Skip to main content
SpringerLink
Log in

Prediction of Infinite Words with Automata

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

In the classic problem of sequence prediction, a predictor receives a sequence of values from an emitter and tries to guess the next value before it appears. The predictor masters the emitter if there is a point after which all of the predictor’s guesses are correct. In this paper we consider the case in which the predictor is an automaton and the emitted values are drawn from a finite set; i.e., the emitted sequence is an infinite word. We examine the predictive capabilities of finite automata, pushdown automata, stack automata (a generalization of pushdown automata), and multihead finite automata. We relate our predicting automata to purely periodic words, ultimately periodic words, and multilinear words, describing novel prediction algorithms for mastering these sequences.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Angluin, D., Fisman, D.: Learning regular omega languages. In: Auer, P., Clark, A., Zeugmann, T., Zilles, S. (eds.) Algorithmic Learning Theory, Lecture Notes in Computer Science, vol. 8776, pp 125–139. Springer International Publishing (2014). doi:10.1007/978-3-319-11662-4_10

  2. Angluin, D., Smith, C.H.: Inductive inference: Theory and methods. ACM Comput. Surv. 15(3), 237–269 (1983). doi:10.1145/356914.356918

    Article  MathSciNet  Google Scholar 

  3. Blackwell, D.: Minimax vs. Bayes prediction. Probab. Eng. Inf. Sci. 9, 53–58 (1995). doi:10.1017/S0269964800003685. http://journals.cambridge.org/article_S0269964800003685

    Article  MathSciNet  MATH  Google Scholar 

  4. Blum, L., Blum, M.: Toward a mathematical theory of inductive inference. Inf. Control. 28(2), 125–155 (1975). doi:10.1016/S0019-9958(75)90261-2. http://www.sciencedirect.com/science/article/pii/S0019995875902612

  5. Broglio, A., Liardet, P.: Predictions with Automata. In: Symbolic Dynamics and Its Applications, Contemporary Mathematics, vol. 135, pp 111–124. American Mathematical Society (1992)

  6. Cerruti, U., Giacobini, M., Liardet, P.: Prediction of binary sequences by evolving finite state machines. In: Collet, P., Fonlupt, C., Hao, J.K., Lutton, E., Schoenauer, M. (eds.) Artificial Evolution, Lecture Notes in Computer Science, vol. 2310, pp 42–53. Springer, Berlin (2002). doi:10.1007/3-540-46033-0_4

    Google Scholar 

  7. Drucker, A.: High-confidence predictions under adversarial uncertainty. TOCT 5(3), 12 (2013). doi:10.1145/2493252.2493257

    Article  MathSciNet  MATH  Google Scholar 

  8. Endrullis, J., Hendriks, D., Klop, J.W.: Degrees of Streams. Integers, Electron. J. Comb. Number Theory 11B(A6), 1–40 (2010). Proc.Leiden Numeration Conf

    MATH  Google Scholar 

  9. Feder, M., Merhav, N., Gutman, M.: Universal prediction of individual sequences. IEEE Trans. Inf. Theory 38, 1258–1270 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ginsburg, S., Greibach, S.A., Harrison, M.A.: One-way stack automata. J. ACM 14(2), 389–418 (1967). doi:10.1145/321386.321403

    Article  MathSciNet  MATH  Google Scholar 

  11. Gold, E.M.: Language identification in the limit. Inf. Control. 10(5), 447–474 (1967). http://groups.lis.illinois.edu/amag/langev/paper/gold67limit.html

    Article  MathSciNet  MATH  Google Scholar 

  12. Hibbard, B.: Adversarial sequence prediction. In: Proceedings of the 2008 Conference on Artificial General Intelligence 2008: Proceedings of the First AGI Conference, pp. 399–403. IOS Press, Amsterdam, The Netherlands, The Netherlands. http://dl.acm.org/citation.cfm?id=1566174.1566212 (2008)

