About One Possible Tool for Analysing Safeness of Discrete Dynamic Systems

The paper is devoted to developing the coalgebraic technique for specifying and analyzing general discrete systems that can be used for modeling distributed, embedded, and cyber-physical ones. The authors ground this technique with an assumption of two specification languages. In more detail, each language specifies the actual state associated with the corresponding language for describing forecasts about system states. An endofunctor in the appropriate category models the relationships between these languages. This assumption leads us to consider the system’s dynamical transformation as a forecasting mechanism and model it with the corresponding coalgebraic operation.The introduction of the concept of an eventual coalgebra is an attempt to provide a coalgebraic formalization of steady system behavior, avoiding reference to the notion of a final coalgebra. This is motivated by the importance of final coalgebras in studying limiting properties of systems and the absence of final coalgebras for practically interesting types of systems. The authors assume that the approach outlined in the article allows for the analysis of practical and significant systems, those for which determining the final coalgebra is not feasible.In the paper as well, the properties of eventual coalgebras are studied, and their functorial nature is demonstrated.