Syntactic Complexity of Circular Semi-Flower Automata

We investigate the syntactic complexity of certain types of finitely generated submonoids of a free monoid. In fact, we consider those submonoids which are accepted by circular semi-flower automata (CSFA). Here, we show that the syntactic complexity of CSFA with at most one `branch point going in' (bpi) is linear. Further, we prove that the syntactic complexity of $n$-state CSFA with two bpis over a binary alphabet is $2n(n+1)$

Automatic sets of rational numbers

The notion of a k-automatic set of integers is well-studied. We develop a new notion - the k-automatic set of rational numbers - and prove basic properties of these sets, including closure properties and decidability.Comment: Previous version appeared in Proc. LATA 2012 conferenc

An efficient algorithm for minimizing time granularity periodical representations

This paper addresses the technical problem of efficiently reducing the periodic representation of a time granularity to its minimal form. The minimization algorithm presented in the paper has an immediate practical application: it allows users to intuitively define granularities (and more generally, recurring events) with algebraic expressions that are then internally translated to mathematical characterizations in terms of minimal periodic sets. Minimality plays a crucial role, since the value of the recurring period has been shown to dominate the complexity when processing periodic sets.

On some generalizations of Shamir’s secret sharing scheme

A Lai-Ding's secret sharing scheme Sigma^{LD}_q(c, i) defined by parameters c = (c_0, ..., c_{k-1}), i and q is a modification of a Shamir's k-threshold scheme in which the share given to a participant x in F_q is computed as the value of P(x) = sum_{j = 0}^{k-1} a_j x^{c_j}, where a_j are confidential while c_j are publicly known, and a_i is the value of the secret. Following the prior research of Spież, Urbanowicz et al., we study access structures realized by such schemes, as well as the behaviour of their admissible sets, where a set of participants is called admissible if the scheme restricted to it is k-threshold.Our main efforts focus on providing asymptotic estimates for the number of admissible (or non-admissible) sets of a given size n in a Lai-Ding's scheme Sigma^{LD}_q(c, i); in these estimates, q is the variable and c, i n are parameters (which may influence the asymptotic constants). Generalizing prior results for the case c = (0, 1, ..., k-1), we show in general that the number of admissible sets of size n is Theta(q^n). As for non-admissible sets, we show that, for fixed c and i, the number of such sets of size k - 1 may be 0 for all q, Theta(q^{k-2}) for all q, or may periodically switch between those two patterns. Moreover, in many cases, we provide computationally tractable lower bounds for q (and for the characteristic of F_q) for which those sets must exist. This takes place in particular when c is an arithmetic progression, or when \hat{c}_i (i.e. c with c_i removed) has the property that every two its consecutive increments are coprime.As an internal step in the above considerations (required by our need to use Weil's theorem), we investigate absolute irreducibility of the classical Schur polynomials over finite fields. Using the arguments of Monge and Rajan, and (partially) translating the latter from C to finite fields, we obtain a new result on irreducibility of a large class of such polynomials. Moreover, by implementing another novel method based on Newton polytopes, we generalize our irreducibility criterion to a large class of perturbations of Schur polynomials.Finally, we make several preliminary observations on Lai-Ding's access structures. First, we show that they are almost as general as in Brickell's schemes; however, our construction of an appropriate Lai-Ding's scheme leads to significantly complex results. Then, we analyze the cases when c or \hat{c}_i are arithmetic. While the former case essentially reduces to Shamir's type schemes, the latter exhibits new examples of access structures, including certain graphic structures; we provide a characterization of graphs which can appear in this context.Schematem Lai-Dinga współdzielenia sekretu (oznaczenie: Sigma^{LD}_q(c, i)) dla parametrów c = (c_0, ..., c_{k-1}), i, q nazywamy modyfikację k-progowego schematu Shamira, w której udziałem uczestnika x w F_q jest wartość wielomianu P(x) = \sum_{j = 0}^{k-1} a_j x^{c_j}, przy czym współczynniki a_j są tajne, zaś wykładniki c_j jawne, zaś wartością sekretu jest współczynnik a_i. Kontynuując wcześniejsze badania Spieża, Urbanowicza i in., badamy struktury dostępu realizowane przez takie schematy, a także zachowanie tzw. zbiorów progowych, gdzie zbiór uczestników nazywamy progowym, jeśli schemat po obcięciu do niego staje się k-progowy.Jednym z naszych ważniejszych celów jest podanie asymptotycznych oszacowań liczby zbiorów progowych (bądź nie-progowych) o danej wielkości n w schemacie Lai-Dinga Sigma^{LD}_q(\mathbf{c}, i), przy czym w oszacowaniach tych rolę zmiennej pełni q, zaś c, i, n są parametrami (mogącymi wpływać na stałe w notacji asymptotycznej). Uogólniając wcześniejsze wyniki dla c = (0, 1, ..., k-1, wykazujemy w ogólności, że liczba zbiorów progowych wynosi Theta(q^n). Odnośnie zbiorów nie-progowych, wykazujemy, że dla ustalonych c oraz i liczba takich zbiorów o wielkości k - 1 może wynosić 0 dla wszystkich q, Theta(q^{k-2}) dla wszystkich q, lub w sposób okresowy przełączać się pomiędzy tymi dwoma wzorcami. Ponadto dla wielu przypadków podajemy rozsądne z obliczeniowego punktu widzenia ograniczenia dolne na q (a także na charakterystykę ciała F_q), powyżej których takie zbiory muszą istnieć. Ma to miejsce w szczególności gdy ciąg c jest arytmetyczny, lub gdy w ciągu \hat{c}_i (powstającym z c przez usunięcie c_i) każde dwa kolejne przyrosty są względnie pierwsze.W ramach powyższego rozumowania (na potrzeby wykorzystywanego w nim twierdzenia Weila) badamy absolutną nierozkładalność klasycznych wielomianów Schura nad ciałami skończonymi. Wykorzystując rozumowania Mongego i Rajana i przenosząc (częściowo) metody Rajana z C nad ciała skończone, otrzymujemy nowy wynik dotyczący nierozkładalności dużej klasy wielomianów Schura. Co więcej, wykorzystując inną, nową metodę, opartą na wielościanach Newtona, uogólniamy powyższe kryterium nierozkładalności na szeroką klasę zaburzeń wielomianów Schura.W ostatnim rozdziale pracy gromadzimy kilka spostrzeżeń dotyczących struktur dostępu w schematach Lai-Dinga. Najpierw wykazujemy, że są one niemal równie ogólne jak w schematach Brickella, choć nasza konstrukcja odpowiedniego schematu Lai-Dinga ma znaczny stopień złożoności. Następnie analizujemy przypadki, gdy ciąg c lub \hat{c}_i jest arytmetyczny. O ile pierwszy z nich zasadniczo sprowadza się do schematów typu Shamira, o tyle w drugim można znaleźć nowe przykłady struktur dostępowych, w tym niektóre struktury grafowe; podajemy charakteryzację grafów uzyskiwalnych w powyższy sposób

