Mathematical Understanding of Interactive Systems

Our research aims the mathematical representation of general interactive systems. The first principle of designing a large system is to “divide and conquer”, which implies that we could possibly reduce human error if we divided a large system into smaller subsystems. Interactive systems are, however, often composed of many subsystems that are “organically” connected to one another and thus difficult to divide. In other words, we cannot apply a framework of set theory to the programming of interactive systems. We can overcome this difficulty by applying a framework of category theory (Kleisli category) to the programming, but this requires highly abstract mathematics, which is not very popular. We introduce the fundamental idea of category theory using only lambda calculus and then demonstrate how it can be used in the practical design of an interactive system. Finally, we mention how this discussion relates to category theory.