Conditional Bisimilarity for Reactive Systems

Reactive systems \`a la Leifer and Milner, an abstract categorical framework for rewriting, provide a suitable framework for deriving bisimulation congruences. This is done by synthesizing interactions with the environment in order to obtain a compositional semantics. We enrich the notion of reactive systems by conditions on two levels: first, as in earlier work, we consider rules enriched with application conditions and second, we investigate the notion of conditional bisimilarity. Conditional bisimilarity allows us to say that two system states are bisimilar provided that the environment satisfies a given condition. We present several equivalent definitions of conditional bisimilarity, including one that is useful for concrete proofs and that employs an up-to-context technique, and we compare with related behavioural equivalences. We consider examples based on DPO graph rewriting, an instantiation of reactive systems.

Exploring Students’ Misconception in the Frame of Graphic and Figural Representation on Projectile Motion Regarding to the COVID-19 Constraints

Remote learning has reformed the normal tradition of physics instruction in the existence of COVID-19 pandemic. The previous documentation of student’s misconception is mostly discovered in the normal condition of physics instruction. Regarding to the pandemic outbreak is necessary since distance learning routines have absolutely influence the physics education outcomes all over the world. To explore the issues, this quantitative study has descriptively identified students' misconception on projectile motion in the frame of graphic and figural representations. The six items of Busyairi and Zuhdi’s three tier test were administered to the 76 students in a Javanese public high school, Indonesia. Overall, the Arslan, Cigdemolu, & Moseley’s categorical framework classified that the whole participants still congregate in the misconception’s groups on projectile motion. The lack of students’ representation on projectile motion might be magnified by the pandemic restrictions in providing appropriate visualization on projectile concept. Finally, the author suggested that the alternative of online learning during the outbreak have not yet been effectively engaged to promote the physics representation on projectile motion.

Fibrational bisimulations and quantitative reasoning: Extended version

Bisimulation and bisimilarity are fundamental notions in comparing state-based systems. Their extensions to a variety of systems have been actively pursued in recent years, a notable direction being quantitative extensions. In this paper we enhance a categorical framework for such extended (bi)simulation notions. We use coalgebras as system models and fibrations for organizing predicates—following the seminal work by Hermida and Jacobs. Endofunctor liftings are crucial predicate-forming ingredients; the first contribution of this work is to extend several extant lifting techniques from particular fibrations to $\textbf {CLat}_\wedge $-fibrations over $\textbf {Set}$. The second contribution of this work is to introduce endolifting morphisms as a mechanism for comparing predicates between fibrations. We apply these techniques by deriving some known properties of the Hausdorff pseudometric and approximate bisimulation in control theory.

Culturology: conceptual and categorical framework in the research process logic

The purpose of the article is to analyze the conditions and stages in the formation of the conceptual-categorical framework of cultural sciences. The methodology is based on general scientific approaches such as terminological and systematic. The purpose and objectives of the publication define the use of such research methods as analysis, synthesis, generalization, which made it possible to clarify the meaning of a number of concepts and terms. Scientific Novelty. The degree of scientific research of the topic is concentrated in identifying the problems of interaction between the conceptual and categorical frameworks in various scientific fields, namely, history, philosophy, philology, aesthetics, ethics, art history, and logic, to shape up the research space. We conceptualize the idea of "formal logical structures": that is not a mechanical combination of concepts or categories, but developing a new system adapted to solving theoretical problems of cultural studies. The question about the validity of the ―borrowing‖ process that allows cultural studies to analyze theoretical problems related to other humanitarian fields is also touched upon. Conclusion. So, among the other Ukrainian human sciences, cultural studies are developing quite actively and dynamically, constantly expanding its research space; its activity and dynamism of the cultural knowledge ―accumulation‖ determine those problems requiring special attention of modern scientists; in particular, the identification of the structural layering of the conceptual and categorical framework of cultural sciences.

Digital economic networks in the context of global transformations

Theoretically, the digital economy (DE) is the core of the modern networked economic system, and in practice – a growing sector of national and world economies. The essential features of IT as a new phenomenon in the socio-economic system are complemented on an interdisciplinary basis by the epistemology of information and computer sciences, electronic technologies and platforms. This refers to the widespread use of margins (marginal costs, marginal capital, marginal labour, etc.) and concepts such as institutions, trust, risk, security, etc. The purpose of the study is to investigate the trends of economic digitalisation, theoretical and methodological changes and applied vision of economic processes. Analysis, comparison, and generalisation were used in the study. The study considers the current trends of digitalisation of the economy, which cause significant theoretical and methodological changes and a new applied vision of economic processes. In the current conditions, the actual economic laws of IT are determined by the prevailing concepts and doctrines – neoclassical and neo-institutional. It was concluded that the synergistic synthesis of the categorical framework of the economic mainstream and computer sciences allows for the development of complexity economics, which is characterised by fundamentally new dimensions and parameters.

