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Multi-letter quantum finite automata: decidability of the equivalence and minimization of states

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Abstract

Multi-letter quantum finite automata (QFAs) can be thought of quantum variants of the one-way multi-head finite automata (Hromkovič, Acta Informatica 19:377–384, 1983). It has been shown that this new one-way QFAs (multi-letter QFAs) can accept with no error some regular languages, for example (a + b)*b, that are not acceptable by QFAs of Moore and Crutchfield (Theor Comput Sci 237:275–306, 2000) as well as Kondacs and Watrous (66–75, 1997; Observe that 1-letter QFAs are exactly measure-once QFAs (MO-1QFAs) of Moore and Crutchfield (Theor Comput Sci 237:275–306, 2000)). In this paper, we study the decidability of the equivalence and minimization problems of multi-letter QFAs. Three new results presented in this paper are the following ones: (1) Given a k 1-letter QFA \({{\mathcal A}_1}\) and a k 2-letter QFA \({{\mathcal A}_2}\) over the same input alphabet Σ, they are equivalent if and only if they are (n 2 m k-1−m k-1 + k)-equivalent, where m = |Σ| is the cardinality of Σ, k = max(k 1,k 2), and n = n 1 + n 2, with n 1 and n 2 being numbers of states of \({{\mathcal A}_{1}}\) and \({{\mathcal A}_{2}}\) , respectively. When k = 1, this result implies the decidability of equivalence of measure-once QFAs (Moore and Crutchfield in Theor Comput Sci 237:275–306, 2000). (It is worth mentioning that our technical method is essentially different from the previous ones used in the literature.) (2) A polynomial-time O(m 2k-1 n 8 + km k n 6) algorithm is designed to determine the equivalence of any two multi-letter QFAs (see Theorems 2 and 3; Observe that if a brute force algorithm to determine equivalence would be used, as suggested by the decidability outcome of the point (1), the worst case time complexity would be exponential). Observe also that time complexity is expressed here in terms of the number of states of the multi-letter QFAs and k can be seen as a constant. (3) It is shown that the states minimization problem of multi-letter QFAs is solvable in EXPSPACE. This implies also that the state minimization problem of MO-1QFAs (see Moore and Crutchfield in Theor Comput Sci 237:275–306, 2000, page 304, Problem 5), an open problem stated in that paper, is also solvable in EXPSPACE.

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Authors and Affiliations

  1. Department of Computer Science, Sun Yat-sen University, Guangzhou, 510006, China

    Daowen Qiu, Lvzhou Li & Xiangfu Zou

  2. SQIG—Instituto de Telecomunicações, Departamento de Matemática, Instituto Superior Técnico, TULisbon, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    Daowen Qiu & Paulo Mateus

  3. The State Key Laboratory of Computer Science, Institute of Software, Chinese Academy of Sciences, Beijing, 100080, China

    Daowen Qiu

  4. Faculty of Informatics, Masaryk University, Brno, Czech Republik

    Jozef Gruska

  5. Department of Mathematics and Physics, Wuyi University, Jiangmen, 529020, China

    Xiangfu Zou

Authors
  1. Daowen Qiu
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  2. Lvzhou Li
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  3. Xiangfu Zou
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  4. Paulo Mateus
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  5. Jozef Gruska
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Correspondence to Daowen Qiu.

Additional information

This work is supported in part by the National Natural Science Foundation (Nos. 60873055, 61073054), the Natural Science Foundation of Guangdong Province of China (No. 10251027501000004), the Fundamental Research Funds for the Central Universities (Nos. 10lgzd12,11lgpy36), the Research Foundation for the Doctoral Program of Higher School of Ministry of Education (Nos. 20100171110042, 20100171120051) of China, the China Postdoctoral Science Foundation project (Nos. 20090460808, 201003375), the grant MSM0021622419 of Czech Republic, and the project of SQIG at IT, funded by FCT and EU FEDER projects projects QSec PTDC/EIA/67661/2006, AMDSC UTAustin/MAT/0057/2008, NoE Euro-NF, and IT Project QuantTel.

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Qiu, D., Li, L., Zou, X. et al. Multi-letter quantum finite automata: decidability of the equivalence and minimization of states. Acta Informatica 48, 271 (2011). https://doi.org/10.1007/s00236-011-0139-6

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