Swinging types=functions+relations+transition systems

Swinging types 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 prototyping and verication. A swinging specication separates from each other visible sorts that denote domains of data identied by their structure; hidden sorts that denote domains of data identied by their behavior in response to observers; -predicates, i.e., least relations representing inductive(ly provable) properties of a system; and -predicates, i.e., greatest relations representing complementary \coinductive" properties, which often describe behavioral aspects \in the innity". Programming paradigms, such as functional, relational or state-oriented ones, and specication formalisms, such as algebraic, set-theoretic, rule-based, net-based, coalgebraic, order-theoretic ones, usually handle either static or dynamic components, either structural or behavioral aspects of a system. Swinging types admit the integrated design and analysis of these components and aspects. An integrated model is obtained naturally if all entities (objects, states, etc.) of the system are presented as terms built up of constructors for visible or hidden sorts, functions are specied in terms of conditional equations (=functional programs), least relations in terms of Horn clauses (= logic programs or transition system specications) and greatest relations in terms of co-Horn clauses. Data equalities are either structural or behavioral, the former being least, the latter being greatest solutions of axioms that are determined by (components of) the type’s signature. This paper mainly presents the theoretical foundations of swinging types, such as standard (term) models, criteria for structural and behavioral consistency, and proof rules. Swinging types admit exible design guidelines, tailored to particular objectives or application elds. Suitable design methods may be based upon this and the companion paper [61] that explores various application areas and illustrates how swinging types may realize dierent programming or specication styles. As to structuring concepts for swinging types, parameterization and genericity are involved in this paper, while [64] deals with extensions and renements. c 2000 Elsevier Science B.V. All rights reserved.