Bounded Languages Meet Cellular Automata with Sparse Communication

Cellular automata are one-dimensional arrays of interconnected interacting finite automata. We investigate one of the weakest classes, the real-time one-way cellular automata, and impose an additional restriction on their inter-cell communication by bounding the number of allowed uses of the links between cells. Moreover, we consider the devices as acceptors for bounded languages in order to explore the borderline at which non-trivial decidability problems of cellular automata classes become decidable. It is shown that even devices with drastically reduced communication, that is, each two neighboring cells may communicate only constantly often, accept bounded languages that are not semilinear. If the number of communications is at least logarithmic in the length of the input, several problems are undecidable. The same result is obtained for classes where the total number of communications during a computation is linearly bounded

DESCRIPTIONAL COMPLEXITY AND PARIKH EQUIVALENCE

The thesis deals with some topics in the theory of formal languages and automata. Speci\ufb01cally, the thesis deals with the theory of context-free languages and the study of their descriptional complexity. The descriptional complexity of a formal structure (e.g., grammar, model of automata, etc) is the number of symbols needed to write down its description. While this aspect is extensively treated in regular languages, as evidenced by numerous references, in the case of context-free languages few results are known. An important result in this area is the Parikh\u2019s theorem. The theorem states that for each context-free language there exists a regular language with the same Parikh image. Given an alphabet \u3a3 = {a1, . . . , am}, the Parikh image is a function \u3c8 : \u3a3^ 17\u2192 N^m that associates with each word w 08\u3a3^ 17, the vector \u3c8(w)=(|w|_a1, |w|_a2, . . . , |w|_am), where |w|_ai is the number of occurrences of ai in w. The Parikh image of a language L 86\u3a3^ 17 is the set of Parikh images of its words. For instance, the language {a^nb^n | n 65 0} has the same Parikh image as (ab)^ 17. Roughly speaking, the theorem shows that if the order of the letters in a word is disregarded, retaining only the number of their occurrences, then context-free languages are indistinguishable from regular languages. Due to the interesting theoretical property of the Parikh\u2019s theorem, the goal of this thesis is to study some aspects of descriptional complexity according to Parikh equivalence. In particular, we investigate the conversion of one-way nondeterministic \ufb01nite automata and context-free grammars into Parikh equivalent one-way and two-way deterministic \ufb01nite automata, from a descriptional complexity point of view. We prove that for each one-way nondeterministic automaton with n states there exist Parikh equivalent one-way and two-way deterministic automata with e^O(sqrt(n lnn)) and p(n) states, respectively, where p(n) is a polynomial. Furthermore, these costs are tight. In contrast, if all the words accepted by the given one-way nondeterministic automaton contain at least two different letters, then a Parikh equivalent one-way deterministic automaton with a polynomial number of states can be found. Concerning context-free grammars, we prove that for each grammar in Chomsky normal form with h variables there exist Parikh equivalent one-way and two-way deterministic automata with 2^O(h^2 ) and 2^O(h) states, respectively. Even these bounds are tight. A further investigation is the study under Parikh equivalence of the state complexity of some language operations which preserve regularity. For union, concatenation, Kleene star, complement, intersection, shuffle, and reversal, we obtain a polynomial state complexity over any \ufb01xed alphabet, in contrast to the intrinsic exponential state complexity of some of these operations in the classical version. For projection we prove a superpolynomial state complexity, which is lower than the exponential one of the corresponding classical operation. We also prove that for each two one-way deterministic automata A and B it is possible to obtain a one-way deterministic automaton with a polynomial number of states whose accepted language has as Parikh image the intersection of the Parikh images of the languages accepted by A and B

On Communication Complexity in Evolution-Communication P Systems

Looking for a theory of communication complexity for P systems, we consider here so-called evolution-communication (EC for short) P systems, where objects evolve by multiset rewriting rules without target commands and pass through membranes by means of symport/antiport rules. (Actually, in most cases below we use only symport rules.) We first propose a way to measure the communication costs by means of “quanta of energy” (produced by evolution rules and) consumed by communication rules. EC P systems with such costs are proved to be Turing complete in all three cases with respect to the relation between evolution and communication operations: priority of communication, mixing the rules without priority for any type, priority of evolution (with the cost of communication increasing in this ordering in the universality proofs). More appropriate measures of communication complexity are then defined, as dynamical parameters, counting the communication steps or the number (and the weight) of communication rules used during a computation. Such parameters can be used in three ways: as properties of P systems (considering the families of sets of numbers generated by systems with a given communication complexity), as conditions to be imposed on computations (accepting only those computations with a communication complexity bounded by a given threshold), and as standard complexity measures (defining the class of problems which can be solved by P systems with a bounded complexity). Because we ignore the evolution steps, in all three cases it makes sense to consider hierarchies starting with finite complexity thresholds. We only give some preliminary results about these hierarchies (for instance, proving that already their lower levels contain complex – e.g., non-semilinear – sets), and we leave open many problems and research issues.Junta de Andalucía P08 – TIC 0420

