The problem of assessing the reliability of clusters patients identified by clustering algorithms is crucial to estimate the significance of subclasses of diseases detectable at bio-molecular level, and more in general to support bio-medical discovery of patterns in gene expression data. In this paper we present an experimental analysis of the reliability of clusters discovered in lung tumor patients using DNA microarray data. In particular we investigate if subclasses of lung adenocarcinoma can be detected with high reliability at bio-molecular level. To this end we apply cluster validity measures based on random projections recently proposed by Bertoni and coworkers. The results show that at least two subclasses of lung adenocarcinoma can be detected with relatively high reliability, confirming and extending previous findings reported in the literature.

We consider the position and number of occurrences of squares in the Thue-Morse sequence, and show that the corresponding sequences are 2-regular. We also prove that changing any finite but nonzero number of bits in the Thue-Morse sequence creates an overlap, and any linear subsequence of the Thue-Morse sequence (except those corresponding to decimation by a power of 2) contains an overlap.

In this note we consider the longest word, which has periods p1,...,pn, and does not have the period gcd(p1,...,pn). The length of such a word can be established by a simple algorithm. We give a short and natural way to prove that the algorithm is correct. We also give a new proof that the maximal word is a palindrome.

We present several solutions to the Firing Squad Synchronization Problem on grid networks of different shapes. The nodes are finite state processors that work in unison with other processors and in synchronized discrete steps. The networks we deal with are: the line, the ring and the square. For all of these models we consider one- and two-way communication modes and we also constrain the quantity of information that adjacent processors can exchange at each step. We first present synchronization algorithms that work in time n2, nlogn, $n\sqrt n$, 2n, where n is a total number of processors. Synchronization methods are described through so called signals that are then used as building blocks to compose synchronization solutions for the cases that synchronization times are expressed by polynomials with nonnegative coefficients.

There exists a bijection between one-stack sortable permutations (permutations which avoid the pattern (231)) and rooted plane trees. We define an edit distance between permutations which is consistent with the standard edit distance between trees. This one-to-one correspondence yields a polynomial algorithm for the subpermutation problem for (231) pattern-avoiding permutations. Moreover, we obtain the generating function of the edit distance between ordered unlabeled trees and some special ones. For the general case we show that the mean edit distance between a rooted plane tree and all other rooted plane trees is at least n/ln(n). Some results can be extended to labeled trees considering colored Dyck paths or, equivalently, colored one-stack sortable permutations.

The paper treats the question whether there always exists a minimal nondeterministic finite automaton of n states whose equivalent minimal deterministic finite automaton has α states for any integers n and α with n ≤ α ≤ 2n. Partial answers to this question were given by Iwama, Kambayashi, and Takaki (2000) and by Iwama, Matsuura, and Paterson (2003). In the present paper, the question is completely solved by presenting appropriate automata for all values of n and α. However, in order to give an explicit construction of the automata, we increase the input alphabet to exponential sizes. Then we prove that 2n letters would be sufficient but we describe the related automata only implicitly. In the last section, we investigate the above question for automata over binary and unary alphabets.

We investigate the complexity of several problems concerning Las Vegas finite automata. Our results are as follows. (1) The membership problem for Las Vegas finite automata is in NL. (2) The nonemptiness and inequivalence problems for Las Vegas finite automata are NL-complete. (3) Constructing for a given Las Vegas finite automaton a minimum state deterministic finite automaton is in NP. These results provide partial answers to some open problems posed by Hromkovič and Schnitger [Theoret. Comput. Sci.262 (2001) 1–24)].

We consider conversions of regular expressions into k-realtime finite state automata, i.e., automata in which the number of consecutive uses of ε-transitions, along any computation path, is bounded by a fixed constant k. For 2-realtime automata, i.e., for automata that cannot change the state, without reading an input symbol, more than two times in a row, we show that the conversion of a regular expression into such an automaton produces only O(n) states, O(nlogn) ε-transitions, and O(n) alphabet-transitions. We also show how to easily transform these 2-realtime machines into 1-realtime automata, still with only O(nlogn) edges. These results contrast with the known lower bound Ω(n(logn)2 / loglogn), holding for 0-realtime automata, i.e., for automata with no ε-transitions.

In this work we study some probabilistic models for the random generation of words over a given alphabet used in the literature in connection with pattern statistics. Our goal is to compare models based on Markovian processes (where the occurrence of a symbol in a given position only depends on a finite number of previous occurrences) and the stochastic models that can generate a word of given length from a regular language under uniform distribution. We present some results that show the differences between these two stochastic models and their relationship with the rational probabilistic measures.

Among Sturmian words, some of them are morphic, i.e. fixed point of a non-identical morphism on words. Berstel and Séébold (1993) have shown that if a characteristic Sturmian word is morphic, then it can be extended by the left with one or two letters in such a way that it remains morphic and Sturmian. Yasutomi (1997) has proved that these were the sole possible additions and that, if we cut the first letters of such a word, it didn't remain morphic. In this paper, we give an elementary and combinatorial proof of this result.

