Characterizing Behavioural Semantics and Abstractor Semantics

In the literature one can distinguish two main approaches to the definition of observational semantics of algebraic specifications. On one hand, observational semantics is defined using a notion of observational satisfaction for the axioms of a specification and on the other hand one can define observational semantics of a specification by abstraction with respect to an observational equivalence relation between algebras. In this paper we present an analysis and a comparative study of the different approaches in a more general framework which subsumes not only the observational case but also other examples like the bisimulation congruence of concurrent processes. Thereby the distinction between the different concepts of observational semantics is reflected by our notions of behavioural semantics and abstractor semantics. Our main results show that behavioural semantics can be characterized by an abstractor construction and, vice versa, abstractor semantics can be characterized in terms of behaviourai semantics. Hence there exists a duality between both concepts which allows to express each one by each other. As a consequence we obtain a sufficient and necessary condition under which behavioural and abstractor semantics coincide. Moreover, the semantical characterizations lead to proof-theoretic results which show that behavioural theories of specifications can be reduced to standard theories (of some classes of algebras).