Skip to main content
SpringerLink
Log in

Characterization of Tree Automata Based on Quantum Logic

  • Published:
International Journal of Theoretical Physics Aims and scope Submit manuscript

Abstract

In this paper, we introduce the concept of non-deterministic tree automata based on quantum-valued logic whose underlying structure is a complete orthomodular lattice. First, we provide some concepts concerning quantum tree automata such as unary quantum acceptability predicate, and unary quantum regularity predicate. Next, some operations on quantum tree automata, including sum, product, concatenation and star are studied, and it is shown that the validity of many closure properties depends heavily upon the commutativity of the underlying logic. After that, the Kleene theorem about the equivalence of quantum regular tree expressions and quantum tree automata is proved. The developed results in this paper, clarify somewhat some essential distinctions between classical computation and quantum one.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Bhatia, A.S., Kumar, A.: Quantum ω-automata over infinite words and their relationships. Int. J. Theor. Phys. 58, 878–889 (2019)

    Article  MathSciNet  Google Scholar 

  2. Bhatia, A.S., Kumar, A.: On the power of two-way multihead quantum finite automata. RAIRO-Theor. Inform. Applic 53(1–2), 19–35 (2019)

    Article  MathSciNet  Google Scholar 

  3. Bianchi, M.P., Mereghetti, C., Palano, B.: Quantum finite automata: advances on Bertoni’s ideas. Theor. Comput. Sci. 664, 39–53 (2017)

    Article  MathSciNet  Google Scholar 

  4. Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936)

    Article  MathSciNet  Google Scholar 

  5. Broadsky, A., Pippenger, N.: Characterizations of 1-way quantum finite automata. SIAM J. Comp. 31, 1456–1478 (2002)

    Article  MathSciNet  Google Scholar 

  6. Cheng, W., Wang, J.: Grammar theory based on quantum logic. Int. J. Theor. Phys. 42, 1677–1691 (2003)

    Article  MathSciNet  Google Scholar 

  7. Comon, H., Dauchet, M., Gilleron, R., Jacquemard, F., Lugiez, D., Loding, C., Tison, S., Tommasi, M.: Tree automata: techniques and applications, Available: http://tata.gforge.inria.fr (2007)

  8. Dai, S.: A note on implication operators of quantum logic. Quantum Mach. Intell. 2, 15 (2020)

    Article  Google Scholar 

  9. Doner, J.E.: Tree acceptors and some of their applications. J. Comput. Syst. Sci. 4, 406–451 (1970)

    Article  MathSciNet  Google Scholar 

  10. Drewes, F.: Grammatical Picture Generation, a Tree-Based Approach, Texts in Theoretical Computer Science. Springer, Berlin (2006)

    MATH  Google Scholar 

  11. Engelfriet, J., Hoogeboom, H.J., Samwel, B.: XML navigation and transformation by tree-walking automata and transducers with visible and invisible pebbles. Theor. Comput. Sci. 850, 40–97 (2021)

    Article  MathSciNet  Google Scholar 

  12. Esik, Z., Kuich, W.: Formal tree series. J. Autom. Lang. Comb. 8(2), 219–285 (2003)

    MathSciNet  MATH  Google Scholar 

  13. Esik, Z., Liu, G.: Fuzzy tree automata. Fuzzy Sets. Syst. 158, 1450–1460 (2007)

    Article  MathSciNet  Google Scholar 

  14. Fülöp, Z., Vogler, H.: Weighted tree automata and tree transducers. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata. Springer (1998)

  15. Fülöp, Z., Maletti, A., Vogler, H.: A Kleene theorem for weighted tree automata over distributive multioperator monoids. Theory Comput. Syst. 44, 455–499 (2009)

    Article  MathSciNet  Google Scholar 

  16. Ghorani, M.: State hyperstructures of tree automata based on lattice-valued logic. RAIRO-Theoretical Informatics and Applications 52(1), 23–42 (2018)

    Article  MathSciNet  Google Scholar 

  17. Ghorani, M.: On characterization of fuzzy tree pushdown automata. Soft. Comput. 23, 1123–1131 (2019)

    Article  Google Scholar 

  18. Ghorani, M., Moghari, S.: Decidability of the minimization of fuzzy tree automata with membership values in complete lattices. Journal of Applied Mathematics and Computing, In press (2021)

