Event structures and trace monoids

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.

The correspondence between sequential program schemes and formal languages is well known (Blikle and Mazurkiewicz (1972), Engelfriet (1974)). The situation is more complicated in the case of parallel program schemes, and trace languages (Mazurkiewicz (1977)) have been introduced to describe them. We introduce the concept of the closure of a language on a so called independence relation on the alphabet of the language, and formulate several theorems about them and the trace languages. We investigate the closedness properties of Chomsky classes under closure on independence relations, and as a special case we derive a new necessary and sufficient condition for the regularity of the commutative closure of a language.

We introduce a meta-model based on formal languages, dubbed formal choreographic languages, to study message-passing systems. Our framework allows us to generalise standard constructions from the literature and to compare them. In particular, we consider notions such as global view, local view, and projections from the former to the latter. The correctness of local views projected from global views is characterised in terms of a closure property. We consider a number of communication properties -- such as (dead)lock-freedom -- and give conditions on formal choreographic languages to guarantee them. Finally, we show how formal choreographic languages can capture existing formalisms; specifically we consider communicating finite-state machines, choreography automata, and multiparty session types. Notably, formal choreographic languages, differently from most approaches in the literature, can naturally model systems exhibiting non-regular behaviour.

We study the relationships between a number of behavioural notions that have arisen in the theory of distributed computing. In order to sharpen the understanding of these relationships we apply the chosen behavioural notions to a basic net-theoretic model of distributed systems called elementary net systems. The behavioural notions that are considered here are trace languages, non-sequential processes, unfoldings and event structures.<p>The relationships between these notions are brought out in the process of establishing that for each elementary net system, the trace language representation of its behaviour agrees in a strong way with the event structure representation of its behaviour.</p>

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.

Models will play a central role in the representation, storage, manipulation, and communication of knowledge in systems biology. Models capable of fulfilling such a role will likely differ from the all familiar styles deployed with great success in the physical sciences. Molecular systems at the basis of cellular decision processes are concurrent and combinatorial. Their behavior is as much constrained by relationships of causality between molecular interactions as it is by chemical kinetics. Understanding how such systems give rise to coherent behavior and designing effective interventions to fight disease will require a notion of model that is akin to the concept of program in computer science. I will discuss recent progress in implementing a platform and tools for formal analysis that bring us closer to this vision. Protein interactions are represented by means of rules expressed in a formal language that captures a very simple, yet effective and biologically meaningful level of abstraction. Models, then, are collections of rules operating on an initial set of agents, in complete analogy to rules of organic chemical reactions. I will describe tools for analyzing and navigating rule collections as well as exploring their dynamics. We draw on concepts familiar to computer science, especially event structures, and adapt them to biological needs with the goal of formalizing the notion of pathway. The challenges are many, but a road map for the future is discernible. Computer science will play a central role in providing an additional foundational layer, both theoretical and practical, that neither physics nor chemistry can offer on their own in the future definition of the biological sciences.

Event Structures for Local Traces

Traditionally, formal languages are defined as sets of words. More recently, the alternative coalgebraic or coinductive representation as infinite tries, i.e., prefix trees branching over the alphabet, has been used to obtain compact and elegant proofs of classic results in language theory. In this article, we study this representation in the Isabelle proof assistant. We define regular operations on infinite tries and prove the axioms of Kleene algebra for those operations. Thereby, we exercise corecursion and coinduction and confirm the coinductive view being profitable in formalizations, as it improves over the set-of-words view with respect to proof automation.

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.

Written for graduate students and advanced undergraduates in computer science, A Second Course in Formal Languages and Automata Theory treats topics in the theory of computation not usually covered in a first course. After a review of basic concepts, the book covers combinatorics on words, regular languages, context-free languages, parsing and recognition, Turing machines, and other language classes. Many topics often absent from other textbooks, such as repetitions in words, state complexity, the interchange lemma, 2DPDAs, and the incompressibility method, are covered here. The author places particular emphasis on the resources needed to represent certain languages. The book also includes a diverse collection of more than 200 exercises, suggestions for term projects, and research problems that remain open.

In Ten Lectures on Event Structure in a Network Theory of Language, Nikolas Gisborne explores verb meaning. He discusses theories of events and how a network model of language-in-the-mind should be theorized; what the lexicon is; how to probe word meaning; evidence for structure in word meaning; polysemy; the lexical semantics of causation; a type hierarchy of events; and event types cross-linguistically. He also looks at the relationship between different classes of events or event types and aktionsarten; transitivity alternations and argument linking. Gisborne argues that the social and cognitive embedding of language, requires a view of linguistic structure as a network where even the analysis of verb meaning can require an understanding of the role of speaker and hearer.

An abstract definition of bisimulation is presented. It enables a uniform definition of bisimulation across a range of different models for parallel computation presented as categories. As examples, transition systems, synchronisation trees, transition systems with independence (an abstraction from Petri nets) and labelled event structures are considered. On transition systems the abstract definition readily specialises to Milner's strong bisimulation. On event structures it explains and leads to a revision of history-preserving bisimulation of Rabinovitch and Traktenbrot, Goltz and van Glabeek. A tie-up with open maps in a (pre)topos, as they appear in the work of Joyal and Moerdijk, brings to light a promising new model, presheaves on categories of pomsets, into which the usual category of labelled event structures embeds fully and faithfully. As an indication of its promise, this new presheaf model has ``refinement'' operators, though further work is required to justify their appropriateness and understand their relation to previous attempts. The general approach yields a logic, generalising Hennessy-Milner logic, which is characteristic for the generalised notion of bisimulation.

The theory of traces belongs to both formal language theory and the theory of concurrent systems. In both these disciplines it is a well-recognized and dynamic research area. Within formal language theory it yields the theory of partially commutative monoids, and provides an important connection between languages and graphs. Within the theory of concurrent systems it provides an important formal framework for the analysis and synthesis of concurrent systems. This monograph covers all important research lines of the theory of traces - each chapter of the book is devoted to one research line and is written by leading experts. It is organized in such a way that each chapter can be read independently - and hence is suitable for advanced courses/seminars on formal language theory and the theory of concurrent systems.

Almost four decades have passed since Formal Grammars first appeared in 1974. At that time it was still possible to rather comprehensively review for (psycho)linguists the relevant literature on the theory of formal languages and automata, on their applications in linguistic theory and in the psychology of language. That is no longer feasible. In all three areas developments have been substantial, if not breathtaking. Nowadays, an interested linguist or psycholinguist opening any text on formal languages can no longer see the wood for the trees, as it is by no means evident which formal, mathematical tools are really required for natural language applications. An historical perspective can be helpful here. There are paths through the wood that have been beaten since decades; they can still provide useful orientation. The origins of these paths can be traced in the three volumes of Formal Grammars, brought together in the present re-edition. In a newly added postscript the author has sketched what has become, after all these years, of formal grammars in linguistics and psycholinguistics, or at least some of the core developments. This chapter may provide further motivation for the reader to make a trip back to some of the historical sources.

In the paper will be presented some advances in crypto-biometric procedures used for encryption and division of secret data in the cloud. A novel solutions of the secret data sharing will be presented, based on the concept of mathematical linguistics, which are used to split and share information between various instances in the cloud. The essence of the presented approach is a linguistic concept stemming from the theory of formal languages, which concept is used in recording, analysing and interpreting various types of data treated as elements of a certain formal language.