Concurrent Parameterized Games

Traditional concurrent games on graphs involve a fixed number of players, who take decisions simultaneously, determining the next state of the game. In this paper, we introduce a parameterized variant of concurrent games on graphs, where the parameter is precisely the number of players. Parameterized concurrent games are described by finite graphs, in which the transitions bear regular languages to describe the possible move combinations that lead from one vertex to another. We consider the problem of determining whether the first player, say Eve, has a strategy to ensure a reachability objective against any strategy profile of her opponents as a coalition. In particular Eve\u27s strategy should be independent of the number of opponents she actually has. Technically, this paper focuses on an a priori simpler setting where the languages labeling transitions only constrain the number of opponents (but not their precise action choices). These constraints are described as semilinear sets, finite unions of intervals, or intervals. We establish the precise complexities of the parameterized reachability game problem, ranging from PTIME-complete to PSPACE-complete, in a variety of situations depending on the contraints (semilinear predicates, unions of intervals, or intervals) and on the presence or not of non-determinism

Gaifman Normal Forms for Counting Extensions of First-Order Logic

We consider the extension of first-order logic FO by unary counting quantifiers and generalise the notion of Gaifman normal form from FO to this setting. For formulas that use only ultimately periodic counting quantifiers, we provide an algorithm that computes equivalent formulas in Gaifman normal form. We also show that this is not possible for formulas using at least one quantifier that is not ultimately periodic. Now let d be a degree bound. We show that for any formula phi with arbitrary counting quantifiers, there is a formula gamma in Gaifman normal form that is equivalent to phi on all finite structures of degree <= d. If the quantifiers of phi are decidable (decidable in elementary time, ultimately periodic), gamma can be constructed effectively (in elementary time, in worst-case optimal 3-fold exponential time). For the setting with unrestricted degree we show that by using our Gaifman normal form for formulas with only ultimately periodic counting quantifiers, a known fixed-parameter tractability result for FO on classes of structures of bounded local tree-width can be lifted to the extension of FO with ultimately periodic counting quantifiers (a logic equally expressive as FO+MOD, i.e., first-oder logic with modulo-counting quantifiers)

Scattered one-counter languges have rank less than ω2\omega^2

A linear ordering is called context-free if it is the lexicographic ordering of some context-free language and is called scattered if it has no dense subordering. Each scattered ordering has an associated ordinal, called its rank. It is known that scattered context-free (regular, resp.) orderings have rank less than $\omega^\omega$ ($\omega$, resp). In this paper we confirm the conjecture that one-counter languages have rank less than $\omega^2$