BISIMULATION AND TRACE SEMANTICS

So far, we only considered models having the property of containing a submodel which is isomorphic to the initial algebra of the current theory (for instance see 2.7.35). In other words we may say that if two finite processes in a model are equal, this equality must be derivable from the theory.

Models that with respect to finite processes correspond to the graph model or the term model are said to be models in bisimulation semantics. Because for every one of our theories, its initial model is a model in bisimulation semantics, our axiom systems are said to be a complete axiomatization of bisimulation semantics. In this chapter we will consider other semantics than bisimulation semantics, and we will present complete axiomatizations of these alternative semantics as well.

Because all operators except for + and · can be eliminated from closed terms, we will mainly restrict ourselves to the theory BPA, with special constant 8. The addition of the special constant τ leads to many interesting observations, and a vastly increased complexity. We do not include it here, in order to focus on a few key issues.

We will discuss semantics that identify more processes than bisimulation semantics does. The advantage of this is clear: calculations become easier, and more simplifications can be made. On the other hand, some differences between processes are disregarded, and as a consequence, in some cases some operators cannot be defined any more.

We start with the repetition of an earlier result.