Abstract-System specifications can be modeled using various types of notations and diagrams regarding applications of the particular model. In this paper, we present an overview of the existing solutions, focusing on UML, BPMN and DMN models and the diagrams provided by these notations. We perform a comparison of these approaches and provide examples of representing system requirements in these notations.

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Development of modern software systems consists of many different phases, the sequence of whom is referred to as the software life cycle. During these phases, business analysts gather requirements from clients and try to design a system in order to fulfil these requirements. Software design of complex systems exploits various notations for representing knowledge about system structure and behaviour, decisions, processes and different cases. These elements are modelled using such graphical notations, maintained by Object Management Group (OMG), such as UML (Unified Modelling Language), DMN (Decision Model and Notation), BPMN (Business Process Model and Notation), and CMMN (Case Management Model and Notation). In this paper, we present our work-in-progress analysis of the current state of the art in knowledge interchange for these notations. Moreover, we identify the integration or interchange approaches in terms of application areas. Our goal is to provide an input for an integrated method of designing systems with the use of these notations.

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Abstract-Process models can specify various aspects of business processes. In this paper, we present an overview of the existing solutions for describing time aspects of such models. We focus on Business Process Model and Notation and provide examples of representing time patterns in this notation. As temporal issues can be specified using temporal logics, we provide a short overview of selected temporal logics which can be used to specify the time patterns in business process models.

The natural transformation constitutes one of the most important entity of category theory and it introduces a piece of sophisticated dynamism to the categorial structures. Each natural transformation forms a unique mapping between the so-called functors, which live between categories. In the most simple contexts, natural transformations may be recognized by commutativity of diagrams, which determine them. In fact, the natural transformation does not form any single mapping, but a pair of two components, which–together with the commutativity condition itself–introduces a kind of a symmetry to the functor diagrams. Meanwhile, the general form of the natural transformation may be predicted by means the so-called Yoneda’s lemma in each scenario based on two-valued logic. Meanwhile, the situation may be radically different if we deal with multi-diagrams (instead of the single ones) and if we exchange the two-valued scenario for a multi-valued or fuzzy one. Due to this background–the paper introduces a new concept of multi-fuzzy natural transformation. Its definition exploits the notion of fuzzy natural transformation. Moreover, a multi-fuzzy Yoneda’s lemma is formulated and proved. Finally, some references of these constructions to coding theory are elucidated in last parts of the paper.