Anticipatory Topoi

摘要

The aim of this paper is to give a topos-theoretical approach to anticipatory aspects of reactive systems. In order to reason about reactive systems we identify the category of transition systems with a category of functors, Set~F, where F is a suitable small category with four objects. As Set~F is a topos we have its internal logic to reason about its objects. In this paper we exhibit the terminal object and the subobject classifier of Set~F, and analyse the way subobjects are classified according to several truth values. We define in the internal language of this topos the notion of approximation between systems. Among all possible approximations of a system, we take a good approximation (to compute how good an approximation is we compare approximations of a system using the ordering between truth values of the topos) to be its predictive model, as it differs from the original one in time and gives information about undesirable paths. With these tools we could complete the formulation of the concept of anticipatory system in a topos-theoretic framework.