Quantum ω-Automata over Infinite Words and Their Relationships

  • Amandeep Singh BhatiaEmail author
  • Ajay Kumar


Inspired by the results of finite automata working on infinite words, we studied the quantum ω-automata with Büchi, Muller, Rabin and Streett acceptance condition. Quantum finite automata play a pivotal part in quantum information and computational theory. Investigation of the power of quantum finite automata over infinite words is a natural goal. We have investigated the classes of quantum ω-automata from two aspects: the language recognition and their closure properties. It has been shown that quantum Muller automaton is more dominant than quantum Büchi automaton. Furthermore, we have demonstrated the languages recognized by one-way quantum finite automata with different quantum acceptance conditions. Finally, we have proved the closure properties of quantum ω-automata.


Quantum finite automata Quantum ω-automata Quantum Muller automaton Quantum Rabin automaton Quantum Streett automaton 



Amandeep Singh Bhatia was supported by Maulana Azad National Fellowship (MANF), funded by Ministry of Minority Affairs, Government of India.


  1. 1.
    Feynman, R.P.: Simulating physics with computers. Int. J. Theor. Phys. 21 (6-7), 467–488 (1982)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Nielsen, M.A., Chuang, I: Quantum computation and quantum information (2002)Google Scholar
  3. 3.
    Büchi, J.R.: On a decision method in restricted second order arithmetic. In: The collected works of J. Richard Büchi, Springer, pp. 425–435 (1990)Google Scholar
  4. 4.
    Muller, D.E.: Infinite sequences and finite machines. In: Proceedings of the 4th Annual Symposium on Switching Circuit Theory and Logical Design, IEEE, 1963, pp. 3–16Google Scholar
  5. 5.
    Landauer, R.: Irreversibility and heat generation in the computing process. IBM J. Res. Dev. 5(3), 183–191 (1961)MathSciNetCrossRefGoogle Scholar
  6. 6.
    McNaughton, R.: Testing and generating infinite sequences by a finite automaton. Inf. Control. 9(5), 521–530 (1966)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Rabin, M.O.: Decidability of second-order theories and automata on infinite trees. Trans. Amer. Math. Soc. 141, 1–35 (1969)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bhatia, A.S., Kumar, A.: Neurocomputing approach to matrix product state using quantum dynamics. Quantum Inf. Process 17(10), 278 (2018)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bhatia, A.S., Kumar, A.: Modeling of rna secondary structures using two-way quantum finite automata. Chaos, Solitons Fractals 116, 332–339 (2018)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Yakaryılmaz, A.: Superiority of one-way and realtime quantum machines. RAIRO-Theoretical Informatics and Applications 46(4), 615–641 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ambainis, A., Yakaryılmaz, A.: Superiority of exact quantum automata for promise problems. Inf. Process. Lett. 112(7), 289–291 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Bhatia, A.S., Kumar, A.: On the power of quantum queue automata in real-time. arXiv:1810.12095 (2018)
  13. 13.
    Wang, Q., Ying, M.: Quantum b∖” uchi automata. arXiv:1804.08982 (2018)
  14. 14.
    Dzelme-Bērziņa, I.: Quantum finite state automata over infinite words. In: International Conference on Unconventional Computation, Springer, pp. 188–188 (2010)Google Scholar
  15. 15.
    Mukund, M.: Finite-state automata on infinite inputs. TCS 96, 2 (1996)Google Scholar
  16. 16.
    Nivat, M., Perrin, D.: Automata on infinite words, vol. 192, Springer Science & Business Media (1985)Google Scholar
  17. 17.
    Perrin, D.: Recent results on automata and infinite words. In: International Symposium on Mathematical Foundations of Computer Science, Springer, pp. 134–148 (1984)Google Scholar
  18. 18.
    Baier, C., Bertrand, N., Größer, M.: Probabilistic automata over infinite words: expressiveness, efficiency, and decidability . arXiv:0907.4760 (2009)
  19. 19.
    Giannakis, K., Papalitsas, C., Andronikos, T.: Quantum automata for infinite periodic words. In: 2015 6th International Conference on Information, Intelligence, Systems and Applications (IISA), IEEE, pp. 1–6 (2015)Google Scholar
  20. 20.
    Dzelme-Bērziṅa, I.: Quantum finite automata and logic, P. h. D. thesis, University of Latvia Riga (2010)Google Scholar
  21. 21.
    Bhatia, A.S., Kumar, A.: Quantifying matrix product state. Quantum Inf. Process. 17(3), 41 (2018)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Rukšane, I., Krišlauks, R., Mischenko-Slatenkova, T., Dzelme-Berzina, I., Freivalds, R., Nagele, I.: Probabilistic, frequency and quantum automata on omega-wordsGoogle Scholar
  23. 23.
    Gruska, J.: Quantum computing, vol. 2005. McGraw-Hill, London (1999)Google Scholar
  24. 24.
    Moore, C., Crutchfield, J.P.: Quantum automata and quantum grammars. Theor. Comput. Sci. 237(1-2), 275–306 (2000)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Brodsky, A., Pippenger, N.: Characterizations of 1-way quantum finite automata. SIAM J. Comput. 31(5), 1456–1478 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Computer ScienceThapar Institute of Engineering & TechnologyPatialaIndia

Personalised recommendations