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 [6] referred as affine automata. First, we present efficient simulations of probabilistic and quantum automata by means of affine automata which allows us to characterize 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 states of any quantum and probabilistic automata cannot be bounded. Finally, we present the characterization of all (regular) unary languages recognized by two-state affine automata.
The omitted proofs can be found in [22].
A.Yakaryılmaz—Yakaryılmaz was partially supported by CAPES with grant 88881.030338/2013-01 and some parts of this work was done while he was visiting Universidad Nacional de Asunción in September 2015.
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Notes
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This way of scanning an input tape is sometimes referred to as “strict realtime.”.
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- 3.
Pures states are vectors in a complex Hilbert space normalized with respect to the \(\ell _2\)-norm.
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A promise problem \(\mathtt L=(L_{yes},L_{no})\) is solved by a machine M, or M solves \(\mathtt L\), if for all \( w \in \mathtt L_{yes}\), M accepts w, and for all \( w \in \mathtt L_{no}\), M rejects w.
References
Ablayev, F., Gainutdinova, A., Khadiev, K., Yakaryılmaz, A.: Very narrow quantum OBDDs and width hierarchies for classical OBDDs. In: Jürgensen, H., Karhumäki, J., Okhotin, A. (eds.) DCFS 2014. LNCS, vol. 8614, pp. 53–64. Springer, Heidelberg (2014)
Ambainis, A., Freivalds, R.: 1-way quantum finite automata: strengths, weaknesses and generalizations. In: FOCS 1998, pp. 332–341 (1998). arXiv:9802062
Ambainis, A., Yakaryılmaz, A.: Superiority of exact quantum automata for promise problems. Inf. Process. Lett. 112(7), 289–291 (2012)
Ambainis, A., Yakaryılmaz, A.: Automata and quantum computing. Technical report 1507.01988, arXiv (2015)
Belovs, A., Montoya, J.A., Yakaryılmaz, A.: Can one quantum bit separate any pair of words with zero-error? Technical report 1602.07967, arXiv (2016)
Díaz-Caro, A., Yakaryılmaz, A.: Affine computation and affine automaton. In: Kulikov, A.S., Woeginger, G.J. (eds.) CSR 2016. LNCS, vol. 9691, pp. 146–160. Springer, Heidelberg (2016). doi:10.1007/978-3-319-34171-2_11
Freivalds, R., Karpinski, M.: Lower space bounds for randomized computation. In: ICALP 1994, pp. 580–592 (1994)
Gainutdinova, A., Yakaryılmaz, A.: Unary probabilistic and quantum automata on promise problems. In: Potapov, I. (ed.) DLT 2015. LNCS, vol. 9168, pp. 252–263. Springer, Heidelberg (2015)
Geffert, V., Yakaryılmaz, A.: Classical automata on promise problems. Discrete Math. Theor. Comput. Sci. 17(2), 157–180 (2015)
Gruska, J., Qiu, D., Zheng, S.: Potential of quantum finite automata with exact acceptance. Int. J. Found. Comput. Sci. 26(3), 381–398 (2015)
Kupferman, O., Ta-Shma, A., Vardi, M.Y.: Counting with automata. Short paper presented at the 15th Annual IEEE Symposium on Logic in Computer Science (LICS 2000) (1999)
Li, L., Qiu, D., Zou, X., Li, L., Wu, L., Mateus, P.: Characterizations of one-way general quantum finite automata. Theor. Comput. Sci. 419, 73–91 (2012)
Macarie, I.I.: Space-efficient deterministic simulation of probabilistic automata. SIAM J. Comput. 27(2), 448–465 (1998)
Moore, C., Crutchfield, J.P.: Quantum automata and quantum grammars. Theor. Comput. Sci. 237(1–2), 275–306 (2000)
Paz, A.: Introduction to Probabilistic Automata. Academic Press, New York (1971)
Rabin, M.O.: Probabilistic automata. Inf. Control 6, 230–243 (1963)
Rabin, M., Scott, D.: Finite automata and their decision problems. IBM J. Res. Dev. 3, 114–125 (1959)
Rashid, J., Yakaryılmaz, A.: Implications of quantum automata for contextuality. In: Holzer, M., Kutrib, M. (eds.) CIAA 2014. LNCS, vol. 8587, pp. 318–331. Springer, Heidelberg (2014)
Say, A.C.C., Yakaryılmaz, A.: Quantum finite automata: a modern introduction. In: Calude, C.S., Freivalds, R., Kazuo, I. (eds.) Gruska Festschrift. LNCS, vol. 8808, pp. 208–222. Springer, Heidelberg (2014)
Shur, A.M., Yakaryılmaz, A.: More on quantum, stochastic, and pseudo stochastic languages with few states. Nat. Comput. 15(1), 129–141 (2016)
Turakainenn, P.: Word-functions of stochastic and pseudo stochastic automata. Ann. Acad. Scientiarum Fennicae, Ser. A. I Math. 1, 27–37 (1975)
Villagra, M., Yakaryılmaz, A.: Language recognition power and succintness of affine automata. Technical report 1602.05432, arXiv (2016)
Yakaryılmaz, A., Say, A.C.C.: Succinctness of two-way probabilistic and quantum finite automata. Discrete Math. Theor. Comput. Sci. 12(2), 19–40 (2010)
Yakaryılmaz, A., Say, A.C.C.: Languages recognized with unbounded error by quantum finite automata. In: Frid, A., Morozov, A., Rybalchenko, A., Wagner, K.W. (eds.) CSR 2009. LNCS, vol. 5675, pp. 356–367. Springer, Heidelberg (2009)
Yakaryılmaz, A., Say, A.C.C.: Languages recognized by nondeterministic quantum finite automata. Quantum Inf. Comput. 10(9&10), 747–770 (2010)
Yakaryılmaz, A., Say, A.C.C.: Unbounded-error quantum computation with small space bounds. Inf. Comput. 279(6), 873–892 (2011)
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We thank the anonymous referees for their helpful comments.
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Villagra, M., Yakaryılmaz, A. (2016). Language Recognition Power and Succinctness of Affine Automata. In: Amos, M., CONDON, A. (eds) Unconventional Computation and Natural Computation. UCNC 2016. Lecture Notes in Computer Science(), vol 9726. Springer, Cham. https://doi.org/10.1007/978-3-319-41312-9_10
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