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arXiv:0903.0050 · Discrete Mathematics & Theoretical Computer Science

Succinctness of two-way probabilistic and quantum finite automata

Abuzer Yakaryılmaz, A. C. Cem Say

Published Jan 1, 2010• 50 citations
arXivHTMLPDF

Abstract

special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications We introduce a new model of two-way finite automaton, which is endowed with the capability of resetting the position of the tape head to the left end of the tape in a single move during the computation. Several variants of this model are examined, with the following results: The weakest known model of computation where quantum computers recognize more languages with bounded error than their classical counterparts is identified. We prove that two-way probabilistic and quantum finite automata (2PFAs and 2QFAs) can be considerably more concise than both their one-way versions (1PFAs and 1QFAs), and two-way nondeterministic finite automata (2NFAs). For this purpose, we demonstrate several infinite families of regular languages which can be recognized with some fixed probability greater than 1 2 by just tuning the transition amplitudes of a 2QFA (and, in one case, a 2PFA) with a constant number of states, whereas the sizes of the corresponding 1PFAs, 1QFAs and 2NFAs grow without bound. We also show that 2QFAs with mixed states can support highly efficient probability amplification.

DOI
10.46298/dmtcs.509

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Abstract. We prove that several variants of the quantum finite automaton (QFA) model are strictly superior to their probabilistic counterparts, in terms of both language recognition power and succinctness, in the onesided unbounded error setting. This is the first case where one-way QFA models have been shown to outperform the corresponding probabilistic automata in both these respects. We give a characterization of the class of languages recognized by these QFA’s with cutpoint 0, demonstrating that it is equal to the class of exclusive stochastic languages. We prove several closure properties of the related classes. 1

