Interpolation in computing science: The semantics of modularization

Abstract

The Interpolation Theorem, first formulated and proved by W. Craig fifty years ago for predicate logic, has been extended to many other logical frameworks and is being applied in several areas of computer science. We give a short overview, and focus on the theory of software systems and modules. An algebra of theories TA is presented, with a nonstandard interpretation of the existential quantifier {\exists} . In TA, the interpolation property of the underlying logic corresponds with the quantifier combination property {\exists{\Sigma} \,{\exists{\Pi} \,{S}} \equiv \exists{(\Sigma \cup \Pi)} \,{S}} . It is shown how the Modularization Theorem, the Factorization Lemma and the Normal Form Theorem for module expressions can be proved in TA.