On Behaviour Equivalence for Probabilistic I/O Automata and its Relationship to Probabilistic Bisimulation

Previous work of the author has developed probabilistic in-put/output automata (PIOA) as a formalism for modeling systems that exhibit concurrent and probabilistic behavior. Central to that work was the notion of the \behavior map" associated with a state of a PIOA. The present paper presents a new, simpler deenition for PIOA behavior maps, investigates the induced \same behavior map" equivalence relation, and compares it with the standard notion of probabilistic bisimulation equivalence. Weighted nite automata are used as a unifying formalism to facilitate the comparison. A general notion of congruence for weighted automata is deened, which relates signed measures on states, rather than just individual states. PIOA are deened as a class of weighted automata, as are the class of probabilistic weighted automata for which the standard deenition of probabilistic bisimulation makes sense. A characterization is obtained of probabilistic bisimulation as the largest congruence that is in a sense generated by its restriction to a relation on states. This characterization is then used as the deenition of weighted bisimulation, which generalizes probabilistic bisimulation to the full class of weighted automata. PIOA behavior equivalence is also shown to deene a weighted automata congruence, which is strictly reened by weighted bisimulation equivalence. The relationship between these congruences and a notion of composition for weighted automata is also examined.