Descriptional complexity of cellular automata and decidability questions

We study the descriptional complexity of cellular automata (CA), a parallel model of computation. We show that between one of the simplest cellular models, the realtime-OCA. and "classical" models like deterministic finite automata (DFA) or pushdown automata (PDA), there will be savings concerning the size of description not bounded by any recursive function, a so-called nonrecursive trade-off. Furthermore, nonrecursive trade-offs are shown between some restricted classes of cellular automata. The set of valid computations of a Turing machine can be recognized by a realtime-OCA. This implies that many decidability questions are not even semi decidable for cellular automata. There is no pumping lemma and no minimization algorithm for cellular automata

A Note on Emergence in Multi-Agent String Processing Systems

We propose a way to define (and, in a certain extent, even to measure) the phenomenon of emergence which appears in a complex system of interacting agents whose global behaviour can be described by a language and whose components (agents) can also be associated with grammars and languages. The basic idea is to identify the "linear composition of behaviours" with "closure under basic operations", such as the AFL (Abstract Families of Languages) operations, which are standard in the theory of formal languages

Regulated Formal Models and Their Reduction

Department of Theoretical Computer Science and Mathematical LogicKatedra teoretické informatiky a matematické logikyFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult

Cooperating Distributed Grammar Systems of Finite Index Working in Hybrid Modes

We study cooperating distributed grammar systems working in hybrid modes in connection with the finite index restriction in two different ways: firstly, we investigate cooperating distributed grammar systems working in hybrid modes which characterize programmed grammars with the finite index restriction; looking at the number of components of such systems, we obtain surprisingly rich lattice structures for the inclusion relations between the corresponding language families. Secondly, we impose the finite index restriction on cooperating distributed grammar systems working in hybrid modes themselves, which leads us to new characterizations of programmed grammars of finite index.Comment: In Proceedings AFL 2014, arXiv:1405.527

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

Capacity Bounded Grammars and Petri Nets

A capacity bounded grammar is a grammar whose derivations are restricted by assigning a bound to the number of every nonterminal symbol in the sentential forms. In the paper the generative power and closure properties of capacity bounded grammars and their Petri net controlled counterparts are investigated

Strictly Locally Testable and Resources Restricted Control Languages in Tree-Controlled Grammars

Tree-controlled grammars are context-free grammars where the derivation process is controlled in such a way that every word on a level of the derivation tree must belong to a certain control language. We investigate the generative capacity of such tree-controlled grammars where the control languages are special regular sets, especially strictly locally testable languages or languages restricted by resources of the generation (number of non-terminal symbols or production rules) or acceptance (number of states). Furthermore, the set theoretic inclusion relations of these subregular language families themselves are studied.Comment: In Proceedings AFL 2023, arXiv:2309.0112

Two-way metalinear PC grammar systems and their descriptional complexity

Besides a derivation step and a communication step, a two-way PC grammar system can make a reduction step during which it reduces the right-hand side of a context-free production to its left-hand side. This paper proves that every non-unary recursively enumerable language is defined by a centralized two-way grammar system, ┌, with two metalinear components in a very economical way. Indeed, ┌'s master has only three nonterminals and one communication production; furthermore, it produces all sentential forms with no more than two occurrences of nonterminals. In addition, during every computation, ┌ makes a single communication step. Some variants of two-way PC grammar systems are discussed in the conclusion of this paper

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