Two-way Probabilistic Finite Automata Research Articles

Lifting query complexity to time-space complexity for two-way finite automata

Time-space tradeoff has been studied in a variety of models, such as Turing machines, branching programs, and finite automata, etc. While communication complexity as a technique has been applied to study finite automata, it seems it has not been used to study time-space tradeoffs of finite automata. We design a new technique showing that separations of query complexity can be lifted, via communication complexity, to separations of time-space complexity of two-way finite automata. As an application, one of our main results exhibits the first example of a language L such that the time-space complexity of two-way probabilistic finite automata with a bounded error (2PFA) is Ω˜(n2), while of exact two-way quantum finite automata with classical states (2QCFA) is O˜(n5/3), that is, we demonstrate for the first time that exact quantum computing has an advantage in time-space complexity comparing to classical computing.

Modeling of RNA secondary structures using two-way quantum finite automata

Quantum finite automata (QFA) play a crucial role in quantum information processing theory. The representation of ribonucleic acid (RNA) and deoxyribonucleic acid (DNA) structures using theory of automata had a great influence in the computer science. Investigation of the relationship between QFA and RNA structures is a natural goal. Two-way quantum finite automata (2QFA) is more dominant than its classical model in language recognition. Motivated by the concept of quantum finite automata, we have modeled RNA secondary structure loops such as, internal loop and double helix loop using two-way quantum finite automata. The major benefit of this approach is that these sequences can be parsed in linear time. In contrast, two-way deterministic finite automata (2DFA) cannot represent RNA secondary structure loops and two-way probabilistic finite automata (2PFA) can parse these sequences in exponential time. To the best of authors knowledge this is the first attempt to represent RNA secondary structure loops using two-way quantum finite automata.

Power of the interactive proof systems with verifiers modeled by semi-quantum two-way finite automata

Interactive proof systems (IP) are very powerful – languages they can accept form exactly PSPACE. They represent also one of the very fundamental concepts of theoretical computing and a model of computation by interactions. One of the key players in IP is verifier. In the original model of IP whose power is that of PSPACE, the only restriction on verifiers is that they work in randomized polynomial time. Because of such key importance of IP, it is of large interest to find out how powerful will IP be when verifiers are more restricted. So far this was explored for the case that verifiers are two-way probabilistic finite automata (Dwork and Stockmeyer, 1990) and one-way quantum finite automata as well as two-way quantum finite automata (Nishimura and Yamakami, 2009). IP in which verifiers use public randomization is called Arthur–Merlin proof systems (AM). AM with verifiers modeled by Turing Machines augmented with a fixed-size quantum register (qAM) were studied also by Yakaryilmaz (2012). He proved, for example, that an NP-complete language Lknapsack, representing the 0–1 knapsack problem, can be recognized by a qAM whose verifier is a two-way finite automaton working on quantum mixed states using superoperators.In this paper we explore the power of AM for the case that verifiers are two-way finite automata with quantum and classical states (2QCFA) – introduced by Ambainis and Watrous in 2002 – and the communications are classical. It is of interest to consider AM with such “semi-quantum” verifiers because they use only limited quantum resources. Our main result is that such Quantum Arthur–Merlin proof systems (QAM(2QCFA)) with polynomial expected running time are more powerful than the models in which the verifiers are two-way probabilistic finite automata (AM(2PFA)) with polynomial expected running time. Moreover, we prove that there is a language which can be recognized by an exponential expected running time QAM(2QCFA), but cannot be recognized by any AM(2PFA), and that the NP-complete language Lknapsack can also be recognized by a QAM(2QCFA) working only on quantum pure states using unitary operators.

Interactive proofs with quantum finite automata

Following an early work of Dwork and Stockmeyer on interactive proof systems whose verifiers are two-way probabilistic finite automata, the authors initiated in 2004 a study on the computational power of quantum interactive proof systems whose verifiers are particularly limited to quantum finite automata. As a follow-up to the authors' early journal publication [J. Comput. System Sci., vol. 75, pp. 255–269, 2009], we further investigate the quantum nature of interactions between provers and verifiers by studying how various restrictions on quantum interactive proof systems affect the language recognition power of the proof systems. In particular, we examine three intriguing restrictions that (i) provers always behave in a classical fashion, (ii) verifiers always reveal to provers the information on their next moves, and (iii) the number of interactions between provers and verifiers is bounded.

SOME LANGUAGES RECOGNIZED BY TWO-WAY FINITE AUTOMATA WITH QUANTUM AND CLASSICAL STATES

Two-way finite automata with quantum and classical states (2QCFA) were introduced by Ambainis and Watrous, and it was shown that 2QCFA have superiority over two-way probabilistic finite automata (2PFA) for recognizing some non-regular languages such as the language Leq = {anbn ∣ n ∈ N} and the palindrome language Lpal = {w ∈ {a,b}* ∣ w=wR}, where xR is x in the reverse order. It is interesting to find more languages like these that witness the superiority of 2QCFA over 2PFA. In this paper, we consider the language Lm = {xcy ∣ ∑ = {a,b,c},x,y ∈ {a,b}*, ∣x∣ = ∣y∣} that is similar to the middle language Lmiddle = {xay ∣ x,y ∈ {a,b}*, ∣x∣ = ∣y∣}. We prove that the language Lm show that Lm can be recognized by 2PFA with bounded error, but only in exponential expected time. Thus Lm is another witness of the fact that 2QCFA are more powerful than their classical counterparts.

An application of quantum finite automata to interactive proof systems

Quantum finite automata have been studied intensively since their introduction in late 1990s as a natural model of a quantum computer working with finite-dimensional quantum memory space. This paper seeks their direct application to interactive proof systems in which a mighty quantum prover communicates with a quantum-automaton verifier through a common communication cell. Our quantum interactive proof systems are juxtaposed to Dwork–Stockmeyer's classical interactive proof systems whose verifiers are two-way probabilistic finite automata. We demonstrate strengths and weaknesses of our systems by studying how various restrictions on the behaviors of quantum-automaton verifiers affect the power of quantum interactive proof systems.

Multihead two-way probabilistic finite automata

We present properties of multihead two-way probabilistic finite automata that parallel those of their deterministic and nondeterministic counterparts. We define multihead probabilistic finite automata withlogspace constructible transition probabilities, and we describe a technique to simulate these automata by standard logspace probabilistic Turing machines. Next, we represent logspace probabilistic complexity classes as proper hierarchies based on corresponding multihead two-way probabilistic finite automata, and we show their (deterministic logspace) reducibility to the second levels of these hierarchies.