Krauthgamer, R. (ed.) Proceedings of 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, pp. 25–39. SIAM (2016).  https://doi.org/10.1137/1.9781611974331.ch3
  11. 11.
    Elmasry, A., Hagerup, T., Kammer, F.: Space-efficient basic graph algorithms. In: Mayr, E.W., Ollinger, N. (eds.) Proceedings of 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015. LIPIcs, vol. 30, pp. 288–301. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015).  https://doi.org/10.4230/LIPIcs.STACS.2015.288
  12. 12.
    Fraigniaud, P., Ilcinkas, D., Peer, G., Pelc, A., Peleg, D.: Graph exploration by a finite automaton. Theoret. Comput. Sci. 345(2–3), 331–344 (2005).  https://doi.org/10.1016/j.tcs.2005.07.014MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Geffert, V., Mereghetti, C., Pighizzini, G.: Converting two-way nondeterministic unary automata into simpler automata. Theoret. Comput. Sci. 295, 189–203 (2003).  https://doi.org/10.1016/S0304-3975(02)00403-6MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Geffert, V., Mereghetti, C., Pighizzini, G.: Complementing two-way finite automata. Inf. Comput. 205(8), 1173–1187 (2007).  https://doi.org/10.1016/j.ic.2007.01.008MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kapoutsis, C.: Removing bidirectionality from nondeterministic finite automata. In: Jȩdrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 544–555. Springer, Heidelberg (2005).  https://doi.org/10.1007/11549345_47CrossRefGoogle Scholar
  16. 16.
    Kapoutsis, C.A.: Two-way automata versus logarithmic space. Theory Comput. Syst. 55(2), 421–447 (2014).  https://doi.org/10.1007/s00224-013-9465-0MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kondacs, A., Watrous, J.: On the power of quantum finite state automata. In: Proceedings of 38th Annual Symposium on Foundations of Computer Science, FOCS 1997, pp. 66–75. IEEE Computer Society (1997).  https://doi.org/10.1109/SFCS.1997.646094
  18. 18.
    Kunc, M., Okhotin, A.: Describing periodicity in two-way deterministic finite automata using transformation semigroups. In: Mauri, G., Leporati, A. (eds.) DLT 2011. LNCS, vol. 6795, pp. 324–336. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22321-1_28CrossRefzbMATHGoogle Scholar
  19. 19.
    Kunc, M., Okhotin, A.: Reversibility of computations in graph-walking automata. In: Chatterjee, K., Sgall, J. (eds.) MFCS 2013. LNCS, vol. 8087, pp. 595–606. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40313-2_53CrossRefzbMATHGoogle Scholar
  20. 20.
    Landauer, R.: Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5(3), 183–191 (1961).  https://doi.org/10.1147/rd.53.0183MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Lange, K., McKenzie, P., Tapp, A.: Reversible space equals deterministic space. J. Comput. Syst. Sci. 60(2), 354–367 (2000).  https://doi.org/10.1006/jcss.1999.1672MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Martynova, O.: Personal Communication, April 2019Google Scholar
  23. 23.
    Morita, K.: A deterministic two-way multi-head finite automaton can be converted into a reversible one with the same number of heads. In: Glück, R., Yokoyama, T. (eds.) RC 2012. LNCS, vol. 7581, pp. 29–43. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-36315-3_3CrossRefzbMATHGoogle Scholar
  24. 24.
    Muscholl, A., Samuelides, M., Segoufin, L.: Complementing deterministic tree-walking automata. Inf. Process. Lett. 99(1), 33–39 (2006).  https://doi.org/10.1016/j.ipl.2005.09.017MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Panaite, P., Pelc, A.: Exploring unknown undirected graphs. J. Algorithms 33(2), 281–295 (1999).  https://doi.org/10.1006/jagm.1999.1043MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Rabin, M.O., Scott, D.S.: Finite automata and their decision problems. IBM J. Res. Dev. 3(2), 114–125 (1959).  https://doi.org/10.1147/rd.32.0114MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rollik, H.: Automaten in planaren graphen. Acta Inform. 13, 287–298 (1980).  https://doi.org/10.1007/BF00288647MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sipser, M.: Halting space-bounded computations. Theoret. Comput. Sci. 10, 335–338 (1980).  https://doi.org/10.1016/0304-3975(80)90053-5MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Thomas, W.: On logics, tilings, and automata. In: Albert, J.L., Monien, B., Artalejo, M.R. (eds.) ICALP 1991. LNCS, vol. 510, pp. 441–454. Springer, Heidelberg (1991).  https://doi.org/10.1007/3-540-54233-7_154CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySaint PetersburgRussia

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