Fixing Zeno gaps

AbstractIn computer science, fixpoints play a crucial role. Most often least and greatest fixpoints are sufficient. However, there are situations where other ones are needed. In this paper, we study, on an algebraic base, a special fixpoint of the function f(x)=a⋅x that describes infinite iteration of an element a. We show that the greatest fixpoint is too imprecise. Special problems arise if the iterated element contains the possibility of stepping on the spot (e.g. skip in a programming language) or if it allows Zeno behaviour. We present a construction for a fixpoint that captures these phenomena in a precise way. The theory is presented and motivated using an example from hybrid system analysis