The Social Pedagogical Discourse of Modernity. The Politics of Human Rights and International Social Work

The basic thesis of this paper is that the helping professions have an indispensable task: to reflect the constitution of society in the context of the perception of the other. This discourse of social pedagogy addresses the conditions under which we encounter each other and under which we recognize each other. It is to be asked to what extent this discourse of social pedagogy could contribute to open the horizons of the common, which have been closed by all conceivable forms of violence. The background of the discourse, mentioned here, is close to the social philosophical discourse of modernity. It reflects the form of modernity in all its moral, social and political dimensions. The social-philosophical reference to the present is accompanied by the indispensable critique of power. It designates stages of reflection of that critique that make possible a theory of society. It thus forms, which is to be shown as aa result, the categorical framework of an analysis that enables a view of the structures of existing power relations. It thematizes the essential approaches to the practical overcoming of these phenomena. For this purpose, it will be reflected by way of introduction which theoretical reference the mentioned discourse has and how it is to be understood in comparison to other discourses - thus, it is first about the connections from social philosophy to social pedagogy (2). Within this framework, it will then be shown that the helping professions have an internal reference to the world, which is of eminent importance for the self-description of the discourse (3). How the language of human rights can be translated into transcultural perspectives of action is a complex question that is answered here from a fundamental anthropological perspective (4).

RECIPROCITY SHEAVES AND THEIR RAMIFICATION FILTRATIONS

We define a motivic conductor for any presheaf with transfers F using the categorical framework developed for the theory of motives with modulus by Kahn, Miyazaki, Saito and Yamazaki. If F is a reciprocity sheaf, this conductor yields an increasing and exhaustive filtration on $F(L)$ , where L is any henselian discrete valuation field of geometric type over the perfect ground field. We show that if F is a smooth group scheme, then the motivic conductor extends the Rosenlicht–Serre conductor; if F assigns to X the group of finite characters on the abelianised étale fundamental group of X, then the motivic conductor agrees with the Artin conductor defined by Kato and Matsuda; and if F assigns to X the group of integrable rank $1$ connections (in characteristic $0$ ), then it agrees with the irregularity. We also show that this machinery gives rise to a conductor for torsors under finite flat group schemes over the base field, which we believe to be new. We introduce a general notion of conductors on presheaves with transfers and show that on a reciprocity sheaf, the motivic conductor is minimal and any conductor which is defined only for henselian discrete valuation fields of geometric type with perfect residue field can be uniquely extended to all such fields without any restriction on the residue field. For example, the Kato–Matsuda Artin conductor is characterised as the canonical extension of the classical Artin conductor defined in the case of a perfect residue field.

THE APPROACH TO UNDERSTANDING RESPONSIBILITY FROM THE POSITION OF THE CONCRETE-UNIVERSAL THEORY OF DEVELOPMENT THEORY

The article deals with the problem of the most fundamental grounds for the analysis of different situations and meanings connected with responsibility. The proposed solution of this problem is based on concept of the concrete-universal. The conclusions of the concrete-universal theory of development make it possible to relate the objective principles of morality to the dynamics of interconnections of the universal moments that have been accumulating in the human nature in the course of the global development. The universal moments, abstracted by man in the form of categories, are organized in human existence in the form of certain aspects and interrelations that, being interwoven in a unique way, construct certain social situations and meanings reflecting thereof. The first stage of the research presents an analysis of the role of the world-inherent key universal moments in the formation of the phenomenon of responsibility. This helps to formulate the most fundamental characteristics of responsibility. The second stage offers a study of the specific forms of interconnections between the universal moments underlying certain aspects of responsibility. As a result, a categorical framework of universal moments representing a basis of responsibility was identified. This framework is inextricably linked with specific and unique content, is always filled with the concrete content of the circumstances existing at a particular period in time, of life situations of individuals. The paper shows how the concrete-universal organized in this way determines the most important situations and meanings related to the problem of responsibility. This approach made it possible to distinguish the substance of responsibility, connected with a certain objective situation, from its manifestations in the form of subjective components related to personal characteristics and states of a person, formulation of agreements, social norms and sanctions for their violation, various forms of accountability.

Mathematics, Social Structure, and Technology

This chapter provides insights gained from a collaborative action research project with university and middle school mathematics faculty. A categorical framework that considers the relationship between technology, mathematics content, and social interaction was used by the researchers to more deeply examine the varied uses and types of technology related to online teaching. In particular was the use of a relatively new category of software, Interactive Mathematics Classroom Builders, which integrates powerful mathematics tools with highly interactive classroom management features. The teachers found opportunities to try several novel uses of technology in their online lessons, advancing their teaching and the student experience, while learning lessons about teaching with technology that may apply to both remote teaching and the face-to-face classroom.