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Automata over Infinite Sequences of Reals

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Language and Automata Theory and Applications (LATA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11417))

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Abstract

Gandhi, Khoussainov, and Liu introduced and studied a generalized model of finite automata able to work over algebraic structures, in particular the real numbers. The present paper continues the study of (a variant) of this model dealing with computations on infinite strings of reals. Our results support the view that this is a suitable model of finite automata over the real numbers. We define Büchi and Muller versions of the model and show, among other things, several closure properties of the languages accepted, a real number analogue of McNaughton’s theorem, and give a metafinite logic characterizing the infinite languages acceptable by non-deterministic Büchi automata over \({\mathbb R}\).

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Notes

  1. 1.

    Note that if the counter has value \(T-1\) also non-deterministic automata have to reset all register values to 0; only the next state can then be chosen non-deterministically.

  2. 2.

    A semi-algebraic set in \({\mathbb R}^n\) is a set that can be defined as a Boolean combination (using finite unions, intersections, and complements) of solution sets of polynomial equalities and inequalities.

References

  1. Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, New York (1998). https://doi.org/10.1007/978-1-4612-0701-6

    Book  MATH  Google Scholar 

  2. Gandhi, A., Khoussainov, B., Liu, J.: Finite automata over structures. In: Agrawal, M., Cooper, S.B., Li, A. (eds.) TAMC 2012. LNCS, vol. 7287, pp. 373–384. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29952-0_37

    Chapter  Google Scholar 

  3. Grädel, E., Gurevich, Y.: Metafinite model theory. Inf. Comput. 140(1), 26–81 (1998)

    Article  MathSciNet  Google Scholar 

  4. Grädel, E., Meer, K.: Descriptive complexity theory over the real numbers. In: Leighton, F.T., Borodin, A. (eds.) Proceedings of the 27th STOC, pp. 315–324 (1995)

    Google Scholar 

  5. Hofmann, M., Lange, M.: Automatentheorie und Logik. Springer, Berlin (2011). https://doi.org/10.1007/978-3-642-18090-3

    Book  MATH  Google Scholar 

  6. Khoussainov, B., Liu, J.: Decision problems for finite automata over infinite algebraic structures. In: Han, Y.-S., Salomaa, K. (eds.) CIAA 2016. LNCS, vol. 9705, pp. 3–11. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40946-7_1

    Chapter  MATH  Google Scholar 

  7. Khoussainov, B., Nerode, A.: Automata Theory and Its Applications. Birkhäuser, Springer, Berlin (2001)

    Book  Google Scholar 

  8. Makowsky, J.A., Meer, K.: Polynomials of bounded tree-width. In: Foundations of Computational Mathematics, Proceedings of the Smalefest, pp. 211–250. World Scientific (2002)

    Google Scholar 

  9. McNaughton, R.: Testing and generating infinite sequences by a finite automaton. Inf. Control 9(5), 521–530 (1966)

    Article  MathSciNet  Google Scholar 

  10. Meer, K., Naif, A.: Generalized finite automata over real and complex numbers. Theor. Comput. Sci. 591, 85–98 (2015)

    Article  MathSciNet  Google Scholar 

  11. Meer, K., Naif, A.: Periodic generalized automata over the reals. In: Dediu, A.-H., Janoušek, J., Martín-Vide, C., Truthe, B. (eds.) LATA 2016. LNCS, vol. 9618, pp. 168–180. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-30000-9_13

    Chapter  MATH  Google Scholar 

  12. Safra, S.: On the complexity of omega-automata. In: 29th Symposium on FOCS, New York, pp. 319–327 (1988)

    Google Scholar 

  13. Thomas, W.: Languages, automata, and logic. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 389–455. Springer, Heidelberg (1997). https://doi.org/10.1007/978-3-642-59126-6_7

    Chapter  Google Scholar 

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Acknowledgement

We thank all reviewers for their very thorough reading and many useful comments.

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Authors and Affiliations

  1. Brandenburg University of Technology Cottbus-Senftenberg, Platz der Deutschen Einheit 1, 03046, Cottbus, Germany

    Klaus Meer & Ameen Naif

Authors
  1. Klaus Meer
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  2. Ameen Naif
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Correspondence to Klaus Meer .

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Editors and Affiliations

  1. Rovira i Virgili University, Tarragona, Spain

    Carlos Martín-Vide

  2. Saint Petersburg State University, St. Petersburg, Russia

    Alexander Okhotin

  3. Ariel University, Ariel, Israel

    Dana Shapira

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Meer, K., Naif, A. (2019). Automata over Infinite Sequences of Reals. In: Martín-Vide, C., Okhotin, A., Shapira, D. (eds) Language and Automata Theory and Applications. LATA 2019. Lecture Notes in Computer Science(), vol 11417. Springer, Cham. https://doi.org/10.1007/978-3-030-13435-8_9

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  • DOI: https://doi.org/10.1007/978-3-030-13435-8_9

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-13434-1

  • Online ISBN: 978-3-030-13435-8

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