JUMPING FINITE AUTOMATA

The present chapter introduces and studies jumping finite automata. In essence, these automata work just like classical finite automata except that they do not read their input strings in a symbol-by-symbol left-to-right way. Instead, after reading a symbol, they can jump in either direction within their input tapes and continue making moves from there. Once an occurrence of a symbol is read, it cannot be re-read again later on. Otherwise, their definition coincides with the definition of standard finite automata. Organized into eight sections, this chapter gives a systematic body of knowledge concerning jumping finite automata. First, it formalizes them (Sect. 17.1). Then, it demonstrates their fundamental properties (Sect. 17.2), after which it compares their power with the power of well-known language-defining formal devices (Sect. 17.3). Naturally, this chapter also establishes several results concerning jumping finite automata with respect to commonly studied areas of formal language theory, such as closure properties (Sect. 17.4) and decidability (Sect. 17.5). In addition, it establishes an infinite hierarchy of language families resulting from these automata (Sect. 17.6). Finally, it studies some special topics and features, such as one-directional jumps (Sect. 17.7) and various start configurations (Sect. 17.8). Throughout its discussion, this chapter points out several open questions regarding these automata, which may represent a new investigation area of automata theory in the future.

Various quantum versions o f the most basic models o f the classical finite automata have already been introduced and various modes of their computations have already started t o be investigated. In this paper we overview basic models, approaches, techniques and results in this promising area of quantum automata that is expected to play an important role also in theoretical computer science. We also summarize some open problems and research directions to pursue in this area.

Interference automata

An efficient algorithm for finding the Kleene closure of regular expressions matrices or fuzzy regular expression matrices is presented and illustrated by examples. Properties of Kleene closure are also investigated. The results may have useful applications in automata theory, pattern recognition, and pictorial information systems.

This paper proposes an improved morphological analyser for Arabic pronominal system using finite state method.The main advantage of the finite state method is very flexible, powerful and efficient.The most important results about FSAs, relates the class of languages generated by finite state automaton to certain closure properties.This result makes the theory of finite-state automata a very versatile and descriptive framework.The main contribution of this work is the full analysis and the representation of morphological analysis of all the inflections of pronoun forms in Arabic.In this paper we build a finite state network for the inflectional forms of the root words, restricted to all the inflections and grammatical properties of generating the dependent and independent forms of pronouns in Arabic language.The results show high score of accuracy in the output with all the needed linguistic features and the evaluation process of output is conducted using f-score test and the achievement is at the rate of 80% to 83%.The results from the study also provide the evidence that Arabic has strong concatenative word formations.

Complementation of finite automata on infinite words is not only a fundamental problem in automata theory, but also serves as a cornerstone for solving numerous decision problems in mathematical logic, model-checking, program analysis and verification. For Streett complementation, a significant gap exists between the current lower bound 2^{Omega(n*log(n*k))} and upper bound 2^{O(n*k*log(n*k))}, where n is the state size, k is the number of Streett pairs, and k can be as large as 2^{n}. Determining the complexity of Streett complementation has been an open question since the late 80's. In this paper we show a complementation construction with upper bound 2^{O(n*log(n)+n*k*log(k))} for k=O(n) and 2^{O(n^{2}*log(n))} for k=Omega(n), which matches well the lower bound obtained in the paper arXiv:1102.2963. We also obtain a tight upper bound 2^{O(n*log(n))} for parity complementation.

A finite automaton is one of the classic models of recognition devices, which is used to determine the type of language a string belongs to. A string is said to be recognized by a finite automaton if the automaton "reads" the string from the left to the right starting from the initial state and finishing at a final state. Another type of automata which is a counterpart of sticker systems, namely Watson-Crick automata, is finite automata which can scan the double-stranded tapes of DNA strings using the complimentary relation. The properties of groups have been extended for the recognition of finite automata over groups. In this paper, two variants of automata, modified deterministic finite automata and modified deterministic Watson-Crick automata are used in the study of Abelian groups. Moreover, the relation between finite automata diagram over Abelian groups and the Cayley table is introduced. In addition, some properties of Abelian groups are presented in terms of automata.

The characterization of the class of FO[+]-definable languages by some generating or recognizing device is still an open problem. We prove that, restricted to word bounded languages, this class coincides with the class of semilinear languages. We also study the closure properties of the classes of languages definable in FO[+1], FO[<], FO[+] and FOC[+] under the main classical operations.

