An Abstract Coalgebraic Approach to Process Equivalence for Well-Behaved
This thesis is part of the programme aimed at finding a mathematical theory of well-behaved structural operational semantics. General and basic results shown in 1997 in a seminal paper by Turi and Plotkin are extended in two directions, aiming at greater expressivity of the framework.
The so-called bialgebraic framework of Turi and Plotkin is an abstract generalization of the well-known structural operational semantics format GSOS, and provides a theory of operational semantic rules for which bisimulation equivalence is a congruence.
The first part of this thesis aims at extending that framework to cover other operational equivalences and preorders (e.g. trace equivalence), known collectively as the van Glabbeek spectrum. To do this, a novel coalgebraic approach to relations on processes is desirable, since the usual approach to coalgebraic bisimulations as spans of coalgebras does not extend easily to other known equivalences on processes. Such an approach, based on fibrations of test suites, is presented. Based on this, an abstract characterization of congruence formats is given, parametrized by the relation on processes that is expected to be compositional. This abstract characterization is then specialized to the case of trace equivalence, completed trace equivalence and failures equivalence. In the two latter cases, novel congruence formats are obtained, extending the current state of the art in this area of research.
The second part of the thesis aims at extending the bialgebraic framework to cover a general class of recursive language constructs, defined by (possibly unguarded) recursive equations. Since unguarded equations may be a source of divergence, the entire framework is interpreted in a suitable domain category, instead of the category of sets and functions. It is shown that a class of recursive equations called regular equations can be merged seamlessly with GSOS operational rules, yielding well-behaved operational semantics for languages extended with recursive constructs.