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Counting with Analog Neurons

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Artificial Neural Networks and Machine Learning – ICANN 2019: Theoretical Neural Computation (ICANN 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11727))

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Abstract

We refine the analysis of binary-state neural networks with \(\alpha \) extra analog neurons (\(\alpha \)ANNs). For rational weights, it has been known that online 1ANNs accept context-sensitive languages including examples of non-context-free languages, while offline 3ANNs are Turing complete. We now prove that the deterministic (context-free) language containing the words of n zeros followed by n ones, cannot be recognized offline by any 1ANN with real weights. Hence, the offline 1ANNs are not Turing complete. On the other hand, we show that any deterministic language can be accepted by a 2ANN with rational weights. Thus, two extra analog units can count to any number which is not the case of one analog neuron.

Research was done with institutional support RVO: 67985807 and partially supported by the grant of the Czech Science Foundation No. 19-05704S.

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Notes

  1. 1.

    Some of the results are valid for more general classes of activation functions [6, 13, 21] including the logistic function [5].

References

  1. Alon, N., Dewdney, A.K., Ott, T.J.: Efficient simulation of finite automata by neural nets. J. ACM 38(2), 495–514 (1991)

    Article  MathSciNet  Google Scholar 

  2. 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)

    Article  MathSciNet  Google Scholar 

  3. 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)

    Article  Google Scholar 

  4. 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). https://doi.org/10.1007/3-540-59042-0_85

    Chapter  Google Scholar 

  5. Kilian, J., Siegelmann, H.T.: The dynamic universality of sigmoidal neural networks. Inf. Comput. 128(1), 48–56 (1996)

    Article  MathSciNet  Google Scholar 

  6. Koiran, P.: A family of universal recurrent networks. Theoret. Comput. Sci. 168(2), 473–480 (1996)

    Article  MathSciNet  Google Scholar 

  7. Minsky, M.: Computations: Finite and Infinite Machines. Prentice-Hall, Englewood Cliffs (1967)

    MATH  Google Scholar 

  8. Orponen, P.: Computing with truly asynchronous threshold logic networks. Theoret. Comput. Sci. 174(1–2), 123–136 (1997)

    Article  MathSciNet  Google Scholar 

  9. Schmidhuber, J.: Deep learning in neural networks: An overview. Neural Netw. 61, 85–117 (2015)

    Article  Google Scholar 

  10. Siegelmann, H.T.: Neural Networks and Analog Computation: Beyond the Turing Limit. Birkhäuser, Boston (1999)

    Book  Google Scholar 

  11. Siegelmann, H.T., Sontag, E.D.: Analog computation via neural networks. Theoret. Comput. Sci. 131(2), 331–360 (1994)

    Article  MathSciNet  Google Scholar 

  12. Siegelmann, H.T., Sontag, E.D.: On the computational power of neural nets. J. Comput. Syst. Sci. 50(1), 132–150 (1995)

    Article  MathSciNet  Google Scholar 

  13. Šíma, J.: Analog stable simulation of discrete neural networks. Neural Netw. World 7(6), 679–686 (1997)

    Google Scholar 

  14. Šíma, J.: Energy complexity of recurrent neural networks. Neural Comput. 26(5), 953–973 (2014)

    Article  MathSciNet  Google Scholar 

  15. 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, Cham (2014). https://doi.org/10.1007/978-3-319-13749-0_21

    Chapter  Google Scholar 

  16. Šíma, J.: Three analog neurons are turing universal. In: Fagan, D., Martín-Vide, C., O’Neill, M., Vega-Rodríguez, M.A. (eds.) TPNC 2018. LNCS, vol. 11324, pp. 460–472. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-04070-3_36

    Chapter  Google Scholar 

  17. Šíma, J.: Subrecursive neural networks. Neural Netw. 116, 208–223 (2019)

    Article  Google Scholar 

  18. Šíma, J., Orponen, P.: General-purpose computation with neural networks: A survey of complexity theoretic results. Neural Comput. 15(12), 2727–2778 (2003)

    Article  Google Scholar 

  19. Šíma, J., Savický, P.: Quasi-periodic \(\beta \)-expansions and cut languages. Theoret. Comput. Sci. 720, 1–23 (2018)

    Article  MathSciNet  Google Scholar 

  20. Šíma, J., Wiedermann, J.: Theory of neuromata. J. ACM 45(1), 155–178 (1998)

    Article  MathSciNet  Google Scholar 

  21. Šorel, M., Šíma, J.: Robust RBF finite automata. Neurocomputing 62, 93–110 (2004)

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Computer Science, Czech Academy of Sciences, P. O. Box 5, 18207, Prague 8, Czech Republic

    Jiří Šíma

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  1. Jiří Šíma
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Correspondence to Jiří Šíma .

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Editors and Affiliations

  1. Helmholtz Zentrum München - Deutsches Forschungszentrum für Gesundheit und Umwelt (GmbH), Neuherberg, Germany

    Igor V. Tetko

  2. Institute of Computer Science, Czech Academy of Sciences, Prague 8, Czech Republic

    Věra Kůrková

  3. Helmholtz Zentrum München - Deutsches Forschungszentrum für Gesundheit und Umwelt (GmbH), Neuherberg, Germany

    Pavel Karpov

  4. Helmholtz Zentrum München - Deutsches Forschungszentrum für Gesundheit und Umwelt (GmbH), Neuherberg, Germany

    Fabian Theis

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Šíma, J. (2019). Counting with Analog Neurons. In: Tetko, I., Kůrková, V., Karpov, P., Theis, F. (eds) Artificial Neural Networks and Machine Learning – ICANN 2019: Theoretical Neural Computation. ICANN 2019. Lecture Notes in Computer Science(), vol 11727. Springer, Cham. https://doi.org/10.1007/978-3-030-30487-4_31

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  • DOI: https://doi.org/10.1007/978-3-030-30487-4_31

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-30486-7

  • Online ISBN: 978-3-030-30487-4

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