Observing Branching Structure Through Probabilistic Contexts

N.A. Lynch, R. Segala and F.W. Vaandrager. Observing Branching Structure Through Probabilistic Contexts. In:Abstract

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 (branching-time) 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 external behavior. Our result shows that
probabilistic contexts together with global schedulers are able to
exhibit the branching structure of PAs.