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Natural Computing

, Volume 17, Issue 2, pp 283–293 | Cite as

Language recognition power and succinctness of affine automata

  • Marcos VillagraEmail author
  • Abuzer Yakaryılmaz
Article

Abstract

In this work we study a non-linear generalization based on affine transformations of probabilistic and quantum automata proposed recently by Díaz-Caro and Yakaryılmaz (in: Computer science—theory and applications, LNCS, vol 9691. Springer, pp 1–15, 2016. ArXiv:1602.04732) referred to as affine automata. First, we present efficient simulations of probabilistic and quantum automata by means of affine automata which characterizes the class of exclusive stochastic languages. Then we initiate a study on the succintness of affine automata. In particular, we show that an infinite family of unary regular languages can be recognized by 2-state affine automata, whereas the number of inner states of quantum and probabilistic automata cannot be bounded. Finally, we present a characterization of all (regular) unary languages recognized by two-state affine automata.

Keywords

Probabilistic automata Quantum automata Affine automata State complexity Stochastic language Bounded-error One-sided error 

Notes

Acknowledgements

We thank to the anonymous referees for their helpful comments. Yakaryılmaz was partially supported by CAPES with Grant 88881.030338/2013-01 and ERC Advanced Grant MQC.

References

  1. Ablayev FM, Gainutdinova A, Khadiev K, Yakaryılmaz A (2014) Very narrow quantum OBDDs and width hierarchies for classical OBDDs. In: Descriptional complexity of formal systems, LNCS, vol 8614. Springer, pp 53–64Google Scholar
  2. Ambainis A, Freivalds R (1998) 1-way quantum finite automata: strengths, weaknesses and generalizations. In: FOCS’98. pp 332–341. ArXiv:9802062
  3. Ambainis A, Yakaryılmaz A (2012) Superiority of exact quantum automata for promise problems. Inf Process Lett 112(7):289–291MathSciNetCrossRefzbMATHGoogle Scholar
  4. Ambainis A, Yakaryılmaz A (2015) Automata and quantum computing. Tech. Rep. ArXiv:1507.01988
  5. Belovs A, Montoya J A, Yakaryılmaz A (2016) Can one quantum bit separate any pair of words with zero-error? Tech. Rep. ArXiv:1602.07967
  6. Díaz-Caro A, Yakaryılmaz A (2016) Affine computation and affine automaton. In: Computer science—theory and applications, LNCS, vol 9691. Springer, pp 1–15. ArXiv:1602.04732
  7. Freivalds R, Karpinski M (1994) Lower space bounds for randomized computation. In: ICALP’94. pp 580–592Google Scholar
  8. Gainutdinova A, Yakaryılmaz A (2015) Unary probabilistic and quantum automata on promise problems. In: Developments in language theory, LNCS, vol 9168. Springer, pp 252–263Google Scholar
  9. Geffert V, Yakaryılmaz A (2015) Classical automata on promise problems. Discrete Math Theor Comput Sci 17(2):157–180MathSciNetzbMATHGoogle Scholar
  10. Gruska J, Qiu D, Zheng S (2015) Potential of quantum finite automata with exact acceptance. Int J Found Comput Sci 26(3):381–398MathSciNetCrossRefzbMATHGoogle Scholar
  11. Hirvensalo M (2010) Quantum automata with open time evolution. Int J Nat Comput 1(1):70–85MathSciNetCrossRefGoogle Scholar
  12. Hirvensalo M, Moutot E, Yakaryılmaz A (2017) On the computational power of affine automata. Lang Automat Theory Appl Lect Notes Comput Sci 10168:405–417MathSciNetCrossRefzbMATHGoogle Scholar
  13. Ibrahimov R, Khadiev K, Prūsis K, Yakaryılmaz A (2017) Zero-error affine, unitary, and probabilistic OBDDs. Tech. Rep. ArXiv:1703.07184
  14. Kupferman O, Ta-Shma A, Vardi M Y (1999) Counting with automata. In: Short paper presented at the 15th annual IEEE symposium on logic in computer science (LICS 2000)Google Scholar
  15. Li L, Qiu D, Zou X, Li L, Wu L, Mateus P (2012) Characterizations of one-way general quantum finite automata. Theor Comput Sci 419:73–91MathSciNetCrossRefzbMATHGoogle Scholar
  16. Macarie II (1998) Space-efficient deterministic simulation of probabilistic automata. SIAM J Comput 27(2):448–465MathSciNetCrossRefzbMATHGoogle Scholar
  17. Moore C, Crutchfield JP (2000) Quantum automata and quantum grammars. Theor Comput Sci 237(1–2):275–306MathSciNetCrossRefzbMATHGoogle Scholar
  18. Nakanishi M, Khadiev K, Prūsis K, Vihrovs J, Yakaryılmaz A (2017) Affine counter automata. In: Proceedings 15th international conference on automata and formal languages, EPTCS, vol 252. pp 205–218Google Scholar
  19. Paz A (1971) Introduction to probabilistic automata. Academic Press, New YorkzbMATHGoogle Scholar
  20. Rabin M, Scott D (1959) Finite automata and their decision problems. IBM J Res Dev 3:114–125MathSciNetCrossRefzbMATHGoogle Scholar
  21. Rabin MO (1963) Probabilistic automata. Inf Control 6:230–243CrossRefzbMATHGoogle Scholar
  22. Rashid J, Yakaryılmaz A (2014) Implications of quantum automata for contextuality. In: Implementation and application of automata, LNCS, vol 8587. Springer, pp 318–331. ArXiv:1404.2761
  23. Say ACC, Yakaryılmaz A (2014) Quantum finite automata: A modern introduction. In: Computing with new resources, LNCS, vol 8808. Springer, pp 208–222Google Scholar
  24. Shur AM, Yakaryılmaz A (2016) More on quantum, stochastic, and pseudo stochastic languages with few states. Nat Comput 15(1):129–141MathSciNetCrossRefGoogle Scholar
  25. Turakainenn P (1975) Word-functions of stochastic and pseudo stochastic automata. Ann Acad Sci Fenn Ser A I Math 1:27–37MathSciNetCrossRefzbMATHGoogle Scholar
  26. Villagra M, Yakaryılmaz A (2016) Language recognition power and succintness of affine automata. In: Unconventional computation and natural computation, LNCS, vol. 9726. Springer, pp 116–129Google Scholar
  27. Yakaryılmaz A, Say ACC (2009) Languages recognized with unbounded error by quantum finite automata. In: Computer science—theory and applications, LNCS, vol 5675. Springer, pp 356–367Google Scholar
  28. Yakaryılmaz A, Say ACC (2010) Languages recognized by nondeterministic quantum finite automata. Quantum Inf Comput 10(9&10):747–770MathSciNetzbMATHGoogle Scholar
  29. Yakaryılmaz A, Say ACC (2010) Succinctness of two-way probabilistic and quantum finite automata. Discrete Math Theor Comput Sci 12(2):19–40MathSciNetzbMATHGoogle Scholar
  30. Yakaryılmaz A, Say ACC (2011) Unbounded-error quantum computation with small space bounds. Inf Comput 279(6):873–892MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Núcleo de Investigación y Desarrollo TecnológicoUniversidad Nacional de Asunción Campus UniversitarioSan LorenzoParaguay
  2. 2.Center of Quantum Computer ScienceUniversity of LatviaRigaLatvia

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