State succinctness of two-way finite automata with quantum and classical states

Zheng, Shenggen; Qiu, Daowen; Gruska, Jozef; Li, Lvzhou; Mateus, Paulo

doi:10.1016/j.tcs.2013.06.005

Cited by 27 publications

(32 citation statements)

References 22 publications

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“…Remark 1. Similar matrices as U a have been first used in [3] and also in [5,6,11,41,43,46]. Our proof has been inspired by the methods introduced in [7].…”

Section: Unary Promise Problemsmentioning

confidence: 97%

“…Quantum finite automata were first introduced by Kondacs and Watrous [29] and by Moore and Crutchfields [34]. Since that time they have been explored in [2,14,27,30,[40][41][42][43][44][45] and in other papers. The state complexity of quantum finite automata is one of the interesting topics in studying the power of quantum finite automata.…”

Section: Preliminariesmentioning

confidence: 99%

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Potential of Quantum Finite Automata with Exact Acceptance

Gruska

Qiu

Zheng

2015

Int. J. Found. Comput. Sci.

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The potential of the exact quantum information processing is an interesting, important and intriguing issue. For examples, it has been believed that quantum tools can provide significant, that is larger than polynomial, advantages in the case of exact quantum computation only, or mainly, for problems with very special structures. We will show that this is not the case.In this paper the potential of quantum finite automata producing outcomes not only with a (high) probability, but with certainty (so called exactly) is explored in the context of their uses for solving promise problems and with respect to the size of automata. It is shown that for solving particular classes {A n } ∞ n=1 of promise problems, even those without some very special structure, that succinctness of the exact quantum finite automata under consideration, with respect to the number of (basis) states, can be very small (and constant) though it grows proportional to n in the case deterministic finite automata (DFAs) of the same power are used. This is here demonstrated also for the case that the component languages of the promise problems solvable by DFAs are non-regular. The method used can be applied in finding more exact quantum finite automata or quantum algorithms for other promise problems.

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“…Remark 1. Similar matrices as U a have been first used in [3] and also in [5,6,11,41,43,46]. Our proof has been inspired by the methods introduced in [7].…”

Section: Unary Promise Problemsmentioning

confidence: 97%

Section: Preliminariesmentioning

confidence: 99%

Potential of Quantum Finite Automata with Exact Acceptance

Gruska

Qiu

Zheng

2015

Int. J. Found. Comput. Sci.

Self Cite

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“…An initial result was given to compare exact quantum and deterministic pushdown automata [19], the former one was shown to be more powerful (see also [20] and [21] for the results in this direction). Then, the result given by Ambainis and Yakaryılmaz [6], the state advantages of exact quantum finite automata (QFAs) over deterministic finite automata (DFAs) cannot be bounded in the case of unary promise problems, has stimulated the topic and a series of papers appeared on the succinctness of QFAs and other models [30,12,13,29,8,1]. In parallel, the new results were given on classical and quantum automata models [23,10]:…”

Section: Introductionmentioning

confidence: 99%

Unary Probabilistic and Quantum Automata on Promise Problems

Gainutdinova

Yakaryılmaz

2015

Developments in Language Theory

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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.

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“…Moreover, Rashid and Yakaryılmaz [33] presented many superiority results of qfas over their classical counterparts. There are also some new results on succinctness of qfas [40,17,18,39,3] and a new result on quantum pushdown automata [30].…”

Section: Introductionmentioning

confidence: 99%

Classical Automata on Promise Problems

Geffert

Yakaryılmaz

2014

Descriptional Complexity of Formal Systems

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Promise problems were mainly studied in quantum automata theory. Here we focus on state complexity of classical automata for promise problems. First, it was known that there is a family of unary promise problems solvable by quantum automata by using a single qubit, but the number of states required by corresponding one-way deterministic automata cannot be bounded by a constant. For this family, we show that even two-way nondeterminism does not help to save a single state. By comparing this with the corresponding state complexity of alternating machines, we then get a tight exponential gap between two-way nondeterministic and one-way alternating automata solving unary promise problems. Second, despite of the existing quadratic gap between Las Vegas realtime probabilistic automata and oneway deterministic automata for language recognition, we show that, by turning to promise problems, the tight gap becomes exponential. Last, we show that the situation is different for one-way probabilistic automata with two-sided bounded-error. We present a family of unary promise problems that is very easy for these machines; solvable with only two states, but the number of states in two-way alternating or any simpler automata is not limited by a constant. Moreover, we show that one-way bounded-error probabilistic automata can solve promise problems not solvable at all by any other classical model.We assume the reader is familiar with the basic standard models of finite state automata (see e.g. [19]). For a more detailed exposition related to probabilistic automata, the reader is referred to [31,4]. Here we only recall some models and introduce some elementary notation.By Σ * , we denote the set of strings over an alphabet Σ. The set of strings of length n is denoted by Σ n and the unique string of length zero by ε. By N + , we denote the set of all positive integers. The cardinality of a finite set S is denoted by S .A one-way nondeterministic finite automaton with ε-moves (1 ε nfa, for short) is a quintuple A = (S, Σ, H, s I , S A ), where S is a finite set of states, Σ an input alphabet, s I ∈ S an initial

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State succinctness of two-way finite automata with quantum and classical states

Cited by 27 publications

References 22 publications

Potential of Quantum Finite Automata with Exact Acceptance

Potential of Quantum Finite Automata with Exact Acceptance

Unary Probabilistic and Quantum Automata on Promise Problems

Classical Automata on Promise Problems

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