•Feb 12, 2009•2 citations
special issue dedicated to the second edition of the conference AutoMathA: from Mathematics to Applications We introduce a new model of two-way finite automaton, which is endowed with the capability of resetting the position of the tape head to the left end of the tape in a single move during the computation. Several variants of this model are examined, with the following results: The weakest known model of computation where quantum computers recognize more languages with bounded error than their classical counterparts is identified. We prove that two-way probabilistic and quantum finite automata (2PFAs and 2QFAs) can be considerably more concise than both their one-way versions (1PFAs and 1QFAs), and two-way nondeterministic finite automata (2NFAs). For this purpose, we demonstrate several infinite families of regular languages which can be recognized with some fixed probability greater than 1 2 by just tuning the transition amplitudes of a 2QFA (and, in one case, a 2PFA) with a constant number of states, whereas the sizes of the corresponding 1PFAs, 1QFAs and 2NFAs grow without bound. We also show that 2QFAs with mixed states can support highly efficient probability amplification.
In this paper, we introduce 1-way and 2-way quantum finite state automata (1qfa's and 2qfa's), which are the quantum analogues of deterministic, nondeterministic and probabilistic 1-way and 2-way finite state automata. We prove the following facts regarding 2qfa's. 1. For any /spl epsiv/>0, there is a 2qfa M which recognizes the non-regular language L={a/sup m/b/sup m/|m/spl ges/1} with (one-sided) error bounded by E, and which halts in linear time. Specifically, M accepts any string in L with probability 1 and rejects any string not in L with probability at least 1-/spl epsiv/. 2. For every regular language L, there is a reversible (and hence quantum) 2-way finite state automaton which recognizes L and which runs in linear time. In fact, it is possible to define 2qfar's which recognize the non-context-free language {a/sup m/b/sup m/c/sup m/|m/spl ges/1}, based on the same technique used for 1. Consequently, the class of languages recognized by linear time, bounded error 2qfa's properly includes the regular languages. Since it is known that 2-way deterministic, nondeterministic and polynomial expected time, bounded error probabilistic finite automata can recognize only regular languages, it follows that 2qfa's are strictly more powerful than these "classical" models. In the case of 1-way automata, the situation is reversed. We prove that the class of languages recognizable by bounded error 1qfa's is properly contained in the class of regular languages.
1-way quantum finite state automata are reversible in nature, which greatly reduces its accepting property. In fact, the set of languages accepted by 1-way quantum finite automata is a proper subset of regular languages. We introduce 2-tape 1-way quantum finite state automaton (2T1QFA(2))which is a modified version of 1-way 2-head quantum finite state automaton(1QFA(2)). In this paper, we replace the single tape of 1-way 2-head quantum finite state automaton with two tapes. The content of the second tape is determined using a relation defined on input alphabet. The main claims of this paper are as follows: (1)We establish that 2-tape 1-way quantum finite state automaton(2T1QFA(2)) can accept all regular languages (2)A language which cannot be accepted by any multi-head deterministic finite automaton can be accepted by 2-tape 1-way quantum finite state automaton(2T1QFA(2)) .(3) Exploiting the superposition property of quantum automata we show that 2-tape 1-way quantum finite state automaton(2T1QFA(2)) can accept the language L=ww.
In automata theory, quantum computation has been widely examined for finite state machines, known as quantum finite automata (QFAs), and less attention has been given to QFAs augmented with counters or stacks. In this paper, we focus on such generalizations of QFAs where the input head operates in one-way or realtime mode, and present some new results regarding their superiority over their classical counterparts. Our first result is about the nondeterministic acceptance mode: Each quantum model architecturally intermediate between realtime finite state automaton and one-way pushdown automaton (one-way finite automaton, realtime and one-way finite automata with one-counter, and realtime pushdown automaton) is superior to its classical counterpart. The second and third results are about bounded error language recognition: for any k > 0, QFAs with k blind counters outperform their deterministic counterparts; and, a one-way QFA with a single head recognizes an infinite family of languages, which can be recognized by one-way probabilistic finite automata with at least two heads. Lastly, we compare the nondeterminictic and deterministic acceptance modes for classical finite automata with k blind counter(s), and we show that for any k > 0, the nondeterministic models outperform the deterministic ones.
In this note, we generalize the results of arXiv:0901.2703v1 We show that all one-way quantum finite automaton (QFA) models that are at least as general as Kondacs-Watrous QFA's are equivalent in power to classical probabilistic finite automata in this setting. Unlike their probabilistic counterparts, allowing the tape head to stay put for some steps during its traversal of the input does enlarge the class of languages recognized by such QFA's with unbounded error. (Note that, the proof of Theorem 1 in the abstract was presented in the previous version (arXiv:0901.2703v1).)
We show that there are quantum devices that accept all regular languages and that are exponentially more concise than deterministic finite automata (DFA). For this purpose, we introduce a new computing model of {\it one-way quantum finite automata} (1QFA), namely, {\it one-way quantum finite automata together with classical states} (1QFAC), which extends naturally both measure-only 1QFA and DFA and whose state complexity is upper-bounded by both. The original contributions of the paper are the following. First, we show that the set of languages accepted by 1QFAC with bounded error consists precisely of all regular languages. Second, we prove that 1QFAC are at most exponentially more concise than DFA. Third, we show that the previous bound is tight for families of regular languages that are not recognized by measure-once (RMO), measure-many (RMM) and multi-letter 1QFA. % More concretely we exhibit regular languages L0(m)L^0(m)L0(m) for mmm prime such that: (i) L0(m)L^0(m)L0(m) cannot be recognized by measure-once, measure-many and multi-letter 1QFA; (ii) the minimal DFA that accepts L0(m)L^0(m)L0(m) has O(m)O(m)O(m) states; (iii) there is a 1QFAC with constant classical states and O(log⁡(m))O(\log(m))O(log(m)) quantum basis that accepts L0(m)L^0(m)L0(m). Fourth, we give a polynomial-time algorithm for determining whether any two 1QFAC are equivalent. Finally, we show that state minimization of 1QFAC is decidable within EXPSPACE. We conclude the paper by posing some open problems.
Abstract. We prove that several variants of the quantum finite automaton (QFA) model are strictly superior to their probabilistic counterparts, in terms of both language recognition power and succinctness, in the onesided unbounded error setting. This is the first case where one-way QFA models have been shown to outperform the corresponding probabilistic automata in both these respects. We give a characterization of the class of languages recognized by these QFA’s with cutpoint 0, demonstrating that it is equal to the class of exclusive stochastic languages. We prove several closure properties of the related classes. 1