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 2 1a6n + 25 states inducing the event ap + b, for constants a > 0, b 65 0, satisfying a + b 64 1. This fact is proved by designing an algorithm which constructs the desired 1qfa in polynomial time. As a consequence, we get that any periodic language of period n can be accepted with isolated cut point by a 1qfa with no more than 2 1a6n + 26 states. Our results give added evidence of the strength of measure-once 1qfa's with respect to classical automata

Note on the succinctness of deterministic, nondeterministic, probabilistic and quantum finite automata

We investigate the succinctness of several kinds of unary automata by studying their state complexity in accepting the family {Lm} of cyclic languages, where Lm = {akm|k 08 N}. In particular, we show that, for any m, the number of states necessary and sufficient for accepting the unary language Lm with isolated cut point on one-way probabilistic finite automata is p1\u3b11 + p2\u3b12 + ef + ps\u3b1s, with p1\u3b11p2\u3b12 ef ps\u3b1s being the factorization of m. To prove this result, we give a general state lower bound for accepting unary languages with isolated cut point on the one-way probabilistic model. Moreover, we exhibit one-way quantum finite automata that, for any m, accept Lm with isolated cut point and only two states. These results are settled within a survey on unary automata aiming to compare the descriptional power of deterministic, nondeterministic, probabilistic and quantum paradigms

The complexity of minimum difference cover

The complexity of searching minimum difference covers, both in (Formula Not Shown) and in (Formula Not Shown), is studied. We prove that these two optimization problems are NP-hard. To obtain this result, we characterize those sets—called extrema—having themselves plus zero as minimum difference cover. Such a combinatorial characterization enables us to show that testing whether sets are not extrema, both in Formula Not Shown and in Formula Not Shown, is NP-complete. However, for these two decision problems we exhibit pseudo-polynomial time algorithms

Threshold circuits for iterated matrix product and powering

The complexity of computing, via threshold circuits, the iterated product and powering of fixed-dimension k 7k matrices with integer or rational entries is studied. We call these two problems IMPk and MPOWk, respectively, for short. We prove that: (i) For k 652, IMPk does not belong to TC0, unless TC0 = NC1. (ii) For stochastic matrices: IMP2 belongs to TC0 while, for k 653, IMPk does not belong to TC0, unless TC0 = NC1. (iii) For any k, MPOWk belongs to TC0

Quantum automata for some multiperiodic languages

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

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

Some formal tools for analyzing quantum automata

Results in the area of compact monoids and groups are useful in the analysis of quantum automata (1qfa's). In this paper: We settle isolated cut point Rabin's theorem in the context of compact monoids, and we prove a lower bound on the state complexity of 1qfa's accepting regular languages.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 Formula Not Shown -tuple of 1qfa's is a classifier of words in Formula Not Shown ; this problem turns out to be undecidable if the completeness of the classifier is required.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 1qfa's exponentially succinct in the period and polynomially succinct in the inverse of the bounded error

Golomb rulers and difference sets for succinct quantum automata

Given a function p : N \u2192 [0,1] of period n, we study the minimal size (number of states) of a one-way quantum finite automaton (Iqfa) inducing the stochastic event ap + b, for real constants a>0, b 650, a+b 641. First of all, we relate the estimation of the minimal size to the problem of finding a minimal difference cover for a suitable subset of Zn. Then, by observing that the cardinality of a difference cover \u394 for a set A Z n, must satisfy $|\Delta| \ge (1 + \sqrt{4|A| - 3})/2$, we investigate the class of sets A admitting difference covers of cardinality exactly $(1 + \sqrt{4|A| - 3})/2$. We relate this problem with the efficient construction of Golomb rulers and difference sets. We design an algorithm which outputs each of the Golomb rulers (if any) of a given set in pseudo-polynomial time. As a consequence, we obtain an efficient algorithm that construct minimal difference covers for a non-trivial class of sets. Moreover, by using projective geometry arguments, we give an algorithm that, for any n=q2+q+1 with q prime power, constructs difference sets for Z n in quadratic time