Order-sorted model theory for temporal executable specifications

An order-sorted, temporal programming paradigm is presented. It consists of a typed, modular, declarative language, its associated order-sorted temporal Horn clause logic basis, and a model theory generalizing order-sorted algebras with predicates to temporal order-sorted structures. The essence of this generalization is in time-dependent interpretation of predicates, so that a temporal order-sorted model amounts to a sequence of order-sorted equational models with predicates, one per each state. The main advantage of the presented paradigm in comparison with paradigms based on Hem-clause logic with equality is that it is more expressive, particularly so in representing properly state transitions, and other event-oriented, temporal behavioral properties of objects. At the same time, the generalized paradigm is proved to have the initial model semantics. The rules for temporal order-sorted deduction are established as an appropriate generalization of the rules for order-sorted Horn-clause logic with equality. The initial model is a quotient temporal order-sorted structure constructed from the initial temporal order-sorted structure and a congruence relation derived from a given set of temporal constraints. Temporal order-sorted model theoretic properties are also naturally established for temporal queries. The temporal constraint language has an execution model, and it is intended to be a basis for a prototyping tool for complex, typed, modular software systems.