Joint work with Erik de Vink (University of Eindhoven)
In this presentation the applicability of an ultrametric version of stochastic kernels, which extend the notion of compact support measures, is studied in the context of a language with random assignment. In the random assignment the value assigned to a variable is selected according to a measure over the ultrametric space of values. To find the operational meaning for a program in this language it is necessary to compose the measures of different random assignments in the program. To be able to do this, the operational semantics is modeled as a kernel.
It is shown that the version of kernels used fits well within the metric framework. The use of ultrametric kernels, however, restrict the possible applications. An important example of infinite probabilistic choices are those made according to a continuous distributions over the real numbers. As the space of real numbers with the usual Euclidean metric is not an ultrametric space, it cannot be used as a space of values. The limitations created by the use of ultrametric spaces as well as a way of modeling spaces like the natural numbers and the real number using ultrametric spaces are illustrated by giving some simple examples.