56 research outputs found

    On equivalence, languages equivalence and minimization of multi-letter and multi-letter measure-many quantum automata

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    We first show that given a k1k_1-letter quantum finite automata A1\mathcal{A}_1 and a k2k_2-letter quantum finite automata A2\mathcal{A}_2 over the same input alphabet Σ\Sigma, they are equivalent if and only if they are (n12+n221)Σk1+k(n_1^2+n_2^2-1)|\Sigma|^{k-1}+k-equivalent where n1n_1, i=1,2i=1,2, are the numbers of state in Ai\mathcal{A}_i respectively, and k=max{k1,k2}k=\max\{k_1,k_2\}. By applying a method, due to the author, used to deal with the equivalence problem of {\it measure many one-way quantum finite automata}, we also show that a k1k_1-letter measure many quantum finite automaton A1\mathcal{A}_1 and a k2k_2-letter measure many quantum finite automaton A2\mathcal{A}_2 are equivalent if and only if they are (n12+n221)Σk1+k(n_1^2+n_2^2-1)|\Sigma|^{k-1}+k-equivalent where nin_i, i=1,2i=1,2, are the numbers of state in Ai\mathcal{A}_i respectively, and k=max{k1,k2}k=\max\{k_1,k_2\}. Next, we study the language equivalence problem of those two kinds of quantum finite automata. We show that for kk-letter quantum finite automata, the non-strict cut-point language equivalence problem is undecidable, i.e., it is undecidable whether Lλ(A1)=Lλ(A2)L_{\geq\lambda}(\mathcal{A}_1)=L_{\geq\lambda}(\mathcal{A}_2) where 0<λ10<\lambda\leq 1 and Ai\mathcal{A}_i are kik_i-letter quantum finite automata. Further, we show that both strict and non-strict cut-point language equivalence problem for kk-letter measure many quantum finite automata are undecidable. The direct consequences of the above outcomes are summarized in the paper. Finally, we comment on existing proofs about the minimization problem of one way quantum finite automata not only because we have been showing great interest in this kind of problem, which is very important in classical automata theory, but also due to that the problem itself, personally, is a challenge. This problem actually remains open.Comment: 30 pages, conclusion section correcte

    Quantum Pushdown Automata

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    Quantum finite automata, as well as quantum pushdown automata (QPA) were first introduced by C. Moore and J. P. Crutchfield. In this paper we introduce the notion of QPA in a non-equivalent way, including unitarity criteria, by using the definition of quantum finite automata of Kondacs and Watrous. It is established that the unitarity criteria of QPA are not equivalent to the corresponding unitarity criteria of quantum Turing machines. We show that QPA can recognize every regular language. Finally we present some simple languages recognized by QPA, not recognizable by deterministic pushdown automata.Comment: Conference SOFSEM 2000, extended version of the pape

    Turing machines based on unsharp quantum logic

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    In this paper, we consider Turing machines based on unsharp quantum logic. For a lattice-ordered quantum multiple-valued (MV) algebra E, we introduce E-valued non-deterministic Turing machines (ENTMs) and E-valued deterministic Turing machines (EDTMs). We discuss different E-valued recursively enumerable languages from width-first and depth-first recognition. We find that width-first recognition is equal to or less than depth-first recognition in general. The equivalence requires an underlying E value lattice to degenerate into an MV algebra. We also study variants of ENTMs. ENTMs with a classical initial state and ENTMs with a classical final state have the same power as ENTMs with quantum initial and final states. In particular, the latter can be simulated by ENTMs with classical transitions under a certain condition. Using these findings, we prove that ENTMs are not equivalent to EDTMs and that ENTMs are more powerful than EDTMs. This is a notable difference from the classical Turing machines.Comment: In Proceedings QPL 2011, arXiv:1210.029

    Another approach to the equivalence of measure-many one-way quantum finite automata and its application

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    In this paper, we present a much simpler, direct and elegant approach to the equivalence problem of {\it measure many one-way quantum finite automata} (MM-1QFAs). The approach is essentially generalized from the work of Carlyle [J. Math. Anal. Appl. 7 (1963) 167-175]. Namely, we reduce the equivalence problem of MM-1QFAs to that of two (initial) vectors. As an application of the approach, we utilize it to address the equivalence problem of {\it Enhanced one-way quantum finite automata} (E-1QFAs) introduced by Nayak [Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, 1999, pp.~369-376]. We prove that two E-1QFAs A1\mathcal{A}_1 and A2\mathcal{A}_2 over Σ\Sigma are equivalence if and only if they are n12+n221n_1^2+n_2^2-1-equivalent where n1n_1 and n2n_2 are the numbers of states in A1\mathcal{A}_1 and A2\mathcal{A}_2, respectively.Comment: V 10: Corollary 3 is deleted, since it is folk. (V 9: Revised in terms of the referees's comments) All comments, especially the linguistic comments, are welcom

    Finite state verifiers with constant randomness

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    We give a new characterization of NL\mathsf{NL} as the class of languages whose members have certificates that can be verified with small error in polynomial time by finite state machines that use a constant number of random bits, as opposed to its conventional description in terms of deterministic logarithmic-space verifiers. It turns out that allowing two-way interaction with the prover does not change the class of verifiable languages, and that no polynomially bounded amount of randomness is useful for constant-memory computers when used as language recognizers, or public-coin verifiers. A corollary of our main result is that the class of outcome problems corresponding to O(log n)-space bounded games of incomplete information where the universal player is allowed a constant number of moves equals NL.Comment: 17 pages. An improved versio
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