Abstract
We continue the systematic investigation of probabilistic and quantum finite automata (PFAs and QFAs) on promise problems by focusing on unary languages. We show that bounded-error QFAs are more powerful than PFAs. But, in contrary to the binary problems, the computational powers of Las-Vegas QFAs and bounded-error PFAs are equivalent to deterministic finite automata (DFAs). Lastly, we present a new family of unary promise problems with two parameters such that when fixing one parameter QFAs can be exponentially more succinct than PFAs and when fixing the other parameter PFAs can be exponentially more succinct than DFAs.
The extended version of the paper that contains the missing proofs is [10].
Abuzer Yakaryılmaz— Yakaryılmaz was partially supported by CAPES with grant 88881.030338/2013-01.
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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)
Ablayev, F., Gainutdinova, A.: On the lower bounds for one-way quantum automata. In: Nielsen, M., Rovan, B. (eds.) MFCS 2000. LNCS, vol. 1893, pp. 132–140. Springer, Heidelberg (2000)
Ablayev, F., Gainutdinova, A.: Complexity of quantum uniform and nonuniform automata. In: De Felice, C., Restivo, A. (eds.) DLT 2005. LNCS, vol. 3572, pp. 78–87. Springer, Heidelberg (2005)
Ablayev, F.M., Gainutdinova, A., Karpinski, M., Moore, C., Pollett, C.: On the computational power of probabilistic and quantum branching program. Information Computation 203(2), 145–162 (2005)
Ambainis, A., Freivalds, R.: 1-way quantum finite automata: strengths, weaknesses and generalizations. In: FOCS 1998, pp. 332–341 (1998)
Ambainis, A., Yakaryılmaz, A.: Superiority of exact quantum automata for promise problems. Information Processing Letters 112(7), 289–291 (2012)
Apostol, T.M.: Introduction to Analytic Number Theory. Springer (1976)
Bianchi, M.P., Mereghetti, C., Palano, B.: Complexity of promise problems on classical and quantum automata. In: Calude, C.S., Freivalds, R., Kazuo, I. (eds.) Gruska Festschrift. LNCS, vol. 8808, pp. 161–175. Springer, Heidelberg (2014)
Condon, A., Lipton, R.J.: On the complexity of space bounded interactive proofs (extended abstract). In: FOCS 1989, pp. 462–467 (1989)
Gainutdinova, A., Yakaryilmaz, A.: Unary probabilistic and quantum automata on promise problems. Technical Report arxiv.org/abs/1502.01462, arXiv (2015)
Geffert, V., Yakaryılmaz, A.: Classical automata on promise problems. In: Jürgensen, H., Karhumäki, J., Okhotin, A. (eds.) DCFS 2014. LNCS, vol. 8614, pp. 126–137. Springer, Heidelberg (2014)
Goldreich, O.: On promise problems: a survey. In: Goldreich, O., Rosenberg, A.L., Selman, A.L. (eds.) Theoretical Computer Science. LNCS, vol. 3895, pp. 254–290. Springer, Heidelberg (2006)
Gruska, J., Qiu, D., Zheng, S.: Generalizations of the distributed Deutsch-Jozsa promise problem. Technical report, arXiv (2014). arXiv:1402.7254
Gruska, J., Qiu, D., Zheng, S.: Potential of quantum finite automata with exact acceptance. Technical Report arXiv:1404.1689 (2014)
Hirvensalo, M.: Quantum automata with open time evolution. International Journal of Natural Computing 1(1), 70–85 (2010)
Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Van Nostrand, Princeton (1960)
Klauck, H.: On quantum and probabilistic communication: las vegas and one-way protocols. In: STOC 2000, pp. 644–651 (2000)
Mereghetti, C., Palano, B., Pighizzini, G.: Note on the succinctness of deterministic, nondeterministic, probabilistic and quantum finite automata. Theoretical Informatics and Applications 35(5), 477–490 (2001)
Moore, C., Crutchfield, J.P.: Quantum automata and quantum grammars. Theoretical Computer Science 237(1–2), 275–306 (2000)
Murakami, Y., Nakanishi, M., Yamashita, S., Watanabe, K.: Quantum versus classical pushdown automata in exact computation. IPSJ Digital Courier 1, 426–435 (2005)
Nakanishi, M.: Quantum pushdown automata with a garbage tape. In: Italiano, G.F., Margaria-Steffen, T., Pokorný, J., Quisquater, J.-J., Wattenhofer, R. (eds.) SOFSEM 2015-Testing. LNCS, vol. 8939, pp. 352–363. Springer, Heidelberg (2015)
Nakanishi, M., Yakaryılmaz, A.: Classical and quantum counter automata on promise problems (2014). (arXiv:1412.6761) (Accepted to CIAA2015)
Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press (2000)
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)
Salomaaa, A., Soittola, M.: Automata-Theoretic Aspects of Formal Power Series. Texts and monographs in computer science. Springer-Verlag (1978)
Cem Say, A.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)
Watrous, J.: Encyclopedia of Complexity and System Science. In: Quantum computational complexity (chapter). Springer (2009). arXiv:0804.3401
Yakaryılmaz, A., Cem Say, A.C.: Languages recognized by nondeterministic quantum finite automata. Quantum Information and Computation 10(9&10), 747–770 (2010)
Yakaryılmaz, A., Cem Say, A.C.: Unbounded-error quantum computation with small space bounds. Information and Computation 279(6), 873–892 (2011)
Zheng, S., Gruska, J., Qiu, D.: On the state complexity of semi-quantum finite automata. In: Dediu, A.-H., Martín-Vide, C., Sierra-Rodríguez, J.-L., Truthe, B. (eds.) LATA 2014. LNCS, vol. 8370, pp. 601–612. Springer, Heidelberg (2014)
Zheng, S., Qiu, D., Gruska, J., Li, L., Mateus, P.: State succinctness of two-way finite automata with quantum and classical states. Theoretical Computer Science 499, 98–112 (2013)
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Gainutdinova, A., Yakaryılmaz, A. (2015). Unary Probabilistic and Quantum Automata on Promise Problems. In: Potapov, I. (eds) Developments in Language Theory. DLT 2015. Lecture Notes in Computer Science(), vol 9168. Springer, Cham. https://doi.org/10.1007/978-3-319-21500-6_20
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DOI: https://doi.org/10.1007/978-3-319-21500-6_20
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