Abstract
We propose the concept of finite stop quantum automata (ftqa) based on Hilbert space and compare it with the finite state quantum automata (fsqa) proposed by Moore and Crutchfield (Theoretical Computer Science 237(1–2), 2000, 275–306). The languages accepted by fsqa form a proper subset of the languages accepted by ftqa. In addition, the fsqa form an infinite hierarchy of language inclusion with respect to the dimensionality of unitary matrices. We introduce complex-valued acceptance degrees and two types of finite stop quantum automata based on them: the invariant ftqa (icftq) and the variant ftqa (vcftq). The languages accepted by icftq form a proper subset of the languages accepted by vcftq. In addition, the icftq form an infinite hierarchy of language inclusion with respect to the dimensionality of unitary matrices. In this way, we establish two proper inclusion relations \({\cal L}\) (fsqa) ⊂ \({\cal L}\) (ftqa) and \({\cal L}\) (icftq) ⊂ \({\cal L}\) (vcftq), where the symbol \({\cal L}\) means languages, and two infinite language hierarchies \({\cal L}_{n}\) (fsqa) ⊂ \({\cal L}_{n+1}\) (fsqa), \({\cal L}_{n}\) (icftq) \({\cal L}_{n+1}\) (icftq).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ambainis, A., Bonner, R., Freivalds, R., and Kikusts, A. (1999). A hierarchy of languages accepted by quantum finite automata. In QCL 99, International Workshop on Quantum Computation and Learning, Riga.
Chomsky, N. (1956). Three models for the description of language. IRE Transactions on Information Theory 2(3), 113–124.
Chomsky, N. (1959). On certain formal properties of grammars. Information and Control 2(2), 137–167.
Chomsky, N. and Miller, G. A. (1958). Finite state languages. Information and Control 1(2), 91–112.
Cohen, D. W. (1989). An Introduction to Hilbert Space and Quantum Logic, Springer, New York.
Cohn, P. M. (1981). F.R.S. Universal Algebra, D. Reidel Publishing Company.
Deutsch, D. (1985). Quantum theory, the church Turing principle and the universal quantum computer. Proceedings of the Royal Society of London, Series A 400, 97–117.
Greechie, R. (1981). A non-standard quantum logic with a strong set of states. In Current Issues in Quantum Logic, Vol. 8, E. G. Beltrametti and B. C. van Fraassen, eds., Plenum Press, New York, pp. 375–380.
Gudder, S. (1999). Basic properties of quantum automata. International Journal of Theoretical Physics 38, 2261–2282.
Hermes, H. (1955). Einfuehrung in die Verbandstheorie, Springer-Verlag, Berlin.
Hopcroft, J. E. and Ullman, J. D. (1979). Introduction to Automata Theory, Languages, and Computation, Addison-Wesley.
Kondacs, A. and Watrous, J. (1997). On the power of quantum finite state automata. In Proceedings of 38th FOCS, pp. 66–75.
Kuroda, S. Y. (1964). Classes of languages and linear bounded automata. Information and Control 7(2), 207–223.
Lindenmayer, A. and Prusinkiewicz, P. (1990). The Algorithmic Beauty of Plants, Springer-Verlag, Berlin.
Lu, R. and Zhang, W. (2002). Generalized L systems. Science in China, Series F 45(3), 220–231.
Lu, R. and Zheng, H. (2003). Lattices of quantum automata. International Journal of Theoretical Physics 42(7), 1425–1449.
Lu, R. and Zheng, H. (2004). Pumping Lemma of quantum automata. International Journal of Theoretical Physics 43(5), 1191–1217.
Mackey, G. W. (1957). Quantum mechanics and Hilbert space. American Mathematics 64, 45–57.
Moore, C. and Crutchfield, J. P. (2000). Quantum automata and quantum grammars. Theoretical Computer Science 237(1–2), 275–306.
Qiu, D. (2003). Automata and grammars theory based on quantum logic. Journal of Software 14, 23–27.
Rawling, J. P. and Selesnick, S. A. (2000). Orthologic and quantum logic: Models and computational elements. JACM 47(4), 721–751.
Reichenbach, H. (1948). Philosophic Foundations of Quantum Mechanics, California.
Rozenberg, G. and Salomaa, A. (1980). The Mathematical Theory of L Systems. Academic Press, New York.
Shannon, C. E. (1956). A universal Turing machine with two internal states. In Automata Studies, Princeton University Press, Princeton, NJ, pp. 129–153.
Turing, A. M. (1936). On computable numbers with an application to the Entscheidungs problem. Proceedings of the London Mathematical Society 2(42), 230–265 (correction on 2(43), 544–546).
Ying, M. S. (2000a). Automata theory based on quantum logic. I. International Journal of Theoretical Physics 39(4), 985–995.
Ying, M. S. (2000b). Automata theory based on quantum logic. II. International Journal of Theoretical Physics 39(11), 2545–2557.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lu, R., Zheng, H. Finite State and Finite Stop Quantum Languages. Int J Theor Phys 44, 1495–1530 (2005). https://doi.org/10.1007/s10773-005-4781-z
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10773-005-4781-z