Swinging Data Types { the Dielectic between Actions and Constructors {

final initial Abstract Initial structures are good for modelling constructor-based data types because they t the intuition about these types and admit resolution-and rewrite-oriented inductive theorem proving. The corresponding speciication and veriication methods do not comply so well with non-free or permutative types such as sets, bags and maps and are still less appropriate when innnite structures like streams or processes come into play. Non-free and innnite structure are better modelled as dynamic objects, which are identiied through reactions upon actions (methods, messages, state transitions) rather than through constructors they might be built of. Extensional, contextual, behavioural, observational or bisimilarity relations model object equality and the suitable domains are nal structures that are conservative with respect to visible subtypes. Consequently, a collection of data types and programs should be designed hierarchically as a \swinging" chain of speciications each of which extends its predecessor by either constructor types or action types. Constructor types introduce the visible domains and come with inductively deened total functions, structural equality and safety predicates with Horn clause axioms, while action types provide the hidden domains together with coinductively deened partial functions, behavioural equality and liveness predicates with liveness axioms that are dual to Horn clauses. A swinging speciication is interpreted as a sequence of initial and nal models. General proof rules capture this semantics and exploit the duality of induction and coinduction to its utmost extent. The deductive tractability is further enhanced by making both constructor and action types amenable to rewrite-oriented proof methods so that we can reason about swinging speciications in the same way we are used to reason about exclusively constructor-based types.