Free objects and equational deduction for partial conditional specifications

Abstract

Partial conditional specifications consist of conditional axioms, with equalities in the (possibly infinite set of) premises and in the consequence, which are interpreted in partial algebras. Equalities may be existential (= e ) or strong ( = ); t = e t′ holds in an algebra iff both t and t′ are defined and equal, while t = t′ holds in an algebra iff either t = e t′ holds or both t and t′ are undefined. Contrary to the well explored case of positive conditional axioms (only existential equalities in the premises), general partial conditional specifications do not always admit free models and the related theory is much more subtle. In this paper we fully investigate and solve the problem of existence of free and initial models, giving necessary and sufficient conditions, first from a model-theoretical and then from a logical deduction viewpoint. In particular we present a deduction system which is complete w.r.t. strong equalities between open terms. Since positive conditional partial specifications and conditional total specifications are special cases of the paradigm investigated here, the presented theory generalizes the related results about free models and the Birkhoff-like deduction theory. The system we exhibit handles also the case of infinitary conjunctions as premises of the axioms; it reduces to a classical one for the positive conditional case by just dropping one rule, and finally it solves the empty-carrier problem without using explicit quantification. The theory presented here also gives the basis for solving, via the usual first-order reduction, the problem of the existence of free and initial models for partial higher-order specifications of term-generated extensional models.