The notion of binary relation is fundamental in logic. What is the correct analogue of this concept in the probabilistic case? I will argue that the notion of conditional probability distribution (Markov kernel, stochastic kernel) is the correct generalization. One can define a category based on stochastic kernels which has many of the formal properties of the ordinary category of relations. Using this concept I will show how to define iteration in this category and give a simple treatment of Kozen's language of while loops and probabilistic choice. I will use the concept of stochastic relation to introduce some of the ongoing joint work with Edalat and Desharnais on Labeled Markov Processes. In my talk I will assume that people do not know what partially additive categories are but that they do know basic category theory and basic notions like measure and probability. This work is mainly due to Kozen, Giry, Lawvere and others. 1 Introduction The notion of binary relation and relation...
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School of Computer Science, McGill University, Montreal