A standard bridge between automata theory and logic is provided by the notion of characteristic formula. This paper investigates this problem for the class of event-recording automata (ERA), a subclass of timed automata in which clocks are associated with actions and that enjoys very good closure properties. We first study the problem of expressing characteristic formulae for ERA in Event-Recording Logic (ERL ), a logic introduced by Sorea to express event-based timed specifications. We prove that the construction proposed by Sorea for ERA without invariants is incorrect. More generally, we prove that timed bisimilarity cannot in general be expressed in ERL for the class of ERA , and study under which conditions on ERA it can be. Then, we introduce the logic WTμ , a new logic for event-based timed specifications closer to the timed logic ℒν that was introduced by Laroussinie, Larsen and Weise. We prove that it is strictly more expressive than ERL , and that its model-checking problem against ERA is EXPTIME -complete. Finally, we provide characteristic formulae constructions in WTμ for characterizing the general class of ERA up to timed (bi)similarity and study the complexity issues.

Closure properties of linear context-free tree languages with an application to optimality theory

This book is the first self-contained exposition of the fascinating link between dynamical systems and dimension groups. The authors explore the rich interplay between topological properties of dynamical systems and the algebraic structures associated with them, with an emphasis on symbolic systems, particularly substitution systems. It is recommended for anybody with an interest in topological and symbolic dynamics, automata theory or combinatorics on words. Intended to serve as an introduction for graduate students and other newcomers to the field as well as a reference for established researchers, the book includes a thorough account of the background notions as well as detailed exposition – with full proofs – of the major results of the subject. A wealth of examples and exercises, with solutions, serve to build intuition, while the many open problems collected at the end provide jumping-off points for future research.

In previous articles we presented a PC based educational software on lexical analysis [1] and semantical analysis [4]. These systems were developed using an authoring system under MS Windows. The user can not change regular expressions or input words for nondeterministic respectively deterministic finite automata. To overcome these restrictions we developed GaniFA, our Java applet for visualization of algorithms from automata theory. It can be downloaded from our web page http://www.cs.uni-sb.de/GANIMAL as a JAR-file and requires a Java Plug-In 1.2. We invite the reader to use it in own web-based exercises, lecture notes or presentations on finite automata. Furthermore this web page gives a short overview, how this applet can be customized and embedded into HTML web pages.

We use the concept of trie (prefix tree) representation of a prefix-closed finite language L to design a simple nondeterministic automaton. Each computation of this trie automaton corresponds to a subword occurrence of a word from L in the input word. The matrix representation of the trie automaton leads to a fairly general extension of the original concept of the Parikh matrix from [7].

Representing recursively enumerable languages by iterated deletion

Logarithmic space bounded complexity classes such as L and NL play a central role in space bounded computation. The study of counting versions of these complexity classes have lead to several interesting insights into the structure of computational problems such as computing the determinant and counting paths in directed acyclic graphs. Though parameterised complexity theory was initiated roughly three decades ago by Downey and Fellows, a satisfactory study of parameterised logarithmic space bounded computation was developed only in the last decade by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). In this paper, we introduce a new framework for parameterised counting in logspace, inspired by the parameterised space bounded models developed by Elberfeld, Stockhusen and Tantau (IPEC 2013, Algorithmica 2015). They defined the operators para_W and para_β for parameterised space complexity classes by allowing bounded nondeterminism with multiple-read and read-once access, respectively. Using these operators, they characterised the parameterised complexity of natural problems on graphs. In the spirit of the operators para_W and para_β by Stockhusen and Tantau, we introduce variants based on tail-nondeterminism, para_{W[1]} and para_{βtail}. Then, we consider counting versions of all four operators applied to logspace and obtain several natural complete problems for the resulting classes: counting of paths in digraphs, counting first-order models for formulas, and counting graph homomorphisms. Furthermore, we show that the complexity of a parameterised variant of the determinant function for (0,1)-matrices is #para_{βtail} L-hard and can be written as the difference of two functions in #para_{βtail} L. These problems exhibit the richness of the introduced counting classes. Our results further indicate interesting structural characteristics of these classes. For example, we show that the closure of #para_{βtail} L under parameterised logspace parsimonious reductions coincides with #para_β L, that is, modulo parameterised reductions, tail-nondeterminism with read-once access is the same as read-once nondeterminism. Initiating the study of closure properties of these parameterised logspace counting classes, we show that all introduced classes are closed under addition and multiplication, and those without tail-nondeterminism are closed under parameterised logspace parsimonious reductions. Also, we show that the counting classes defined can naturally be characterised by parameterised variants of classes based on branching programs in analogy to the classical counting classes. Finally, we underline the significance of this topic by providing a promising outlook showing several open problems and options for further directions of research.