Behavioural Theories and The Proof of Behavioural

Behavioural theories are a generalization of rst-order theories where the equality predicate symbol is interpreted by a behavioural equality of objects (and not by their identity). In this paper we rst consider arbitrary behavioural equalities determined by some (partial) congruence relation and we show how to reduce the behavioural theory of any class of-algebras to (a subset of) the standard theory of some corresponding class of algebras. This reduction is the basis of a method for proving behavioural theorems whenever an axiomatization of the behavioural equality is provided. Then we focus on the important special case of (partial) observational equalities where two elements are observationally equal if they cannot be distinguished by observable computations over some set of input values. We provide general conditions under which an obvious innnite axiomatization of the observational equality can be replaced by a nitary one and we provide method-ological guidelines for nding such nitary axiomatizations. As a consequence, any proof system for rst-order logic can be used to prove the behavioural validity of rst-order formulas w.r.t. a given (partial) observational equality.