On Correct Program Schemas

Extended Abstract 1 Introduction A program schema is an abstraction of a class of actual programs, in the sense that it represents their data-ow and control-ow, but does not contain (all) their actual computations or (all) their actual data structures. Program schemas have been shown to be useful in a variety of applications, such as proving properties of programs, teaching programming to novices, guiding manual and/or (semi-)automatic synthesis of programs, debugging programs, transforming programs, and so on, both within and without logic programming. An overview of schemas and their applications can be found in 5, 6]. In this paper, we present our work on three aspects of schemas: representation, correctness, and usage, in the context of logic program synthesis. In logic programming, most researchers represent their schemas as higher-order expressions, sometimes augmented by extra-logical annotations and features, so that actual ((rst-order) programs are obtained by applying higher-order substitutions to the schema. We shall take a diierent approach and show that schemas can also be expressed as open rst-order programs, viz. programs in which some of the relations are open, i.e. left undeened. One advantage of this approach is that it simpliies the semantics of schemas and of their manipulations. We shall endow schemas with a formal (model-theoretic) semantics by setting them in the context of a framework, i.e. an axiomatisation of the (possibly open) problem domain. This allows us to deene a meaningful notion of correctness for schemas. Indeed, we show that correct program schemas can be expressed as steadfast open programs, i.e. programs that are always correct provided their open relations, i.e. their parameters, are computed correctly. Steadfastness is a priori correctness, and therefore correct schemas are a priori correctly reusable. We shall also discuss how to use correct schemas in practice. Using any kind of schemas requires suitable strategies, and we shall present some ideas on such strategies for correct schemas. 2 Schemas as Open Programs Our approach to schemas (and program synthesis) is set in the context of a (fully) rst-order axiomati-sation F of the problem domain in question, which we call a framework F. Speciications are given in F, i.e. written in the language of F. We adopt a model-theoretic semantics for F, and for speciications and (deenite logic) programs in F. This declarative approach enables us to deene program correctness wrt 1 speciications not only for closed programs but also for open programs, i.e. programs …