Formal Foundations for Controlled Stochastic Activity Networks

Abstract: We introduce Controlled Stochastic Activity Networks (Controlled SANs), a formal extension of classical Stochastic Activity Networks that integrates explicit control actions into a unified semantic framework for modeling distributed real-time systems. Controlled SANs systematically capture dynamic behavior involving nondeterminism, probabilistic branching, and stochastic timing, while enabling policy-driven decision-making within a rigorous mathematical framework. We develop a hierarchical, automata-theoretic semantics for Controlled SANs that encompasses nondeterministic, probabilistic, and stochastic models in a uniform manner. A structured taxonomy of control policies, ranging from memoryless and finite-memory strategies to computationally augmented policies, is formalized, and their expressive power is characterized through associated language classes. To support model abstraction and compositional reasoning, we introduce behavioral equivalences, including bisimulation and stochastic isomorphism. Controlled SANs generalize classical frameworks such as continuous-time Markov decision processes (CTMDPs), providing a rigorous foundation for the specification, verification, and synthesis of dependable systems operating under uncertainty. This framework enables both quantitative and qualitative analysis, advancing the design of safety-critical systems where control, timing, and stochasticity are tightly coupled.