In this paper, we consider a class of scheduling problems that are among the fundamental optimization problems in operations research. More specifically, we deal with a particular version called job shop scheduling with unit length tasks. Using the results of Hromkovič, Mömke, Steinhöfel, and Widmayer presented in their work Job Shop Scheduling with Unit Length Tasks: Bounds and Algorithms, we analyze the problem setting for 2 jobs with an unequal number of tasks. We contribute a deterministic algorithm which achieves a vanishing delay in certain cases and a randomized algorithm with a competitive ratio tending to 1. Furthermore, we investigate the problem with 3 jobs and we construct a randomized online algorithm which also has a competitive ratio tending to 1.

Frequency hopping sequences sets are required in frequency hopping code division multiple access systems. For the anti-jamming purpose, frequency hopping sequences are required to have a large linear span. In this paper, by using a permutation polynomial δ(x) over a finite field, we transform several optimal sets of frequency hopping sequences with small linear span into ones with large linear span. The exact values of the linear span are presented by using the methods of counting the terms of the sequences representations. The results show that the transformed frequency hopping sequences are optimal with respect to the Peng-Fan bound, and can resist the analysis of Berlekamp-Massey algorithm.

In this paper, we define k-counting automata as recognizers forω-languages, i.e. languages of infinite words. Weprove that the class of ω-languages they recognize is a proper extensionof the ω-regular languages. In addition we prove that languagesrecognized by k-counting automata are closed under Boolean operations. Itremains an open problem whether or not emptiness is decidable fork-counting automata. However, we conjecture strongly that it is decidableand give formal reasons why we believe so.

In the paper we study abelian versions of the critical factorization theorem. We investigate both similarities and differences between the abelian powers and the usual powers. The results we obtained show that the constraints for abelian powers implying periodicity should be quite strong, but still natural analogies exist.

Starting from late 90’s the public administration has started to employ a quite relevantamount of its budget in developing ICT solutions to better deliver services to citizens.In spite of this effort many statistics show that the mere availability of ICT basedservices does not guarantee per se their usage. Citizens have continued to largely accessservices through “traditional” means. In our study we suggest that the highlightedsituation is partly due to the fact that relevant domain dependent requirements, mainlyrelated to the delivery process of e-government digital services, are often ignored in thedevelopment of e-government solutions. We provide here a domain related quality frameworkand encoded it in a set of formal statements, so that we can apply automatic verificationtechniques to assess and improve ICT solutions adopted by public administrations. Thepaper discusses both the defined quality framework and the tool chain we developed toenable automatic assessment of ICT solutions. The tool chain is based on a denotationalmapping of business process modeling notation elements into process algebraic descriptionsand to the encoding of quality requirements in linear temporal logic formulas. Theresulting approach has been applied to real case studies with encouraging results.

A square is the concatenation of a nonempty word with itself. A word has period p if its letters at distance p match. The exponent of a nonempty word is the quotient of its length over its smallest period. In this article we give a proof of the fact that there exists an infinite binary word which contains finitely many squares and simultaneously avoids words of exponent larger than 7/3. Our infinite word contains 12 squares, which is the smallest possible number of squares to get the property, and 2 factors of exponent 7/3. These are the only factors of exponent larger than 2. The value 7/3 introduces what we call the finite-repetition threshold of the binary alphabet. We conjecture it is 7/4 for the ternary alphabet, like its repetitive threshold.

LetLϕ,λ = {ω ∈ Σ∗| ϕ(ω) > λ} be thelanguage recognized by a formal seriesϕ:Σ∗ → ℝ with isolated cut pointλ. We provide new conditions that guarantee the regularity of thelanguage Lϕ,λ in the case thatϕ is rational or ϕ is a Hadamard quotient of rationalseries. Moreover the decidability property of such conditions is investigated.

Taking the view that infinite plays are draws, we study Conwaynon-terminating games and non-losing strategies. These admit asharp coalgebraic presentation, where non-terminating games are seen as afinal coalgebra and game contructors, such as disjunctivesum, as final morphisms. We have shown, in a previous paper,that Conway’s theory of terminating games can be rephrased naturally in terms of game(pre)congruences. Namely, various conceptually independent notions ofequivalence can be defined and shown to coincide on Conway’sterminating games. These are the equivalence induced by the ordering on surrealnumbers, the contextual equivalence determined by observingwhat player has a winning strategy, Joyal’s categoricalequivalence, and, for impartial games, the denotationalequivalence induced by Grundy semantics. In this paper, wediscuss generalizations of such equivalences to non-terminating games andnon-losing strategies. The scenario is even more rich and intriguing inthis case. In particular, we investigate efficient characterizations of the contextualequivalence, and we introduce a category of fair strategies and acategory of fair pairs of strategies, both generalizing Joyal’s categoryof Conway games and winning strategies. Interestingly, the category of fair pairs capturesthe equivalence defined by Berlekamp, Conway, Guy on loopy games.

