Modular Swinging Types

Swinging types 18] provide an integrated framework for specifying software on the basis of many-sorted logic in terms of \static" functions and relations as well as \dynamic" transition systems. Swinging types combine equational, Horn and modal logic for the purpose of using evaluation and proof rules from all three logics for rapid prototyp-ing and veriication. A swinging speciication separates from each other visible sorts that denote domains of data identiied by their structure; hidden sorts that denote domains of data identiied by their behavior in response to observers;-predicates (least relations) that represent in-ductive(ly provable) properties; and-predicates (greatest relations) that represent complementary \coinductive" properties. The paper at hand deals with structured speciications with swinging components. Vertical structuring is supported by a deduction-oriented reenement criterion that admits, for instance, to implement visible sorts by hidden sorts and structural by behavioral equalities. Horizontal struc-turing is achieved by employing speciications with import, which admit the axiomatization of functions in terms of arbitrary-or-predicates. In order to obtain simple term models and proof rules we provide a translation of swinging into implicational speciications whose \Horn" axioms may involve universal quantiiers in the premise.