Cut Languages in Rational Bases

  • Jiří ŠímaEmail author
  • Petr Savický
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10168)


We introduce a so-called cut language which contains the representations of numbers in a rational base that are less than a given threshold. The cut languages can be used to refine the analysis of neural net models between integer and rational weights. We prove a necessary and sufficient condition when a cut language is regular, which is based on the concept of a quasi-periodic power series. For a nonnegative base and digits, we achieve a dichotomy that a cut language is either regular or non-context-free while examples of regular and non-context-free cut languages are presented. We show that any cut language with a rational threshold is context-sensitive.


Grammars Quasi-periodic power series Cut language 


  1. 1.
    Adamczewski, B., Frougny, C., Siegel, A., Steiner, W.: Rational numbers with purely periodic \(\beta \)-expansion. Bull. Lond. Math. Soc. 42(3), 538–552 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Allouche, J.P., Clarke, M., Sidorov, N.: Periodic unique beta-expansions: the Sharkovskiĭ ordering. Ergod. Theory Dyn. Syst. 29(4), 1055–1074 (2009)CrossRefzbMATHGoogle Scholar
  3. 3.
    Alon, N., Dewdney, A.K., Ott, T.J.: Efficient simulation of finite automata by neural nets. J. ACM 38(2), 495–514 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Balcázar, J.L., Gavaldà, R., Siegelmann, H.T.: Computational power of neural networks: a characterization in terms of Kolmogorov complexity. IEEE Trans. Inf. Theory 43(4), 1175–1183 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chunarom, D., Laohakosol, V.: Expansions of real numbers in non-integer bases. J. Korean Math. Soc. 47(4), 861–877 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Glendinning, P., Sidorov, N.: Unique representations of real numbers in non-integer bases. Math. Res. Lett. 8(4), 535–543 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hare, K.G.: Beta-expansions of Pisot and Salem numbers. In: Proceedings of the Waterloo Workshop in Computer Algebra 2006: Latest Advances in Symbolic Algorithms, pp. 67–84. World Scientic (2007)Google Scholar
  8. 8.
    Horne, B.G., Hush, D.R.: Bounds on the complexity of recurrent neural network implementations of finite state machines. Neural Netw. 9(2), 243–252 (1996)CrossRefGoogle Scholar
  9. 9.
    Indyk, P.: Optimal simulation of automata by neural nets. In: Mayr, E.W., Puech, C. (eds.) STACS 1995. LNCS, vol. 900, pp. 337–348. Springer, Heidelberg (1995). doi: 10.1007/3-540-59042-0_85 CrossRefGoogle Scholar
  10. 10.
    Komornik, V., Loreti, P.: Subexpansions, superexpansions and uniqueness properties in non-integer bases. Periodica Mathematica Hungarica 44(2), 197–218 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Minsky, M.: Computations: Finite and Infinite Machines. Prentice-Hall, Englewood Cliffs (1967)zbMATHGoogle Scholar
  12. 12.
    Parry, W.: On the \(\beta \)-expansions of real numbers. Acta Math. Hung. 11(3), 401–416 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rényi, A.: Representations for real numbers and their ergodic properties. Acta Math. Acad. Scientiarum Hung. 8(3–4), 477–493 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Schmidt, K.: On periodic expansions of Pisot numbers and Salem numbers. Bull. Lond. Math. Soc. 12(4), 269–278 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Sidorov, N.: Expansions in non-integer bases: Lower, middle and top orders. J. Number Theory 129(4), 741–754 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Siegelmann, H.T.: Recurrent neural networks and finite automata. J. Comput. Intell. 12(4), 567–574 (1996)CrossRefGoogle Scholar
  17. 17.
    Siegelmann, H.T.: Neural Networks and Analog Computation: Beyond the Turing Limit. Birkhäuser, Boston (1999)CrossRefzbMATHGoogle Scholar
  18. 18.
    Siegelmann, H.T., Sontag, E.D.: Analog computation via neural networks. Theoret. Comput. Sci. 131(2), 331–360 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Siegelmann, H.T., Sontag, E.D.: On the computational power of neural nets. J. Comput. Syst. Sci. 50(1), 132–150 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Šíma, J.: Energy complexity of recurrent neural networks. Neural Comput. 26(5), 953–973 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Šíma, J.: The Power of Extra Analog Neuron. In: Dediu, A.-H., Lozano, M., Martín-Vide, C. (eds.) TPNC 2014. LNCS, vol. 8890, pp. 243–254. Springer, Heidelberg (2014). doi: 10.1007/978-3-319-13749-0_21 Google Scholar
  22. 22.
    Šíma, J.: Neural networks between integer and rational weights. Technical report, no. V-1237, Institute of Computer Science, The Czech Academy of Sciences, Prague (2016)Google Scholar
  23. 23.
    Šíma, J., Orponen, P.: General-purpose computation with neural networks: A survey of complexity theoretic results. Neural Comput. 15(12), 2727–2778 (2003)CrossRefzbMATHGoogle Scholar
  24. 24.
    Šíma, J., Savický, P.: Cut languages in rational bases. Technical report, no. V-1236, Institute of Computer Science, The Czech Academy of Sciences, Prague (2016)Google Scholar
  25. 25.
    Šíma, J., Wiedermann, J.: Theory of neuromata. J. ACM 45(1), 155–178 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Institute of Computer ScienceThe Czech Academy of SciencesPrague 8Czech Republic

Personalised recommendations