ON EXPERIMENTAL PROOF OF "P VERSUS NP"THEOREM

Lecture Notes in Computer Science, 2001

This paper considers finite-automata based algorithms for handling linear arithmetic with both real and integer variables. Previous work has shown that this theory can be dealt with by using finite automata on infinite words, but this involves some difficult and delicate to implement algorithms. The contribution of this paper is to show, using topological arguments, that only a restricted class of automata on infinite words are necessary for handling real and integer linear arithmetic. This allows the use of substantially simpler algorithms and opens the path to the implementation of a usable system for handling this combined theory. ⋆ This work was partially funded by a grant of the "Communauté française de Belgique-Direction de la recherche scientifique-Actions de recherche concertées".

In this paper we discuss P automata, constructs combining properties of classical automata and P systems being in interaction with their environments. We describe the most important variants and their properties, and propose new topics and open problems for future research.

2019

We summarize basic concepts, important results, and recent developments on P automata, variants of P systems combining features of classical automata and complex systems being in interaction with their environments. We also discuss topics for future research. brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Repository of the Academy's Library and investigated, which differ from each other in the main ingredients of these systems: the objects the P system operates with, the way of defining the acceptance, the way of communication with the environment, the types of the communication rules used by the regions, the types of the rules associated with the regions (whether or not evolution rules are allowed to be used), and whether or not the membrane structure changes during the computation. Summaries on these constructs and their properties can be found in [36, 8, 14, 48, 9-11, 21]. Due to the power of the underlying P system, a lot of P automaton variants are able to accept any recursively enumerable language, even with size parameters bounded by a small constant. However, P automata with significantly less computational power are of special interest as well. For example, the generic variant which is based on antiport rules with promoters or inhibitors and accepts with final states, using the sequential working mode and some well-chosen mappings for defining its language, describes a language class with sub-logarithmic space complexity. In this way, a "natural description" of this particular complexity class is provided. In the following sections we describe the most important variants of P automata and their properties. Special emphasis is put on non-standard features of P automata, namely, that the same construct is able to operate over both finite and infinite alphabets, languages of P automata can be defined on the base of finite and infinite run (computation) of the system, and that to obtain large computational power they do not need workspace overhead. We also discuss dP automata, i.e. distributed systems of P automata, a finite collection of P automata communicating not only with their joint environment but with each other. We demonstrate further concepts and methods for describing parallelism, cooperation, communication, and distribution in terms of purely communicating accepting P systems. We also suggest possible new topics and problems for future research.

Theoretical Computer Science, 2009

Watson-Crick automata are finite state automata working on double-stranded tapes, introduced to investigate the potential of DNA molecules for computing. In this paper, we continue the investigation of descriptional complexity of Watson-Crick automata initiated by Păun et al. [A. Păun, M. Păun, State and transition complexity of Watson-Crick finite automata, in: G. Ciobanu, G. Paun (Eds.), Fundamentals of Computation Theory, FCT'99, in: LNCS, vol. 1684. In particular, we show that any finite language, as well as any unary regular language, can be recognized by a Watson-Crick automaton with only two, and respectively three, states. Also, we formally define the notion of determinism for these systems. Contrary to the case of non-deterministic Watson-Crick automata, we show that, for deterministic ones, the complementarity relation plays a major role in the acceptance power of these systems.

2009

Automata theory lies at the foundation of computer science, and is vital to a theoretical understanding of how computers work and what constitutes formal methods. This treatise gives a rigorous account of the topic and illuminates its real meaning by looking at the subject in a variety of ways. The first part of the book is organised around notions of rationality and recognisability. The second part deals with relations between words realised by finite automata, which not only exemplifies the automata theory but also illustrates the variety of its methods and its fields of application. Many exercises are included, ranging from those that test the reader, to those that are technical results, to those that extend ideas presented in the text. Solutions or answers to many of these are included in the book.

Lecture Notes in Computer Science, 2016

We survey recent results on the descriptional complexity of self-verifying finite automata. In particular, we discuss the cost of simulation of self-verifying finite automata by deterministic finite automata, and the complexity of basic regular operations on languages represented by self-verifying finite automata.

Communications in Mathematics and Applications, 2022

Automaton is a system that spontaneously gives an output from an input. The input may be energy, information, materials, etc. The system works without the intervention of man. Simply automaton (plural: automata or automatons) is a self-operating machine. Its synonym is ROBOT. In this paper, an attempt has been made to exhibit the relation between linear congruence and automata theory. Also, an effort has been put to solve certain problems of the Chinese Remainder theorem using the Cartesian product of finite automata theory. In deterministic finite automata, the acceptable strings give the solutions of the Chinese Remainder Theorem (CRT). The main result of the paper is that residue classes can be recognized by finite automaton. This is the novelty of the article. Finally, we conclude with certain examples and non-examples alike!

2022

A deterministic finite automaton (DFA) with a set of states Q is completely reachable if every non-empty subset of Q is the image of the action of some word applied to Q. The concept was first introduced by Bondar and Volkov (2016), who also raised the question of the complexity of deciding if an automaton is completely reachable. We develop a polynomial-time algorithm for this problem, which is based on a complement-intersecting technique for finding an extending word for a subset of states. Additionally, we prove a weak Don's conjecture for this class of automata: a subset of size k is reachable with a word of length at most 2(|Q| − k)|Q|. This implies a quadratic upper bound in |Q| on the length of the shortest synchronizing words (reset threshold) and generalizes earlier upper bounds derived for subclasses of completely reachable automata. 2012 ACM Subject Classification Theory of computation → Formal languages and automata theory; Mathematics of computing → Discrete mathematics

arXiv (Cornell University), 2023

We study the complexity relationship between three models of unbounded memory automata: nu-automata (ν-A), Layered Memory Automata (LaMA)and History-Register Automata (HRA). These are all extensions of finite state automata with unbounded memory over infinite alphabets. We prove that the membership problem is NP-complete for all of them, while they fall into different classes for what concerns non-emptiness. The problem of non-emptiness is known to be Ackermann-complete for HRA, we prove that it is PSPACE-complete for ν-A.

Computers Ieee Transactions on, 1991

We investigate the local testability problem of deterministic finite automata. A locally testable language is a language with the property that, for some nonnegative integer FE, whether or not a word w is in the language depends on 1) the prefix and sufix of w of length IC, and 2) the set of substrings of w of length k + 1, without regard to the order in which these substrings occur. The local testability problem is, given a deterministic finite automaton, to decide whether or not it accepts a locally testable language. Until now, no polynomial time algorithm for this problem has appeared in the literature. We present an 0 (n2) time algorithm for the local testability problem based on two simple properties that characterize locally testable automata.