Beatrice Palano

Quantum computing: 1-way quantum automata

In this paper we analyze several models of 1-way quantum finite automata, in the light of formal power series theory. In this general context, we recall two well known constructions, by proving: 1. Languages generated with isolated cut-point by a class of bounded rational formal series are regular. 2. If a class of formal series is closed under f-complement, Hadamard product and convex linear combination, then the class of languages generated with isolated cut-point is closed under boolean operations. We introduce a general model of 1-way quantum automata and we compare their behaviors with those of measure-once, measure-many and reversible 1-way quantum automata.

Small size quantum automata recognizing some regular languages

Given a class {pα | α ∈ I} of stochastic events induced by M-state 1-way quantum finite automata (1qfa) on alphabet Σ, we investigate the size (number of states) of 1qfas that δ-approximate a convex linear combination of {pα | α ∈ I}, and we apply the results to the synthesis of small size 1qfas. We obtain: • An O((Md/δ3) log2(d/δ2)) general size bound, where d is the Vapnik dimension of {pα(w) | w ∈ Σ*}. • For commutative n-periodic events p on Σ with |Σ| = H, we prove an O((H log n/δ2)) size bound for inducing a δ-approximation of ½ + ½ p whenever ||F(p)||1 ≤nH, where F(p) is the discrete Fourier transform of (the vector p associated with) p. • If the characteristic function χL of an n-periodic unary language L satisfies ||F(χL))||1 ≤ n, then L is recognized with isolated cut-point by a 1qfa with O(log n) states. Vice versa, if L is recognized with isolated cut-point by a 1qfa with O(log n) state, then ||F(χL))||1 = O(n log n).

More concise representation of regular languages by automata and regular expressions

We consider two formalisms for representing regular languages: constant height pushdown automata and straight line programs for regular expressions. We constructively prove that their sizes are polynomially related. Comparing them with the sizes of finite state automata and regular expressions, we obtain optimal exponential and double exponential gaps, i.e., a more concise representation of regular languages.

On the Size of One-way Quantum Finite Automata with Periodic Behaviors

We show that, for any stochastic event p of period n, there exists a measure-once one-way quantum finite automaton (1qfa) with at most 26n + 25 states inducing the event ap + b, for constants a > 0, b ≥ 0, satisfying a + b < 1. This fact is proved by designing an algorithm which constructs the desired lqfa in polynomial time. As a consequence, we get that any periodic language of period n can be accepted with isolated cut point by a lqfa with no more than 26n+26 states. Our results give added evidence of the strength of measure-once 1qfas with respect to classical automata.

Quantum finite automata with control language

Bertoni et al.  introduced in Lect. Notes Comput. Sci. 2710 (2003) 1–20 a new model of 1-way quantum finite automaton (1qfa) called 1qfa with control language (1qfc) . This model, whose recognizing power is exactly the class of regular languages, generalizes main models of 1qfas proposed in the literature. Here, we investigate some properties of 1qfcs. In particular, we provide algorithms for constructing 1qfcs accepting the inverse homomorphic images and quotients of languages accepted by 1qfcs. Moreover, we give instances of binary regular languages on which 1qfcs are proved to be more succinct ( i.e. , to have less states) than the corresponding classical (deterministic) automata.

Some formal tools for analyzing quantum automata

Results in the area of compact monoids and groups are useful in the analysis of quantum automata (lqfas). In this paper: (1) We settle isolated cut point Rabins theorem in the context of compact monoids, and we prove a lower bound on the state complexity of lqfas accepting regular languages. (2) We use a method pointed out by Blondel et al. [Decidable and undecidable problems about quantum automata, Technical Report RR2003-24, LIP, ENS Lyon, 2003] based on compact groups theory to design an algorithm for testing whether a k-tuple of lqfas is a classifier of words in Σ*; this problem turns out to be undecidable if the completeness of the classifier is required. (3) In the unary case, we give an exponential time algorithm for computing the descriptional complexity of periodic languages. Moreover, we present a probabilistic method to construct lqfas exponentially succinct in the period and polynomially succinct in the inverse of the bounded error.

Behaviours of Unary Quantum Automata

We study the stochastic events induced by MM-qfas working on unary alphabets. We give two algorithms for unary MM-qfas: the first computes the dimension of the ergodic and transient components of the non halting subspace, while the second tests whether the induced event is d-periodic. These algorithms run in polynomial time whenever the MM-qfa given in input has complex amplitudes with rational components. We also characterize the recognition power of unary MM-qfas, by proving that any unary regular language can be accepted by a MM-qfa with constant cut point and isolation. Yet, the amount of states of the resulting MM-qfa is linear in the size of the corresponding minimal dfa. We also single out families of unary regular languages for which the size of the accepting MM-qfas can be exponentially decreased.

Quantum automata for some multiperiodic languages

We exhibit small size measure-once one-way quantum finite automata (mo-1qfas) inducing multiperiodic stochastic events. Moreover, for certain classes of multiperiodic languages, we exhibit: (i) isolated cut point mo-1qfas whose size logarithmically depends on the periods; (ii) Monte Carlo mo-1qfas whose size logarithmically depends on the periods and polynomially on the inverse of the error probability.

The size-cost of Boolean operations on constant height deterministic pushdown automata

We study the size-cost of Boolean operations on constant height deterministic pushdown automata, i.e., on the traditional pushdown automata with a built-in constant limit on the height of the pushdown. We design a simulation showing that a complement can be obtained with a polynomial tradeoff. For intersection and union, we show an exponential simulation, and prove that the exponential blow-up cannot be avoided.

Descriptional complexity of two-way pushdown automata with restricted head reversals

Two-way nondeterministic pushdown automata (2PDA) are classical nondeterministic pushdown automata (PDA) enhanced with two-way motion of the input head. In this paper, the subclass of 2PDA accepting bounded languages and making at most a constant number of input head turns is studied with respect to descriptional complexity aspects. In particular, the effect of reducing the number of pushdown reversals to a constant number is of interest. It turns out that this reduction leads to an exponential blow-up in case of nondeterministic devices, and to a doubly-exponential blow-up in case of deterministic devices. If the restriction on boundedness of the languages considered and on the finiteness of the number of head and pushdown turns is dropped, the resulting trade-offs are no longer bounded by recursive functions, and so-called non-recursive trade-offs are shown.