A Second Course in Formal Languages and Automata Theory

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

Engaging and maintaining student engagement with the content of a course in formal language theory and the theory of computation is notoriously difficult. A major factor is student's perception of the relevance of the required mathematical material to computer science as they have studied the discipline. Usually, the mathematical material discourages students and for many it forms an insurmountable barrier. This paper describes the construction of a course using the discovery learning technique known as the Moore Method that directly addresses student engagement with this material. The results of the delivery of this course were that students remained engaged with the course throughout the semester and enthusiastically received the pedagogical approach employed. While the course described was delivered to a small class in a liberal arts college, it may be adapted to larger class sizes and taught in any computer science program.

This is a paper on Stone duality in computer science with special focus on topics with applications in formal language theory. In Section 2 we give a general overview of Stone duality in its various forms: for Boolean algebras, distributive lattices, and frames. For distributive lattices, we discuss both Stone and Priestley duality. We identify how to move between the different dualities and which dual spaces carry the Scott topology. We then focus on three themes. The first theme is additional operations on distributive lattices and Boolean algebras. Additional operations arise in denotational semantics in the form of predicate transformers. In verification they occur in the form of modal operators. They play an essential role in Eilenberg's variety theorem in the form of quotient operations. Quotient operations are unary instantiations of residual operators which are dual to the operations in the profinite algebras of algebraic language theory. We discuss additional operations in Section 3. The second theme is that of hyperspaces, that is, spaces of subsets of an underlying space. Some classes of algebras may be seen as the class of algebras for a functor. In the case of predicate transformers the dual functors are hyperspace constructions such as the Plotkin, Smyth, and Hoare powerdomain constructions. The algebras-for-a-functor point of view is central to the coalgebraic study of modal logic and to the solution of domain In the algebraic theory of formal languages various hyperspace-related product constructions, such as block and Schutzenberger products, are used to study complexity hierarchies. We describe a construction, similar to the Schutzenberger product, which is dual to adding a layer of quantification to formulas describing formal languages. We discuss hyperspaces in Section 4. The final theme is that of equations. These are pairs of elements of dual spaces. They arise via the duality between subalgebras and quotient spaces and have provided one of the most successful tools for obtaining decidability results for classes of regular languages. The perspective provided by duality allows us to obtain a notion of equations for the study of arbitrary formal languages. Equations in language theory is the topic of Section 5.

Introductory notions - preliminaries words and languages regular and context-free languages - regular languages rewriting systems and grammars context-free languages pushdown automata algorithm aspects - partial recursive functions recursively enumerable languages context-sensitive languages applications - codes biological applications.

В цифровую эпоху в процессе автоматизации, роботизации различных отраслей производства и внедрения в него искусственного интеллекта используется уже трудно обозримое множество языков программирования. В результате в подготовке будущих инженеров все большое значение приобретает теория формальных языков, лежащая в основе разработки языков программирования. Незнание теории формальных языков стало причиной многих ошибок в разработке и использовании языков программирования в высокотехнологичных отраслях производства. Поэтому необходимо внедрение в содержание подготовки будущих инженеров дисциплины «Элементы теории формальных языков» для овладения ими теорией и практикой языков программирования. Установлено, что в методической литературе не разработаны эффективные методы обучения будущих инженеров формальным языкам. Цель статьи заключается в исследовании важных математических и методических аспектов обучения теории формальных языков будущих инженеров. Ведущую роль в исследовании играли методология дискретной математики и абстрактной алгебры, культурологический подход и методика преемственности в обучения между школой и вузом в содержании подготовки будущих инженеров. Результаты исследования. Охарактеризована культурологическая роль дискретной математики в современной теории формальных языков. На основе анализа избранных трудов В. М. Глушкова, Д. Кнута и других ученых исследована фундаментальная роль формальных языков абстрактной алгебры в обучении теории формальных языков. Охарактеризованы алгебраические аспекты методики обучения теории формальных языков и грамматик и роль в этой методике классических понятий абстрактной алгебры и их свойств. На основе этого изложены элементы методики преемственности в обучении элементам теории формальных языков будущих инженеров. В этих элементах важную роль играют различные трактовки понятия «язык» и примеры языков из математики и информатики. Раскрыта важная роль грамматик формальных языков в дальнейшем обучении этой теории. Результаты статьи имеют важное теоретическое и практическое значение в дальнейшем исследовании проблем методологии обучения будущих инженеров элементам теории формальных языков. Они представляют интерес для преподавателей, ведущих профессиональную подготовку студентов инженерных направлений в сфере автоматизации, роботизации отраслей производства и внедрения в него Искусственного интеллекта. In the digital era, in the process of automation, robotization of various industries and the introduction of Artificial Intelligence into it, numerous programming languages are widely applied. As a result, the theory of formal languages ​that underlies the development of programming languages is becoming increasingly important in the training of engineering students. The nescience of formal languages theory has caused many mistakes in the development and use of programming languages ​in high-tech industries. Therefore, it is necessary to introduce the discipline "Elements of the Theory of Formal Languages" into the content of the training of future engineers in order to master the theory and practice of programming languages. At the same time, it was established that the methodological literature hasn’t developed effective methods for teaching future engineers to formal languages. The purpose of the article is to investigate important mathematical and methodological aspects of teaching the theory of formal languages ​to engineering students. The leading role in the study was played by the methodology of discrete math and abstract algebra. Also, the cultural approach and the methodology of pre-graduation in training between school and university in the content of training future engineers. Findings.The cultural role of discrete mathematics in the modern theory of formal languages is determined in the article. Based on the analysis of selected works of V.M. Glushkov, D. Knut, and other scientists, the article investigates the fundamental role of formal languages ​of abstract algebra in teaching the theory of formal languages. Algebraic aspects of teaching methodology of the theory of formal languages ​and grammars, and the role in this methodology of classical concepts of abstract algebra and their propertieshave also become the subjects of the study. Basing on this, the elements of the methodology of continuity in teaching the elements of the theory of formal languages of future engineers are set out in the article. Various interpretations of the concept of "language" and examples of languages from mathematics, and computer science play an important role in these elements. The research also states the important role of grammars of formal languages in the further training of this theory. The findings of the research have important theoretical and practical significance in the further study of the issues of the methodology for teaching future engineers to the elements of the theory of formal languages. They are of interest to teachers conducting the professional training of engineering students in the field of automation, robotization of production industries, and the introduction of Artificial Intelligence into it.

