We design Latvian quantum finite state automata (LQFAs for short) recognizing unary regular languages with isolated cut point 1 2 .From an architectural point of view, we combine two LQFAs recognizing with isolated cut point, respectively, the finite part and the ultimately periodic part of any given unary regular language L. In both modules, we use a component addressed in the literature and here suitably adapted to the unary case, to discriminate strings on the basis of their length.The number of basis states and the isolation around the cut point of the resulting LQFA for L exponentially depends on the size of the minimal deterministic finite state automaton for L.
We design Latvian quantum finite state automata (LQFAs for short) recognizing unary regular languages with isolated cut point 1 2 .From an architectural point of view, we combine two LQFAs recognizing with isolated cut point, respectively, the finite part and the ultimately periodic part of any given unary regular language L. In both modules, we use a component addressed in the literature and here suitably adapted to the unary case, to discriminate strings on the basis of their length.The number of basis states and the isolation around the cut point of the resulting LQFA for L exponentially depends on the size of the minimal deterministic finite state automaton for L.
We design Latvian quantum finite state automata (lqfas) recognizing unary regular languages with isolated cut point [Formula: see text]. From an architectural viewpoint, we suitably combine two lqfas recognizing with isolated cut point, respectively, the finite part and the ultimately periodic part any given unary regular language [Formula: see text] consists of. In particular, both these lqfas incorporate a sub-module discriminating strings on the basis of their length. Both the number of basis states and the isolation around the cut point of the resulting lqfa for [Formula: see text] exponentially depend on the size of the minimal deterministic finite state automaton for [Formula: see text]. Moreover, the recognition of [Formula: see text] tends to becoming deterministic as the number of the basis states employed in the length-discriminating sub-module grows.
We study quantum finite automata with control language (qfcs), a theoretical model for finite memory hybrid systems coupling a classical computational framework with a quantum component. We constructively show how to simulate measure-once, measure-many, reversible, and Latvian qfas by qfcs, emphasizing the size cost of such simulations. Next, we prove the decidability of testing the periodicity of the stochastic event induced by a given qfc. Thanks to our qfa simulations, we can extend such a decidability result to measure-once, measure-many, reversible, and Latvian qfas as well. Finally, we focus on comparing the size efficiency of quantum and classical finite state automata on unary regular language recognition. We show that unary regular languages can be recognized by isolated cut point qfcs for which the size is generally quadratically smaller than the size of equivalent dfas.
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.
The computational model of Quantum Finite Automata has been introduced by multiple authors (e.g. [38, 44]) with some variations in definition. The objective of this thesis is to understand what class of languages can be recognized by these different variations, and how many states are required. We begin by showing that we can use algebraic automata theory to characterize the language recognition power of QFAs. Algebraic automata theory associates to each language a canonical syntactic monoid, and the algebraic structure of this monoid becomes a meaningful parameter in describing language classes. We show that the class of languages recognized by Latvian QFAs [3] corresponds exactly to boolean combinations of languages recognized by Brodsky and Pippenger's QFA model [20], which correspond exactly to those languages whose syntactic monoid is in the class BG. Known results give us a decision procedure for testing membership in this language class. We also use algebraic automata theory to give nearly tight upper and lower bounds on the class of languages recognized by Brodsky and Pippenger's QFAs. We then extend a number of lower bound techniques known for Kondacs and Watrous' 1-way QFA model to Nayak's Generalized QFA. Both of these models are related in that they are permitted to halt before reading the entire input, allowing them to recognize certain languages whose syntactic monoid lies outside of BG. Finally, we investigate the question of QFA succinctness. It is known that QFAs can recognize some languages using exponentially fewer states compared to deterministic finite automata. We extend results from [16] to show that the word problem over abelian groups has this property. We also give example of interesting noncommutative languages with this property.
One-way quantum finite automata (1QFA) proposed by Moore and Crutchfield and by Kondacs and Watrous accept only subsets of regular languages with bounded error. In this paper, we develop a new computing model of 1QFA, namely, one-way quantum finite automata together with classical states (1QFAC for short). In this model, a component of classical states is added, and the choice of unitary evolution of quantum states at each step is closely related to the current classical state. 1QFAC can accept all regular languages with no error, and in particular, 1QFAC can accept some languages with essentially less number of states than deterministic finite automata (DFA). The main technical results are as follows. (1) We prove that the set of languages accepted by 1QFAC with bounded error consists precisely of all regular languages. (2) We show that, for any prime number m ≥ 2, there exists a regular language L0(m) whose minimal DFA needs O(m) states and that can not be accepted by the 1QFA proposed by Moore and Crutchfield and by Kondacs and Watrous, but there exists 1QFAC accepting L0(m) with only constant classical states and O(log(m)) quantum basis states. Analogous results for multi-letter automata are also established. (3) By a bilinearization technique we prove that any two 1QFAC A1 and A2 are equivalent if and only if they are (k1n1) 2 + (k2n2) 2 − 1-equivalent, and there exists a polynomial-time O([(k1n1) 2 + (k2n2) 2]4) algorithm for determining their equivalence, where k1 and k2 are the numbers of classical states of A1 and A2, as well as n1 and n2 are the numbers of quantum basis states of A1 andA2, respectively. (4) We show that the minimization problem of 1QFAC is in EXPSPACE. As a corollary of this result, we obtain that the minimization problem of MO-1QFA is also in EXPSPACE. Finally, we draw some conclusions and point out open ∗Corresponding author. E-mail address: issqdw@mail.sysu.edu.cn (D. Qiu). †E-mail address: {paulo.mateus,amilcar.sernadas}@math.ist.utl.pt.
We consider the model of one-way automata with quantum and classical states (QCFAs) introduced in [28]. We show, by a direct approach, that QCFAs with isolated cut-point accept regular languages only, thus characterizing their computational power. Concerning descriptional power, we quickly overview a size lower bound for QCFAs accepting regular languages, and address its optimality. Then, we explicitly build QCFAs accepting the word quotients and inverse homomorphic images of languages accepted by given QCFAs with isolated cut-point, maintaining the same cut-point, isolation, and only polynomially increasing the size.