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More on quantum, stochastic, and pseudo stochastic languages with few states

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Abstract

Stochastic languages are the languages recognized by probabilistic finite automata (PFAs) with cutpoint over the field of real numbers. More general computational models over the same field such as generalized finite automata (GFAs) and quantum finite automata (QFAs) define the same class. In 1963, Rabin proved the set of stochastic languages to be uncountable presenting a single 2-state PFA over the binary alphabet recognizing uncountably many languages depending on the cutpoint. In this paper, we show the same result for unary stochastic languages. Namely, we exhibit a 2-state unary GFA, a 2-state unary QFA, and a family of 3-state unary PFAs recognizing uncountably many languages; all these numbers of states are optimal. After this, we completely characterize the class of languages recognized by 1-state GFAs, which is the only nontrivial class of languages recognized by 1-state automata. Finally, we consider the variations of PFAs, QFAs, and GFAs based on the notion of inclusive/exclusive cutpoint, and present some results on their expressive power.

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Notes

  1. Whether Moore-Crutchfield quantum finite automata define more languages when the cutpoint is changed from 0 to a value from the open interval (0, 1).

  2. MC stands for Moore and Crutchfield who introduced the model (Moore and Crutchfield 2000).

  3. A GFA with \(v_0=0\) or \(f=0\) recognizes either \(\varnothing \) or \(\varSigma ^*\). The same effect can be achieved by setting all transition numbers to 0. Hence we assume w.l.o.g. \(v_0,\,f \ne 0\). We also use the standard convention that \(0^0=1\).

  4. From the geometric point of view, a 1-state positive GFA defines a hyperplane in \({\mathbb {R}}^n\) and accepts exactly the words having the ends of their Parikh vectors on the prescribed side of this hyperplane.

References

  • Ambainis A, Freivalds R (1998) 1-way quantum finite automata: strengths, weaknesses and generalizations. In: FOCS’98: Proceedings of the 39th annual symposium on foundations of computer science, pp 332–341. (http://arxiv.org/abs/quant-ph/9802062)

  • Bertoni A, Carpentieri M (2001) Analogies and differences between quantum and stochastic automata. Theor Comput Sci 262(1–2):69–81

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Bukharaev RG (1967) Probabilistic methods and cybernetics. V, Gos. Univ. Uchen. Zap., vol 127:3, chap. On the representability of events in probabilistic automata, pp 7–20. Kazan (Russian)

  • Hirvensalo M (2010) Quantum automata with open time evolution. Int J Nat Comput 1(1):70–85

    Article  Google Scholar 

  • Macarie I (1993) Closure properties of stochastic languages. Technical report 441, University of Rochester

  • Moore C, Crutchfield JP (2000) Quantum automata and quantum grammars. Theor Comput Sci 237(1–2):275–306

    Article  MathSciNet  MATH  Google Scholar 

  • Parikh RJ (1966) On context-free languages. J ACM 13(4):570–581

    Article  MathSciNet  MATH  Google Scholar 

  • Paz A (1971) Introduction to Probabilistic Automata. Academic Press, New York

    MATH  Google Scholar 

  • Rabin MO (1963) Probabilistic automata. Inf Control 6:230–243

    Article  MATH  Google Scholar 

  • Salomaa A, Soittola M (1978) Automata-theoretic aspects of formal power series. Texts and monographs in computer science. Springer, New York

    Book  Google Scholar 

  • Say ACC, Yakaryılmaz A (2015) Quantum finite automata: a modern introduction. In: Gruska Festschrift, LNCS, vol 8808. Springer, pp 208–222

  • Shur AM, Yakaryilmaz A (2014) Quantum, stochastic, and pseudo stochastic languages with few states. In: Unconventional computation and natural computation, LNCS, vol 8553. Springer, Switzerland, pp 327–339

  • Turakainen P (1968) On stochastic languages. Inf Control 12(4):304–313

    Article  MathSciNet  MATH  Google Scholar 

  • Turakainen P (1969) Generalized automata and stochastic languages. Proc Am Math Soc 21:303–309

    Article  MathSciNet  MATH  Google Scholar 

  • Turakainen P (1975) Word-functions of stochastic and pseudo stochastic automata. Ann Acad Sci Fenn 1:27–37

    Article  MathSciNet  Google Scholar 

  • Yakaryılmaz A, Say ACC (2009) Languages recognized with unbounded error by quantum finite automata. In: CSR’09: Proceedings of the fourth international computer science symposium in Russia, LNCS, vol 5675, pp 356–367

  • Yakaryılmaz A, Say ACC (2010) Languages recognized by nondeterministic quantum finite automata. Quantum Inf Comput 10(9, 10):747–770

    MathSciNet  MATH  Google Scholar 

  • Yakaryılmaz A, Say ACC (2011) Unbounded-error quantum computation with small space bounds. Inf Comput 279(6):873–892

    Article  Google Scholar 

  • Yamakami T, Yao AC (1999) \( {\text{ NQP }}_{\mathbb{C}} = \text{ co-C}_{=}\text{ P }\). Inf Process Lett 71(2):63–69

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

Arseny M. Shur was partially supported under the Agreement 02.A03.21.0006 of 27.08.2013 between the Ministry of Education and Science of the Russian Federation and Ural Federal University. Abuzer Yakaryılmaz was partially supported by CAPES with Grant 88881.030338/2013-01, ERC Advanced Grant MQC, and FP7 FET Projects QALGO.

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Authors and Affiliations

  1. Ural Federal University, Ekaterinburg, Russia

    Arseny M. Shur

  2. National Laboratory for Scientific Computing, Petrópolis, RJ, 25651-075, Brazil

    Abuzer Yakaryılmaz

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  1. Arseny M. Shur
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  2. Abuzer Yakaryılmaz
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Correspondence to Arseny M. Shur.

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Shur, A.M., Yakaryılmaz, A. More on quantum, stochastic, and pseudo stochastic languages with few states. Nat Comput 15, 129–141 (2016). https://doi.org/10.1007/s11047-015-9511-8

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