Indexed grammars are a generalization of context-free grammars and recognize a proper subset of context-sensitive languages. The class of languages recognized by indexed grammars are called indexed languages and they correspond to the languages recognized by nested stack automata. For example indexed grammars can recognize the language which is not context-free, but they cannot recognize which is context-sensitive. Indexed grammars identify a set of languages that are more expressive than context-free languages, while having decidability results that lie in between the ones of context-free and context-sensitive languages. In this work we study indexed grammars in order to formalize the relation between indexed languages and the other classes of languages in the Chomsky hierarchy. To this end, we provide a fixpoint characterization of the languages recognized by an indexed grammar and we study possible ways to abstract, in the abstract interpretation sense, these languages and their grammars into context-free and regular languages.

This article shows a general result about finite monoids and weight reducing string rewriting systems. As a consequence it proves a long standing conjecture in formal language theory: All regular languages are Church-Rosser congruential. The class of Church-Rosser congruential languages was introduced by McNaughton, Narendran, and Otto in 1988. A language L is Church-Rosser congruential if there exists a finite, confluent, and length-reducing semi-Thue system S such that L is a finite union of congruence classes modulo S . It was known that there are deterministic linear context-free languages which are not Church-Rosser congruential, but the conjecture was that all regular languages are of this form. The article offers a stronger statement: A language is regular if and only if it is strongly Church-Rosser congruential. It is the journal version of the conference abstract which was presented at ICALP 2012.

An operation of concatenation is defined for graphs. This allows strings to be viewed as expressions denoting graphs, and string languages to be interpreted as graph languages. For a class \(K\) of string languages, \({\rm Int}(K)\) is the class of all graph languages that are interpretations of languages from \(K\). For the classes REG and LIN of regular and linear context-free languages, respectively, \({\rm Int}({\rm REG}) = {\rm Int}({\rm LIN})\). \({\rm Int}({\rm REG})\) is the smallest class of graph languages containing all singletons and closed under union, concatenation and star (of graph languages). \({\rm Int}({\rm REG})\) equals the class of graph languages generated by linear HR (= Hyperedge Replacement) grammars, and \({\rm Int}(K)\) is generated by the corresponding \(K\)-controlled grammars. Two characterizations are given of the largest class \(K'\) such that \({\rm Int}(K') = {\rm Int}(K)\). For the class CF of context-free languages, \({\rm Int}({\rm CF})\) lies properly inbetween \({\rm Int}({\rm REG})\) and the class of graph languages generated by HR grammars. The concatenation operation on graphs combines nicely with the sum operation on graphs. The class of context-free (or equational) graph languages, with respect to these two operations, is the class of graph languages generated by HR grammars.

Polyhedral and Algebraic Methods in Computational Geometry provides a thorough introduction into algorithmic geometry and its applications. It presents its primary topics from the viewpoints of discrete, convex and elementary algebraic geometry. The first part of the book studies classical problems and techniques that refer to polyhedral structures. The authors include a study on algorithms for computing convex hulls as well as the construction of Voronoi diagrams and Delone triangulations. The second part of the book develops the primary concepts of (non-linear) computational algebraic geometry. Here, the book looks at Grbner bases and solving systems of polynomial equations. The theory is illustrated by applications in computer graphics, curve reconstruction and robotics. Throughout the book, interconnections between computational geometry and other disciplines (such as algebraic geometry, optimization and numerical mathematics) are established. Polyhedral and Algebraic Methods in Computational Geometry is directed towards advanced undergraduates in mathematics and computer science, as well as towards engineering students who are interested in the applications of computational geometry.

