PsProduct

Psproduct is an operator used in theoretical computer science and systems modeling to form a single, joint representation of two stochastic or probabilistic systems. It is designed to capture both synchronized interactions and independent progression of its components, enabling analysis of the combined behavior under probabilistic choices.

In formal terms, Psproduct applies to two probabilistic labeled transition systems or stochastic models, each described by a state space and a set of transition probabilities. The result is a product model whose state space is the Cartesian product of the two component state spaces. Transitions in the product are defined according to a specified synchronization policy. Under strict synchronization, actions that appear in both components cause paired transitions to be taken together, with the probability determined by the product or a related combination of the two component distributions. Under asynchronous or mixed policies, actions can occur independently in one component while the other component either remains in its current state or advances according to its own distribution.

Variants of the Psproduct differ in how they handle timing and concurrency, leading to different product state spaces and probability distributions. The construction is designed to preserve essential properties and to support compositional verification, enabling model checking of the product against probabilistic temporal logics such as PCTL. This makes Psproduct useful for studying the reliability, performance, and safety of interacting stochastic systems, including network protocols, distributed control systems, and cyber-physical systems.

Limitations include the potential exponential growth of the product state space and the need to select a synchronization policy appropriate to the modeling goals. The concept sits alongside other product constructions used in probabilistic model checking and automata theory.