This paper deals with the decidability of semigroup freeness. More precisely, the freeness problem over a semigroup S is defined as: given a finite subset X ⊆ S, decide whether each element of S has at most one factorization over X. To date, the decidabilities of the following two freeness problems have been closely examined. In 1953, Sardinas and Patterson proposed a now famous algorithm for the freeness problem over the free monoids. In 1991, Klarner, Birget and Satterfield proved the undecidability of the freeness problem over three-by-three integer matrices. Both results led to the publication of many subsequent papers. The aim of the present paper is (i) to present general results about freeness problems, (ii) to study the decidability of freeness problems over various particular semigroups (special attention is devoted to multiplicative matrix semigroups), and (iii) to propose precise, challenging open questions in order to promote the study of the topic.

Multiwords are words in which a single symbol can be replaced by a nonempty set of symbols. They extend the notion of partial words. A word w is certain in a multiword M if it occurs in every word that can be obtained by selecting one single symbol among the symbols provided in each position of M. Motivated by a problem on incomplete databases, we investigate a variant of the pattern matching problem which is to decide whether a word w is certain in a multiword M. We study the language CERTAIN(w) of multiwords in which w is certain. We show that this regular language is aperiodic for three large families of words. We also show its aperiodicity in the case of partial words over an alphabet with at least three symbols.

Joint 2-adic complexity is a new important index of the cryptographic security for multisequences. In this paper, we extend the usual Fourier transform to the case of multisequences and derive an upper bound for the joint 2-adic complexity. Furthermore, for the multisequences with pn-period, we discuss the relation between sequences and their Fourier coefficients. Based on the relation, we determine a lower bound for the number of multisequences with given joint 2-adic complexity.

We study translations of dyadic first-order sentences into equalities between relationalexpressions. The proposed translation techniques (which work also in the conversedirection) exploit a graphical representation of formulae in a hybrid of the twoformalisms. A major enhancement relative to previous work is that we can cope with therelational complement construct and with the negation connective. Complementation ishandled by adopting a Smullyan-like uniform notation to classify and decompose relationalexpressions; negation is treated by means of a generalized graph-representation offormulae in ℒ+, and through a series of graph-transformation rules whichreflect the meaning of connectives and quantifiers.

We investigate the possibility of extending Chrobak normal form to the probabilisticcase. While in the nondeterministic case a unary automaton can be simulated by anautomaton in Chrobak normal form without increasing the number of the states in thecycles, we show that in the probabilistic case the simulation is not possible by keepingthe same number of ergodic states. This negative result is proved by considering thenatural extension to the probabilistic case of Chrobak normal form, obtained by replacingnondeterministic choices with probabilistic choices. We then propose a different kind ofnormal form, namely, cyclic normal form, which does not suffer from the same problem: weprove that each unary probabilistic automaton can be simulated by a probabilisticautomaton in cyclic normal form, with at most the same number of ergodic states. In thenondeterministic case there are trivial simulations between Chrobak normal form and cyclicnormal form, preserving the total number of states in the automata and in theircycles.

We prove that the subsets of that are S-recognizable for all abstract numeration systems S are exactly the 1-recognizable sets. This generalizes a result of Lecomte and Rigo in the one-dimensional setting.

The Parikh finite word automaton (PA) was introduced and studied in 2003 by Klaedtke andRueß. Natural variants of the PA arise from viewing a PA equivalently as an automaton thatkeeps a count of its transitions and semilinearly constrains their numbers. Here we adoptthis view and define the affine PA, that extends the PA by having eachtransition induce an affine transformation on the PA registers, and the PA onletters, that restricts the PA by forcing any two transitions on the sameletter to affect the registers equally. Then we report on the expressiveness, closure, anddecidability properties of such PA variants. We note that deterministic PA are strictlyweaker than deterministic reversal-bounded counter machines.

A closed λ-term M is easy if, for anyother closed term N, the lambda theory generated byM = N is consistent. Recently, it has been introduceda general technique to prove the easiness of λ-terms through thesemantical notion of simple easiness. Simple easiness implies easiness and allows to proveconsistency results via construction of suitable filter models ofλ-calculus living in the category of complete partial orderings: givena simple easy term M and an arbitrary closed term N, itis possible to build (in a canonical way) a non-trivial filter model which equates theinterpretation of M and N. The question whether easinessimplies simple easiness constitutes Problem 19 in the TLCA list of open problems. In thispaper we negatively answer the question providing a non-empty co-r.e. (complement of arecursively enumerable) set of easy, but not simple easy, λ-terms.

Probabilistic operational semantics for a nondeterministic extension of pure λ-calculus is studied. In this semantics, a term evaluates to a (finite or infinite) distribution of values. Small-step and big-step semantics, inductively and coinductively defined, are given. Moreover, small-step and big-step semantics are shown to produce identical outcomes, both in call-by-value and in call-by-name. Plotkin’s CPS translation is extended to accommodate the choice operator and shown correct with respect to the operational semantics. Finally, the expressive power of the obtained system is studied: the calculus is shown to be sound and complete with respect to computable probability distributions.