Compositional methods for probabilistic systems

Abstract. We present acompositional trace-based modelfor probabilistic systems. The behavior of a system with probabilistic choice is a stochastic process, namely, a probability distribution on traces, or \bundle." Consequently, the semantics of a system with both nondeterministic and probabilistic choice is a set of bundles. The bundles of a composite system can be obtained by combining the bundles of the components in a simple mathematical way. Re nement between systems is bundle containment. Weachieve assume-guarantee compositionality for bundle semantics by introducing two scoping mechanisms. The rst mechanism, which is standard in compositional modeling, distinguishes inputs from outputs and hidden state. The second mechanism, which arises in probabilistic systems, partitions the state into probabilistically independent regions.