### A General Framework for Probabilistic Characterizing Formulae [chapter]

Joshua Sack, Lijun Zhang
2012 Lecture Notes in Computer Science
Recently, a general framework on characteristic formulae was proposed by Aceto et al. It offers a simple theory that allows one to easily obtain characteristic formulae of many non-probabilistic behavioral relations. Our paper studies their techniques in a probabilistic setting. We provide a general method for determining characteristic formulae of behavioral relations for probabilistic automata using fixed-point probability logics. We consider such behavioral relations as simulations and
more » ... lations, probabilistic bisimulations, probabilistic weak simulations, and probabilistic forward simulations. This paper shows how their constructions and proofs can follow from a single common technique. This paper focuses on behavioral relations over probabilistic automata and their characteristic formulae. The semantics of our languages involve fixed-points, which provide us with a natural facility for expressing various kinds of infinite behavior, such as those that are infinite or have loops. We present a single method, adapted from , that allows one to easily obtain characteristic formulae of many behavioral relations, including simulations and bisimulations, probabilistic bisimulations, probabilistic weak simulations, and probabilistic forward simulations. The strength of this technique is its generality: we can construct a variety of characteristic formulae and prove their correctness using a single simple method. Relation to Related Work: Our theory builds on a recent paper by Aceto et al.  , where a general framework is introduced for constructing non-probabilistic characteristic formulae over transition systems. It allows one to directly obtain the characteristic formulae for many behavioral relations, which have traditionally involved technicaleven if not difficult -proofs. Their main result (an earlier version of Theorem 1 in this paper), in its generality, can be used for all the behavioral relations we consider, except for probabilistic forward simulation. We thus provide a modest generalization of this theorem to address forward simulation. A more universally relevant extension to the overall setting of  is to involve in its applications (previously developed) liftings of relations. Liftings are discussed in [10, 23] , and employed in  for fixed-point characterizations of (bi)simulations and probabilistic (bi)simulations. As they are central to probabilistic behavioral relations, liftings play a key role in adapting the framework of  to a probabilistic setting. Another difference between our work and  is with the language used. The languages in  are fixed-point variants of Hennessy-Milner logic. For all our behavioral relations except the probabilistic forward simulation, we use a fixed-point variant of a two-sorted probability logic given in  . This allows us to interpret the characteristic formulae over states, as in , but to also have formulae over distributions that better fit with the setting of probabilistic automata. For probabilistic forward simulation, we involve a language, as in  , only interpreted over distributions rather than states. In  , Deng and van Glabbeek study characteristic formulae for all the behavioral relations over probabilistic automata that we consider, though they restrict their automata to being finite. For all their behavioral relations, their characteristic formulae use a more complex one-sorted language over distributions than the one we use for probabilistic forward simulation, and the form of their formulae are different (reflecting their different but equivalent approach to lifting) and somewhat simpler (our characteristic formula for probabilistic bisimulation involve an infinitary disjunction). But the difference that we emphasize is that they use a separate technique for proving correctness of characteristic formulae for each preorder considered, while our framework provides characteristic formulae which are correct by construction. Organization of the paper: In Section 2, we provide definitions to be used later in the paper. In Section 3, we present a slight adaptation of the framework developed in  . In Section 4, we recall the definition of probabilistic automata, the fixed-point characterization of bisimulation and simulation relations, and the weak bisimulations, and then we clarify the relationship between liftings used in  and in  . In Section