Stuttering for Abstract Probabilistic Automata

Probabilistic Automata (PAs) are a widely-recognized mathematical framework for the specification and analysis of systems with non-deterministic and stochastic behaviors. In a series of recent papers, we proposed Abstract Probabilistic Automata (APAs), a new abstraction framework for representing possibly infinite sets of PAs. We have developed a complete abstraction theory for APAs, and also proposed the first specification theory for them. APAs support both satisfaction and refinement operators, together with classical stepwise design operators.One of the major drawbacks of APAs is that the formalism cannot capture PAs with hidden actions – such actions are however necessary to describe behaviors that shall not be visible to a third party. In this paper, we revisit and extend the theory of APAs to such context. Our first main result takes the form of proposal for a new probabilistic satisfaction relation that captures several definitions of PAs with hidden actions. Our second main contribution is to revisit all the operations and properties defined on APAs for such notions of PAs. Finally, we also establish the first link between stochastic modal logic and APAs, hence linking an automata-based specification theory to a logical one.