Mathematical modeling of ubiquitous systems

Ubiquitous systems built in the environment of distributed or parallel computing are more complicated than conventional digital systems. This paper describes how ubiquitous systems are modeled mathematically or in a formal way using the incrementally modular abstraction hierarchy. Firstly, the system requirements represented by event sequences are mathematically expressed by the Cartesian product of actors and events using a fiber bundle. Then, the fiber bundles is lifted by the homotopy lifting property to the set of subspaces, each of which describes the behavior of a part of the system. This property is used for modeling the ubiquitous system in a bottom-up way. Assembling behaviors distributed in parts of the system, the behavior of an actor is defined by the homotopy extension property for modeling the system in a top-down way. Finally, the behaviors of the actors are adjoined together by attaching functions to express the system behavior, which is equivalent to the process obtained by process algebra. The problem of process algebra not having the methodology of how the system is modeled from system requirements to formal description is solved by the proposed incrementally modular abstraction hierarchy.