  13. Holzer, M., Kutrib, M., Malcher, A.: Complexity of multi-head finite automata: Origins and directions. Theor. Comput. Sci. 412(1-2), 83–96 (2011). doi:10.1016/j.tcs.2010.08.024

    Article  MathSciNet  MATH  Google Scholar 

  14. Hromkovič, J.: One-way multihead deterministic finite automata. Acta Inform. 19(4), 377–384 (1983). doi:10.1007/BF00290734

    Article  MathSciNet  MATH  Google Scholar 

  15. Johansen, P.: Inductive inference of ultimately periodic sequences. BIT Numer. Math. 28(3), 573–580 (1988). doi:10.1007/BF01941135

    Article  MathSciNet  MATH  Google Scholar 

  16. Leblanc, B., Lutton, E., Allouche, J.P.: Inverse problems for finite automata: A solution based on genetic algorithms. In: Hao, J.K., Lutton, E., Ronald, E., Schoenauer, M., Snyers, D. (eds.) Artificial Evolution, Lecture Notes in Computer Science, vol. 1363, pp 157–166. Springer, Berlin (1998). doi:10.1007/BFb0026598

    Google Scholar 

  17. Legg, S.: Is there an elegant universal theory of prediction?. In: Algorithmic Learning Theory, 17th International Conference, ALT 2006, Barcelona, Spain, October 7-10, 2006, Proceedings, pp 274–287 (2006). doi:10.1007/11894841_23

  18. Narendra, K.S., Thathachar, M.A.L.: Learning automata: an introduction. Prentice-hall, Inc., Upper Saddle River, NJ, USA (1989)

  19. O’Connor, M.G.: An unpredictability approach to finite-state randomness. J. Comput. Syst. Sci. 37(3), 324–336 (1988). doi:10.1016/0022-0000(88)90011-6

    Article  MathSciNet  MATH  Google Scholar 

  20. Sedgewick, R., Szymanski, T.G., Yao, A.C.: The complexity of finding cycles in periodic functions. SIAM J. Comput. 11(2), 376–390 (1982). doi:10.1137/0211030

    Article  MathSciNet  MATH  Google Scholar 

  21. Shubert, B.: Games of Prediction of Periodic Sequences. Tech. rep., United States Naval Postgraduate School (1971)

  22. Smith, T.: On Infinite Words Determined by Stack Automata. doi:10.4230/LIPIcs.FSTTCS.2013.413 (2013)

  23. Smith, T.: Prediction of Infinite Words with Automata. In: Computer Science — Theory and Applications, CSR 2016. To appear. Springer, Berlin (2016)

    Google Scholar 

  24. Solomonoff, R.: A formal theory of inductive inference. part i. Inf. Control. 7(1), 1–22 (1964). doi:10.1016/S0019-9958(64)90223-2. http://www.sciencedirect.com/science/article/pii/S0019995864902232

  25. Wagner, K., Wechsung, G.: Computational Complexity. Mathematics and its Applications Springer (1986)

Download references

Acknowledgments

I would like to thank my Ph.D. advisor at Northeastern, Rajmohan Rajaraman, for his helpful comments and suggestions. The continuation of this work at Marne-la-Vallée was supported by the Agence Nationale de la Recherche (ANR) under the project EQINOCS (ANR-11-BS02-004).

Author information

Authors and Affiliations

  1. Northeastern University, Boston, MA, USA

    Tim Smith

  2. Université Paris-Est Marne-la-Vallée, Champs-sur-Marne, France

    Tim Smith

Authors
  1. Tim Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tim Smith.

Additional information

This journal paper comprises the full version of the author’s conference paper [23].

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Smith, T. Prediction of Infinite Words with Automata. Theory Comput Syst 62, 653–681 (2018). https://doi.org/10.1007/s00224-016-9739-4

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-016-9739-4

Keywords

Search

Navigation