29 research outputs found
A novel cryptosystem based on Gluškov product of automata
The concept of Gluškov product was introduced by V. M. Gluškov in 1961. It was intensively studied by several scientists (first of all, by Ferenc Gécseg and the automata-theory school centred around him in Szeged, Hungary) since the middle of 60’s. In spite of the large number of excellent publications, no application of Gluškov-type products of automata in cryptography has arisen so far. This paper is the first attempt in this direction
Additive Cellular Automata Over Finite Abelian Groups: Topological and Measure Theoretic Properties
We study the dynamical behavior of D-dimensional (D >= 1) additive cellular automata where the alphabet is any finite abelian group. This class of discrete time dynamical systems is a generalization of the systems extensively studied by many authors among which one may list [Masanobu Ito et al., 1983; Giovanni Manzini and Luciano Margara, 1999; Giovanni Manzini and Luciano Margara, 1999; Jarkko Kari, 2000; Gianpiero Cattaneo et al., 2000; Gianpiero Cattaneo et al., 2004]. Our main contribution is the proof that topologically transitive additive cellular automata are ergodic. This result represents a solid bridge between the world of measure theory and that of topology theory and greatly extends previous results obtained in [Gianpiero Cattaneo et al., 2000; Giovanni Manzini and Luciano Margara, 1999] for linear CA over Z_m i.e. additive CA in which the alphabet is the cyclic group Z_m and the local rules are linear combinations with coefficients in Z_m. In our scenario, the alphabet is any finite abelian group and the global rule is any additive map. This class of CA strictly contains the class of linear CA over Z_m^n, i.e.with the local rule defined by n x n matrices with elements in Z_m which, in turn, strictly contains the class of linear CA over Z_m. In order to further emphasize that finite abelian groups are more expressive than Z_m we prove that, contrary to what happens in Z_m, there exist additive CA over suitable finite abelian groups which are roots (with arbitrarily large indices) of the shift map.
As a consequence of our results, we have that, for additive CA, ergodic mixing, weak ergodic mixing, ergodicity, topological mixing, weak topological mixing, topological total transitivity and topological transitivity are all equivalent properties. As a corollary, we have that invertible transitive additive CA are isomorphic to Bernoulli shifts. Finally, we provide a first characterization of strong transitivity for additive CA which we suspect it might be true also for the general case
Uniform Random Expressions Lack Expressivity
In this article, we question the relevance of uniform random models for algorithms that use expressions as inputs. Using a general framework to describe expressions, we prove that if there is a subexpression that is absorbing for a given operator, then, after repeatedly applying the induced simplification to a uniform random expression of size n, we obtain an equivalent expression of constant expected size. This proves that uniform random expressions lack expressivity, as soon as there is an absorbing pattern. For instance, (a+b)^* is absorbing for the union for regular expressions on {a,b}, hence random regular expressions can be drastically reduced using the induced simplification
Synchronizing automata over nested words
We extend the concept of a synchronizing word from deterministic finite-state automata (DFA) to nested word automata (NWA): A well-matched nested word is called synchronizing if it resets the control state of any configuration, i. e., takes the NWA from all control states to a single control state.
We show that although the shortest synchronizing word for an NWA, if it exists, can be (at most) exponential in the size of the NWA, the existence of such a word can still be decided in polynomial time. As our main contribution, we show that deciding the existence of a short synchronizing word (of at most given length) becomes PSPACE-complete (as opposed to NP-complete for DFA). The upper bound
makes a connection to pebble games and Strahler numbers, and the lower bound goes via small-cost synchronizing words for DFA, an intermediate problem that we also show PSPACE-complete. We also characterize the complexity of a number of related problems, using the observation that the intersection nonemptiness problem for NWA
is EXP-complete
Decidable Classes of Tree Automata Mixing Local and Global Constraints Modulo Flat Theories
We define a class of ranked tree automata TABG generalizing both the tree
automata with local tests between brothers of Bogaert and Tison (1992) and with
global equality and disequality constraints (TAGED) of Filiot et al. (2007).
