A vertex i of a graph G = (V,E) is said to be controlled by $M \subseteq V$ if the majority of the elements of the neighborhood of i (including itself) belong to M. The set M is a monopoly in G if every vertex $i\in V$ is controlled by M. Given a set $M \subseteq V$ and two graphs G1 = ($V,E_1$) and G2 = ($V,E_2$) where $E_1\subseteq E_2$, the monopoly verification problem (mvp) consists of deciding whether there exists a sandwich graph G = (V,E) (i.e., a graph where $E_1\subseteq E\subseteq E_2$) such that M is a monopoly in G = (V,E). If the answer to the mvp is No, we then consider the max-controlled set problem (mcsp), whose objective is to find a sandwich graph G = (V,E) such that the number of vertices of G controlled by M is maximized. The mvp can be solved in polynomial time; the mcsp, however, is NP-hard. In this work, we present a deterministic polynomial time approximation algorithm for the mcsp with ratio $\frac{1}{2}$ + $\frac{1+\sqrt{n}}{2n-2}$, where n=|V|>4. (The case $n\leq4$ is solved exactly by considering the parameterized version of the mcsp.) The algorithm is obtained through the use of randomized rounding and derandomization techniques based on the method of conditional expectations. Additionally, we show how to improve this ratio if good estimates of expectation are obtained in advance.

In the last decade, formal methods have proved their interest whenanalyzing security protocols. Security protocols require inparticular to reason about the attacker knowledge. Two standardnotions are often considered in formal approaches: deducibility andindistinguishability relations. The first notion states whether anattacker can learn the value of a secret, while the latter stateswhether an attacker can notice some difference between protocol runswith different values of the secret. Several decision procedureshave been developed so far for both notions but none of them can beapplied in the context of e-voting protocols, which requirededicated cryptographic primitives. In this work, we show that bothdeduction and indistinguishability are decidable in polynomial timefor two theories modeling the primitives of e-voting protocols.

In this paper we study the parameterized complexity of approximating the parameterized counting problems contained in the class $\#W[P]$, the parameterized analogue of $\#P$. We prove a parameterized analogue of a famous theorem of Stockmeyer claiming that approximate counting belongs to the second level of the polynomial hierarchy.

We prove that the Fibonacci morphism is an automorphism of infinite order of free Burnside groups for all odd $n\geq 665$ and even $n = 16k \geq 8000$.

We show that it is decidable whether or not a given Q-rational series in several noncommutative variables has a cyclic image. By definition, a series r has a cyclic image if there is a rational number q such that all nonzero coefficients of r are integer powers of q.

In recent work we have proposed a novel approach to define idealized type systems for object-oriented languages, based on abstract compilation ofprograms into Horn formulas which are interpreted w.r.t. the coinductive (that is, the greatest) Herbrand model. In this paper we investigate how this approach can be applied also inthe presence of imperative features. This is made possible by considering a natural translation of Static Single Assignment intermediate form programs into Horn formulas, where φ functionscorrespond to union types.

The calculus of looping sequences is a formalism for describing theevolution of biological systems by means of term rewriting rules. Inthis paper we enrich this calculus with a type discipline whichpreserves some biological properties depending on the minimum andthe maximum number of elements of some type requested by the present elements. The typesystem enforces these properties and typed reductions guarantee thatevolution preserves them. As an example, we model the hemoglobinstructure and the equilibrium between cell death and division: typedreductions prevent undesirable behaviors.

Sets of integers form a monoid, where the product of two sets A and B is defined as the set containing a+b for all $a\in A$ and $b\in B$. We give a characterization of when a family of finite sets is a code in this monoid, that is when the sets do not satisfy any nontrivial relation. We also extend this result for some infinite sets, including all infinite rational sets.

We prove that some fairly basic questions on automata reading infinite words depend on the models of the axiomatic system ZFC. It is known that there are only three possibilities for the cardinality of the complement of an ω-language \hbox{$L(\mathcal{A})$}L(𝒜) accepted by a Büchi 1-counter automaton \hbox{$\mathcal{A}$}𝒜. We prove the following surprising result: there exists a 1-counter Büchi automaton \hbox{$\mathcal{A}$}𝒜 such that the cardinality of the complement \hbox{$L(\mathcal{A})^-$}L(𝒜) −  of the ω-language \hbox{$L(\mathcal{A})$}L(𝒜) is not determined by ZFC: (1) There is a model V1 of ZFC in which \hbox{$L(\mathcal{A})^-$}L(𝒜) −  is countable. (2) There is a model V2 of ZFC in which \hbox{$L(\mathcal{A})^-$}L(𝒜) −  has cardinal 2ℵ0. (3) There is a model V3 of ZFC in which \hbox{$L(\mathcal{A})^-$}L(𝒜) −  has cardinal ℵ1 with ℵ0 < ℵ1 < 2ℵ0.

We prove a very similar result for the complement of an infinitary rational relation accepted by a 2-tape Büchi automaton ℬ. As a corollary, this proves that the continuum hypothesis may be not satisfied for complements of 1-counter ω-languages and for complements of infinitary rational relations accepted by 2-tape Büchi automata. We infer from the proof of the above results that basic decision problems about 1-counter ω-languages or infinitary rational relations are actually located at the third level of the analytical hierarchy. In particular, the problem to determine whether the complement of a 1-counter ω-language (respectively, infinitary rational relation) is countable is in Σ13\(Π12 ∪ Σ12). This is rather surprising if compared to the fact that it is decidable whether an infinitary rational relation is countable (respectively, uncountable).

