Behavioural reasoning for conditional equations

The behavioral equivalence of hidden terms in an equational specification logic is not itself specifiable in general (Buss and Roşu 2000). But much recent work has been done on its partial specification, in particular using coinduction. In this paper we consider the more general notion of conditional behavioral equivalence introduced by Reichel in 1984. We investigate the behavioral proof theory of a general class of equational specification logics, the hidden equational logics. Among other things we characterize the behaviorally valid conditional equations of a hidden equational logic as those conditional equations which, in a natural sense, do not increase the deductive power of the logic when they are added as new rules of inference. For a special kind of hidden equational logic (the equivalential logics) we obtain methods for proving behavioral validity that work well in practice. Those hidden equational logics whose behavioral is specifiable by a (non-hidden) equational logic are characterized in terms of a special class of equivalential logics—equivalently as those hidden equational logics that have a cobasis (Roşu and Goguen 2001) of a special form.