PageRank is a ranking method that assigns scores to web pages using the limit distribution of a random walk on the web graph. A fibration of graphs is a morphism that is a local isomorphism of in-neighbourhoods, much in the same way a covering projection is a local isomorphism of neighbourhoods. We show that a deep connection relates fibrations and Markov chains with restart, a particular kind of Markov chains that include the PageRank one as a special case. This fact provides constraints on the values that PageRank can assume. Using our results, we show that a recently defined class of graphs that admit a polynomial-time isomorphism algorithm based on the computation of PageRank is really a subclass of fibration-prime graphs, which possess simple, entirely discrete polynomial-time isomorphism algorithms based on classical techniques for graph isomorphism. We discuss efficiency issues in the implementation of such algorithms for the particular case of web graphs, in which O(n) space occupancy (where n is the number of nodes) may be acceptable, but O(m) is not (where m is the number of arcs).

We consider the four families of recognizable, synchronous, deterministic rational and rational subsets of a direct product of free monoids. They form a strict hierarchy and we investigate the following decision problem: given a relation in one of the families, does it belong to a smaller family? We settle the problem entirely when all monoids have a unique generator and fill some gaps in the general case. In particular, adapting a proof of Stearns, we show that it is recursively decidable whether or not a deterministic subset of an arbitrary number of free monoids is recognizable. Also we exhibit a single exponential algorithm for determining if a synchronous relation is recognizable.

We consider the family UREC of unambiguous recognizable two-dimensional languages. We prove that there are recognizable languages that are inherently ambiguous, that is UREC family is a proper subclass of REC family. The result is obtained by showing a necessary condition for unambiguous recognizable languages. Further UREC family coincides with the class of picture languages defined by unambiguous 2OTA and it strictly contains its deterministic counterpart. Some closure and non-closure properties of UREC are presented. Finally we show that it is undecidable whether a given tiling system is unambiguous.

Given a finite alphabet Σ and a language L ⊆ ∑+, the centralizer of L is defined as the maximal language commuting with it. We prove that if the primitive root of the smallest word of L (with respect to a lexicographic order) is prefix distinguishable in L then the centralizer of L is as simple as possible, that is, the submonoid L*. This lets us obtain a simple proof of a known result concerning the centralizer of nonperiodic three-word languages.

In this paper we analyze some intrusion detection strategies proposed in the literature and we show that they represent the various facets of a well known formal languages problem: computing the distance between a string x and a language L. In particular, the main differences among the various approaches adopted for building intrusion detection systems can be reduced to the characteristics of the language L and to the notion of distance adopted. As a further contribution we will also show that from the computational point of view all these strategies are equivalent and they are amenable to efficient parallelization.

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 1qfa's proposed in the literature. Here, we investigate some properties of 1qfc's. In particular, we provide algorithms for constructing 1qfc's accepting the inverse homomorphic images and quotients of languages accepted by 1qfc's. Moreover, we give instances of binary regular languages on which 1qfc's are proved to be more succinct (i.e. , to have less states) than the corresponding classical (deterministic) automata.

The main goal of this paper is the investigation of a relevant property which appears in the various definition of deterministic topological chaos for discrete time dynamical system: transitivity. Starting from the standard Devaney's notion of topological chaos based on regularity, transitivity, and sensitivity to the initial conditions, the critique formulated by Knudsen is taken into account in order to exclude periodic chaos from this definition. Transitivity (or some stronger versions of it) turns out to be the relevant condition of chaos and its role is discussed by a survey of some important results about it with the presentation of some new results. In particular, we study topological mixing, strong transitivity, and full transitivity. Their applications to symbolic dynamics are investigated with respect to the relationships with the associated languages.

A number of methodological papers published during the last years testify that a need for a thorough revision of the research methodology is felt by the operations research community – see, for example, [Barr et al., J. Heuristics1 (1995) 9–32; Eiben and Jelasity, Proceedings of the 2002 Congress on Evolutionary Computation (CEC'2002) 582–587; Hooker, J. Heuristics1 (1995) 33–42; Rardin and Uzsoy, J. Heuristics7 (2001) 261–304]. In particular, the performance evaluation of nondeterministic methods, including widely studied metaheuristics such as evolutionary computation and ant colony optimization, requires the definition of new experimental protocols. A careful and thorough analysis of the problem of evaluating metaheuristics reveals strong similarities between this problem and the problem of evaluating learning methods in the machine learning field. In this paper, we show that several conceptual tools commonly used in machine learning – such as, for example, the probabilistic notion of class of instances and the separation between the training and the testing datasets – fit naturally in the context of metaheuristics evaluation. Accordingly, we propose and discuss some principles inspired by the experimental practice in machine learning for guiding the performance evaluation of optimization algorithms. Among these principles, a clear separation between the instances that are used for tuning algorithms and those that are used in the actual evaluation is particularly important for a proper assessment.

We propose a heuristic for solving the maximum independent set problem for a set of processors in a network with arbitrary topology. We assume an asynchronous model of computation and we use modified Hopfield neural networks to find high quality solutions. We analyze the algorithm in terms of the number of rounds necessary to find admissible solutions both in the worst case (theoretical analysis) and in the average case (experimental Analysis). We show that our heuristic is better than the greedy one at 1% significance level.

We study the problem of designing a distributed voting scheme for electing a candidate that maximizes the preferences of a set of agents. We assume the preference of agent i for candidate j is a real number xi,j, and we do not make any assumptions on the mechanism generating these preferences. We show simple randomized voting schemes guaranteeing the election of a candidate whose expected total preference is nearly the highest among all candidates. The algorithms we consider are designed so that each agent has to disclose only a few bits of information from his preference table. Finally, in the important special case in which each agent is forced to vote for at most one candidate we show that our voting scheme is essentially optimal.