Observing Branching Structure through Probabilistic Contexts

Probabilistic automata (PAs) constitute a general framework for modeling and analyzing discrete event systems that exhibit both nondeterministic and probabilistic behavior, such as distributed algorithms and network protocols. The behavior of PAs is commonly defined using schedulers (also called adversaries or strategies), which resolve all nondeterministic choices based on past history. From the resulting purely probabilistic structures, trace distributions can be extracted, whose intent is to capture the observable behavior of a PA. However, when PAs are composed via an (asynchronous) parallel composition operator, a global scheduler may establish strong correlations between the behavior of system components and, for example, resolve nondeterministic choices in one PA based on the outcome of probabilistic choices in the other. It is well known that, as a result of this, the (linear-time) trace distribution precongruence is not compositional for PAs. In his PhD thesis from ’95, Segala has shown that the (branchingtime) probabilistic simulation preorder is compositional for PAs. In this paper, we establish that the simulation preorder is in fact the coarsest refinement of the trace distribution preorder that is compositional. We prove our characterization result by providing (1) a context of a given PA A, called the tester, that may announce the state of A to the outside world, and (2) a specific global scheduler, called the observer, which ensures that the state information that is announced is actually correct. Now when another PA B is composed with the tester, it may generate the same external behavior as the observer only when it is able to simulate A in the sense that whenever A goes to some state s, B can go to a corresponding state u from which it may generate the same ∗A preliminary version of this paper appeared as [24]. †Supported by AFOSR contract #F49620-00-1-0097, AFRL Award #FA9550-04-1-0121, NSF grants #CCR0121277 and #CCR-0326277, NSF Award #CNS-0614414, NSF/ITR grants #CCR-0121277 and #CCR-0326277, USAF/AFRL Award #FA9550-04-1-0121, and DARPA/AFOSR MURI #F49620-02-1-0325. ‡Supported by MURST projects MEFISTO and CoVer, MIUR project AIDA, Inria project ProNoBis. §Supported by PROGRESS project TES4999: Verification of Hard and Softly Timed Systems (HaaST) and DFG/NWO bilateral cooperation project Validation of Stochastic Systems (VOSS and VOSS2).