Behavioural Theories

Behavioural theories are a generalization of standard theories where the equality predicate is interpreted by an arbitrary (possibly partial) congruence relation, called behavioural equality. In this paper we first show how to reduce the behavioural theory of any class C of Σ-algebras to (a subset of) the standard theory of some corresponding class of algebras. Then we consider infinitary axiomatizations of the behavioural equality and we define a general condition under which the infinitary axiomatization can be replaced by a finitary one. As a consequence, behavioural theorems over different kinds of specifications can be proved using standard proof techniques. All results are stated in an abstract way, i.e. independent of a particular behavioural equality. As an important example we can apply our results to observational theories where the behavioural equality is given by an observational equality between elements.