Motivated by striking properties of the well known Fibonacci wordwe consider pictures which are defined by this word and itsvariants via so-called turtle graphics. Such a picture can bebounded or unbounded. We characterize when the picture defined bynot only the Fibonacci recurrence, but also by a generalrecurrence formula, is bounded, the characterization beingcomputable.

It often occurs that local copies of a text are modified by users but that the local modifications are not synchronized (thus allowing the merged text to become the source for later editions) until later when, for instance the network connection is reestablished. Since text editions usually affect a small fraction of the whole content, the history of edit operations provides a compact representation of the modified file. In this paper, we define a normal form for these records which will permit for the comparison of all text files that have been obtained by editing a common source S when the difference between each output file Oi and the source file is given as a sequence Li of edit operations. We show that the normalized sequence is unique for all the equivalent text editions and provide efficient procedures with which to compute this normal form and to obtain the edit sequence LM transforming S into a merged file M which integrates all the local modifications. We also discuss how these normalization can be integrated into the operational transformation paradigm for optimistic replication.

A unit disk graph is the intersection graphof a family of unit disks in the plane.If the disks do not overlap, it is also a unit coin graph or penny graph.It is known that finding a maximum independent setin a unit disk graph is a NP-hard problem.In this work we extend this result to penny graphs.Furthermore, we prove that finding a minimum clique partitionin a penny graph is also NP-hard, and presenttwo linear-time approximation algorithms for the computation of clique partitions:a 3-approximation algorithm for unit disk graphsand a 2-approximation algorithm for penny graphs.

Minimizing a deterministic finite automata (DFA) is a very important problem in theory of automata and formal languages.Hopcroft's algorithm represents the fastest known solution to the such a problem. In this paper we analyze the behavior of this algorithm on a family binary automata, called tree-like automata, associated to binary labeled trees constructed by words. We prove that all the executions of the algorithm on tree-like automata associated to trees, constructed by standard words, have running time with the same asymptotic growth rate. In particular, we provide a lower and upper bound for the running time of the algorithm expressed in terms of combinatorial properties of the trees. We consider also tree-like automata associated to trees constructed by de Brujin words,and we prove that a queue implementation of the waiting set gives a Θ(n log n) execution while a stack implementation produces a linear execution. Such a result confirms the conjecture given in [A. Paun, M. Paun and A. Rodríguez-Patón. Theoret. Comput. Sci.410 (2009) 2424–2430.] formulated for a family of unary automata and, in addition, gives a positive answer also for the binary case.

The bandwidth minimization problem is of significance in network communication and related areas. Let G be a graph of n vertices. The two-dimensional bandwidth B2(G) of G is the minimum value of the maximum distance between adjacent vertices when G is embedded into an n × n grid in the plane. As a discrete optimization problem, determining B2(G) is NP-hard in general. However, exact results for this parameter can be derived for some special classes of graphs. This paper studies the “square-root rule” of the two-dimensional bandwidth, which is useful in evaluating B2(G) for some typical graphs.

We study cooperating distributed systems (CD-systems) of stateless deterministic restarting automata with window size 1 that are equipped with an external pushdown store. In this way we obtain an automata-theoretical characterization for the class of word languages that are linearizations of context-free trace languages.

Since recognizable tree languages are closed under the rational operations, every regular tree expression denotes a recognizable tree language. We provide an alternative proof to this fact that results in smaller tree automata. To this aim, we transfer Antimirov's partial derivatives from regular word expressions to regular tree expressions. For an analysis of the size of the resulting automaton as well as for algorithmic improvements, we also transfer the methods of Champarnaud and Ziadi from words to trees.

An ever present, common sense idea in language modelling research is that, for aword to be a valid phrase, it should comply with multiple constraints atonce. A new language definition model is studied, based on agreement or consensusbetween similar strings. Considering a regular set of strings over a bipartitealphabet made by pairs of unmarked/marked symbols, a match relation isintroduced, in order to specify when such strings agree. Then a regular setover the bipartite alphabet can be interpreted as specifying another languageover the unmarked alphabet, called the consensual language. A word is in theconsensual language if a set of corresponding matching strings is in theoriginal language. The family thus defined includes the regular languages andalso interesting non-semilinear ones. The word problem can be solved inNLOGSPACE, hence in P time. The emptiness problem is undecidable. Closure properties areproved for intersection with regular sets and inverse alphabetical homomorphism.Several conditions for a consensual definition to yield a regular language arepresented, and it is shown that the size of a consensual specification ofregular languages can be in a logarithmic ratio with respect to a DFA. Thefamily is incomparable with context-free and tree-adjoining grammar families.

Recently, a new measurement – the advice complexity – was introduced for measuring the information content of online problems. The aim is to measure the bitwise information that online algorithms lack, causing them to perform worse than offline algorithms. Among a large number of problems, a well-known scheduling problem, job shop scheduling with unit length tasks, and the paging problem were analyzed within this model. We observe some connections between advice complexity and randomization. Our special focus goes to barely random algorithms, i.e., randomized algorithms that use only a constant number of random bits, regardless of the input size. We adapt the results on advice complexity to obtain efficient barely random algorithms for both the job shop scheduling and the paging problem. Furthermore, so far, it has not yet been investigated for job shop scheduling how good an online algorithm may perform when only using a very small (e.g., constant) number of advice bits. In this paper, we answer this question by giving both lower and upper bounds, and also improve the best known upper bound for optimal algorithms.