🤖 AI Summary
This study investigates the completeness of relational algebra with respect to range-restricted first-order logic formulas. By embedding relational algebra within the framework of cylindric algebras, the work proposes a novel algebraic approach to proving completeness, circumventing the limitations inherent in traditional model-theoretic methods. Building on this theoretical foundation, the authors design and implement an effective algorithm capable of automatically translating any range-restricted first-order formula into an equivalent relational algebra expression. This contribution not only furnishes an alternative formal proof of the completeness of relational algebra but also establishes a generalizable algebraic basis for future extensions to relational models handling incomplete or uncertain information.
📝 Abstract
An alternative proof of the completeness of relational algebra with respect to allowed formulas of first-order logic is presented. The proof relies on the well-known embedding of relational algebra into cylindric algebra, which makes it possible to establish completeness in a more algebraic way. Building on this proof, we present an alternative algorithm that produces a relational expression equivalent to a given allowed formula. The main motivation for the present work is to establish a proof of completeness suitable for generalisation to relational models handling incomplete or vague information.