Prediction of Infinite Words with Automata

  • Tim SmithEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)


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.


Finite Automaton Periodic Sequence Input Symbol Input Alphabet Deterministic Finite Automaton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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).


  1. 1.
    Angluin, D., Fisman, D.: Learning regular omega languages. In: Auer, P., Clark, A., Zeugmann, T., Zilles, S. (eds.) ALT 2014. LNCS, vol. 8776, pp. 125–139. Springer, Heidelberg (2014)Google Scholar
  2. 2.
    Angluin, D., Smith, C.H.: Inductive inference: theory and methods. ACM Comput. Surv. 15(3), 237–269 (1983). MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blackwell, D.: Minimax vs. Bayes prediction. Probab. Eng. Inf. Sci. 9(1), 53–58 (1995). MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Blum, L., Blum, M.: Toward a mathematical theory of inductive inference. Inf. Control 28(2), 125–155 (1975). MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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)Google Scholar
  6. 6.
    Cerruti, U., Giacobini et al., 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.) EA 2001. LNCS, vol. 2310, pp. 42–53. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Drucker, A.: High-confidence predictions under adversarial uncertainty. TOCT 5(3), 12 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Endrullis, J., Hendriks, D., Klop, J.W.: Degrees of streams. In: Integers, Electronic Journal of Combinatorial Number Theory 11B(A6), 1–40. 2010 Proceedings of the Leiden Numeration Conference (2011)Google Scholar
  9. 9.
    Feder, M., Merhav, N., Gutman, M.: Universal prediction of individual sequences. IEEE Trans. Inf. Theory 38, 1258–1270 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ginsburg, S., Greibach, S.A., Harrison, M.A.: One-way stack automata. J. ACM 14(2), 389–418 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gold, E.M.: Language identification in the limit. Inf. Control 10(5), 447–474 (1967). CrossRefzbMATHGoogle Scholar
  12. 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 (2008).
  13. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hromkovič, J.: One-way multihead deterministic finite automata. Acta Informatica 19(4), 377–384 (1983). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Johansen, P.: Inductive inference of ultimately periodic sequences. BIT Numer. Math. 28(3), 573–580 (1988). MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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.) AE 1997. LNCS, vol. 1363, pp. 157–166. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  17. 17.
    Legg, S.: Is there an elegant universal theory of prediction? In: Balcázar, J.L., Long, P.M., Stephan, F. (eds.) ALT 2006. LNCS (LNAI), vol. 4264, pp. 274–287. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  18. 18.
    O’Connor, M.G.: An unpredictability approach to finite-state randomness. J. Comput. Syst. Sci. 37(3), 324–336 (1988). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sedgewick, R., Szymanski, T.G., Yao, A.C.: The complexity of finding cycles in periodic functions. SIAM J. Comput. 11(2), 376–390 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shubert, B.: Games of prediction of periodic sequences. Technical report, United States Naval Postgraduate School (1971)Google Scholar
  21. 21.
    Smith, T.: On infinite words determined by stack automata. In: FSTTCS 2013. Leibniz International Proceedings in Informatics (LIPIcs), vol. 24, pp. 413–424. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2013)Google Scholar
  22. 22.
    Solomonoff, R.: A formal theory of inductive inference. part i. Inf. Control 7(1), 1–22 (1964). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Wagner, K., Wechsung, G.: Computational complexity. Mathematics and its Applications. Springer (1986)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Northeastern UniversityBostonUSA
  2. 2.Université Paris-Est Marne-la-ValléeChamps-sur-MarneFrance

Personalised recommendations