TABG can test for equality and disequality modulo a given flat equational
theory between brother subterms and between subterms whose positions are
defined by the states reached during a computation. In particular, TABG can
check that all the subterms reaching a given state are distinct. This
constraint is related to monadic key constraints for XML documents, meaning
that every two distinct positions of a given type have different values. We
prove decidability of the emptiness problem for TABG. This solves, in
particular, the open question of the decidability of emptiness for TAGED. We
further extend our result by allowing global arithmetic constraints for
counting the number of occurrences of some state or the number of different
equivalence classes of subterms (modulo a given flat equational theory)
reaching some state during a computation. We also adapt the model to unranked
ordered terms. As a consequence of our results for TABG, we prove the
decidability of a fragment of the monadic second order logic on trees extended
with predicates for equality and disequality between subtrees, and cardinality.Comment: 39 pages, to appear in LMCS journa
Complexity and modeling power of insertion-deletion systems
SISTEMAS DE INSERCIĂ“N Y BORRADO: COMPLEJIDAD Y
CAPACIDAD DE MODELADO
El objetivo central de la tesis es el estudio de los sistemas de inserciĂłn y borrado y su
capacidad computacional. Más concretamente, estudiamos algunos modelos de
generación de lenguaje que usan operaciones de reescritura de dos cadenas. También
consideramos una variante distribuida de los sistemas de inserciĂłn y borrado en el
sentido de que las reglas se separan entre un nĂşmero finito de nodos de un grafo.
Estos sistemas se denominan sistemas controlados mediante grafo, y aparecen en
muchas áreas de la Informática, jugando un papel muy importante en los lenguajes
formales, la lingĂĽĂstica y la bio-informática. Estudiamos la decidibilidad/
universalidad de nuestros modelos mediante la variación de los parámetros de tamaño
del vector. Concretamente, damos respuesta a la cuestión más importante
concerniente a la expresividad de la capacidad computacional: si nuestro modelo es
equivalente a una máquina de Turing o no. Abordamos sistemáticamente las
cuestiones sobre los tamaños mĂnimos de los sistemas con y sin control de grafo.COMPLEXITY AND MODELING POWER OF
INSERTION-DELETION SYSTEMS
The central object of the thesis are insertion-deletion systems and their computational
power. More specifically, we study language generating models that use two string
rewriting operations: contextual insertion and contextual deletion, and their
extensions. We also consider a distributed variant of insertion-deletion systems in the
sense that rules are separated among a finite number of nodes of a graph. Such
systems are refereed as graph-controlled systems. These systems appear in many
areas of Computer Science and they play an important role in formal languages,
linguistics, and bio-informatics. We vary the parameters of the vector of size of
insertion-deletion systems and we study decidability/universality of obtained models.
More precisely, we answer the most important questions regarding the expressiveness
of the computational model: whether our model is Turing equivalent or not. We
systematically approach the questions about the minimal sizes of the insertiondeletion
systems with and without the graph-control
Nondeterministic Instance Complexity and Proof Systems with Advice
Motivated by strong Karp-Lipton collapse results in bounded arithmetic, Cook and KrajĂÄŤek [1] have recently introduced the notion of propositional proof systems with advice. In this paper we investigate the following question: Given a language L , do there exist polynomially bounded proof systems with advice for L ? Depending on the complexity of the underlying language L and the amount and type of the advice used by the proof system, we obtain different characterizations for this problem. In particular, we show that the above question is tightly linked with the question whether L has small nondeterministic instance complexity
New algorithms for Steiner tree reoptimization
Reoptimization is a setting in which we are given an (near) optimal solution of a problem instance and a local modification that slightly changes the instance. The main goal is that of finding an (near) optimal solution of the modified instance.
We investigate one of the most studied scenarios in reoptimization known as Steiner tree reoptimization. Steiner tree reoptimization is a collection of strongly NP-hard optimization problems that are defined on top of the classical Steiner tree problem and for which several constant-factor approximation algorithms have been designed in the last decade. In this paper we improve upon all these results by developing a novel technique that allows us to design polynomial-time approximation schemes. Remarkably, prior to this paper, no approximation algorithm better than recomputing a solution from scratch was known for the elusive scenario in which the cost of a single edge decreases. Our results are best possible since none of the problems addressed in this paper admits a fully polynomial-time approximation scheme, unless P=NP
Derivative Based Extended Regular Expression Matching Supporting Intersection, Complement and Lookarounds
Regular expressions are widely used in software. Various regular expression
engines support different combinations of extensions to classical regular
constructs such as Kleene star, concatenation, nondeterministic choice (union
in terms of match semantics). The extensions include e.g. anchors, lookarounds,
counters, backreferences. The properties of combinations of such extensions
have been subject of active recent research.
In the current paper we present a symbolic derivatives based approach to
finding matches to regular expressions that, in addition to the classical
regular constructs, also support complement, intersection and lookarounds (both
negative and positive lookaheads and lookbacks). The theory of computing
symbolic derivatives and determining nullability given an input string is
presented that shows that such a combination of extensions yields a match
semantics that corresponds to an effective Boolean algebra, which in turn opens
up possibilities of applying various Boolean logic rewrite rules to optimize
the search for matches.
In addition to the theoretical framework we present an implementation of the
combination of extensions to demonstrate the efficacy of the approach
accompanied with practical examples