  19. Ghorani, M., Zahedi, M.M.: Characterization of complete residuated lattice-valued finite tree automata. Fuzzy Set. Syst. 199, 28–46 (2012)

    Article  MathSciNet  Google Scholar 

  20. Ghorani, M., Zahedi, M.M.: Coding tree languages based on lattice-valued logic. Soft. Comput. 21, 3815–3825 (2017)

    Article  Google Scholar 

  21. Gudder, S.: Basic properties of quantum automata. Found. Phys. 30, 301–319 (2000)

    Article  MathSciNet  Google Scholar 

  22. Jakobi, S., Meckel, K., Mereghetti, C., Palano, B.: The descriptional power of queue automata of constant length. Acta Informatica 58, 335–356 (2021)

    Article  MathSciNet  Google Scholar 

  23. Jurvanen, E., Steinby, M.: Fuzzy deterministic top-down tree automata, arXiv:1911.11529 (2019)

  24. Lu, R.Q., Zheng, H.: Lattices of quantum automata. Int. J. Theor. Phys. 42, 1425–1449 (2003)

    Article  MathSciNet  Google Scholar 

  25. Moore, C., Crutchfield, J.P.: Quantum automata and quantum grammars. Theor. Comput. Sci. 237, 275–306 (2000)

    Article  MathSciNet  Google Scholar 

  26. Nakanishi, M.: Quantum pushdown automata with garbage tape. Int. J. Found. Comput. Sci. 29(3), 425–446 (2018)

    Article  MathSciNet  Google Scholar 

  27. Ptak, P., Pulmannová, S.: Orthomodular Structures as Quantum Logics. Kluwer, Dordrecht (1991)

    MATH  Google Scholar 

  28. Qiu, D.: Characterization of sequential quantum machines. Int. J. Theor. Phys. 41, 811–822 (2002)

    Article  MathSciNet  Google Scholar 

  29. Qiu, D.: Quantum pushdown automata. Int. J. Theor. Phys. 41, 1627–1639 (2002)

    Article  MathSciNet  Google Scholar 

  30. Qiu, D.: Automata theory based on quantum logic: Some characterizations. Inf. Comput. 190, 179–195 (2004)

    Article  MathSciNet  Google Scholar 

  31. Qiu, D.: Automata theory based on quantum logic: reversibilities and pushdown automata. Theor. Comput. Sci. 386, 38–56 (2007)

    Article  MathSciNet  Google Scholar 

  32. Qiu, D., Liu, F.: Fuzzy discrete-event systems under fuzzy observability and a test algorithm. IEEE Trans. Fuzzy Syst. 17(3), 578–589 (2009)

    Article  MathSciNet  Google Scholar 

  33. Qiu, D., Ying, M.S.: Characterization of quantum automata. Theor. Comput. Sci. 312, 479–489 (2004)

    Article  MathSciNet  Google Scholar 

  34. Thatcher, J.W., Wright, J.B.: Generalized finite automata theory with an application to a decision-problem of second order logic. Math. Syst. Theory 2, 57–81 (1968)

    Article  MathSciNet  Google Scholar 

  35. Wang, H., Zhao, L., Li, P.: Nondeterministic finite automata based on quantum logic: Language equivalence relation and robustness. Int. J. Approx. Reason. 129, 20–40 (2021)

    Article  MathSciNet  Google Scholar 

  36. Ying, M.: Automata theory based on quantum logic (I). Int. J. Theor. Phys. 39, 985–996 (2000)

    Article  MathSciNet  Google Scholar 

  37. Ying, M.: Automata theory based on quantum logic (II). Int. J. Theor. Phys. 39, 2545–2557 (2000)

    Article  MathSciNet  Google Scholar 

  38. Ying, M.: A theory of computation based on quantum logic (I). Theor. Comput. Sci. 344, 134–207 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Funding

No funding was received for conducting this study.

Author information

Authors and Affiliations

  1. Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Iran

    M. Ghorani

Authors
  1. M. Ghorani
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Ghorani.

Ethics declarations

Conflict of Interests

The author has no conflicts of interest to declare that are relevant to the content of this article.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ghorani, M. Characterization of Tree Automata Based on Quantum Logic. Int J Theor Phys 61, 13 (2022). https://doi.org/10.1007/s10773-022-04974-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10773-022-04974-6

Keywords

Search

Navigation