Geometric decision procedures and the VC dimension of linear arithmetic theories

This paper resolves two open problems on linear integer arithmetic (LIA), also known as Presburger arithmetic. First, we give a triply exponential geometric decision procedure for LIA, i.e., a procedure based on manipulating semilinear sets. This matches the running time of the best quantifier elimination and automata-based procedures. Second, building upon our first result, we give a doubly exponential upper bound on the Vapnik–Chervonenkis (VC) dimension of sets definable in LIA, proving a conjecture of D. Nguyen and I. Pak [Combinatorica 39, pp. 923–932, 2019]. These results partially rely on an analysis of sets definable in linear real arithmetic (LRA), and analogous results for LRA are also obtained. At the core of these developments are new decomposition results for semilinear and -semilinear sets, the latter being the sets definable in LRA. These results yield new algorithms to compute the complement of (-)semilinear sets that do not cause a non-elementary blowup when repeatedly combined with procedures for other Boolean operations and projection. The existence of such an algorithm for semilinear sets has been a long-standing open problem.</p

Complexity and modeling power of insertion-deletion systems

SISTEMAS DE INSERCIÓN Y BORRADO: COMPLEJIDAD Y CAPACIDAD DE MODELADO El objetivo central de la tesis es el estudio de los sistemas de inserción 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 inserción y borrado en el sentido de que las reglas se separan entre un número 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 lingüística 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 mínimos 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

Computing with cells: membrane systems - some complexity issues.

Membrane computing is a branch of natural computing which abstracts computing models from the structure and the functioning of the living cell. The main ingredients of membrane systems, called P systems, are (i) the membrane structure, which consists of a hierarchical arrangements of membranes which delimit compartments where (ii) multisets of symbols, called objects, evolve according to (iii) sets of rules which are localised and associated with compartments. By using the rules in a nondeterministic/deterministic maximally parallel manner, transitions between the system configurations can be obtained. A sequence of transitions is a computation of how the system is evolving. Various ways of controlling the transfer of objects from one membrane to another and applying the rules, as well as possibilities to dissolve, divide or create membranes have been studied. Membrane systems have a great potential for implementing massively concurrent systems in an efficient way that would allow us to solve currently intractable problems once future biotechnology gives way to a practical bio-realization. In this paper we survey some interesting and fundamental complexity issues such as universality vs. nonuniversality, determinism vs. nondeterminism, membrane and alphabet size hierarchies, characterizations of context-sensitive languages and other language classes and various notions of parallelism

P Systems and Topology: Some Suggestions for Research

Lately, some studies linked the computational power of abstract computing systems based on multiset rewriting to Petri nets and the computation power of these nets to their topology. In turn, the computational power of these abstract computing devices can be understood just looking at their topology, that is, information flow. This line of research is very promising for several aspects: its results are valid for a broad range of systems based on multiset rewriting; it allows to know the computational power of abstract computing devices without tedious proofs based on simulations; it links computational power to topology and, in this way, it opens a broad range of questions. In this note we summarize the known result on this topic and we list a few suggestions for research together with the relevance of possible outcomes

Commutative Languages and their Composition by Consensual Methods

Commutative languages with the semilinear property (SLIP) can be naturally recognized by real-time NLOG-SPACE multi-counter machines. We show that unions and concatenations of such languages can be similarly recognized, relying on -- and further developing, our recent results on the family of consensually regular (CREG) languages. A CREG language is defined by a regular language on the alphabet that includes the terminal alphabet and its marked copy. New conditions, for ensuring that the union or concatenation of CREG languages is closed, are presented and applied to the commutative SLIP languages. The paper contributes to the knowledge of the CREG family, and introduces novel techniques for language composition, based on arithmetic congruences that act as language signatures. Open problems are listed.Comment: In Proceedings AFL 2014, arXiv:1405.527

Operational State Complexity under Parikh Equivalence

We investigate, under Parikh equivalence, the state complexity of some language operations which preserve regularity. For union, concatenation, Kleene star, complement, intersection, shue, and reversal, we obtain a polynomial state complexity over any xed alphabet, in contrast to the intrinsic exponential state complexity of some of these operations in the classical version. For projection we prove a superpolynomial state complexity, which is lower than the exponential one of the corresponding classical operation. We also prove that for each two deterministic automata A and B it is possible to obtain a deterministic automaton with a polynomial number of states whose accepted language has as Parikh image the intersection of the Parikh images of the languages accepted by A and B