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Completeness of Relational Algebra via Cylindric Algebra

Jan Laštovička

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.

Completeness of Relational Algebra via Cylindric Algebra

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.
Paper Structure (10 sections, 23 theorems, 65 equations, 1 figure)

This paper contains 10 sections, 23 theorems, 65 equations, 1 figure.

Table of Contents

  1. Introduction
  2. First-order logic
  3. Syntax.
  4. Semantics.
  5. Relational algebra
  6. Syntax.
  7. Semantics.
  8. Semantic equivalence
  9. Allowed formulas
  10. Normalization

Key Result

Lemma 2.1

The following holds.

Theorems & Definitions (44)

  • Lemma 2.1
  • proof
  • Theorem 3.1
  • Theorem 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • ...and 34 more