GSOS for probabilistic transition systems

We introduce PGSOS, an operator specification format for (reactive) probabilistic transition systems which bears similarity to the known GSOS format for labelled (nondeterministic) transition systems. Like the standard one, the format is well behaved in the sense that on all models bisimilarity is a congruence and the up-to-context proof principle is valid. Moreover, guarded recursive equations involving the specified operators have unique solutions up to bisimilarity. These results generalize well-behavedness results given in the literature for specific operators that turn out to be definable by our format. PGSOS arose from the following procedure: Turi and Plotkin proposed to model specifications in the (standard) GSOS format as natural transformations of a type they call abstract GSOS. This formulation allows for simple proofs of several well-behavedness properties, such as bisimilarity being a congruence on all models of such a specification. First, we give a full proof of Turi and Plotkin's claim about the correspondence of abstract GSOS and standard GSOS for labelled transition systems. Next, we instantiate their categorical framework to yield a specification format for probabilistic transition systems. The main contribution of the present paper is the derivation of the PGSOS format as a rule-style representation of the natural transformations obtained this way. We benefit from the fact that some parts of our argument for the nondeterministic case can be reused. The well-behavedness results for abstract GSOS immediately carry over to the new concrete format.