Carlo Mereghetti

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.

Optimal Simulations between Unary Automata

We consider the problem of computing the costs---{ in terms of states---of optimal simulations between different kinds of finite automata recognizing unary languages. Our main result is a tight simulation of unary n-state two-way nondeterministic automata by

Complementing two-way finite automata

O({{\rm e}^{\sqrt{{n}\ln{n}}}})

Small size quantum automata recognizing some regular languages

-state one-way deterministic automata. In addition, we show that, given a unary n-state two-way nondeterministic automaton, one can construct an equivalent O(n2)-state two-way nondeterministic automaton performing both input head reversals and nondeterministic choices only at the ends of the input tape. Further results on simulating unary one-way alternating finite automata are also discussed.

More concise representation of regular languages by automata and regular expressions

We study the relationship between the sizes of two-way finite automata accepting a language and its complement. In the deterministic case, for a given automaton (2dfa) with n states, we build an automaton accepting the complement with at most 4n states, independently of the size of the input alphabet. Actually, we show a stronger result, by presenting an equivalent 4n-state 2dfa that always halts. For the nondeterministic case, using a variant of inductive counting, we show that the complement of a unary language, accepted by an n-state two-way automaton (2nfa), can be accepted by an O(n^8)-state 2nfa. Here we also make 2nfas halting. This allows the simulation of unary 2nfas by probabilistic Las Vegas two-way automata with O(n^8) states.

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

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).

TESTING THE DESCRIPTIONAL POWER OF SMALL TURING MACHINES ON NONREGULAR LANGUAGE ACCEPTANCE

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.

Quantum finite automata with control language

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.

Some formal tools for analyzing quantum automata

We study lower bounds on space and input head reversals for deterministic, nondeterministic, and alternating Turing machines accepting nonregular languages. Three notions of space, namely strong, middle, weak are considered, and another notion, called accept, is introduced. In all cases, we obtain tight lower bounds. Moreover, we show that, while for determinism and nondeterminism such lower bounds are optimal even with respect to unary languages, for alternation optimal lower bounds for unary languages turn out to be strictly higher than those for languages over alphabets with two or more symbols.

Strong Optimal Lower Bounds for Turing Machines that Accept Nonregular Languages

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.