Living systems process information using chemistry. Computations can be viewed as language recognition problems where both languages and automata recognizing them form an inclusive hierarchy. Chemical realizations, without using biochemistry, of the main classes of computing automata, Finite Automata (FA), 1-stack Push Down Automata (1-PDA) and Turing Machine (TM) have recently been presented. These use chemistry for the representation of input information, its processing and output information. The Turing machine uses the Belousov-Zhabotinsky (BZ) oscillatory reaction to recognize a representative Context-Sensitive Language (CSL), the 1-PDA uses a pH network to recognize a Context Free Language (CFL) and a FA for a Regular Language (RL) uses a precipitation reaction. By chemically reconfiguring them to recognize representative languages in the lower classes of the Chomsky hierarchy we illustrate the inclusiveness of the hierarchy of native chemical automata. These examples open the door for chemical programming without biochemistry. Furthermore, the thermodynamic metric originally introduced to identify the accept/reject state of the chemical output for the CSL, can equally be used for recognizing CFL and RL by the automata. Finally, we point out how the chemical and thermodynamic duality of accept/reject criteria can be used in the optimization of the energetics and efficiency of computations.

We introduce geometric consideration into the theory of formal languages. We aim to shed light on our understanding of global patterns that occur on infinite strings. We utilise methods of geometric group theory. Our emphasis is on large scale geometries. Two infinite strings have the same large scale geometry if there are colour preserving bi-Lipschitz maps with distortions between the strings. Call these maps quasi-isometries. Introduction of large scale geometries poses several questions. The first question asks to study the partial order induced by quasi-isometries. This partial order compares large scale geometries; as such it presents an algebraic tool for classification of global patterns. We prove there is a greatest large scale geometry and infinitely many minimal large scale geometries. The second question is related to understanding the quasi-isometric maps on various classes of strings. The third question investigates the sets of large scale geometries of strings accepted by computational models, e.g. Buchi automata. We provide an algorithm that describes large scale geometries of strings accepted by Buchi automata. This links large scale geometries with automata theory. The fourth question studies the complexity of the quasi-isometry problem. We show the problem is Σ 3 0-complete thus providing a bridge with computability theory. Finally, the fifth question asks to build algebraic structures that are invariants of large scale geometries. We invoke asymptotic cones, a key concept in geometric group theory, defined via model-theoretic notion of ultra-product. Partly, we study asymptotic cones of algorithmically random strings thus connecting the topic with algorithmic randomness.

There is much discussion about the undergraduate computer science curriculum. With the rapid changes in computer science, the need to stay current is mandatory while the ability to develop and implement a contemporary undergraduate curriculum is extremely time-consuming and costly. Four schools have each developed a different model to undergraduate computer science education. Each school believes the work done has been successfully implemented at the respective institutions. They also believe that these works can be successfully adapted by others. An overview is given of the laboratory models and topics developed for a laboratory activity.

The dissecting power of regular languages

There has been much debate in the past few years about the appropriate mathematics requirements for an undergraduate computer science major. The discussion has focused primarily on two issues: (1) the underlying mathematical content of computer science courses and (2) the content of mathematics courses which would serve as appropriate cognate requirements for computer science major programs. While this discussion has been helpful, it has been too narrowly focused--it has not started from an understanding of the relationship between the disciplines of mathematics and computer science, but rather has sought to identify mathematical prerequisites that computer science majors need in order to take existing computer science courses. This paper is a small step in seeking to apply an understanding of the relationship between the disciplines of mathematics and computer science to the undergraduate computer science curriculum.

We investigate the descriptional and computational complexity of boundary sets of regular and context-free languages. The right (left, respectively) a-boundary set of a language L are those words that belong to L, where the a-predecessor or the a-successor of these words w.r.t. the prefix (suffix, respectively) relation is not in L. For regular languages described by deterministic finite automata (DFAs) we give tight bounds on the number of states for accepting boundary sets. Moreover, the question whether the boundary sets of a regular language is finite is shown to be NL-complete for DFAs, while it turns out to be PSPACE-complete for nondeterministic devices. Boundary sets for context-free languages are not necessarily context free anymore. Here we find a subtle difference of right and left a-boundary sets. While right a-boundary sets of deterministic context-free languages stay deterministic context free, we give an example of a deterministic context-free language the left a-boundary set of which is already non context free. In fact, the finiteness problem for a-boundary sets of context-free languages becomes undecidable.KeywordsTuring MachineRegular LanguageInput SymbolLanguage TreePushdown AutomatonThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.