Recursively defined metric spaces without contraction

摘要

In this paper we use the theory of accessible categories to find fixed points of endofunctors on the category of 1-bounded complete metric spaces and nonexpansive functions. In contrast to previous approaches, we do not assume that the endofunctors are locally contractive, and our results do not depend on Banach’s fixed-point theorem. Our approach is particularly suitable for constructing models of systems that feature quantitative data. For instance, using the Kantorovich metric on probability measures we construct a quantitative model for probabilistic transition systems. The metric in our model can reasonably be seen as measuring the behavioural distance between states of the system; it depends exclusively on the transition probabilities and not on an arbitrary discount factor.